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 THE PROBLEM OF LOGIC
 
 Br THE SAME AUTHOR 
 
 THE PROBLEM OF FREEDOM IN ITS 
 RELATION TO PSYCHOLOGY 
 
 ( 'IN PERSONAL IDEALISM." EDITED BY HENRY STURT) 
 
 LONDON : MACMILLAN AND CO., LTD. I902 
 
 A PHILOSOPHICAL INTRODUCTION 
 TO ETHICS 
 
 LONDON: SWAN SONNENSCHEIN AND CO., LTD. 1904 
 
 RUDOLF EUCKEN'S PHILOSOPHY OF 
 
 LIFE 
 
 Third Edition 
 
 With Frontispiece Portrait of Rddolf Eccken 
 
 Crown 8vo. , cloth, price 3s. 6d. net 
 
 LONDON : A. AND C. BLACK, LTD., SOHO SQUARE. I912 
 
 GOD WITH US 
 
 A STUDY IN RELIGIOUS IDEALISM 
 Crown 8vo., cloth, price 3s. 6d. net 
 
 LONDON : A. AND C. BLACK, LTD., SOHO SQDARE. IQCg 
 
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 AUilEAiAELA . . . OXFORD UNIVERSITY PRESS 
 
 x>i FLINDERS Lane, MELBOURNE
 
 THE 
 
 PROBLEM OF LOGIC 
 
 BY 
 
 W. R. BOYCE GIBSON, M.A., D.Sc. (Oxox.) 
 
 PROFESSOK (elect) OF MENTAL AND MORAL PHILOSOPHY IN T-HE 
 UNIVERSITY OF MELBOURNE 
 
 WITH THE CO-OPERATION OF 
 
 AUGUSTA KLEIN 
 
 LONDON 
 A. AND C. BLACK, LIMITED 
 
 1914 
 
 o o f) n
 
 Published Septrmher, 1908 
 Reprinted Novenibei-, 1914 
 
 ccetcct*(* t (o 
 
 .,..£<■ ■.'■'■<■ 
 
 C 
 
 
 B « •
 
 3a 
 
 7/ 
 
 TO 
 
 MY WIFE
 
 /
 
 PREFACE. 
 
 The present volume has grown up and taken shape under the 
 
 chasteninpf influences of College teacliing. No teacher of Logic 
 
 would wdsh to underestimate the value of the education he receives 
 
 from his students ; and since my education has for nine years been 
 
 advancing along these lines, my claim to have learnt Logic through 
 
 ^ teaching it may be accepted in sincerest good faith. A first and 
 
 ^ most grateful acknowledgment is due to my many fellow-workers 
 
 ^ at Hampstead (at the New College Centre, and at Westfield College) 
 
 ^ ,,who by their doubts, difficulties, criticisms, and suggestions have 
 
 had so much to do with the shaping of this book. 
 
 But there is a still more intimate sense in which the book is the 
 work of many rather than of one. Prom the time wlien it was first 
 jj decided to reconstruct the College lectures with a view to publica- 
 ■^\ tion, I was privileged to enjoy the invaluable sympathy and assist- 
 ^ ance of Professor G. F. Stout. Professor Stout most kindly consented 
 to read through these lectures, and returned them to me shortly 
 afterwards accompanied by a small volume of criticisms. It would 
 be hard to exaggerate the value of these criticisms. On such funda- 
 5 mental heads as the Laws of Thought, the interrelation of Cate- 
 ^gorical, Disjunctive, and Hypothetical Propositions, and the 
 •^essential meaning of the Disjunctive and Hvpothetical Judgments, 
 4 the substance of Professor Stout's contentions was adopted, and 
 f will be easily recognized by all who are familiar with the Professor's 
 logical views. Many extracts from these criticisms will be found 
 in the present volume. Professor Stout has also allowed me to 
 look through a large part of his own Class lectures in Logic, and 
 has helped me in many other ways, not least through certain con- 
 versations which we have had together over fundamental logical 
 principles. 
 
 Miss Klein's collaboration dates from the first revision of the 
 work — from the spring of 1905. Since that date, every change in 
 the treatment — and the reconstructions have been drastic — has 
 been subjected to the friendliest but most unsparing criticism. 
 No point of divergence between us but has been thorouglily 
 discussed, and transmuted into a point of common agreement.
 
 % iii PREFACE 
 
 If, in addition to the reading of the proofs, the verifying of 
 the quotations and the elaboration of the scientific illustra- 
 tions and allusions, I may single out two respects in which Miss 
 Ivlcin's co-operation has been particularly valuable, I would 
 mention her revision of the work in the interest of consistency, 
 and her revision of it in the interest of clear expression. To 
 show the importance of these revisions, it would be necessary 
 to pubUsh the original draft side by side with the final product ; 
 but as this course is not practicable, I can only assure the reader 
 that, however he may suffer from the defects of the present treatise, 
 his sufferings, but for these revisions, would have been incalculably 
 worse. It has, indeed, become increasingly evident to me, as the 
 work proceeded, that it could no longer be honestly regarded as 
 one man's work. The original draft was the work of one ; the 
 reconstruction is the work of three. 
 
 With regard to the help derived from published treatises on 
 Logic, my heaviest obligation has been to the works of Mill and of 
 Sigwart, to Professor Bosanquet's ' Logic,' and to Mr. Joseph's 
 ' Introduction to Logic' My indebtedness to Mr. Joseph is in- 
 direct rather than direct, our points of view being quite different. 
 But though I have been unable to assimilate either the Aristotelian 
 or the Baconian elements which figure so prominently in Mr. 
 Joseph's treatment, I have every reason to be grateful that his 
 work appeared early enough to allow of my making full use of it 
 in revising my own. Among other works which have been par- 
 ticularly useful to me, I would specially mention Professor Minto's 
 treatise, ' Logic, Inductive and Deductive ' (notably the Introduc- 
 tion to Book II., dealing with the Logic of Science), and Mr. Alfred 
 Sid.gwick's books, notably ' The Use of Words in Reasoning '; but I 
 have also profited much by the treatises of Dr. Keynes, Dr. Mellone, 
 Professor Carveth Read, Mr. St. George Stock, Dr. Venn, and 
 Professor Welton. I would, in addition, gratefully acknowledge 
 the help given me by Miss Strudwick, of the Goldsmiths' College, 
 New Cross, in connexion with the scientific illustrations on pj). 59-62. 
 In conclusion, I would add that if I have appeared to ignore the 
 work of such writers as Professor Dewey or Dr. Schiller, it is not 
 through any lack of sympathy or appreciation. I am, indeed, 
 persuaded that the drift of the present work is convergent with 
 that of the Pragmatic Reformation, and that the stress laid on 
 relevancy is a vital bond of union between ourselves and the 
 Pragmatiflts. But the central contentions of Pragmatism concern 
 the Logic of Experience, and cannot, therefore, be appropriately 
 or adequately treated in the pages that follow. We hope to 
 consider thern in a later work. 
 
 The present volume aspires to be the first part of a ' complete ' 
 treatise on Logic, of which the second will deal, or attempt to deal, 
 with the Logical Problem in its more philosophical aspect. Some
 
 PREFACE ix 
 
 brief indication as to this programme will be found in the Intro- 
 duction. Here it may be enough to state that the Religious 
 Idealism in which the author's own conviction culminates seems 
 to him to call imperatively for a frank and fruitful co-operation 
 between the Idealism of the Hegelian School on the one hand, 
 and the Psychologism of the Pragmatic and Genetic movements 
 on the other. In attempting this reconciliation, so far as it is 
 relevant to the requirements of a logical treatise, the author 
 ventures to hope that he may be found working in the service of 
 that liberating movement in Philosophy which, in liis own mind, 
 is centrally associated with the work and personality of Professor 
 Eucken. 
 
 Tlie promise of a sequel is no doubt a convenient shield for 
 sheltering an author — though, indeed, only temporarily — from any 
 charge of incompleteness in his treatment. I would claim this 
 shelter as regards the discussion of the principles of Mathematics 
 in their logical bearing. I hope to deal with this important problem 
 in the sequel. 
 
 I am much more doubtful with regard to the general problem of 
 Symbolic Logic. Whether, in postponing the discussion of this 
 department of Logic, I am or am not shelving its consideration 
 altogether, I am not now j)repared to say. In no case would I 
 contest the interest and importance of Symbolic Logic ; but whether 
 the limitations of my programme — or of my own powers — may not 
 render its discussion irrelevant — or impracticable — is, perhaps, a 
 pardonable question. 
 
 The distinctive feature of the present volume will, I tliink, 
 be found in the dominating position assigned to the idea of 
 relevancy. The fundamental concepts of Truth and Reality have 
 been defined in the light of this category, and the principle of 
 Fidelity to Relevant Fact has been adopted as the master-key to all 
 the main positions, including the central problem of a Formal 
 treatment, and its relation to a material treatment of Logic. I 
 would also draw attention to the distinction between the functions 
 respectively assigned to the Laws of Non-Contradiction and Ex- 
 cluded ]\Iiddle. This distinction will be found to be directly con- 
 nected \^ith that between a Formal and a material treatment of 
 the logical problem. 
 
 In conclusion, I would gratefully acknowledge the work done by 
 Miss Klein in the framing of the Index. The Index is her work, 
 and she alone is responsible for it. 
 
 W. R. BOYCE GIBSON. 
 CnAKDoxxE sfR Yevey, 
 May 10, 1908.
 
 ERRATA, 
 
 Page 337, line 30, instead of 'Nejitnne's irregularities,' read 'The 
 irre;;ularities of Uranua.'
 
 CONTENTS 
 
 BFCTION PAOB 
 
 I. INTRODUCTION 1 
 
 II. LOGIC IN ITS RELATION TO LANGUAGE : 
 
 (i.) Words, their function and ripi;ht use (Ch. I.) - - 13 
 
 (ii.) Definition and the Predicablos (Ch. II.) - - - 17 
 
 (iii.) The Testing of Definitions (Ch. III.) - - - 32 
 
 (iv.) Definition and Division: Logical Division (Ch. IV.) - 39 
 
 (v.) Classification (Ch. V.) - " - - - - 56 
 
 (vi.) Scientific Terminology and Nomenclature (Ch. VI.) - 66 
 
 (vii.) Connotation and Denotation (Ch. VII.) - - - 70 
 
 (viii.) Concrete and Abstract Terms (Ch. VIII.) • - - 85 
 
 III. THE LOGICAL PROPOSITION : 
 
 (i.) The Judgment or Proposition : Introductory Statement 
 
 (Ch. IX.) 93 
 
 (ii.) The Laws of Thought (Ch. X.) : 
 
 (a) The Law of Logical Identity in its relation to the 
 
 Proposition - - - - - 95 
 
 {b) The Laws of Non-Contradiction and Excluded 
 
 Middle - - - - - - 98 
 
 (c) The Inviolability of the Laws of Thought - - 104 
 
 IV. ANALYSIS OF THE LOGICAL PROPOSITION AS STATE- 
 
 MENT OF MEANING : 
 
 (i.) Kinds of Proposition (Qi. XI.) - - - - 111 
 
 (ii.) Analysis of the Categorical Proposition (Ch. XII.) - 115 
 
 (iii.) The Meaning of PossibiHty (Ch. XIII.) - - - 127 
 
 (iv.) The Disjunctive Proposition (Ch. XIV.) - - - 131 
 
 (v.) The Hypothetical Proposition (Ch. XV.) - - - 138 
 
 V. THE FORMAL TREATMENT OF THE LOGICAL PRO- 
 POSITION : 
 
 (i.) Transition to the Formal Treatment of Logic (Cli. XVI.) - 145 
 (ii.) The Formal Import of the Categorical Proposition 
 
 (Ch. XVIL) 146 
 
 (iii.) The Reduction of Categorical Propositions to Strict 
 
 Logical Form (Ch. XVIII.) .... ir.l 
 
 (iv.) The Opposition of Propositions (Ch. XIX.) - - 171 
 
 VI. BmEDIATE INFERENCE (Ch. XX.) - - - - 1S7 
 
 xi
 
 Xll 
 
 CONTENTS 
 
 SKCTION" PAGE 
 
 Vn. THE SDIPLE CATEGORICAL SYLLOGISM : 
 
 (i.) Formal Preliminarv (Qi. XXI.) - - - 213 
 
 (ii.) The Rules of the Syllogism : the Valid Forms (Ch. XXII.) 216 
 (iii.) Exercises on the Structure of the S.C.D. Syllogism 
 
 (Ch. XXIIL) 227 
 
 (iv.) The Analysis of Syllogisms, and the Reduction of Argu- 
 ments into Syllogistic Form (Ch. XXIV.) - - 230 
 (v.) Uses and Characteristics of the Four Figures : the 
 
 Special Rules (Ch. XXV.) - - - - 235 
 
 (vi.) The Dicta (Ch. XXVI.) - - - - 239 
 
 (vii.) The Problem of Reduction (Ch. XXVII.) - - 247 
 
 (viii.) Unorthodox Syllogisms (Ch. XXVIII.) - - 250 
 
 VIIL OTHER FORMS OF SYLLOGISM : 
 
 (i.) Complex Categorical Syllogisms : Sorites and Epi- 
 
 cheirema (Ch. XXIX.) - - - - 255 
 
 (ii.) Tlie Disjunctive Syllogism (Ch. XXX.) - - - 262 
 
 (iii.) The Hypothetical Syllojjism (Ch. XXXI.) - - 2.63 
 
 (iv.) The Dilemma (Ch. XXXlI.) - - - - 271 
 
 IX. FALLACIES (Ch. XXXIII.) 
 
 281 
 
 X. THE PROBLEM OF INFERENCE : 
 
 (i.) Mill's Estimate of the Syllogism (Ch. XXXIV.) - 299 
 
 (ii.) The Function and Value of a Formal DiscipUne 
 
 (Ch. XXXV.) 
 
 (iii.) Truth-Inference, formal and real ((Hi. XXXVI.) 
 
 XL INDUCTION AND THE INDUCTIVE PRINCIPLE : 
 
 (i-) 
 
 (ii.) 
 (iii.) 
 
 General Theory of Induction (Ch. XXXVII. ): 
 
 (a) The Pure Inductive Method 
 
 (b) The Essentials of Induction 
 
 (c) Induction and ' Inductive Inference ' 
 Hypothesis (Ch. XXXVIII.) - 
 Generalization (Ch. XXXIX.) - 
 
 304 
 306 
 
 SIS*-'! 
 3i6 I 
 326 I 
 328'— 
 339 
 
 XII. APPLICATION OF THE INDUCTIVE PRINCIPLE TO 
 ' INDUCTIONS IMPROPERLY SO-CALLED,' AND 
 TO ' IMPERFECT INDUCTIONS ' : 
 
 (i.) Inductive Inferences improperly so-called (Ch. 
 
 (ii.) The 'Imperfect Inductions': 
 
 (a) Enumerative Induction (Ch. XLL) 
 (6) Argument from Analogy (Ch. XLII.) 
 
 XL.) 
 
 347 
 
 351- 
 358 
 
 XIIL THE GOAL OF INDUCTION : CAUSAL EXPLANATION : 
 
 (i.) Cause and Causal Law (Ch. XLIII.) - - - 367 
 
 (ii.) The Process of Scientific Observation (Ch. XLIV.) - 389* 
 
 (iii.) The Method of Causal Explanation (Ch. XLV.) - 395 
 
 (iv.) Illustrations of the Application of Inductive Method 
 
 (Ch. XLVI.) 422 
 
 XIV. THE INDUrTIVE POSTULATE (Ch. XLVII.) 
 Index, Verbal and Analytic 
 
 447 
 463
 
 THE PROBLEM OF LOGIC 
 
 I. 
 
 INTRODUCTION. 
 
 Logic is the mind's systematic attempt to understand the nature 
 and the conditions of the search after Truth. To the question, 
 ' What is Truth V we would answer by suggesting the following 
 provisional definition : 
 
 Truth is the Unity of ideas as systematically organized through 
 the control exercised by relevant fact. 
 
 Or— 
 
 Truth is the Unitj^ of Thought as systematically organized 
 througli the control exercised by that aspect of Reahty 
 which is relevant to the purpose of the thinker. 
 
 With a view to bringing out the meaning of these definitions, 
 we must state in the fij:st place that we do not regard Truth as a 
 datum, but as a problem. The truth we seek cannot be that from 
 which we start, for were truth already attained at the outset, no 
 sufficient reason could be assigned for proceeding any further with 
 the quest. We might, of course, regard the Truth as given, and 
 devote our energies to its systematic exposition and application. 
 But, in that case, we should have radically to alter our definition 
 of Logic. Logic would no longer deal with the Search after Truth, 
 but would be busied solely with the question of its consistent 
 presentation. Logic would just mean Consistency-Logic, and might 
 be defined as the mind's systematic attempt to understand the 
 nature and the conditions of a correct presentation of the Truth. 
 But, valuable as such a Consistency-Logic would be, its logical value 
 would lie, not in its relation to a sj'stem of given truth, but in its 
 analysis and development of the laws of consistent thinking. 
 
 We would draw attention, in the second place, to the fact that 
 Truth is defiiied as a Unity, and that to define Truth as a Unity is 
 to ground logical inquiry on a monistic basis. We cannot, of course, 
 justify monistic faith by merely asserting it, nor, by asserting it, 
 
 ]
 
 THE PROBLEM OF LOGIC [I. 
 
 make our meanin!: clear. ' Monism ' is a catchword as dear to 
 the Rationahsm of Hegel as it is to that of Haeckel, and we suspect 
 that much that calls itself Pluralism is but Monism in the making. 
 It is indeed a much-abused word, and we introduce it thus bluntly, 
 at the outset of our inquiry, not as a dogma but as a problem. 
 To justify our monistic faith we need here do no more than justify 
 our riglit to accept the struggle for complete unity of thought as 
 the fundamental mark of the truth-seeker, and the attempt to define 
 the nature and conditions of such a struggle as the distinctive 
 function of Logic. 
 
 We might justify this right by presenting it as a necessity of 
 our logical reason, and contend that it is meaningless to suppose 
 that Unity of Thought and Purpose can be ultimately satisfied by 
 an\i:hing short of the Lenity of the L^niverse. Or we might defend 
 our monistic faith as a postulate limiting the scope of our inquiry, 
 and proceed confidently with our venture, more than content with 
 the perfect freedom conferred upon us by our own self-limitation. 
 We would prefer, however, to point quite simply to a certain 
 insatiabiUty of logical appetite as the best justification for our 
 Monism. For if we forgo or evade the struggle after Unity, we 
 really do hmit ourselves in quite a literal and painful sense. We 
 renounce the hope of a logical conquest that shall leave us nothing 
 foreign or unsubdued to mock us with its ahen nature. We abdicate 
 a fraction of our empire, and must five in perpetual dread of border 
 troubles, of disturbances emanating from those shadowy entities — 
 the dim hosts of the ununifiable. And can one imagine thought 
 surveying such a chaos from the edge of its own self-limited domain 
 and still dehberately disclaiming a redemptive mission ? Is it not 
 thought's nature rather to weep because, hke Alexander, it sees no 
 further worlds to conquer ? Our sufficient aipology, then, for 
 regarding the Truth-problem as a search after Unity is that logical 
 ambition can be satisfied with nothing less, and cannot endure 
 the sight of chaos battening for lack of its two-edged sword. 
 
 We turn, in the tliird place, to our contention that relevant 
 fact is the agency which controls the process through which our 
 thinking becomes systematized. The precise function of the 
 expression ' relevant fact ' is to indicate that truth implies at once 
 a reference to purpose and a reference to reahty ; and the second of 
 the two definitions of Truth that we have given explicitly brings out 
 this implication. 
 
 Thought submits itself to fact as the experimenter submits himself 
 to the object experimented on. As the experimenter determines 
 the conditions under which the experiment shall take place, so 
 thought selects and determines the aspect under which the facts 
 shall be thought. The purpose of the inquir\^ be it that of the 
 physicist, the biologist, the artist, or the mystic, determines the 
 range of fact within which the student of Nature recognizes an
 
 Intro.] INTRODUCTION 3 
 
 objective control. It is true that to conquer Nature we must 
 obey her ; but we must know clearly what it is tliat we obey, and 
 to this end must first select and mark out the domain that we 
 have then to conquer through submission. Tlie investigator of 
 Nature is thus at once self-controlled by his own purpose, and 
 outwardly controlled by the facts in so far as they are germane to 
 that purpose. In a word, he is controlled throughout by ' relevant 
 fact ' — i.e., by the object! v'^e nature of that aspect of the universe 
 wliich is relevant to his subjective interest. 
 
 We thus reach the conclusion that the conception of Truth from 
 which we set out itself determines the principle which must domi- 
 nate and inform o;ir whole attempt to realize it. If the Unity of our 
 thought is to be shaped through the pressure of relevant fact, then 
 fidelity to relevant fact must be the fundamental principle through 
 which growth in Truth is determined, and it must also figure as the 
 standard or criterion of any inquiry into the conditions of its 
 attainment. So we take it as our guiding clue through the mazes of 
 the logical problem.* We shall realize its determinative influence, 
 not only in the problems of definition and division, where it operates 
 in the interests of Order and non-ambiguity, but even in fixing what 
 we mean by meaning — the ' meaning ' which these processes serve 
 to develop. Again, the reference which the principle imphes to 
 purpose, and through purpose to reality, will be found to enter into 
 the very conception of a complete logical judgment ; whilst, in 
 methodology and the problem of scientific explanation, tliis principle 
 of fidelity to relevant fact will be exphcitly sustained as the funda- 
 mental principle and standard of Induction, and rendered deter- 
 minate in the fight of the Inductive Postulate. 
 
 Let us now apply these general considerations to the special case 
 of the present inquiry. The truth we have in view is Truth in so far 
 as it can relevantly serve as an Ideal for a pre-pliilosopliical Logic. 
 When preparing for more difficult flights — e.g., for a truth-journey 
 doTvii the abysmal depths of personality — Logic might reasonably 
 desire to equip itself with a more penetrating conception of Truth 
 than is required for its more preliminary labours. If Truth is, in 
 all cases, the Ideal which we aim at progressively realizing through 
 Knowledge, and is conceived as that which can adequately satisfy 
 the thinker's will to know, then the Truth-Ideal will vary with the 
 view we take of Ivnowledge, and also with the depth of this will to 
 know. By Knowledge we may understand Self-Knowledge, and the 
 depth of the truth-interest will then be measured by the depth of the 
 
 * The relation of this Principle to the Laws of Thought may be stated in the 
 simplest way by sayiug that the former presupposes the latter. But it is only in the 
 
 inuely Formal treatment of the logical problem, in connexion with the problem of 
 inference, that the Laws of Thought, as we understand them, can be accepted as an 
 adequate logical standard. "Where the truth-interest is present, a concretcr principle 
 — operating, of course, in conformity with the Laws of Thought — is required to give 
 positive direction to our thinking. 
 
 1—2
 
 4 THE PROBLEM OF LOGIC [I. 
 
 self that is seeking for truth-satisfaction ; we should then be con- 
 cerned with the profoundest questions — with Freedom, Personality, 
 Perfection, Immortalit}', God — questions which spring from the 
 unrest and dissatisfaction of our deepest self. But if by Knowledge 
 we understand, not Self-Knowledge, but Knowledge about Things, 
 Knowledge of that which we apprehend through the senses, we 
 may well be content with a less intimate specification of the meaning 
 of Truth. We reach this more restricted conception of Truth 
 through marking out the realm of fact which we take to be relevant 
 to the limited requirements of a pre-philosophical treatment — in 
 a word, by defining what we here mean by Reahty. 
 
 L'nder ' Reality ' we shall include two main aspects of Fact : 
 
 1. The world as common sense understands it (or some con- 
 
 ventionally restricted fragment of it). 
 
 2. Nature, understood as the subject-matter of Science. 
 
 In bringing the worlds of Science and Common Sense thus closely 
 together, we are making an assumption which it is important to 
 notice. We are assuming that the attitude of common sense to 
 the more or less fragmentary world within which its interest is 
 restricted is, on its own humbler level, similar to the attitude of 
 Science towards Nature. It may, however, be objected with good 
 reason that in thus characterizing the common-sense attitude 
 towards reality as pre-scientific, we are doing injustice to the 
 ordinary consciousness, which, over and above its interest in a world 
 external to it, has interests of a personal and social kind. The 
 objection in itself is perfectly legitimate. The ordinary conscious- 
 ness is religious as well as practical, and has inward as well as 
 outward looking views as to tlie nature of truth. If sense-ex- 
 perience rests its beUefs on an ' I have seen,' the intuitionism of the 
 moral and religious consciousness rests its behefs on an ' I have 
 felt.' In the one case truth is taken to be the truth about an object, 
 the truth about a fact ; in the other, it is taken as the truth for a 
 subject, the truth for a person. 
 
 The imphcations of the more inward conception of the meaning 
 of truth are of fundamental importance, and, at a more advanced 
 stage of logical inquiry, their discussion becomes imperative. But 
 for our present purpose — i.e., for the purposes of a pre-philosophical 
 Logic — we propose to ignore this personal, inward interpretation of 
 the truth-problem, and the deeper view of Reahty which would 
 correspond to it. At the same time we must remember that the 
 definition we have provisionally laid down does not do full justice 
 to the truth as it is presented to common sense. It imposes a 
 restriction which reduces common sense to an infra-scientific stand- 
 point. Only when common sense is thus restricted can Science 
 be regarded as its completion and rectification. Only when we 
 have eliminated as irrelevant the relation of truth to personal
 
 Intro.] INTRODUCTION 6 
 
 experience can we fairly describe Science as organized Common 
 Sense. 
 
 The deliberate exclusion of the personal clement from the defini- 
 tion of truth may appear to some to be unjustifiable even when 
 the definition is given solely from the scientific and infra-.scientific 
 points of view. The objection may be raised that, since the reference 
 to reality which is implied in all truth-seeking, whether scientific 
 or infra-scientific, can be characterized and defined only through 
 relation to logical purpose, we cannot study Reality at any stage 
 without introducing the personal element. It is quite true that the 
 truth-definition which we have adopted explicitly includes a refer- 
 ence to purpose. But this mere reference to purpose in no way 
 commits us to a personalistic view of truth or of reaUty. On the 
 contrary, it may so define the reference to reality as to render such 
 a view irrelevant and impossible. How this reality-reference has 
 been defined, in the interest of a pre-philosophical treatment, we 
 have already seen. The limitation ensures that Truth shall be 
 truth about fact, and not the truth of personal realization. To 
 reach the pliilosophical conception of truth, we must study Fact in 
 the light of a philosophical truth-interest, and adopt a correspond- 
 ingly philosophical conception of reality. 
 
 It is true that reference to purpose implies reference to a deeper 
 reality than that reality of nature the conception of which it serves 
 to define, and that in this important sense the scientific point of 
 view implies and presupposes the philosophic ; but the implication 
 remains latent, and the scientific and pre-scientific conceptions of 
 truth and reality correspondingly impersonal and objective. 
 
 There are, we may say, three main stages in the life of Logic. In 
 its first, formal, or common-sense stage, Logic presents itself as a 
 propaedeutic, or preliminary discipline, and the truth-ideal which it 
 then presents to thought is truth as involving the relation of thought 
 not to the reality of the Natural Order, but to a reality of a more or 
 less restricted and conventional kind. The point of view, in a 
 word, is essentially formal in the sense of conventional. There is 
 no reference to a permanent order like that of Nature as conceived 
 by Science, but only to such conventionally restricted aspects of 
 it as answer to the requirements of some particular purpose. 
 
 In the second, real, or scientific stage, the casual, disconnected 
 grasp on Reality which these conventional restrictions involve is 
 definitely abandoned. Thought ceases to play with Reality in the 
 interests of discussion, or of other requirements of practical inter- 
 course. Armed with the idea of natural law, it now disposes itself 
 to face the full force of that great realm of fact which has no limit 
 but thi3t of the api^licability of the idea itself. 
 
 And yet this second stage is not final. It presupposes a relation 
 of externality between fact and idea, and is broken through when 
 this externality is done away with, and Truth shows itself as the
 
 6 THE PROBLEM OF LOGIC [I 
 
 intimate oneness of idea vnth fact. The complete setting forth of 
 this unity is the function of a philosophical Logic. Briefly, it 
 amounts to the idealizing of fact and the realizing of ideas within 
 
 CI? O 
 
 a conception of experienced fact larger than is possible to Scicitice 
 or appropriate to its restricted point of view. In this third stage, 
 Thought, as Hegel would say, finds itself at home with itself, freed 
 from all fettering abstraction, and at the very heart of the reality 
 it is its mission to understand. What remains is then just the sys- 
 tematic articulation of the structure of this experienced fact, at 
 once most real and most ideal — the Logic of spiritual experience. 
 
 Tliis Personalistic Logic, as already stated, lies beyond the scope 
 of the present treatise ; the following course covers only the first 
 two stages. The earher stages, however, are essential to the proper 
 grasp of the third and last. For the lessons of each earlier stage 
 are taken up into the succeeding one in a form determined by the 
 richer, concreter conditions of the latter. Thus, what is gained at 
 the one level is not lost at the next, but transcended and redeemed. 
 The ' Reason ' of Philosophy must have assimilated the ' Under- 
 standing ' of Science, the passion for distinctness and precision, 
 which is characteristic of the scientific attitude, and its loyalty 
 to relevant fact. Loj^alty to ideals can bestead Philosophy but 
 little if it does not, in its own appropriate way, include reverence 
 for fact as an integral requisite of all true spiritual exj)erience. 
 
 Li the foregoing attempt to define the x>oint of view adopted in 
 the following treatise, the meaning of the word ' formal ' deserves 
 particular consideration. For it is more customary to identify the 
 term ' Formal Logic ' with a Logic of Validity than with a treat- 
 ment in which is imphed a merely ' formal ' reference to reality. 
 In particular, the word ' formal ' is associated with the so-called 
 Forms or Formal Laws of Thought as the principles upon which 
 consistent thinking ultimately depends. Thus, in using this 
 ambiguous term, it is essential that we should not confuse the two 
 meaning-. We propose, therefore, in the interests of clearness, to 
 adopt the following device. When ' Formal ' is being used in its 
 fundamental sense of ' abstractly valid,' we shall employ a capital 
 F ; when it is being used in the sense of ' conventional,' we shall 
 write the word with a small ' f .' Should tlie word open the sentence, 
 and the capital letter be indispensable, we shall leave it to the 
 context to decide in which of its two senses the word is being used.* 
 
 The distinction between a formal and a real logical treatment is 
 a distinction witliin a unity. Both methods equally imply a funda- 
 
 • Perhaps the strongest reason for retaining two such closely similar words to 
 designate meanings apparently so dillerent is that the meanings are not so unrelated 
 as they apfK-ar to be. A ' Formal ' treatment of Logic might be considered as a 
 ' limiting case ' of a ' formal ' treatment of the subject — the case, namely, where the 
 conventional restriction put upon the meaning of Reality is such as to reduce it to 
 an essentially hypothetical status (vule p. 145).
 
 iNTBo.] INTRODUCTION 7 
 
 mental respect for consistency, and they both involve a reference to 
 reality, though the reference is occasional in the one case and syste- 
 matic in the other. We do not, then, propose to keep the two 
 methods separate. We propose, on the contrary, to discuss the 
 real in close connexion with the formal aspect, and thereby to secure 
 a unity of treatment which would be forfeited by the attempt to 
 deal with the two aspects successively and in isolation from each 
 other. When we are interested in emphasizing what is common to 
 these two types of logical treatment, we propose to use the word 
 ' material ' to cover both. Thus, a material logical treatment may 
 be either formal or real. 
 
 In contrast with a material treatment of Logic, we have what is 
 customarily known as a purely Formal treatment. We shall find 
 that at a certain stage in the development of our subject it becomes 
 essential to abstract entirely from the reference of thought to reality 
 as we have defined it {vide p. 4),* and to concentrate our whole 
 attention on the logical conditions of valid thinking. When our 
 logical interest is thus rigidly restricted, and reduced to an interest 
 in validit}^, the treatment ceases to be material, and becomes 
 Formal. 
 
 The chapters on the Laws of Thought and their application to the 
 problems of Opposition, Eduction, and Syllogism are tlic chapters 
 essential to a strictly Formal treatment. The ideal of (material) 
 truth, which alone gives meaning to the distinction between ' formal ' 
 and ' real,' here gives place to the ideal of validity. The reference 
 to reahty implied in all reasoning whatsoever is tacitly ignored as 
 ' accidental,' and the primary logical requisite, the requisite of 
 validity, monopohzes the attention. Whatever reference to truth 
 or falsity there is in Formal Logic is wholly hypothetical. If the 
 statements ' All donkeys are daffodils ' and ' All dragons are 
 donkeys ' are both accepted, accepted as though they were true 
 (whether, as a matter of fact, they are true is here a completely 
 irrelevant question), then Formal Logic insists that the statement 
 ' All dragons are daffodils ' must also be accepted, accepted as 
 though it were true. 
 
 The Validity-Ideal, which is regulative of a Formal logical treat- 
 ment, implies the twofold requisite of Self-consistency and of Inter- 
 consistency. A statement or an argument is self-consistent when 
 it so hangs together that thought may pass through it, as it were, 
 from beginning to end without falHng into contradiction with itself 
 by the way. The statement, ' Square tables are round ' violate/ 
 this fundamental requirement. So does the following argument : 
 
 ' All men are rational animals. 
 Nebuchadnezzar was a man. 
 Therefore, Nebuchadnezzar was not a rational animal.' 
 
 ♦ Vide note, p. 9.
 
 8 THE PROBLEM OF LOGIC [I. 
 
 We cannot maintain, Avitliout illegitimate variation in our use of 
 words, that all men are rational, and that one is not so. 
 
 The Interconsistency of our statements is as important as their 
 Self-Consistency. The diligent reader may discover on different 
 pages of a connected treatise statements which no charity can con- 
 strue as interconsistent. The statements may be separated by more 
 than a hundred pages, but the requisite of Interconsistency will still 
 compel a logical readjustment of the passages such as will make 
 them maintainable together by one and the same thinker in one and 
 the same discourse. The coherency of our thinking is essentially 
 dei^endent upon a faithful observance of the requisite of inter- 
 consistenc}'. 
 
 Logical Consistency should be carefully distinguished from 
 Material Compatibility. Whether the assertion that my friend 
 takes no regular exercise is compatible with the statement that he 
 continues to enjoy robust health, and is in that sense ' consistent ' 
 with it, is a question that concerns material truth. A treatment 
 wliich ignores all considerations of truth and falsity* cannot possibly 
 say anj'tliing relevant upon the matter. 
 
 Logical Consistency should be distinguished from logical Validity. 
 The meaning of the former is at once wider and more negative than 
 that of the latter. Consistency impHes mere freedom from self- 
 contradiction ; VaUdity, a connexion so close that the severing of 
 it would involve a contradiction. If we say ' Some people are 
 reasonable,' it is quite consistent to add ' Some people are not 
 reasonable '; but, as we shall see {vide p. 174), we could not validly 
 infer that some peop^.e are not reasonable from the statement that 
 some people are. An argument is said to be valid when the con- 
 clusion dra\\ii from the premisses is such that we must accept it, 
 once the premisses have been accepted. A conclusion dra^Ti in tliis 
 way from its premisses is said to be draA^-n from them with logical 
 necessity, and is known as a valid conclusion. So, again, the pro- 
 position ' If all men are mortals, some mortals are men ' is a vaHd 
 proposition, since the accejotancc of the ' if ' clause necessitates 
 our accepting its consequent. The statement ' If all men are 
 mortals, all mortals are men,' is invalid if taken as asserting a 
 logical connexion, though it is not inconsistent. 
 
 We should also note the distinctively negative character of 
 Logical Consistency. Logical Consistency does not amount to 
 systematic coherency. The coherency of a scientific system means 
 much more tlian mere freedom from self-contradiction. 
 
 We conclude this Introductory Chapter with the following brief 
 resume of its main points : 
 
 Logic is the Science of Right Thinking. 
 
 To think rightly we must think both consistently and truly. 
 
 * Fide note, p. 9.
 
 Intro.] IXTKODUCTION 9 
 
 To think consistently is to avoid all self-contradiction. If wo 
 think as logical necessity requires, our thought is said to 
 be valid. 
 
 Consistent = not involving contradiction = not inconsistent. 
 
 Valid = involving logical necessity. 
 
 Inconsistent = involving self-contradiction. 
 
 Invalid = not involving logical necessity. 
 
 To think truly is to think under the control exercised by that 
 aspect of Reality which is relevant to the purpose of our 
 thinking. 
 
 Under Reality, as relevant to the truth-interest of a pre-philo- 
 sophical discipline, we include the world of Common Sense 
 — the world in relation to our various practical interests — 
 and Nature as understood by Science. 
 
 In cither case, this reahty is conceived as having a nature suffi- 
 ciently stable to control our tentative thought about it. 
 
 When the reahty we have in view is limited by some practical 
 interest, the logical ideal is satisfied in proportion as our 
 ideas adjust themselves to the control exercised by this 
 conventionally hmited reality. Ideas so adjusted may be 
 said to be formally or conventionally true, true in relation 
 to our restricted practical purpose. 
 
 When the reality is Nature as conceived by Science, the con- 
 trolling of our ideas through reahty is said to give us real 
 or scientific truth. 
 
 Finally, when our sole interest is in the validity of our think- 
 ing, the question whether the reference of our thought to 
 reahty is formal or real ceases to be relevant ; for we are 
 here no longer concerned that our thought shall be true, 
 but only that it shall be vahd. 
 
 The treatment of right thinking which is thus exclusively regu- 
 lated by the Ideal of Vahdity is known as Formal Logic. 
 Whatever reference there is to truth or falsity in Formal 
 Logic is wholly hypothetical. 
 
 The Formal treatment of right thinking should be carefull}" dis- 
 tinguished from a formal reference to reality, a Formal 
 treatment being a treatment m accordance with the Formal 
 Laws of Thought, the laws of logical Validity. 
 
 By ' Formal ' we mean dominated by the Ideal of Validity. 
 
 By ' formal ' we mean ' conventional.' 
 
 Note. — Tlicre is a certain misconception witli regard to our use 
 of the term ' Formal,' which our very definition of a Formal treat- 
 ment may have served to foster. We have stated that a logical 
 treatment can be called Formal only in so far as we abstract from 
 all reference to truth or reality ; and if the definitions which, in the 
 interest of a pre-pliilosophical treatment, we have given of these
 
 10 THE PROBLEM OF LOGIC [L 
 
 same terms are not borne carefully in mind, the reader may be left 
 with a very poor opinion as to the status of Formal Logic. Formal 
 Logic will seem to be concerned essentially with some abstract 
 department of Xon-Being. 
 
 If we turn, however, to the definitions of truth and reality, as 
 given on pj). 1, 4, or in the resume, p. 9, we shall readily see 
 that no such disparagement of a Formal treatment is intended or 
 implied. In abstracting from all reference to reality as we have 
 defined it, we do not abstract from all reference to all reality. It 
 is only when the pre-philosopliical definition of reality which we 
 have adopted is mistaken for the ultimate meaning of reality that 
 a Formal treatment of Thought appears unreal, and, in its detailed 
 application, tends to degenerate into mere mechanical drudgery, 
 on the one hand, or, on the other, into irresponsible explorations 
 within a purely artificial world. 
 
 The abstraction from all reference to material reahty and truth 
 still leaves us with the reference of thought to itself ; and when this 
 self-reference of thought, together with the problem of Validity 
 which it involves, is studied under the redeeming conditions of 
 philosophical insight, Formal Thinking gains a vital, a spiritually 
 vital significance. Assuming a philosophical definition of Truth — ■ 
 as we understand the term ' philosophical ' — the interest in Validity 
 is itself an interest in Truth.
 
 II. 
 
 LOGIC IN ITS RELATION TO LANGUAGE. 
 
 (i.) "Words, their function and right use (eh. i.). 
 (ii. ) Definition and the Predicables (eh. ii.). 
 (iii.) The Testing of Definitions (ch. iii.). 
 (iv.) Definition and Division : Logical Division (ch. iv.). 
 (v.) Classification (ch. v.). 
 
 (vi.) Scientific Terminology and Nomenclature (ch. vi.) 
 (vii.) Connotation and Denotation (ch. vii.). 
 (viii.) Concrete and Abstract Terms (ch. viii.).
 
 CHAPTER I. 
 
 II. (i.) WORDS. THEIR FUNCTION AND RIGHT USE. 
 
 The Function of Words. 
 
 PRorosiNG as we do to start in the humblest and most methodical 
 way in our investigation of the nature and conditions of Truth 
 wo look first to the tool or instrument we shall be dependent on 
 all through — namely, Language. 
 
 Logic, like every other science, depends on language, written or 
 spoken, as its only suitable instrument. In Grammar, which 
 considers words in themselves and in relation to each other, 
 Language is the subject-matter treated of as well as the instrument ; 
 but it is not so in Logic. Logic is cor^erned with language only 
 as an instrument of thought, and its aim is so to handle the 
 instrument as to make it a help and not a hindrance to correct 
 thinking. Since thought can be handled only in verbal form, the 
 regulative function of Logic, directed primarily upon thought itself, 
 is inevitably pressed upon language as well. Language must reveal 
 thought and not falsify it. 
 
 Rhetoric, too, is concerned with language and the right use of 
 words. But whereas Logic aims at the right use of words with a 
 view to correct thinking. Rhetoric aims at the right use of words 
 with a view to persuasion. The purpose of Rhetoric is to prove 
 practically effective, and its appeal is therefore made to the whole 
 man, to his emotions and humours as well as to his reason. As a 
 science, at any rate, Logic is concerned with theoretical soundness 
 rather than with practical efficiency. As an art it may be said to 
 aim at practical efficiency, but its appeal is still made exclusively 
 to the reason. 
 
 Over this instrument. Speech, Logic proposes to exercise appro- 
 priate supervision. But supervision, to be logical, must be in 
 accordance with the nature of what is supervised. Before we 
 consider the right use of words, we must learn something of their 
 natural function in relation to thought. 
 
 The main function of words is to fix meanings or ideas both 
 in our own minds and in those of our fellows. If I wish to see an 
 object clearly, I bring it into the focus of vision. This I do 
 instinctively through the help of a number of delicate eye move- 
 
 13
 
 14 THE PROBLEM OF LOGIC [II. i. 
 
 ments. There are movements of convergence of the two eyes, of 
 accommodation to near or far vision, and focusing movements. 
 These contractions of the eye-muscles enable us to fixate the 
 objects we look at. Similarh% we fixate smells by setting our 
 nasal muscles in action, and so inhaling or sniffing upwards. We 
 fixate a taste by setting the muscles of the palate in action, and 
 pressing the food on to the palate. So with the ear-muscles in 
 hearing. A horse will ' prick its ears ' to fix a sound. It is in a 
 perfectly analogous sense that we utilize the muscles of lips, 
 tongue, larj^nx, for toning and articulating our breath into sounds 
 that bring our meaning fixedly before us. Thus, we control the 
 utterance of our thought by means of a certain special set of 
 muscles, the muscles involved in controlling the breath so as to 
 produce articulate sounds. 
 
 The function of words is to fix ideas, and this in a twofold sense. 
 For not only do they serve to impress meanings on ourselves who 
 think ; they also serve to express our meanings to others, and are 
 then known as expressive signs. 
 
 These should be distinguished from substitute signs. An ex- 
 pressive sign is meant to express meaning, whereas a substitute 
 sign is a counter which can be manipulated without our knowing 
 what idea it stands for (c/. Stout, ' Analytic Psychology,' p. 193). 
 Thus, algebraical symbols are used as substitute signs. I may 
 start by jDositing that x shall stand for the number of cows a 
 certain farmer bought ; but I may go on to solve the equation 
 z^ + 3z + 2 = 20 without thinking any more about the cows. I 
 am concerned solely with the algebraical laws according to which I 
 may profitably operate upon the sign. It is only when the value 
 of X is found that I think about the cows again. 
 
 Such substitute signs are not words. If I say ' S is P,' or ' All 
 S is P,' S and P are not words. They would be ' words ' only if 
 they were intended to fixate attention on the letters of the alphabet 
 indicated. They are mere symbols, and do not call attention to 
 their meaning. ' A word,' it has been well said, ' is an instrument 
 for thinking about the meaning which it expresses ; a substitute 
 sign is a means of not thinking about the meaning which it 
 symbolizes ' (ibid., p. 194). 
 
 But to return to the natural function of expressive signs, which 
 is to fix meanings with a view to rendering them unambiguous 
 and stable. Meanings are naturally volatile ; in Hegel's expressive 
 phrase, they have hands and feet. It is indeed no easy task for 
 words to keep even pace with the march of thought. While the 
 meaning runs through a succession of changes, the word has a way 
 of remaining unchanged. 
 
 The change in the meaning of a word tends to take place in one 
 of two opposite directions : it may become more generalized, or 
 it may become more specialized.
 
 Chap. I.] WORDS : THEIR FUNCTION AND USE 15 
 
 Instances of Generalization : 
 
 (a) ' Journej- ' and 'journal.' 'Journey' (Fr. journee) was 
 originally one day's march. ' Journal,' originally a daily paper, 
 has been generalized to include ' weekly ' as well. 
 
 (b) ' Charm ' and ' enchant.' From Lat. carmen, ' song or in- 
 cantation,' and ' incantare.' In Elizabethan English both words 
 involved the notion of ' spell, magical power.' Portia says to 
 Brutus : ' I charm you. . . . That you unfold to me why you are 
 heavy' ('Julius Caesar,' II. i. 271). Here 'charm' means 'lay 
 a spell upon,' and so ' adjure.' Cf. Milton's ' Samson Agonistes,' 934 : 
 ' Thy fair enchanted cup and warbling charms.' As the belief in 
 magic declined, the meanings of both words widened, so as to 
 include influences other than magical. Cf. also ' villain ' and ' clerk.' 
 
 Instances of Specialization : 
 
 (a) ' Success.' In Elizabethan English its usual sense is ' result,' 
 ' fortune,' whether good or bad, Cf. ' Troilus and Cressida,' II. 
 ii. 117 : ' Nor fear of bad success in a bad cause,' 
 
 (6) ' Stare,' ' to stiffen, stand on end,' is used in Shakespeare of 
 hair as well as eyes. Brutus says to Caesar's ghost : ' Thou mak'st 
 my hair to stare ' (' Juhus Caesar,' IV, iii, 280). 
 
 (c) ' Knave ' was originally ' boy ' (German ' Knahe '). The word 
 seems to have been speciaHzed in so far as it now implies dishonesty, 
 and at the same time generalized to include man as well as boy. 
 
 But the intrinsic vitality of thought presses in a still more funda- 
 mental way against the pretensions of language to fix it. The 
 meaning of words is always tending to vary with the context. 
 Adopting Professor Stout's terminology, we may conveniently refer 
 to meaning fixed by context as ' occasional ' meaning, and oppose 
 it to meam'ng fixed by usage, to what we may call the ' dictionary ' 
 meaning of a word. We may look upon the usual interpretation 
 as a sort of fictitious mean position about which the meaning of the 
 term oscillates, and the occasional meanings as the shghtly divergent 
 positions where the balance has oscillated somewhat from the mean 
 position. Thus, if we compare together the following expressions : 
 ' the Queen of Sheba,' ' the Queen of the May,' ' the Quaen of the 
 hive,' ' the Queen of Hearts,' ' the Queen of puddings,' we shall 
 notice that the word ' Queen ' rings differently in the different 
 phrases. Its hving meaning varies from phrase to phrase : a queen 
 in Solomon's palace is not a queen in the same sense as in a pack 
 of cards or even in a hive. 
 
 As an illustration of the influence which context exercises over 
 meaning. Professor Bosanquet's analogy (' Essentials of Logic,' 
 p. 55) may be appropriately cited. He is speaking of a very fine 
 Turner landscape which in 1892 was in the ' Old Masters' Exhibi- 
 tion ' at Burlington House — the jpicture of the two bridges at V^alton-
 
 IG THE PROBLEM OF LOGIC [11. i. 
 
 on-Thamcs. The picture is full of detail — figures, animals, trees, 
 and a curving river-bed. Experts tell us that the organic unity of 
 the parts of that picture is such that, if we were to cut out the 
 smallest appreciable fragment of all this detail, the whole effect of 
 the picture would be destroyed. Now consider this patch of colour 
 wliich we will suppose has been cut out. If seen on a piece of paper 
 by itself, it might be devoid of all significance ; but put it back into 
 its proper place, and it shares at once in the whole beauty and 
 meaning of the picture, takes its part in the picture's life. So a 
 word (colourless enough when seen by itself in its usual meaning 
 as conventionalized by definition), when placed in an appropriate 
 setting, takes on at once the glow of the context. 
 
 The Right Use of Words [Logical Aspect). 
 
 The essential function of words being to fix meanings, the super- 
 vision which Logic exercises over them must consist in guiding and 
 rectifying this intrinsic tendency of language so as to make it the 
 best possible medium for expressing the truth. The essential fact 
 we have to reckon with in this regulation of the function of language 
 as the expression of ideas is that ideas show an intrinsic plasticity 
 and indefiniteness, that meanings grow and vary with the context. 
 Hence, any policy which tends ruthlessly to stereotype the meaning 
 of words would obviously run counter to the proper fulfilling of the 
 essential function of language, which is to express thought. If such 
 definite fixity is imposed upon the use of a word, it will be for special 
 purposes, as when, in the case of the elaborate technology of Science, 
 every other requisite of expression is subordinated to the paramount 
 desideratum of precision. 
 
 This natural tendency of words to fix the meanings they express 
 receives its true logical guidance from the Principle of Non- Am- 
 biguity. This is not the same as the Principle of Identity to be 
 discussed further on, and if we venture to call it the first law of 
 correct and consistent thinking, it is first not for thought itself, but 
 for us who are making our way gradually towards the more inward 
 principles that express most truly the nature of our thinking. It 
 is essentially a limiting or negative principle. It insists, in the 
 interests of right thinking, that the natural indefiniteness and fluency 
 of our meaning shall never reach the point of ambiguity. But it 
 has no quarrel witli an appropriate indefiniteness in the use of words, 
 provided this indefiniteness is definite enough for the p'-irpose — i.e., 
 does not amount to ambiguity. In this sense we see the truth of 
 the saying that Logic is the medicine of the mind. It is only when 
 ambiguity is felt that Logic presses upon us its remedy of definitions 
 
 In interpreting and regulating the tendency in language to render 
 our thinking determinate, Logic has not infrequently to unfix in 
 order to fix better. It unfixes the casual non-purposive association.
 
 Chap. II.] DEFINITION AND THE PREDICABLES 17 
 
 that have grown up at random, undisciplined by reference to any 
 self-consciously held ideal, practical or theoretical. Language, if 
 unthinkingly used, plays the tyrant over our thinking. We may 
 easily become the slaves of words. We may allow a word to gather 
 about it a cluster of subjective associations with which we insist on 
 investing it whenever it is used, never troubling to inquire whether 
 the word in the new context, or as used b}^ the author we are study- 
 ing, docs not mean something quite different from such meaning as 
 we have come to attach to it. In the interests of right thinking, 
 words should stand loose from such associations, so as to take on 
 any desired meaning, the logical ideal requiring only that the mean- 
 ing shall not involve any ambiguity or unreasoned inconsistency. 
 
 CHAPTER II. 
 
 II. (ii.) DEFINITION AND TPIE TREDICABLES. 
 
 Definition per Genus et Differentiam. 
 
 Is ordinary talk we are not over-careful of the right use of words, 
 provided we can make ourselves suificicntly intelligible for practical 
 purposes. If a friend happens to use a word with wliich we are not 
 familiar, we ask him what he means by it ; but we are, as a rule, 
 quite satisfied with his answer if it be sufficiently definite to show 
 us what he is referring to. We are satisfied if he describes to us the 
 meaning of the unfamiliar word. Mr. Alfred S'dgwick has given a 
 name to this kind of information. He calls it ' translation.' ' De- 
 scription ' seems, however, a simpler and more satisfactory term. 
 Description in this sense consists in giving a general account of a 
 word's meaning. It gives us the rough meaning of the word. ^Ir. 
 Sidgwick is anxious, and rightly so, th;at we should not confuse 
 description (or unelaborated definition) with definition proper. 
 Etj^mologically, definition means marking out the limits or boun- 
 daries of the use of words, and tliis, as a rule, we never trouble to 
 do in ordinar}' discourse. We arc content to speak with a certain 
 amount of useful vagueness. The words we use are clear enough 
 at their centre, but they have misty edges. Indeed, apart from a 
 certain inherent indefiniteness of contour, they would cease to be 
 really useful ; for it is the very indefiniteness of words which permits 
 of their taking on different shades of meaning according to context. 
 But, as Mr. Sidgwick points out, indefiniteness does not mean 
 ambiguity, though it is a precondition of it. If a word were definite 
 through and through, with clear-cut edges in addition to a well- 
 marked centre, it could never bo ambiguous. Words become 
 
 2
 
 IS THE TROBLEM OF LOGIC [11. ii. 
 
 ambiguous when their inherent indefinitencss has become such that 
 it perplexes the meaning of what we say. Take the word ' Liberal.' 
 ' The indofiniteness,' sa3^s ^Ir. Sidgwick,* ' which was latent in the 
 name up to the beginning of April, 188G, became a few months 
 afterwards so patent as to cause ambiguity ; witliin what used to 
 be called the Liberal party there had come to hght two sub-classes, 
 each of which denied to the other the right to the name.' The single 
 meaning had split in two ; the word had no longer one well-marked 
 centre, but two ; and so long as wc were not told, on being spoken 
 to about Liberals, whether C^ or C was being referred to, ambiguity 
 would ari:?e. 
 
 We conclude, then, tliat if we would use our words rightly, we 
 must be able — (1) to recognize the point at which definition becomes 
 necessary ; and (2) to know how to set about discovering the defini- 
 tion when required. 
 
 To sum up as regards (1), we have to recognize that, even when 
 there is doubt as to the meaning of a term in an assertion, a defini- 
 tion is not necessarily called for. To define a word formally is to 
 mark off its edges from the encroachments of other words, and 
 there is no point in being precise about the edges if there is uncer- 
 tainty about the centre. A definition, in fact, is rarely wanted 
 unless the rough meaning of a word is already known. If the 
 difficulty in grasping the meaning of a sentence arises from un- 
 familiarity with any word, description is called for, not definition ; 
 but if an actual difficulty is felt in applying a familiar word correctly 
 in a given case — that is, whenever the latent indefinitencss natural 
 to the word is actually causing ambiguity — then definition is called 
 for.t 
 
 If it is called for, how are we to set about the work of defining ? 
 The natural answer is : Through a process of Comparison. Words 
 at their outer edges are in contact with other words, and the respec- 
 tive sphere of influence of each can be marked out only by com- 
 paring and adjusting the meanings. To define a word, we must 
 compare it with such words as are most closely related to it in 
 meaning. This gives us the Genus and Differentia. The genus 
 includes the marks which the word has in common with the rest ; 
 the differentia those which distinguish it from them. 
 
 We may express this result in a slightly different form. Defini- 
 tion, we may say, is the process whereby we assign to a word — (1) its 
 class-designation, and (2) the specific difference which serves to 
 distinguish it from all other words that share the same class- 
 designation. 
 
 Experience siiows that, though nothing is in all respects like any 
 otlier thing, yet things can be separated out into groups, each group 
 comprising all those different objects which resemble each other in 
 
 * A. Sidgwick, ' The Use of Worda in Reasoning,' p. 196, 
 t IhvJ,., p. 49.
 
 Ciur. II.] DEFINITION /VND THE PREDICABLES 19 
 
 certain points — Pj, Pg, P3, P4. The objects are then said to be 
 classed, and the class-name defined, by these common marks — 
 Pp P„, P3, P^. Anything that possesses tliese common marks is 
 then designated by the class-name, also called the general name. 
 Further, the class-name, as such, cannot possibly specify distinc- 
 tions between the included sub-classes. The name ' horse ' cannot 
 inform me whether a cart-horse or a race-horse is in question. If 
 I wish, therefore, to specif}^ a particular section of a class, or, in 
 other words, to differentiate a species from the genus, I must add a 
 qualifying mark, or differentia. Thus, if I wish to define the 
 kind or species of vehicle known as ' omnibus,' I ask myself : What 
 is the genus or class under which this species falls, and what is the 
 differentia, or specific mark, whereby it is distinguishable from 
 whatever other species fall under the same genus ? Now, practi- 
 cally, as we have seen, we answer this question by bringing together 
 as many words with closely related meanings as possible, and com- 
 paring them. Let us compare ' omnibus,' for example, with ' tram.' 
 The terms agree in designating four-wheeled pubhc veliicles ; they 
 differ essentially in this : that, whereas the one designates such 
 vehicles of tliis kind as are confined to rails, the other designates 
 such as are not confined to rails. 
 
 Genus : Four-wheeled public vehicle. 
 
 Species : Omnibus. Tram. 
 
 Differentia : Not confined to rails. Confined to rails. 
 
 If we had compared the two terms ' omnibus ' and ' cab,' we should 
 have had some such result as tliis : 
 
 Genus : Four-wheeled public vehicle, not confined to rails. 
 
 Species : Omnibus. Cab. 
 
 Differentia : Keeping to well- Not keeping to well- 
 
 defined routes. defined routes. 
 
 If we had compared ' omnibus,' ' cab,' ' tram ' together, we should 
 have had some such result as this : 
 
 Genus : Four-wheeled public vehicle. 
 
 Species : Omnibus. Tram. Cab. 
 Differentia : Keeping to Keeping to well defined Not keeping to well- 
 well-defined routes, and routes, and confined defined routes, and not 
 not confined to rails. to rails. confined to rails. 
 
 This defining by direct comparison, and by assigning genus and 
 differentia, is by far the most convenient for practical purposes ; 
 for it is of the essence of practical requirement that it should adapt 
 itself to the exgiencies of the specific occasion. The definition found
 
 20 THE PROBLEM OF LOGIC [II. ii. 
 
 by consulting a dictionarj^ is likely to have this defect : that it will 
 not precisely suit the occasion. The only way in which to make 
 defirition relevant is to select for ourselves the kindred terms with 
 which the term in question is in risk of being confused, and then 
 to note, from the point of view that happens to be interesting us, 
 the differentia which distinguishes its use from that of all these 
 kindred terms. 
 
 The Relation of Genus to Differentia. 
 
 Taken together, Genus and Differentia state the marks essential 
 to the definition. They include just those features which are logically 
 indispensable for the imambiguous statement of our meaning. 
 
 The relation, however, between the two types of definition-mark 
 — the generic and the specific — cannot be adequately represented 
 by placing them side by side as though they were of co-ordinate 
 significance. The differentia, as the specific mark, specifies, and 
 therefore logically presupposes, the generic mark or genus : it is a 
 specification of the genus. And, though the process of comparison 
 through which our occasional definitions are framed does not 
 explicitly bring out this connexion, the connexion is none the less 
 definitely implied. It is concealed only by the logical incomplete- 
 ness of the comparison process as we conduct it. Were this process 
 thorough-going, the marks of agreement between two terms would 
 include not only determinate, but also indeterminate marks, so far 
 as these latter were relevant to our purpose in defining ; and the 
 differentia would tlien reveal itself quite naturally as a specification 
 of one or other of these indeterminate marks of agreement. To 
 define ' tram,' we compare it with ' omnibus,' from the point of 
 view, say, of public transit. The two terms agree determinately in 
 signifying four-wheeled public vehicles, but they also agree inde- 
 terminately in requiring some distinctive method of proceeding from 
 starting-point to destination. The differentia ' confined to rails ' 
 just specifies what this distinctive method must be in the case of a 
 tram. It is thus only in relation to the indeterminate elements of 
 the genus that we could endorse Mr. Joseph's contention that ' the 
 genus is the general type or plan, the differentia the " specific " 
 mode in which that is reahzed or developed.'* 
 
 Let us take an illustration suggested by Mr. Joseph himself. The 
 genus of A and N might be taken as ' plane rectilinear three-sided 
 construction, possessing some specifiable arrangement of the three 
 sides.' The differentia of the term ' triangle ' — namely, ' enclosing 
 a space ' — would then be a specification of the above indeterm'nate 
 mark ; in the case of a triangular construction, the sides are so 
 arranged as to enclose a space, f 
 
 * Joseph, 'An Introduction to Logic,' p. 6S. Cf. p. 70. 
 
 t Mr. .Joaeph points out that the conception of ' species ' as the specification of 
 the ' genus ' forbids our describing a genus as a larger class including the smaller
 
 Chap. II.] DEFINITION AND THE PREDICABLES 21 
 
 If further justification be required for the admission of tlie in- 
 determinate mark into the structure of a definition, we may find it 
 in the fact that it is necessitated by the very nature of the generali- 
 zation process through whicli our definition is reached. The process 
 of Generahzation — or of its main feature, Abstraction — may be so 
 understood as to stultify the attempt to connect genus and species 
 vitally together. We may understand by it a process whereby 
 differences are ruthlessly eliminated, and points of agreement 
 reduced to mere identities — identities disengaged from all relation 
 to difference. But if the abstraction of genus from species implies 
 this logical isolation of the marks of agreement from the marks of 
 difference, it is manifestly impossible to consider the species as 
 specifications of the genus. If in mounting, through generalization, 
 from species to genus, we sever the vital bond between the lower 
 and the higher class, we cannot, when descending, through differ- 
 entiation, from genus to species, behave as though tliis bond were 
 still unsevered. 
 
 But it is surely gratuitous to suppose that generalization (or 
 abstraction) is a devitalizing process of this kind. It is, of course, 
 possible to conceive it after this fasliion, and the Formal Logician 
 has almost invariably done so. But just in so far as we embrace a 
 true conception of identity, and abandon the old static view of it 
 as typified in the formula ' A is A,' we are compelled to entertain 
 new ideas about Abstraction. To abstract agreement from differ- 
 ence, we find, is not to isolate them one from another, but to connect 
 them in a new way. It is through the Abstraction process itself 
 that the difference becomes a specification of the agreement — the 
 agreement a generalization of the difference. Abstraction does not 
 take us from differences that have no identical element to identities 
 that are out of all relation to differences : it takes us from the deter- 
 minate to the relatively indeterminate. But the indeterminate 
 so reached still points back to the specifications from which it has 
 been abstracted. ' Colour ' does not mean that which is neither 
 violet, nor red, nor blue, nor any other colour ; it means ' colour of 
 some kind,' and, when its meaning is pressed a little further, it is 
 seen to signify ' violet, or red, or blue, or some other colour.' As 
 abstracted from these differences, it still stands to them in what 
 
 classes or species within it, and consequently renders the attempt to represent the 
 relation by means of two circles, one within tlie other, entirely misleading. ' The 
 word "class,"' he says {ibid., p. 69), 'suggests a collection, whereas the genus of 
 anything is not a collection to which it belongs, but a scheme which it realizes.' 
 Now. in so far as we are reading the class in intension or conno-denotation (vide 
 p. 72), it is undoubtedly necessary, in the sense above described, to consider it 'as 
 something realized in its various members in a particular way ' {ibid., p. 71) ; but 
 from the point of view of extension {vide p. 158) it is at least reasonable, and may be 
 
 {)urposive, to depict the objects indicated by the class-term as included witliin the 
 argor number of objects indicated by a second class-term. But to admit this is to 
 adiuit that the one class (extensively defined) can be included within the other class 
 (also extensively defined).
 
 22 THE PROBLEM OF LOGIC [II. ii. 
 
 is at least a pre-di'^jiinctive relation. The genus, as abstracted 
 from the species, still points back to the species from which it has 
 been abstracted. A man is a rational (animal of some kind) ; an 
 animal is a sentient (organism of some kind). We conclude, then, 
 that Generalization (or Abstraction), when properly interpreted, 
 works in the service of the logical evolution of meaning. The genus, 
 qua abstract vestige, is potentially the rudiment or germ of wliich 
 the species are the specifications. It requires but the interest in 
 the logical development of meaning to transform it actually from 
 the one to the other.* 
 
 The Predicahles. 
 
 The theory of practical definition, as outlined in the foregoing 
 discussion, is closely connected with the Aristotelian doctrine of 
 the Predicablcs. The Predicahles, for Aristotle, were the various 
 kinds of attribute which might be predicated of a subject. If I 
 make the statement ' S is P ' (where S is a class-concept), P may 
 stand to S in any one of five possible relations. It may be its 
 definition — i.e., it may give the genus and differentia of S. Or it 
 may be the genus alone or the differentia alone. Finally, it may 
 give a proj^erty or proprium of S, or else an accident. These ' heads 
 of predicahles,' as they are sometimes called, ' have passed,' to 
 quote Mr. Joseph again, ' into the language of science and of ordinary 
 conversation. We ask how to define virtue, momentum, air, or a 
 triangle ; we say that the pansy is a species of Viola, limited 
 monarchy a species of constitution ; that one genus contains more 
 species than another ; that the crab and the lobster are generically 
 different ; that man is differentiated from tlie lower animals by 
 the possession of reason ; that quinine is a medicine with many 
 valuable properties ; that the jury brought in a verdict of acci- 
 dental death ; and so forth ' {ibid., p. 54). 
 
 There is a later scheme of Predicahles connected with the name 
 of Porphyry, a logician who wrote some six hundred years after 
 Aristotle. Superficially, the sole difference between Porphyry's 
 scheme of Predicahles, as given in his JLlaayar/i], and the older 
 scheme of Aristotle himself, appears to be the substitution of the 
 predicable of ' species ' for the prcdicablc of ' definition.' The 
 predicahles, for Porphyry, are genus, species, differentia, proprium, 
 and accidens. But the substitution in question conceals a more 
 fundanif-ntal disagreement between the two schemes. In the case 
 of Aristotle the subject-term meant a common nature, a kind, species, 
 or universal, and not the individual object as such. The predi- 
 cahles were, therefore, one and all, predicated about a species, and 
 
 * Cy. with the above the discussion in Chapter VIII., p. 88. The distinction 
 bi^tween the abstraction implied in generalization and tlic abstraction which results 
 in ' abstract terms ' should be noted.
 
 CuAP. II.] DEFINITION AND THE PREDICABLES 23 
 
 it would have been obviously tautological to predicate the species 
 of itself, and therefore illogical to include the ' species ' among the 
 predicables. Witli Porphyry the subject about which the predicable 
 was predicated might not only be a species, but an individual object. 
 In this latter case it was reasonable to predicate the species under 
 which it stood, and so ' species ' found its place among the predi- 
 cables. 
 
 In adopting the Aristotelian scheme of predicables, we at the 
 same time reinterpret it ; for the point of view from which we regard 
 the whole problem of the predicables is essentially different from 
 Aristotle's. Aristotle's outlook was objective. He considered the 
 content of the object as such, and not in its relation to the intent — 
 i.e., the intention — of the subject. To define a thing was to state 
 that which made it what it was, and was therefore essential to its 
 existence. But if we admit that ' essential ' necessarily means 
 ' essential from a certain point of view,' and thus admit the principle 
 that definition is strictly relative to purpose, we have qualified the 
 Aristotelian standpoint in a way so vital as to preclude any appeal 
 to the authority of Aristotle. 
 
 With a view to bringing out the positive significance of the position 
 which we have adopted in regard to the problem of definition, we 
 turn now to the vexed question of the Object of Defijiition. Mean- 
 ing, we would say, is the direct object of definition. What, then, 
 do we understand by Meaning ? 
 
 Meaning, as we conceive it, is, in the first place, a product of 
 thought in its relation to reahty, or of reality in relation to thought. 
 Meaning, in other words, is the meaning of an object for a subject ; 
 or, more specifically, it tells us what an object is in relation to a 
 specified interest or purpose. 
 
 Meaning is thus a product of objective Nature and subjective 
 interest, or, if we prefer it, of objective content and subjective intent. 
 It must not only be the meaning of what is, and so objective in 
 regard to content ; it must be our meaning, and so subjective in 
 regard to the dcfiner's intention or intent. 
 
 Again, in defining meaning we may have in view either some 
 restricted practical purpose or the broader interests of Science. 
 This distinction we may appropriately equate to the familiar dis- 
 tinction between formal and real definition. The formal definition 
 is a conventional definition framed to fit a specific interest that 
 involves no more than a merely fragmentary hold on objective 
 reality. The framing of real defuiitions, on the other hand, is 
 ultimatel}'' controlled by one and the same unvarying ideal — 
 namely, that of bringing the greatest possible simplicity and order 
 into our grasp of Nature. 
 
 Meanings, again, are fixed and made definite through the use of 
 words. Hence, to define the meaning of an object is at the same 
 time to define the meaning of the word which symbolizes it. We
 
 24 THE PROBLEM OF LOGIC [II. ii. 
 
 do not, of course, define words apart from their meaning. What 
 is defined by the term ' rational animal ' is not the mere sound-sign 
 ' man,' but its meaning. If bj^ words we mean the mere sound- 
 signs in themselves, we cannot be said to define words, nor even to 
 ' describe ' them, but only to utilize them as sensory supports for 
 meanings which can be defined. 
 
 The question ' What is it that we define, things, meanings, or 
 words ?' has been the theme of immemorial controversy. There 
 have been three rival parties. The realists have maintained that 
 it is things that we define ; the conce})tualists, that we define 
 meanings ; the nominahsts, that we define words and names. Tiie 
 controversy hinged on the meaning of the ' universal.' The realists 
 held that tilings had, in all those relations in which they resembled 
 each other, a common or universal nature, and that, in defining this 
 common nature, we were defining what was at least as genuine and 
 indispensable a constituent of reahty as was the individual nature 
 of objects. The conceptuahsts held that the universal element 
 existed, not in the objects themselves, but only in the thought which 
 conceived tliem ; the true universal was the concept. Finally, 
 the nominalists held that things called by the same name had 
 nothing in common but the name. The universal was thus a mere 
 convenience of language. The only true existent, whether in reality 
 or in thought, was the individual, and the individual was conceived 
 by the nominalist in a sense which excluded the presence within it 
 of any universal nature. 
 
 The conflict between these rival views was a conflict between ab- 
 stractions which, far from being intrinsically hostile to each other, 
 were, in reality, mutually complementary and indispensable. 
 
 We have already suggested that the definition of meaning is 
 always at the same time the definition of an object, and to this 
 extent the definition is realistic : definition is always definition of 
 objective content. On the other hand, such objective content, wc 
 hold, is definable only in relation to subjective intent, so that, 
 in defining the object, we arc defining it as conceived in the light 
 of this or that specific interest. To this extent our point of view 
 might be cliaracterized as conceptualistic. Still, it is not abstract, 
 but, shall we say, concrete conceptualism. The conceptualism we 
 have ado];>ted is simply reahsm tempered by the requisite of reference 
 to purpose. 
 
 According to tlic interest or purpose engaged, this plastic con- 
 ceptuahsm may bear any shade of meaning, from the limiting case 
 of a mare conceptualism to an ideahsm in which the realistic element 
 is completely transfigured. If what is essential to me in defining a 
 term is primarily and predominantly this, that my meaning shall be 
 clearly and unambiguously understood, the nature of the object 
 counts for little in the definition, and my meaning has but a vanish- 
 ing reference to objective reality. This is logical conceptuahsm in a
 
 Chap. IL] DEFINITION AND THE PREDICABLES 25 
 
 strict but still intelligible sense. It is governed by an interest in 
 the logical purity of meanings as such. If, on the other hand, my 
 interest in the meaning of an object — the interest that it has for me, 
 the subject — Hcs primarily in discovering what that object means, 
 or tends to mean, within the spiritual unity of the universe, the 
 conceptualism is transformed into idealism, and my definition will 
 answer to the logical requirements of ideaUstic conviction. 
 
 We have finally to add that the true logical nominahsm, in its 
 relation to the problem of definition, is indistinguishable from con- 
 ceptualism. To define a word is to define its meaning : we do not 
 define as a mere sound-complex the aggregate of vowels and con- 
 sonants which make up a word. When we say ' Man is a rational 
 animal,' we are not defining the mere verbal label or sign repre- 
 sented by the three letters m, a, n, arranged in a certain order. All 
 definition of meaning is at the same time verbal definition, and 
 vice versa. The distinction between nominahsm and conceptualism, 
 in definition, is a distinction without a difference. 
 
 The statement that we do not define mere sound-complexes as 
 such may easily be misunderstood. It may be taken to mean that 
 we do not even define the meanings of symbols qua symbols. But 
 this is by no means implied in the statement. Any and every 
 meaning, as we hope eventually to sliow, is definable in some true 
 sense of the word. The meanings of symbols as such are indeed 
 definable. I define the conventional symbol ' man ' when I say : 
 ' " Man " is a conventional verbal symbol representing the concept 
 " rational animal." ' Every symbol has, in fact, a twofold mean- 
 ing : the meaning of the symbol qua symbol, and the meaning of 
 the idea which is symbolized by the symbol. The meaning, in a 
 word, may be the meaning either of the sign or of the signification. 
 When I say ' man ' means a rational animal, I am defining the 
 meaning of the sign ; when I say ' man ' is a rational animal, I am 
 defining the meaning of the significate.* 
 
 We now jDroceed to apply the logical doctrine of meaning and of 
 definition, as we have just been formulating it, to the non-defining 
 predicables, property and accident. 
 
 A property or proprium is an attribute wliich, though not neces- 
 sary to the definition itself, is still relevant to the defining interest. 
 It is thus already present by implication in the meaning wliich an 
 object has for us in the light of a specified interest. 
 
 Thus, in the geometrical proposition ' The equilateral triangle is 
 equiangular,' the predicate states a proprium of the subject. 
 ' Triangle ' is the genus, ' equal-sided ' the differentia, ' equiangular ' 
 the proprium. The equiangularity of an equilateral triangle is 
 implied in the system of spatial relations, apart from which an 
 equilateral triangle has no geometrical meaning, and our geometrical 
 interest no real object. The geometrical interest in an equilateral 
 
 * For a further development of this point, cf. pp. 115, 121.
 
 -0 THE PROBLEM OF LOGIC [II. ii. 
 
 triangle presupposes this reference to the nature of Space, and the 
 equilateral triangle is conceived as constructed in Space as Geometry 
 treats of it. The very construction furnishes the definition. We 
 trace out a plane rectilinear figure with three equal sides, enclosing 
 a space — i.e.. a triangle with tliree equal sides. But when we come 
 to examine the ' properties ' of the triangle as thus constructed, 
 we discover that one of these is ' equiangularity.' As a further 
 property of an equilateral triangle, qua triangle, we have the fact 
 that the three internal angles are collectively equal to two right 
 angles. 
 
 Let us look a little more closely at the relation between proprium 
 and defuiition. A definition, as we have seen, is the definition of an 
 objective nature qua related to some definite interest or point cf 
 view. It would, however, be irrelevant to include within the 
 definition whatever was relevant to the interest ; for the function of 
 Definition does not extend beyond the removal of ambiguity, and 
 there may be much that is jDerfectly relevant to the interest, but 
 which, so far as mere non-ambiguity is concerned, need not be 
 explicitly stated. The propria, therefore, develop, from the point 
 at which Definition stops, the meaning of the objective nature that 
 is being defined. What the definition states is only that fraction 
 of the essence which its own logical principle — the principle of non- 
 ambiguity — requires it to state. The residue is developed in the 
 form of propria. 
 
 We must distinguish between two tj^^es of propria — two at 
 least, for we may eventually find it convenient to add a third. 
 Properties may be either ' implied ' or ' characteristic' They are 
 ' implied ' when they are deducible with logical necessity from the 
 nature we are interpreting, as fixed by the definition in strict 
 relevance to the defining interest. Thus ' equiangular ' is an 
 ' implied ' property of ' equilateral triangle,' for it can be deduced 
 with logical necessity from the geometrical space-construction 
 defined by ' three-sided plane rectihnear figure enclosing a space ' 
 and by the differentia of ' equal-sidedness.' 
 
 A property is ' characteristic ' when it predicates of the nature 
 we are interpreting an attribute which, without being ' implied,' 
 can be shown by observation or experience to be both typical of 
 that nature and relevant to our interpreting interest. Thus, from 
 the point of \iew of biological science, such attributes as ' con- 
 tractile,' ' irritable,' ' assimilating food,' ' reproducing itself after its 
 kind,' would be characteristic properties of an ' organism.' 
 
 The Meaning of ' Essence.^ 
 
 By ' essence ' or ' essential meaning ' we aim at expressing the 
 contact between an objective nature and a subjective interest. 
 What is indispensable to the conception of ' essence ' is this interplay
 
 Chap. II.] DEFINITION AND THE PREDICABLES 27 
 
 between content and intent. It will thus be seen that, from the 
 logical point of view, the point of view of right-thinking, ' essence ' 
 and ' meaning ' are synonymous terms. All meaning is essential 
 meaning, tiiough some types of meaning are more intimately 
 essential than others. From the point of view we have adopted, 
 the non-essenticxl or accidental — that which implies no interplay 
 between content and intent — is logically meaningless. It is meaning- 
 less for the interest in question, and therefore meaningless for right- 
 thinking, wliich is so constructed as to be unable to assimilate the 
 irrelevant as such. 
 
 Sonae types of meaning, we have just said, are more intimately 
 essential than others. In so far as the intent is an interest in 
 defining the content up to the point required for satisfying the 
 principle of non-ambiguity, the essence of our meaning is given by 
 genus and differentia. In so far as the intent takes us beyond genus 
 and differentia to other marks which are still relevant to it and 
 characteristic of the content, the essence of our meaning is given 
 more inclusively by propria as well, by ' implied ' or ' characteristic ' 
 properties. But there is yet a third form of interplay between 
 content and intent. The essence of our meaning becomes still more 
 inclusive if .we reckon among the marks which are relevant to our 
 intent, and in tliis sense essential to it, features wliich, though 
 relevant, are problematic. Thus a building may he a palace, a 
 palace may he the palace of a king. From the point of view of a 
 general interest in buildings as edifices for social uses, the possi- 
 bility of being a palace is a perfectly relevant mark of a building, 
 and the possibility of a palace being a royal palace a perfectly 
 relevant mark of a palace. Such ' problematic ' properties, as we 
 may call them, need not be actuallj^ realized in any concrete in- 
 stances of the meaning or nature in question. Any type of building 
 which the architect could imagine, plan, and realize if need be, 
 would be a problematic property of ' building.' It might be con- 
 venient to give a special name to such problematic properties as 
 were not only capable of reahzation, but actual^ reahzed in at least 
 one concrete instance or occasion. We might refer to these as 
 ' occasional ' properties. Thus, from the architect's point of view, 
 it would be an occasional property of a building to be a palace or a 
 country-house. Problematic properties which were not occasional 
 in this sense might be referred to as ' purely problematic' It might 
 be possible to build a house which should have the precise shape of 
 an elephant or of an icosahedron ; but, until such houses are actually 
 built, the device in question remains a purely problematic property. 
 
 Problematic properties should not be identified with accidents 
 or accidental marks, as we have defined them above. The 
 genuine accidents, from the general point of view we have adopted, 
 must be marks which are irrelevant to our intent, and so entirely 
 outside the interplay of intent and content. Thus, in a flower, the
 
 28 THE PROBLEM OF LOGIC [11. ii. 
 
 colour, which to the artist is essential, is to the botanist relatively 
 accidenta,l, whilst the microscopic characters so important to the 
 botanist are, from the artist's point of view, entirely negligible. 
 Again, if my interest lies in the assuaging of my thirst, tumbler, 
 mug, and other appropriate vessels are all ahke to me : the handle 
 of the mug and its absence in the tumbler are mere accidents, for 
 they do not in any way affect the fulfilling of my interest. So, 
 again, despite the fact that the burning of wood and the rusting of 
 iron are both processes of oxidation, and so chemically akin, they 
 are still essentially different for the person who is seeking warmth. 
 To such an one the resemblances which interest the chemist are 
 purely irrelevant, and in this sense accidental. 
 
 It may be objected that accidents as pure ' irrelevants ' are not 
 predicables at all, for no one can logically predicate of a subject 
 what is irrelevant to it. Subject and predicate are united in the 
 interest which prompts the making of the statement, and, as so 
 united, are relevant to each other. This may very well be granted, 
 in which case the ' accidents ' of Aristotle's scheme become identical 
 with the ' problematic properties ' of the scheme that we have 
 adopted, and the accident, in the guise of a realizable possibility, 
 enters, in an intelligible way, into the essence of our meaning. The 
 predicables are then reducible to jour — definition, genus, differentia, 
 and property ; a property being either ' imphed,' ' characteristic,' 
 or ' problematic,' and a problematic property being either ' pure ' 
 or ' occasional.' 
 
 One word more on the problem of Essence. Once the intent or 
 defining purpose is determined, and the content limited to what is 
 strictly relevant to the intent, the meaning of Essence is logically 
 clear. But in ordinary irreflective thought we are, as a rule, neither 
 self-conscious of our defining purj^ose, nor do we consistently apply 
 it to the deciphering of a given content. We are largely the slaves 
 of suggestion and habit. When we habitually experience certain 
 things together, we come, in accordance with well-known laws of 
 mental association, to conceive them as inherently belonging to 
 each other. Indeed, we show independence of mind just in propor- 
 tion as we cease to be the slaves of such association. I quote the 
 following from Dr. Watts's ' Logic ' : 
 
 ' A court lady, born and bred amongst pomp and equipage and 
 the vain notions of birth and quality, constantly joins and mixes all 
 these with the idea of herself, and she imagines these to be essential 
 to her nature, and, as it were, necessary to her being. Thence she is 
 tempted to look upon menial servants and the lowest rank of man- 
 kind as another species of beings, quite distinct from herself. A 
 ploughboy that has never travelled beyond his own village, and has 
 seen nothing but thatched houses and his parish church, is naturally 
 led to imagine that thatch belongs to the very nature of a house, 
 and that that must be a church which is built of stone, and es-
 
 Chap. II.] DEFINITION AND THE PREDICABLES 29 
 
 pecially if it has a spire upon it. A child, again, wliose uncle has 
 been excessiv^ely fond, and his schoolmaster very severe, easily be- 
 lieves that fondness always belongs to uncles, and that severity is 
 essential to masters or instructors. He has seen also soldiers with 
 red coats, or ministers with long black gowns, and therefore he 
 persuades himself that these garbs are essential to the character, 
 and that he is not a minister who has not a long black gown, nor 
 can he be a soldier who is not dressed in red. It would be well if all 
 such mistakes ended in childhood.' I can add an instance from my 
 own experience. I was taken as a child to see the Crystal Palace. 
 From that day onwards right on to advanced boyhood I firmly 
 believed that a palace was not a palace unless it was made of crj'stal. 
 Palace and stone were two ideas that would not blend in my mind 
 until my further reading gave the necessary shocks to this old super- 
 stition, and the power of reflective thought at length slowly dis- 
 solved it. 
 
 Real or Scientific Definition. 
 
 Of all the spc-cial purposes we have in view in framing definitions, 
 one stands out pre-eminently above all others — that of meeting the 
 requirements of Science. The logical function of Definition is here 
 adjusted to the ideal of a systematized knowledge of Nature, and 
 consists in the removal of all ambiguities which arise in the pursuit 
 of this ideal. 
 
 It will be readily understood that the definitions which are 
 required for ordering our meanings within the vastly complex net- 
 work of relations which subserve the organization of Science cannot 
 be reached in quite so simple a manner as can the occasional defini- 
 tions which subserve our varied practical interests. 
 
 Thus, the mere process of comparing one concept with another will 
 not in any way suffice to define a fundamental physical concept such 
 as that of inertia, weight, mass, or gravitation. In each of these 
 concepts we have the condensed expression of great scientific dis- 
 coveries, the embodiment of higlily elaborated theory ; hence the 
 path to definition here lies not in a process of simple comparison, but 
 in a searching analysis of the interactions and interrelations of the 
 facts of Nature. 
 
 In Geometry such analysis proceeds by the help of construction, 
 and it is by ideally constructing its concepts — e.g., those of straight 
 line and circle — that the definitions of Geometry are reached. Hero 
 the specifj'ing mark is genetic, a mark embodying a rule of con- 
 struction. Thus, ' The circumference of a circle is a line traced by 
 a point which moves in one plane at a constant distance from a 
 fixed point in that plane.' Cf. also the definition of a circle as a 
 section of a cone drawn square to its axis. 
 
 Outside Geometry the genetic definition is not usual, though it
 
 30 THE TROBLEM OF LOGIC [II. ii 
 
 is common in Clicmistry, wlion wc wish to define compounds as 
 made up of their elements. The main interest which Science has 
 in defining the terms it uses is in connexion with the problem of 
 Classification. Order is here the dominating need, and the work of 
 definition is therefore dominated by this general requirement of 
 order. 
 
 Thus the relatively simple and schematic requirements of formal 
 definition are quite inadequate for the purposes of Science : the 
 distinction between formal and scientific definition is inevitable ; 
 but the main value of a distinction of this kind would be lost if, by 
 insisting on it, we were in any way to obscure the essential unity of 
 the defining process at whatever stage of thought we choose to 
 consider it. 
 
 In formal and in scientific definition alike we have necessarily to 
 define by relations, and in reference to a purpose stated or implied. 
 
 In formal definition the subjective reference to purpose is more 
 conspicuous than the objective relatedness to a system of kindred 
 meanings. But tlie connexion of the defined meaning, through its 
 verj' definition, with a system of interrelated meanings is none the 
 less present for not being so obvious. If, in the interest of some 
 restricted purpose, we find it sufficient to define ' Man ' as ' rational 
 animal,' we have still three closely related meanings — those of 
 humanity, rationality, and animahty — systematically involved in 
 the definition. Thus formal definition is essentially relational in 
 character, though in some cases the relational reference is more 
 ai">parent than in others. ' King ' can hardly be defined Avitliout 
 explicit reference to the relations in which Kingsliip stands to the 
 government of the country ruled ; and in a whole class of cases — the 
 so-called class of correlatives {e.g., ' Whole and part,' ' Genus and 
 species ') — the definition of either term involves the statement of 
 its relation to the other. In scientific definition, where meanings 
 are so much more systematically interconnected, the relatedness of 
 the defined meaning, as defined, to a system of Idndred meanings 
 is a much more patent characteristic of the definition than is the 
 reference to purpose, which here comes more defuiitely under 
 objective control. It is true that different sciences have different 
 points of view, but the reference to purpose which this distinction 
 in view-point involves is implied rather than expressed, whereas the 
 relatedness of the meaning to be defined to a whole system of other 
 meanings tends to enter more and more explicitly into the very 
 structure of the definition itself. 
 
 The essential unity of the defining process, whether formal or 
 real, practical or scientific, is perhaps brought out most clearly by 
 the consideration that the process of ' comparison ' through which 
 our practical or occasional definitions are obtained is only a special, 
 simple case of the more general procedure of analysis and synthesis, 
 which we utilize in all defmition processes of a scientific character.
 
 Chap. II.] DEFINITION AND THE PREDICABLES 31 
 
 To have defined a term or concept scientifically is to have analyzed 
 its relations to other concepts characteristic of the same scientific 
 system, and to have then synthesized these relations in the simplest 
 and most relevant way possible. But this involves just those very 
 processes of criticism and reconstruction which we shall find indis- 
 pensable in formal definition when we endeavour to remodel certain 
 given definitions in a methodical manner {vide Chapter III.). 
 
 Note on the Categories. 
 
 It has for long been customary to preface a doctrine of Terms 
 with a statement about ' Categories.' There are forms of thinking 
 about reality wliich ai'e, in a certain important sense, irreducible. 
 Activity is not passivity, time is not place, nor quantity quality, nor 
 substance relation. If we add to these eight varieties the concepts 
 of ' state ' and ' situation,' we have before us Aristotle's complete 
 list of Categories — that is, of ' predicates one or other of which must 
 in the last resort be affirmed of any subject, if we ask what in itself it 
 is ' (Joseph, ibid., p. 38). 
 
 In liis excellent chapter on the Categories (ibid., ch.iii.), Mr. Joseph 
 insists on the importance of this ancient enumeration of the 
 ultimate forms of being. Of the importance of a theory of 
 Categories from the point of view of an analysis of knowledge there 
 can be no possible doubt. The subject is, indeed, so important 
 that, were the discussion once broached, a thorough-going treat- 
 ment would be indispensable. Mr. Joseph has done invaluable 
 service in so lucidly connecting the Aristotelian and Kantian 
 doctrines of the Categories ; but, by the very necessities of an 
 ' introductory ' discussion, the story of the Categories is made to 
 end with tliis reconcilement of Aristotle and Kant, and the great 
 attempt of Hegel to systematize the Categories afresh from the 
 point of view of Thought's own logical development is completely 
 ignored. But even Hegel has not said the last word. There are 
 Neo-Hegelian improvements, post-Hegelian developments, and anti- 
 Hegehan reactions ; there are even some who choose to ignore 
 Hegel altogether. The Categories are, in fact, ' Uving oi^tions,' and 
 cannot be adequately discussed as monuments, however imperish- 
 able, of a past that has no longer any relation to the present. 
 
 It seemed better, therefore, not to enter upon any systematic 
 discussion of the Categories. At this initial stage, at any rate, 
 logical propriety required that the Categories should yield pre- 
 cedence to the Predicables, and that the discussion of terms ' accord- 
 ing to the nature of their meaning ' should make way for that more 
 relevant discussion of them which is ' based upon tlie relation in 
 which a predicate may stand to the subject of which it is predicated ' 
 {ibid., p. 53). 
 
 [On the whole subject of Predicables and of Categories, ^h\
 
 32 THE PROBLEM OF LOGIC [II. iii. 
 
 Joseph's masterly treatment (' An Litroduction to Logic,' ch. iii, 
 and iv.) cannot be too strongly recommended, even by those who 
 venture to dilTer from tlie Aristotehan standpoint from which those 
 two chapters are written.] 
 
 CHAPTER III. 
 
 II. (iii.) THE TESTING OF DEFINITIONS. 
 
 Rules towards securing Soundness in Definition. 
 
 1. We must distinguish definitions from translations and derivations. 
 E.g., if we have two equivalent symbols for one and the same idea, 
 we do not define the one symbol by substituting the other for it. 
 To say that ' dyspepsia ' is indigestion, or that a laundress is a 
 washerwoman, is not to state what dyspepsia or laundress means. 
 Such statements are sometimes called circular definitions ; but why 
 call them definitions at all ? They have as httle title to be called 
 definitions as have the statements, ' Anima is the soul,' ' Mere is 
 mother.' So, again, such statements as ' Sycophant means fig- 
 shewer ' [avKov (paiico) suggest mere derivations. They answer the 
 question, ' What did the word mean once V not, ' What does it 
 mean now V They derive but do not define the term. Still a 
 derivation is in a sense a fossil definition, and so has more right to 
 the name than a mere translation. It might be reasonable to refer 
 to it as an etymological definition. The statement ' Assiduity is 
 sitting close to one's work ' is an etymological definition, so far as 
 ' sitting close ' is concerned, 
 
 2, We must see that the definition fits — that it is neither too 
 narrow nor too wide, that it exactly expresses the meaning we wish 
 to convey by the term we use. In other words, definiendum and 
 definition must be commensurate with each other — i.e., whatever can 
 be relevantly predicated of the object defined must be predicable 
 of the definition also, and vice versa. This is, perhaps, the most 
 important rule of all, and can best be observed by always adopting 
 the natural method of defining, which consists in comparing the 
 word or class to be defined with those other words or classes which 
 approach it most closely in sense. This natural method of defining 
 by simple comparison of what is most aUied in meaning ensures a 
 proximate genus being reached instead of some remoter genus , 
 and, further, the differentia can be so chosen as to cover just the 
 one species, and exclude all the sister-species, the class-terms most 
 liable to be confused with it. If the genus is not proximate, the 
 definition is Ukely to be too wide. Suppose I wish to define
 
 Chap. III.] THE TESTING OF DEFINITIONS 33 
 
 ' sqiuare.' I compare it with ' rhoml^us,' and fuid at once, as genus, 
 equilateral quadrilateral, and as differentia, rectangular ; or I 
 compare it with ' oblong,' and find at once, as genus, rectangular 
 quadrilateral, and as differentia, ec|uilateral. But if I reach my 
 definition through comparison with terms less closely allied in 
 meaning, the definition is less likely to fit well. Thus if I compare 
 ' square ' with ' circle,' the obvious genus is ' plane figure.' The 
 square is then quite sufficiently distinguished from the circle by 
 means of the differentia ' rectilinear.' But the resulting definition, 
 ' A square is a rectilinear plane figure,' is very much too wide. 
 
 As an important corollary from this second rule we have the 
 requirement that a definition should contain nothing superfluous. 
 Thus, the following attempt at defining a ' tip ' obviously needs 
 pruning : ' A tip is an extra gratuity paid out of goodwill over and 
 above what can be demanded by contract.' Here ' extra,' 
 ' gratuity,' ' over and above ' all involve the same idea. When 
 reduced to the more economical and ' fitting ' form, ' A tip is a gift 
 paid out of goodwill,' the definition, though still faulty, is much 
 improved. 
 
 When practically applying this rule, we may profitably guide 
 ourselves by the following test-questions : 
 
 (i.) Do all the kinds of objects denoted by the term possess the 
 differentia given ? If not, the definition is, to this extent, too 
 narrow. Example : A dog is a domestic animal. Are all kinds of 
 dogs domestic ? No ; ' dingoes ' are wild. Therefore, the definition 
 fails to include all kinds of dogs, and is consequently too narrow. 
 
 (ii.) Having ascertained that the definition is not too narrow, 
 we ask, ' Is it too wide V This test-question we may state in two 
 equivalent forms : (a) Are there no other terms that satisfy the 
 same definition ? (6) Is the definition simply convertible ? — i.e., 
 given that A is B, is it equally true that B is A ? Example : The 
 house-dog is a domestic animal that barks. Is it equally true to 
 say that a domestic animal that barks is a house-dog ? 
 
 3. The terms of a definition must be of the same order as the 
 term defined. They must not be figurative or metaphorical. A 
 metaphor is* ' the use of a word in a transferred sense, the trans- 
 ference being from the order to which it properly belongs to some 
 other order.' Thus, if I define ' faith ' as ' the eye of the soul,' I 
 am transferring to the spiritual order the word ' eye,' which belongs 
 to the physical order, and primarily means an organ of the body. 
 So in the definition of a camel as ' the ship of the desert,' the term 
 ' ship ' is transferred from the inorganic to the organic order. The 
 definition, in fact, must be homogeneous throughout with the term 
 defined. 
 
 Example. — Logic is the medicine of the mind. 
 
 This is metaphorical. Logic and medicine are not of the same 
 
 * Father Clarke, 'Logic,' p. 222. 
 
 o
 
 34 THE PROBLEM OF LOGIC [II. iii. 
 
 order. One is a discipline to be assimikited by the mind, the other 
 a drug to be absorbed by the body. 
 
 4. Tlie definition must consist of terms more elementary than the 
 term detined — i.e., it must be such that no one can reasonably 
 expect to understand the term to be defined without first under- 
 standing wliat the defining terms themselves mean. Tliis rule must 
 be apphed with reference to the given interest — e.g., that of 
 Geometry. I may be quite right in defining a circle as follows : 
 ' A circle is a plane figure contained b}'' a line of which all the points 
 are equidistant from a fixed point within it,' since the specific mark 
 contains only such terms as ' line,' ' point,' ' equidistant,' all of 
 which express more elementary geometrical ideas than that of the 
 circle. Hence a definition is not invalidated because the untrained 
 mind finds its terms less simple than the term it defines. ' Man ' 
 is to most people, no doubt, a much simpler and more familiar term, 
 much easier to understand, than its definition ' rational animal,' but 
 these defining words are more elementary than the more obvious 
 term they serve to define. 
 
 Example. — A fine is a pecuniary mulct. 
 
 This is, scientifically, a correct definition, as a mulct is any for- 
 feiture or penalty. But from the purely practical point of view it 
 would be a breach of this fourth rule, or, in technical language, 
 an ' ignotum per ignotius.' 
 
 5. A term should not be defined by the aid of terms which 
 cannot themselves be appropriately defined without first defining 
 the original term. To break this rule is to commit a ' circulus in 
 definiendo ' or ' vicious circle.' E.g. : 
 
 ' Man is a human being.' 
 ' The sun is the centre of the solar system.' 
 ' Network is a reticulated system of cordage.' 
 ' An archdeacon is an ecclesiastical dignitary, whose business 
 it is to perform archidiaconal functions.' 
 
 Example. — Cheese is a caseous preparation of milk. 
 
 ' Here caseous ' means ' cheesy,' and we still want the definition of 
 ' cheesy.' We wish to know by what kind of preparation cheese 
 can be obtained out of milk. The differentia should indicate the 
 recipe for transforming milk into cheese. 
 
 A vicious circle in definition is more than a mere blemish. It 
 destroys not only the value of the definition, but the definition itself. 
 The ' statement ' that ' cheese is cheesy ' is, in fact, no statement 
 at all. It does not predicate anything of cheese, but stops at the 
 concept whicli is to be defined. ' Cheese is cheesy ' takes us no 
 further than ' cheese.' The definition is, therefore, to this extent 
 non-existent. 
 
 We must be careful, however, not to be too hasty in accusing a 
 definition of involving a vicious circle.
 
 Chap. III.] THE TESTING OF DEFINITIONS 3u 
 
 Example. — A Lilliputian is an inhabitant of the island of 
 Lilliput. 
 
 Taking ' Lilliputian ' in its primary sense (in its derived sen.se it 
 is a synonym for ' dwarf '), we should have to meet the objection 
 that if Lilliput is defined as the land of the Lilliputians, then to 
 define the Lilliputian as an inhabitant of Lilliput is to shut oneself 
 up within a vicious circle. But if ' Lilliput ' is defined in such a 
 way that its definition does not introduce the Lilliputian — e.g., by 
 its geographical position — then there is no vicious circle at all, and 
 the definition is correct. Such definitions as ' A sovereign is a gold 
 coin equal in value to twenty shillings ' and ' A day is a period of 
 time consisting of twenty-four hours ' are liable under similar 
 limitations to the fallacy of vicious circle, the former if a shilling is 
 defined as the twentieth part of a sovereign, the latter if an hour 
 is defined as the twenty-fourth part of a day. 
 
 As a particular case of circular definition, we have the attempt 
 to define a term by means of its correlative. In the case of corre- 
 latives — in the case, that is, of such terms as ' whole ' and ' part,' 
 ' genus ' and ' species,' ' first ' and ' second,' ' cause ' and ' effect,' 
 the two terms must be defined together. We cannot define one by 
 the other. A whole cannot be logically defined as ' an aggregate of 
 parts,' if by ' a part ' we mean ' a fraction of a whole.' The defini- 
 tion here is, in fact, the definiendum itself. To define a ' whole ' is 
 to defuie a ' whole of parts.' It is a unity of some kind, of which 
 the nature varies with the form of relation between whole and part. 
 In specifying this form of relation, whether spatial, organic, or 
 spiritual, we define the type of unity we have in mind, and specify 
 the general meaning of ' whole.' We may therefore define a 
 ' whole ' or ' whole of parts ' by means of the genus ' unity ' and the 
 indeterminate differentia ' possessing some kind and some degree of 
 self -coherence.' 
 
 Mill, in his ' Logic ' (Bk. I., eh, ii., § 7), clearly points out why it is 
 that certain words go in pairs, as in the case of the instances men- 
 tioned above. It is because the meaning of both terms is derived from 
 the same fact or set of facts. Thus, taking the relation of ' father ' 
 to ' son,' he writes : ' The paternity of A and the filiet}^ of B are not 
 two facts, but two modes of expressing the same fact.' The terms 
 ' father ' and ' son,' however, are not strictly correlatives as are the 
 terms ' parent ' and ' child.' They are semi-correlatives. Father- 
 hood does not necessarily imply sonship, though sonship imphes 
 fatherhood. ' Sheep ' and ' shepherd ' are semi-correlatives in a 
 precisely similar sense. There can be no shepherd unless there are 
 sheep to be herded and tended, but there can be sheep without a 
 shepherd. So, again, a third implies a first and a second, but these 
 do not imply a third. Hence no circle is committed by defining a 
 shepherd as ' a person who looks after sheep,' for we may very well 
 define a sheep without introducing its relation to a shepherd. But 
 
 3—2
 
 36 THE PROBLEM OF LOGIC [II. iii. 
 
 ■we cannot, without a circle, define a sheep as ' the kind of animal 
 which a shepherd looks after.' 
 
 G. A detmition should not be given in a negative form if a 
 positive idea is intended. As Professor Read reminds us, a natural 
 historian would not define a lion by saying that it was not a 
 vegetarian. So, in the positive interests of Geometry, it would be 
 better to define a curve as ' a line that is always changing its 
 direction ' than to defme it as ' a line in no part straight.' On the 
 other hand, where the word to be defined stands for a distinctly 
 negative idea, this form of definition — i.e., negative definition — is 
 to be preferred to any other. E.g., 'An alien is a person who is 
 not a citizen,' ' A bachelor is a man who is not married.' 
 
 Examples on the Testing of Definitions. 
 
 I. A circle is a figure of which all the points are equidistant from 
 its centre. 
 Purpose of the definition : To give a geometrical definition 
 
 of a circle. 
 Criticism of the definition as given. 
 
 (a) The word ' centre ' is not more elementary 
 than the term ' circle,' therefore should be 
 avoided. 
 Correction : A circle is a figure of which all 
 points are equidistant from a certain fixed 
 point within the figure. 
 {b) It is not true that all points of a circular area are 
 equidistant from the centre ; one point of the 
 area is, in fact, the centre itself. 
 
 Correction : A circle is a figure enclosed by 
 one line, the circumference, of which all 
 points, etc. 
 If by ' line ' we understand ' continuous 
 line,' this correction should quiet the 
 suspicion that the circumference might be 
 punctiform, a discontinuous aggregate of 
 points, 
 (c) The ' one line ' of this definition may still meander 
 freely over any surface of which all jooints are 
 equidistant from a certain point within the 
 figure. In mathematical phrase, its locus may 
 be the surface of a sphere. 
 Final Reconstruction : A circle is a plane figure, en- 
 closed by one line (the circumference), of which all 
 points arc equidistant from a certain fixed i>oint within 
 the figure.
 
 Ciup. III.] THE TESTING OF DEFIXITIOXS 37 
 
 II. Work is the salt of life. 
 
 Verbal Division :* By ' work ' we may understand citlier 
 an activity or its product. Tlie former sense is evidently 
 intended here. 
 Purpose : To define Work as an activity having relation to 
 
 moral life. 
 Criticism : The definition is metaphorical. We must get 
 
 rid of the metaphor. 
 Reconstruction : (i.) Work is a type of purposive activity 
 (genus) which stimulates, purifies, and sustains the life 
 (differentia). 
 
 Query : Is ' work ' here sufficiently distinguished 
 from ' play ' 1 
 (ii.) Work is a purposive activity which, when regarded 
 in the light of a moral obUgation, stimulates, purifies 
 and sustains the life. 
 
 III. A chair is an article of furniture with four legs and a back. 
 
 Purpose : To define a chair by the use to wliicli it is put. 
 Criticism : [a] Proximate genus not given. 
 Correction : A chair is a seat. 
 (6) The differentia is not satisfactorily given. 
 
 Correction : If we compare a chair wdth a stool, 
 we obtain as genus ' moveable seat,' and as 
 differentia ' having a back.' If we compare a 
 chair with a sofa, the differentia is ' intended 
 to seat one person.' 
 (c) ' Four legs ' is a mere ' accident ' or problematical 
 property of the occasional type. 
 Beconstruction : Proximate genus (of chair, sofa, stool) — 
 ' ^Moveable seat.' 
 Differentia : ' Intended to accommodate one 
 person at a time, and having a back.' 
 
 IV. A cow is a ruminant with cloven feet and sweet-smelling breath. 
 
 Purpose : To define a cow zoologically. 
 Criticism : (a) Proximate genus wanted. 
 
 Comparing ' cow ' with ' bull,' we obtain as genus 
 ' ox ' (in the ordinary generic sense of that term), 
 and as differentia ' female.' 
 (h) ' With cloven feet ' is a characteristic property. 
 ' Sweet-smelling breath ' is a problematic property of 
 the occasional type. A cow may or may not have 
 sweet-smelling breath. The breath might de- 
 teriorate without the creature ceasing to be a cow. 
 
 Reconstruction : A cow is a female ox. 
 
 o 
 
 * By ' Verbal Division ' we understand tlic division of an equivocal or niany- 
 meaninged word into its various alternative significations. Thus the division of 
 •box' into 'covered case, partition in a theatre, blow with the fist, shrub, or drirer'a 
 scat ' would be a verbal division. 
 
 ■^0
 
 33 THE PROBLEM OF LOGIC [II. iii, 
 
 V. A candle is a kind of light used before gas was invented. 
 
 Purpose : To define a eandle from the point of view of 
 its use as a hght and the structure wliich subserves 
 that use. 
 Criticism : (a) Genus inexact ; a eandle is not a kind but 
 a means of light. 
 (6) The specifying mark leaves the definition in one 
 respect too icide, for other things besides candles 
 were used for lighting before gas came into use. In 
 another respect it is too narrow, for candles are still 
 used, though gas has been ' invented.' ' Used before ' 
 implies that candles ceased to be used when gas came 
 into fasliion. 
 (c) Further, gas was not ' invented ' but ' manufactured.' 
 The specific mark must, therefore, be cancelled as 
 flat and irrelevant, and a radical reconstruction 
 is called for. 
 Reconstruction : A candle is a means of lighting, consisting 
 of a stick of fatty matter traversed by a wick. 
 
 VI. The Sun is the star that shines by day. 
 
 Purpose : To define the sun from the point of view of its 
 
 appearance (Ptolemaic point of view). 
 Criticism : Can ' day ' be defined without involving a 
 
 vicious circle ? Is not ' day ' that time during wliich 
 
 the sun is above the horizon ?* 
 Peconsiruction : Comparing the sun with moon and stars, 
 
 which agree in giving forth no perceptible heat, we 
 
 obtain : 
 
 The sun is a celestial luminary which warms the earth. 
 
 VII. ' A soldier is a brave man who is ready to die for his country. 
 
 Purpose : To define a soldier as such — i.e., from the 
 
 point of view of his mihtary office. 
 Objection : (a) ' Brave ' superfluous, as the essential kind 
 
 of bravery that a soldier requires is implied in 
 
 ' ready to die.' 
 Correction : (i.) A soldier is a man who is ready to die 
 
 for his country. 
 Objection : (6) ' Man ' makes the definition too narrow. 
 
 Cf. Amazons and drummer-boys. 
 Correction : (ii.) A soldier is a person who is ready to 
 
 die for his country. 
 Objection : (c) The definition is still too narrow. It 
 
 excludes mercenaries, organized revolutionists, etc. 
 Correction : (iii.) A soldier is a person who is ready to 
 
 die for country, cause, or material reward. 
 
 * Thi3 criticism, as Mr. Joseph points out (ibid., p. 100), is given by Aristotle 
 hinistlf.
 
 Chap. IV.] DEFINITION AND DIVISION 39 
 
 Objection : (d) Many units in an army are nol ready so 
 to die. In this respect the definition is again too 
 narrow. 
 Correction : (iv.) A soldier is a person pledged to fight 
 — to the death if need be — for country, cause, or 
 material reward. 
 
 N.B. — In the above, it has been found convenient to 
 merge the two stages of criticism and reconstruction 
 under a single process of reconstructive criticism 
 through successive objections and corrections. A 
 tendency in this same direction may already have 
 been noticed in connexion with the discussion of 
 some of the preceding definitions. 
 
 CHAPTER IV. 
 
 II. (iv.) DEFINITION ANT) DI\^SION : LOGICAL DIVISION. 
 
 The process of comparison by which definitions are framed to 
 meet the logical need of the occasion gives, as a result, a genus 
 with two or more species included under it. In a word, the defini 
 tion of the species through a process of comparison results in the 
 division of a genus into two or more of its species. Definition and 
 Division are thus closely cormected from the point of view of logical 
 origin. They are also closely connected from the point of view of 
 logical function. Definition and Division are both necessary to 
 the full understanding of the meaning of a word. Definition 
 gives us — 
 
 (a) The more general class under which the class in question 
 
 faUs. 
 (6) A specific distinguishing mark. 
 
 Division continues the process of supplying information by giving 
 us the alternative sub-classes. 
 
 The problem of meaning, then, covers both Definition and 
 Division, and the principle of Non-Ambiguity is regulative of both 
 processes. {Cf. the illustration of p. 18 borrowed from Mr. Sidg- 
 wiek.) Hence, if we identify the Principle of Non-Ambiguity with 
 the principle of Defi:iition, we must understand the term ' Definition' 
 in that wider sense of a complete definition of meaning which 
 includes Division as well.
 
 40 THE PROBLEM OF LOGIC [II. iv. 
 
 Logical Division. 
 
 The term ' Division,' which is the established designation of the 
 procedure we have now to examine, is not happily chosen. We 
 cannot appropriately speak of dividing a word, or the meaning of a 
 word, for meanings are ' diflfcrentiated ' rather than divided. The 
 very term 'Division' (as also such other metaphorical expressions 
 as ' parts,' ' joints,' etc.) seems almost to imply a physical division, 
 a division of some individual thing into its component parts.* The 
 use of the word has the further disadvantage of prejudicing the 
 interpretation to be put ui^on the process in its logical aspect. 
 For this process essentially concerns the relation between a genus 
 and its species, and the term ' Division ' in this connexion naturally 
 suggests that logical Division consists in the splitting up of a genus 
 into its constituent species. If this is the way in which we are to 
 conceive the process, then the true formula for the relation between 
 genus G and species S^, So, S3 is G = S^ and Sg and S3. Plane triangles, 
 we should have to say, are divided into equilateral, isosceles, and 
 scalene. These are the parts of which ' plane triangle ' is the whole. 
 But when I say that ABC is a plane triangle, I certainly do not 
 mean to say that it is an equilateral triangle and an isosceles triangle 
 and a scalene triangle, that it is S^ and S2 and S3 ; I mean that it 
 is Si or S, or S3. It is this disjunctive formulation which alone truly 
 represents the nature of logical Division. 
 
 Logical Division is in no sense a splitting up of tilings into their 
 parts. For the thing is not a genus, nor are its parts species. The 
 division of an animal (mentally, of course) into head, trunk, and 
 limbs, or of a book into parts or chapters, is a purely physical 
 division. The part here does not stand to the whole in the relation of 
 species to genus. We cannot say that the head or trunk or limb 
 of an animal is itself a sort of animal. But in logical Division the 
 genus divided must be predicable of each of the species into which it 
 is divided. If we divide ' human being ' into ' man or woman,' 
 each of the two species into which the genus ' human being ' is 
 divided is itself a sort or kind of human being. 
 
 There is another species of non-logical Division usually referred to 
 as ' Metaphysical Division.' This is the mental division of an object 
 into its several attributes, as when I analyze ' organism ' into its 
 genus, differentia, and various properties. These are not parts of 
 the concept ' organism ' in the sense in which head, trunk, and 
 limbs are parts of an animal, for the qualities could not really be 
 separated from each other as head or limb could be separated from 
 the trunk, nor are they collectively equivalent to the object 
 divided. 
 
 * Cf. Plato's admonition that ' the philosopher must divide by the joints, and 
 not hack anywhere liiie a clumsy cook ' ; and Seneca's remark that a genus ' should 
 be divided, not cut into shreds.'
 
 Chap. IV. ] BEFIXITIOX AND DIVISION 41 
 
 The true significance of logical Division can best be gauged by 
 considering the relation of Division to Definition in connexion with 
 what we may call the logical development of meaning. To this 
 development, as we have seen, both processes are essential, and we 
 niaj^ define their respective functions within this development by 
 saying that Division serves to render determinate tiiose elements 
 of moaning in the definition which are still left indeterminate, and 
 therefore capable of further specification. Division, in a word, is 
 just the further differentiation of the definition in so far as it 
 contains indeterminate elements. Given the definition of a plane 
 triangle as a three-sided rectilinear plane figure, the relations between 
 the three sides are not determined except to this extent — that we 
 know, from the geometrical definition of ' figure,' that the three 
 sides must include an area ; there is otherwise an indcterminateness 
 in the side-relations, an indcterminateness which is rendered 
 determinate by the division or differentiation into equilateral, 
 isosceles, or scalene. 
 
 Illustration of the Logical Develo})ment of Meaning through 
 Definition and Division. 
 
 A government may be defined as the ruling power in a society 
 consolidated through some dominating interest, the form of rule 
 varying in every case with the structural character of the body 
 wherein the ultimate authority is vested. 
 
 The consolidating interest may be either political or non-political. 
 If non-political, it may be either ecclesiastical (Church-government) 
 or non-ecclesiastical. We restrict ourselves to developing, through 
 division, the meaning of a State-government. 
 
 In the case of a State-government, the structural character of the 
 ruling body may take any one of three forms : it may consist of an 
 individual,* or it may consist of a privileged class, or of the com- 
 munity itself. A State-government, that is, may be either an 
 Autocracy, an Oligarchy, or a Democracy. If it is autocratic, the 
 form of government will vary according as the ' rule by one ' is 
 limited or unlimited. An Autocracy, that is, may be either a 
 Limited or a Constitutional Monarchy, or else an Absolute Monarchy 
 or Despotism, passing, when degenerate, into a Tyrannj-. 
 
 If the governziient is a class-government, the form will vary 
 according to the nature of the ruling qualification. If this is rank, 
 the government will be an Aristocracy ; if wealth, a Plutocracy. 
 
 If the government is a government by the people, its form will 
 varv witli the method of self-government. Tliis mav be direct, 
 as in the Citizen-Rule of ancient Athens, or representative, as in 
 the case of modern Democracies, the form of rex^resentative govern- 
 
 * Perhaps two or three, as in the case of the two Kings of Sparta, or of the 
 Roman Triumvirates.
 
 42 THE PROBLEM OF LOGIC [II. iv, 
 
 ment varying again with the conditions of the franchise and the 
 number and nature of the representative bodies. 
 
 Thus we see that a logical division is not necessarily exhausted 
 by a single division of a genus into its alternative species. The 
 interest which prompts the division may require for its fulfilment 
 the further division of the species into sub-species, and these, again, 
 may require to be divided. These further divisions of species 
 and sub-species would at the same time be subdivisions of the 
 genus. 
 
 The conception or genus with which the division starts is known 
 as the summum genus of the division ; the ultimate subdivisions of 
 this genus — ultimate, that is, in respect to the purpose of the 
 division — are its infimae species. The intermediate classes are 
 sometimes called ' subaltern genera ' — genera, because ever}' species 
 except the infima species is a genus to the classes into which it is 
 divided. Just as the infima species is a species which is not also a 
 genus, so the summum genus is a genus which is not also a species. 
 Every subaltern class in a continued division is at once species and 
 genus. 
 
 The logical interest which prompts and guides a division may be 
 either formal or real. It is ' formal ' (with a small ' f ' ; vide p. 16) 
 when it is ' practical ' and ' occasional ' in character. It is ' real ' 
 when the divisions are drafted in the s( le interest of scientific 
 research. Tliis distinction between formal and real may be apphed 
 to the divisions themselves. A real division might then be regarded 
 as a Scientific Classification. There is, however, a reason for not 
 identifj'ing the two terms ' Classification ' and ' Real Division.' 
 Real Division proceeds always do"OTiwards from ' genus ' to ' species.' 
 In the process of Classification, on the other hand, we may move in 
 either of two directions : we may move from the ' species ' upwards, 
 or from the ' genus ' downwards. 
 
 Every separate classification has its own summum genus, so that 
 a summum genus cannot profitably denote anything absolute, as 
 the ' being ' of Porphyry's tree is not unusually supposed to do. 
 Thus the summum genus of the classification scheme in Zoology is 
 tiie kingdom ' Animal,' and not ' Living Being,' which would include 
 Plants as well, and might even be extended to Metals, if we may 
 trust certain recent scientific research. 
 
 The ' infima species,' again, is by no means a fixed distinction 
 in any given system of classification, but is relative to the limit of 
 purposiveness in the making of class distinctions. The African 
 Lion, which is classed as a ' variety ' in Animal Classification, may 
 be regarded as an infima species, but if it became useful to distin- 
 guish sub- varieties, these latter would in their turn become the 
 infimse species. 
 
 Logical Division must be carefully distinguished from Enumera- 
 tion. Enumeration is a summing up of the individuals which
 
 Chap. IV.] DEFINITION AND DIVISION 43 
 
 answer to a given class-designation, whether that class be a summum 
 genus, subaltern genus, or inlima species. It is therefore a process 
 which runs parallel to the development of meaning through logical 
 Division. At any stage of that development it may be purposive 
 to turn from the conceptual ordering of fact to the counting up of 
 the individual units which the concepts serve to include under 
 classes. When we consider facts from the point of view of their 
 number or quantity, the process is an Enumeration. From the 
 logical point of view the interest in Enumeration centres mainly, 
 as we shall see, in questions relating to its completeness or its 
 incompleteness. 
 
 Basis of Division, or Fundamentum Divisionis. 
 
 Every division is based upon and guided by a jundam&ntum 
 divisionis — i.e., by some character of the group or genus which is 
 a source of difference amongst its members. Thus, in the botanical 
 division of Angiosperms into Monocotyledons and Dicotyledons, the 
 fundamentum divisionis, or F.D., is the number of primary leaves 
 possessed by the plant-embryo. It will readily be seen that the 
 fundamenta divisionis are simply indeterminate attributes of the 
 genus. If ' Man ' is divided into ' White man,' ' Black man,' 
 ' Yellow man,' ' Brown man,' ' Red man,' the F.D. is ' skin-colour.' 
 But the genus ' Man ' is here relevantly defined as ' a rational 
 animal (det.) possessing a skin-colour of some kind (indet.).' The 
 F.D. cannot be a determinate attribute of the genus, qua deter- 
 minate, for the simple reason that, in so far as it is determinate, it 
 ceases to be specifiable. At the same time, most so-called deter- 
 minate attributes are only partially determinate, and, in so far as 
 they are indeterminate, may serve as fundamenta divisionis or bases 
 of division. 
 
 From the point of view of the interest we have in dividing or 
 differentiating the meaning of a concept these fundamenta divisionis 
 are essential characteristics of the concept, and must therefore be 
 included within its definition. Thus, suppose we desire to define the 
 statistical unit from tlie point of view of a statistical inquirj^ which 
 purposes to class the citizens of a country according to means and 
 occupation. The definition would take some such form as this : 
 The ' statistical man ' is ' a citizen of a certain means and occupa- 
 tion ' ; and the full meaning of this unit can be made clear only 
 when we specify the divisions we intend to draw under these two 
 heads. Thus the ' statistical man ' may be regarded as (1) 'a 
 citizen who has an income that is cither under £50 a year or under 
 £500, or over that amount ' ; and as (2) ' a citizen who is an artisan 
 or is engaged in business, or is in a profession, or falls outside these 
 three classes.'
 
 44 THE PROBLEM OF LOGIC [II. 
 
 IV. 
 
 The Rules of Logical Division. 
 
 I. There should be one fundamentum divisioni?, and one only, 
 for each complete act of division. 
 
 II. The species or alternatives into which a genus is divided 
 must be mutually exclusive. 
 
 III. If the division involves more than one step, it should proceed 
 gradually from the summum genus towards the infimse species. 
 Divisio ne fiat j:)er saltum. 
 
 IV. The division, within the limits of relevancy, must be 
 disjunctively exhaustive. 
 
 Rule I. — There should be one fundamentum divisionis, and one 
 only, for each complete act of division.* 
 
 The division of a genus is complete when the genus has been 
 differentiated, and the process of successive dififerentiation con- 
 tinued until the degree of distinction required by the purpose of the 
 division has been precisely attained. In this process each sub- 
 differentiation, or subdivision, should help to develop, more and yet 
 more distincth^ that one indeterminate aspect of the genus of which 
 the differentiation was the original aim in dividing. 
 
 The principle which is here involved is that the F.D. must be a 
 mark of the meaning that we aim at developing through division. 
 We may find it convenient to change the F.D. after a first division, 
 and to carry out the ' subdivisions ' upon fresh bases. But in this 
 case our division is no longer a single process, but a chain of divisions, 
 and the term ' subdivision ' becomes a misnomer. For, in assuming 
 a fresh basis, we have started a fresh division. A division of the 
 species is therefore not necessarily a subdivision of the genus. 
 
 And yet we must not misinterpret the function of the F.D. in 
 Division b}^ in.sisting that it is itself incapable of any development. 
 Discontinuity between one basis and another implies, indeed, a 
 corresponding change of the interest whicli gives unity and direction 
 to the dividing process, and so implies also a corresponding break 
 in the division. But there is an important via media between dis- 
 continuity and a static continuity. The F.D. may legitimately be 
 changed, provided the change is a change within its own original 
 meaning. Thus, after dividing ' human being ' into ' male or 
 female,' the F.D. being ' sex,' we do not necessarily abandon this 
 F.D. when we proceed to subdivide ' male ' into ' man or boy,' 
 and ' female ' into ' woman or girl,' for the age-basis may be here 
 brought forward in its bearing on sex differences. What is essential 
 i.s that the sex interest should dominate the division into its most 
 detailed differentiations, and that all variations in divisional basis 
 
 * It is, of course, possible (as in the last illustration) to divide a genus according 
 to more than one principle of division, provided that we keep the divisions distinct. 
 We then have what is called co-division. Thus, again, adopting the fundamenta 
 of ag • and sex, we may co-divide ' human being ' into ' young, middle-aged, or 
 old,' and into ' man or woman.'
 
 Chap. IV.] DEFINITION ANT) DIVISION 45 
 
 should be variations on the sex-theme. It is in this sense that the 
 F.D. must be one and constant throurihout the dev^elopment of any 
 given division. There may be many sub-f undamenta, but the.ge must 
 themselves be developed in the service of the original fundamentum. 
 
 In so far as the ' sub-fundamenta ' are developed on their own 
 account, each initiating a new interest, the division is broken up 
 into component parts, which arc only loosely and, as it were, exter- 
 nally connected with each other. The organic unity of the division 
 is lost. Moreover, overlapping is almost certain to ensue, for the 
 supreme preventive against the overlapping of the various parts of 
 a division lies in making sure that the parts stand for the various 
 modes in which a single general meaning — e.g., the sex of a human 
 being — can be differentiated or developed. 
 
 When two or more bases of division Bhte simultaneously adopted and 
 developed, the resulting overlapping is known as cross-division. The 
 different divisions cross each other, and the confusion which ensues 
 bears witness to the importance of the first rule of logical Division, 
 
 Rule II. — The second rule of logical Division follows naturally 
 upon the first rule. It is directed against the errors which result 
 in overlapping, whether of the cross-division kind or not. The 
 species or alternatives into ivhich a genus is divided must be mutually 
 exclusive — i.e., no part of the division must overlap or be included 
 under any other part. The only security for observing this rule 
 lies in holding to a single fundamentum. If we divide ' human 
 being ' into ' male or female or young or old,' employing simul- 
 taneously the two fundamenta of sex and age, we obviously break 
 this rule. It is possible, however, to break Rule II. without breaking 
 Rule I. — namely, through carelessness in the statement of alter- 
 natives. Thus I may divide ' man ' (F.D. ' means ') into ' rich, 
 easy, or poor,' but may define ' easy ' in such a way as to cause it 
 to overlap with ' rich ' or ' poor,' or both. 
 
 Bule III. — If the division involves more than one step, it should 
 proceed gradually from the highest genus towards the lowest species. 
 Divisio ne flat per saltum. 
 
 In each step of the division the species must stand in the same 
 order or rank of generality. Let G be divided into S^, S.,, S3 ; and 
 S, again into S'^ S'.,, S'3. Were we to divide G into S^, S',, S3, we 
 should have two ranks of generality under one and the same genus. 
 The division would clearly be inadequate, since no account would 
 have been taken of S'^ or S'3. 
 
 Consider the old-fashioned division of ' Digitigrade ' into ' weasel, 
 civet, hya?na, the cat-kind, fox, wolf, dog.' Here the species are 
 not in the same order of generality. Thus ' fox ' and ' wolf ' are 
 species of the genus Canis (the dog kind), just as ' lion,' ' tiger,' 
 etc. are species of the genus Felis (the cat kind). Had we given 
 the genus Canis, and thereby kept in the same order of generaUty 
 the members of one step in the division, we should have been
 
 iQ THE PROBLEM OF LOGIC [II iv. 
 
 secure against omitting the jackal, wliich would have been included 
 as being under that genus. 
 
 Buk IV. — The division, within the limits of relevancy, must be 
 disjunctive!}'' exhaustive. 
 
 We have already had occasion to point out the essentially dis- 
 junctive character of Division. When we divide G into S^, Sg, S3, 
 we mean that G may be developed either into S^ or into So or into 
 S3 ; we do not mean that G ma}- be developed into Sj^ and Sg and S3. 
 Hence, when we say that the division must be disjunctively ex- 
 haustive, we mean that S^, Sj, S3 must — within the limits of rele- 
 vancy — exhaust the alternatives. 
 
 The meaning of the word ' exhaustive ' can, in fact, be defined 
 only in relation to the requirement of relevancy. When we say 
 that a division of a genus into its species must be exhaustive, we 
 mean that it must give all the differentiations of the genus wliich are 
 at once possible and relevant. The limit of relevance will be given 
 b}^ the purpose of the division. In the case of the divisions which 
 figure witliin the classification of the natural sciences, the exhaustive- 
 ness cannot be other than provisional, for further investigations may 
 reveal new species, or call for the revision of divisions as previously 
 carried out. Moreover, only those species would be relevant that 
 are also actual, for scientific classifications are not concerned with 
 the laying out of possibilities as such, but only with the ordering of 
 such possibilities as Nature has realized. Thus a division of Man, 
 according to skin-colour, which included blue man and green man, 
 would include irrelevant items, since anthropological science studies 
 not mere possibihties, but facts. It would be more than exhaustive, 
 and break this fourtli rule of Division just as much as a division 
 into ' white man or black man ' which would be under-exhaustive. 
 
 In Division by Dichotomy [vide p. 47) the division will be seen to 
 be implicitly, though not determinately, exhaustive. 
 
 In cormexion with this rule of exhaustiveness in Division 
 Mr. Joseph {ibid., p. 103) gives an instructive illustration which I 
 take the liberty of quoting in full : ' Suppose that an income-tax 
 is introduced ; it is necessary that the Act imposing it should state 
 what forms of wealth are to be regarded as income, and taxed 
 accordingly. The rent of land and houses is clearly a form of 
 income, and would be included in the division of that genus ; but 
 if the owner of a house Lives in it instead of letting it, he receives no 
 rent. Nevertheless, he enjoys an income, in the shape of the annual 
 value of the house he lives in, just as truly as if he had let that 
 house, and received for it a sum of mone}'^ sufficient to hire himself 
 another ; and he ouglit to be taxed if he lives in his own house as 
 much as if he lets it. But if the income-tax Act omitted to include 
 among the species of income the annual value of houses occupied 
 by their owners, he would escape payment on that head altogether. 
 Such is the practical importance of making a division exhaustive.
 
 CiiAi'. IV.] DEFINITION AND DIVISION 47 
 
 Division by Dichotomy. 
 
 In the process known as Dichotomy {Si-y^^a, in two; rifivco, I cut) 
 we divide the genus into two alternative species — ' x or not-.r ' : 
 X is commonly called the positive, and not-x the negative species ; 
 but, as the negative species proves on analysis to be negative only 
 in the name, we propose to substitute for the words ' positive ' and 
 ' negative ' the words ' definite ' and ' indefinite.' Thus we may 
 divide ' Animal ' into ' vertebrate or non-vertebrate,' when by ' non- 
 vertebrate ' we mean ' some animal other than vertebrate.' We then 
 systematically subdivide on the same principle, and continue dicho- 
 tomizing in this way until it ceases to be purposive to go further. 
 
 What is known as Porphyry's Tree* illustrates the process in 
 that incomplete form in which only the definite terms are di- 
 chotomized. 
 
 Being. 
 
 Corporeal. Incorporeal. 
 
 Animate. Inanimate. 
 
 Sensible. Insensible. 
 
 Rational. Irrational. 
 
 I I I 
 
 Socrates. Plato. Etc. 
 
 * As Mr. Stock points out ('Logic,' ed. 1903, p. 94), the 'Tree of Porphyry' 
 is ' a device added by later writers.' In Porphyry's treatise there is no division 
 by dichotomy, but simply the logical development of the single category of Sub- 
 stance taken as summum genus : 
 
 Substance. 
 
 I 
 Body. 
 
 Living body. 
 
 I 
 
 Animal. 
 
 I 
 
 Rational animal. 
 
 I 
 Man. 
 
 1 I I 
 
 Socrates. Plato. Etc. 
 
 Mr. Stock adds the folio-wing interesting footnote : ' We might suppose that 
 "thing" or ''being" could be predicated of "substance," but Porphyry, fol- 
 lowing Aristotle, regards each of the ten categories as a distinct summum genus. 
 He will not allow that " being " is prcdicable of them all in the same sense.'
 
 48 THE PR0BLE3I OF LOGIC [II. iv. 
 
 Tliis rejection of the indefinite terra at eacli step of tlie division 
 is technically known as an ' abscissio infiniti,' the ' infinitum ' or 
 ' indeterminate ' being here the indefinite term. 
 
 The definite and indefinite terms in their relation to each o'her 
 are sometimes referred to as Contradictory Opposites, Contra- 
 dictory Relatives, or Contradictories. Thus ' cold ' and ' not-cold ' 
 are said to be contradictory opposites. But the name is unfortunate 
 and apt to mislead. A definite term and its counter-indeterminate 
 are not contradictory in the sense of contradicting each other. It 
 is only statements that can contradict or be contradicted. It is 
 true that when such terms are predicated of the same subject in 
 the same relation the assertions within which they thus function as 
 the respective predicates contradict each other ; but it is the 
 opposition of the two statements, and not that of the two predicates 
 as such, which constitutes the contradiction. 
 
 We shall, in fact, see, when we come to consider what we mean 
 by an indefinite term, that these so-called contradictory opposites 
 are complementary rather than antithetic. They should therefore 
 be carefully distinguished from contrary opposites or contrary 
 relatives, which may be defined as terms markedly opposed under 
 the same head. We say ' markedly ' and not ' most,' since under 
 any given head — e.g., that of temperature — we may have more 
 than one pair of contraries. Thus ' cold ' and ' hot ' are con- 
 traries ; but so also are ' freezing ' and ' broiling.'* It will be seen 
 that each of a given pair of contrary terms is itself a positive term 
 with well-defined positive reference. ' Black ' is just as positive 
 in meaning as ' white,' ' miserable ' as positive as ' happy,' ' hard ' 
 as positive as ' soft.' 
 
 A term is, of course, a ' contradictory ' or a ' contrary,' not per se, 
 but only in relation to its opposite. In particular the indefinite 
 term ' not-x ' is not in itself ' a contradictory term.' It is contra- 
 dictory only in relation to the complementary definite term ' a:,' 
 and that only in the derivative sense already indicated. 
 
 The Meaning of the Indefinite Term. 
 
 Tlie logical significance of Dichotomy depends primarily on the 
 meaning we assign to the indefinite term. We must, therefore, 
 carefully consider what this meaning may be. 
 
 An indefinite term is a term of the form ' not-a; ' or ' non-a:.' It 
 indicates what is other than a; in a sense that we must now proceed 
 to determine. Some logicians insist that it must be, in character, 
 perfectly and illimitably indefinite. Not-a:, they say, must surely 
 take up all that is excluded from x. Out of the sum-total of think- 
 
 • This indefiniteness does not extend to contrary propositions. There the op- 
 position exists unambiguously between ' all ' and ' none,' between ' All S is P ' 
 and ' No S is P.'
 
 Chap. IV.] DEFINITION AND DIVISION 4'J 
 
 able existence we subtract x : all tliat is left must be not-a;. Not-a; = 
 everything - a;. Thus if .r= Europeans, not-x stands for 'the 
 unlimited myriads of entities which people the heterogeneous 
 domain ' of ' everything - European.' It is in this sense that not-x 
 has been called an Infinite Term. This, however, is a useless logical 
 figment, and only worth mentioning as a warning concerning what 
 not-x should not be made to mean.* 
 
 It seems clear that in the interests of logical science not-.r cannot 
 be indefinite in this inimitable sense. This brings us to what we 
 may call the disjunctive, or the suppositional, use of the indefinite 
 term not-.r. 
 
 The ordinary use of terms is limited by some ' Suppositio,'t some 
 Topic, some Universe or Subject of Discourse. In so far as a man's 
 interest is not that of pure negation — in which case the denial will 
 take the form ' S is not P,' and not the form ' S is not-P ' — his mind 
 is always moving within some assignable suppositio, and the sig- 
 nificance of the indefinite terms he uses is limited by reference to 
 this suppositio. 
 
 It is, moreover, important to realize that the term not-x requires 
 to be disjunctively differentiated. Let us take, by way of illustra- 
 tion, the following division : 
 
 Colour. 
 
 Red. Not-red. 
 
 Here ' not-red ' has the implicitly disjunctive meaning of ' some 
 colour other than red ' — i.e., ' either blue or green or yellow,' etc. 
 It does not stand conjunctively for the sum of colours other than 
 red. Were this it? meaning, not-red would be a term fulfilling a 
 merely epitomic or abbreviative function, and ' red ' and ' not-red ' 
 taken together would conjunctively exhaust the suppositio of 
 colour. It is true that the division of colour into red or not-red 
 is also exhaustive, but it is exhaustive in a disjunctive and not in a 
 conjunctive sense. 
 
 This view of the implicitly disjunctive meaning of not-.r in 
 Dichotomy supports the more general view that we have taken of 
 logical Division as the progressive differentiation of the meaning 
 of a concept. On this disjunctive view the division of ' colour ' 
 into ' red ' and ' not-red ' precisely means tliat colour is either red 
 
 * ' Aristotle long ago pointed out that ovk ivSpwwos was not properly a name 
 nt all ; and he perhaps extended his countenance too much to it when he said 
 that, if we were to call it anything, we must call it a "name indeterminate" 
 {6i'o/j.a ddpLO-Tov) because, being the name of nothing positive and in particular, 
 it had a purely indeterminate signification ' (Joseph, ibid., pp. 29, 30 ; cf. also 
 footnote, p. 30). 
 
 t ' Suppositio ' is an earlier name for ' the universe of discourse,' a name recently 
 revived by Venn and Carveth Read. It means ' the range of subject matter 
 about which we consider ourselves to be speaking.' Mr. Joseph, following Do 
 Morgan, prefers the term ' limited universe.' 
 
 4
 
 :0 THE PROBLEM OF LOGIC [II. 
 
 IV. 
 
 or not-red. Hence, when we proceed to differentiate ' not-red ' 
 into ' blue or green or yellow, etc.,' we are simply carrying on the 
 very same principle of disjunctive differentiation which we applied 
 to the division of the concept ' colour,' 
 
 When the indefinite term is understood in the sense which we 
 have attempted to define, the main objections which have been 
 levelled against Dichotomy as a process of division fall entirely 
 away. Thus Mr. Joseph {ibid., p. 106, sq.) maintains, in the first 
 instance, that in the subdivision of the ' negative ' class or ' concep- 
 tion,' the essential nature of division as a process which exhibits its 
 membra dividentia as ' alternative developments of a common 
 notion ' {ibid., p. 107) is consistently violated. Mr. Joseph holds 
 this objection to be fatal and decisive {ibid., p. 109). But it depends 
 entirely for its force on what we conceive to be a misinterpretation 
 of the meaning of the ' negative ' term. Mr. Joseph takes ' land ' 
 as the meaning or conception to be divided. He divides it by 
 dichotomy into ' building-land ' and ' land not used for building.' 
 Each of these conceptions he subdivides. Thus ' land not used for 
 building ' is di\ided into ' farm-land ' and ' non-farm-land,' and so 
 on. He then points out {ibid., p. 109) that ' to farm land is not a 
 way of not building on it,' and, generally, that the division of a 
 ' negative ' conception is necessarily a division in which the 
 species is no longer a specification of the genus — a division, there- 
 fore, which fails to respect the true logical relation between genus 
 and differentia. 
 
 Now it is undeniable that ' to farm land is not a way of not 
 building on it,' but the ' negative ' term ' land not used for building ' 
 has, as we have seen, a certain positive meaning of an indeterminate 
 predi.sjunctive kind. It stands for ' land used for some purpose 
 other than that of building,' and the farming of land is precisely a 
 specification of this indeterminate generic idea. A ' negative ' con- 
 ception affords, therefore, as sound a basis for subdivision as does 
 a positive or definite conception. It is just as sound to specify ' land 
 u.sed for some purpose other than that of building ' by ' farm-land ' 
 as it is to specify ' land used for building ' as urban or suburban. 
 
 The objection may perhaps be raised that if we are proposing to 
 divide the genus ' land ' into the two alternative species ' land 
 used for building purposes ' or ' land used for jiurposes other than 
 building,' we do not really carry out what we propose to do. For 
 what we are so dividing, it may be said, is not ' land,' but ' land as 
 subserving a human purpose.' Hence ' waste-land,' the land that 
 subserves no human purpose, is excluded from the division, though 
 it i.s as genuine a species of land as building-land or farm-land. 
 
 This objection has a certain point and directness which challenges 
 close consideration. We must admit the justice of the plea that 
 it is not ' land as such ' which can be divided into ' building-land,' 
 or ' land used for some purpose other than building.' This division
 
 Chap. IV.] DEFINITION AND DIVISION 51 
 
 is undoubtedly a division of ' land as subserving a human purpose.' 
 But, from our point of view, ' land as such ' is not a suitable genus 
 for logical division. Meaning is necessarily the meaning of an object 
 for a subject, and can be made unambiguous or logically definite only 
 when the subjective interest which goes to meet the object is first 
 clearly specified. Indeed, Ave could go a step further, and maintain 
 that the object which we propose to define and divide is first con- 
 stituted as a logically definable and divisible object through the 
 selective, abstracting activity of a subjective interest. The object 
 to be logically divided is always a ' genus,' and, as such, its meaning 
 will be variously differentiated according as the dividing-interest is 
 variously specified. 
 
 We admit, then, that ' waste-land ' is not included in our division 
 of ' land as subserving a human purpose,' but hasten to add that, 
 in so far as ' waste-land ' means ' land that subserves no human 
 purpose,' it would be irrelevant, and therefore logically meaningless, 
 to include it.* The ' negative ' term ' not used for building,' there- 
 fore, does not mean ' used for some purpose other than building, 
 or else not used for any purpose at all '; for the addendum wliich 
 the words ' or else ' introduce is irrelevant to the genus we are 
 dividing, and cannot therefore be included within the meaning of 
 the ' negative ' term. ' Not used for building ' must therefore mean, 
 as already stated, ' used for some purpose other than building.' By 
 first defining the object to be divided, through the limiting activity 
 of a definite subjective interest, we cut off from the outset, at one 
 logical stroke, all differentiations of the object's meaning which do 
 not positively subserve the development of that interest. Negating 
 addenda of the type of that just considered have no longer any 
 raison d'etre. The whole race of them is excluded ah initio. 
 
 IMr. Joseph further accuses a dichotomic division of not proceed- 
 ing on a single fundamentum. ' In the proper division of land,' he 
 says [i.e., the division of land into building-land, farm-land, forest, 
 means of communication, pleasure-ground, waste), ' the basis taken 
 was the use to which land is put, and that was retained throughout ; 
 but in the division by dichotomy, the basis taken was, first, the use 
 of land for building, by which it was divided into building-land and 
 the rest ; and the rest was divided on a different basis — viz., the u-e 
 of land for farming, and so on ' {ibid., p. 109). But once the in- 
 definite term is understood in the sense we have adopted, it is no 
 longer true to say that the first F.D. in the process by dichotomy 
 was ' the use of land for building.' When we divide land into 
 * building-land or not-buikh'ng land,' we are dividing it into ' build- 
 ing-land or land used for purposes other than building.' Our F.D. 
 is therefore ' the use to which land is put,' just as in the case of 
 
 * In a division of ' land ' from the point of view of scientific intent, such as 
 that of the geologist, there would be no waste-land, just as to the botanist there 
 is no such species as ' weed.' 
 
 4—2
 
 ol THE PROBLEM OF LOGIC [11. iv. 
 
 ' the proper division of land.' So, again, when we proceed to 
 divide ' land used for purposes other than building ' into ' farm-land 
 or non-farm land,' we do not adopt a diflcrcnt basis of division. The 
 basis still remains, as before, ' the use to which land is put.' When 
 land is used for purposes other than building, it maybe used either for 
 farming purposes or for non-building purposes other than farming. 
 
 We would point out, in conclusion, that Dichotomy is by no means 
 a purely Formal process, which can be carried out independently of 
 material knowledge. As Mr. Joseph convincingly insists [ibid.,]). 110, 
 footnote),* we have no right to divide x into the species a and 7iot-a 
 unless we know tliat, as a matter of fact, a is a species of x. Thus, 
 it is absurd to divide circle into rectilinear circle and non-rectilinear 
 circle, though we are, of course, perfectly justified in saying that 
 every circle (here, as in Analytical Geometry, identified with its cir- 
 cumference) must be either rectilinear or not. We cannot develop 
 the meaning of ' circle ' by assigning to it as one of its species the 
 rectilinear circle. Li dividing G into S or not-S, S must be a possible 
 and relevant differentiation of the genus G. 
 
 The Testing of Given Divisions. 
 
 In the testing of given divisions we have first to decide whether 
 the division is logical or non-logical, and, if the latter, whether it ia 
 physical, metaphysical, or verbal. 
 
 If the division is logical in form, we must test its observance of the 
 four rules. This we may conveniently do by means of the following 
 test-questions : 
 
 I. (a) Is there more than one F.D. ? 
 
 (6) If only one, is it appropriately chosen ? 
 II. (a) Is there overlap i^ing of the classes ? 
 
 (6) If so, to what cause is it due — to a confusion of funda- 
 menta (giving cross-division), to careless definition, 
 or to a confusion of the ranks of generality ? 
 
 III. If due to the latter cause — i.e., if the membra dividentia 
 
 are not ' cognate ' — what is the remedy ? Answer : 
 Subdivision. 
 
 IV. Is the division adequately exhaustive ? 
 
 Examples. — Test the following divisions : 
 
 (i.) ' Living being ' into ' moral or immoral.' 
 
 We must begin by defining ' moral.' If we mean genuinely, 
 actively moral, then the class of indiffcrents in morality is left out. 
 If under ' moral ' we mean to include all creatures capable of 
 morality that are not positively immoral, then the division is sound. 
 But in any case we have omitted the non-moral in the sense of 
 
 * Vf. also ilellonc, 'An Introductory Text-Book of Logic,' ch. vi., § 10.
 
 Chap. IV.] DEFIXTTION AND DIVISION 53 
 
 ' moral incapables ' — plants and animals, human infants, and 
 pathological cases of adult human beings. The division, therefore, 
 is not exhaustive (breach of Rule IV.). Moreover, the division as 
 applied to ' living being ' is unsatisfactory, as most men are some- 
 times moral, sometimes immoral. It would better apply to ' act.' 
 
 (ii.) ' Man ' into — (a) ' timid or rash '; 
 
 (b) ' avaricious or prodigal.' 
 
 The F.D.'s are not specified. They may be taken to be (a) ' be- 
 haviour with regard to danger,' and (6) ' behaviour with regard to 
 money.' 
 
 Again, these divisions are not exhaustive. There is in each case 
 a twofold mean to be introduced — 
 
 (a) ' cautious or valiant '; 
 {b) ' economical or liberal.' 
 
 So long as we ' cut well at the joints,' the more distinctions we can 
 relevantly make, the better. Indeed, it is well to avoid the habit of 
 fancying that between two extremes there can only be one mean. In 
 the case of (a) the objection must be raised that the division much 
 more naturally applies to the act than to the man ; for most people 
 are timid in certain respects and not in others, much depending 
 on habit. Further, the timid may, when their emotions are suffi- 
 ciently roused, become rash, or even really brave (c/. the maternal 
 instinct of protection in ordinarily timid women, or the moral 
 courage of the convinced but naturally timid reformer, or the courage 
 of the martyr for faith's sake). 
 
 (iii.) ' Students ' into ' idle, athletic, and diligent.' 
 
 Criticism (1). — The dividendum is not expressed in logical form. 
 The plural term ' students ' necessitates an ' extensive ' interpreta- 
 tion. In so far as Division is differentiation of meaning, we must 
 adopt the singular form, and restate our dividendum a^ ' student.' 
 A corresponding alteration must be made in the form of the division 
 itself. We must substitute the disjunctive ' or ' for the conjunctive 
 ' and.' The division, then, which we have now to discuss is that of 
 ' student ' into ' idle, athletic, or diligent.' 
 
 Criticism (2). — The F.D. is twofold : work-status and games- 
 status. A co-division is here required to remedy overlapping 
 (Breach of Rules I. and II.), and to ensure adec^uate exhaustiveness 
 (Rule IV.). The division excording to work-status may be given 
 briefly as follows : 
 
 Student. 
 
 Idle. Not-idle. 
 
 I 
 
 Perfunctory. Diligent.
 
 54 THE PROBLEM OF LOGIC [11. iv. 
 
 The attempt to divide ' student ' according to playrstatus raises, 
 however, a fundamental difficulty. For the F.D. ' play-status ' 
 cannot be included, even as an indeterminate mark, in the definition 
 of ' student.' A student cannot be defined as one who patronizes 
 some form of play or takes some form of recreation. Problematic 
 properties cannot, even at the call of the dividing-interest, be trans- 
 figured into differentiae. There is certainly a difficulty here, but the 
 logical remedy is simple and direct. The genus or dividendum 
 may be altered so as to answer appropriately to the requirements 
 of the case. We cannot accept ' play-status ' as an F.D. of ' student,' 
 but we can accept it as an F.D. of ' student who is interested in 
 games.' It should not, however, be supposed that this procedure 
 is a mere subterfuge or dodge. We are not infrequentlj'^ asked to 
 perform operations on inappropriate objects. We might be asked, 
 for instance, to multiply 8 cows by 15 sheep, or to divide 15 
 sheep by 5 sheep. We might be asked to decide upon the specific 
 spiritual quality of a ghost's body or a comet's tail. We may even 
 be asked to convert an proposition. Against all such questions 
 as these we safeguard ourselves by pointing out that the requirement 
 cannot be met, and that the nature of the object resents the subjec- 
 tive demand inconsiderately made upon it. A number can be 
 divided by another number, but not a sheep by a sheep, nor so many 
 cows by so many horses. A comet's tail cannot grow spiritual by 
 the simple process of becoming sufficiently thin. Similarly a student 
 cannot put on a games-interest in order to suit the caprices of a 
 question in logical Division. A c^uestion in Logic may itself be 
 illogical. When we are asked, then, to divide ' student ' according 
 to play-status, we answer that it is only the plaj^-student that has 
 a play-status, and that, from the point of view of play, the student 
 who does not play must be cancelled, not, indeed, as a ' skulk,' a 
 ' shirker,' or a ' book-worm ' — for these pretty labels do not express 
 feeUngs controlled by logical interests — but as an irrelevance — an 
 irrelevance to the limited interests of the play-topic. 
 
 We may adopt, then, as our division according to play-status, 
 some such classification as the following : 
 
 Playing student. 
 
 I I 
 
 Athletic. Non-athletic. 
 
 Shaping badly. Shaping well. 
 
 (iv.j * Quadrilateral figure ' into ' square, rectangle, parallelo- 
 gram, or rhomboid.' 
 
 We take ' quadrilateral figure ' to mean ' plane rectilinear quad- 
 rilateral figure.' 
 The classes overlap, with breach of Rule III. The correction
 
 CuAP. IV.] DEFINITION AKD DIVISION 55 
 
 needed here is therefore not that of co-division (the cross-division 
 remedy), but that of subdivision. Species and sub-species are con- 
 fused together, the division taking leaps along the predicamental 
 line. We may correct the division thus : 
 
 Quadrilatoral figure. 
 
 Parallelogram Non-parallelogram 
 
 (bi- parallel). (F.D. parallelism of sides). 
 
 Rectangle. Non-rectangle. Trapezium Trapezoid. 
 I I (mono-parallel). 
 
 Square. Oblong. Rhombus. Rhomboid. 
 
 We now see very clearly that, in the original diWsion, the classes 
 are not mutually exclusive (Breach of Rule II.). The square is a 
 rectangle, and the rectangle a parallelogram ; and, further, the 
 rhomboid is a parallelogram. Thus, what the given division tells 
 us is, briefly, that the quadrilateral figure is a parallelogram. No 
 account, therefore, is taken of the non-parallelogram — the trapezium 
 and trapezoid. The division, then, is not exhaustive (Breach of 
 Rule IV.). 
 
 It may be worth while to consider this division more closely 
 from the point of view of the f undamenta involved. Unless the first 
 rule of Division is to be broken, the fundamentum must remain 
 generically the same throughout. Now the division according to 
 parallelism of sides and the subsequent division of the parallelo- 
 gram — (1) according to angle-relations, (2) according to side- 
 relations — introduce fundamenta which do not at once appear to 
 be modifications of one and the same generic idea. But on closer 
 scrutiny they are seen to be so. We may accept side-relation as 
 the generic fundamentum, and characterize our three specific 
 fundamenta as (1) side-paralleUsm, (2) side-inclination, (3) (rela- 
 tive) side-length. With regard to (2), we see that the angle-relation 
 is itself a specification of the side-relation ; it is that relation of one 
 side to another which is measured by the inclination of each to the 
 other. According as the two sides which contain the angle are 
 more or less inclined to each other, the angle itself is greater or less. 
 We conclude, then, that Rule I. is not in any way broken, but 
 rather legitimately applied. 
 
 (v.) ' Plane Triangle ' into ' equilateral, obtuse-angled, or right- 
 angled.' 
 
 Identifying ' equilateral ' with the commensurate term ' equi- 
 angular,' we notice that the division is not exhaustive. The acute- 
 angled triangle which is not equiangular is not included (Breach of 
 Rule IV.). There is no overlapping.
 
 56 THE PROBLEM OF LOGIC [11. iv. 
 
 But the division, as given, involves two fundamenta, which, 
 though not generically, are still specifically different — namely, ' side- 
 relation of relative magnitude ' and ' angle-relation.' And it is a 
 breach of Rule II L (though not necessarily of Rule I.) to utihze 
 simultaneously at any given stage two fundamenta that are specifi- 
 colli/ different. Proper subdivision, then, might seem to be the 
 natural remedy, and we might present the corrected division as 
 follows : 
 
 Triangle. 
 
 Acute-angled. Right-angled. Obtuse-angled. 
 
 ^ I I 
 
 Equilateral. Non-equilateral. 
 
 Isosceles. Scalene. 
 
 But the division, so framed, seems to require to be completed by 
 the subdivision of tlie two remaining members of the first division, 
 and the total result is unnecessarily complex. It would be simpler 
 in this case to institute a co-divusion wliich, when completed, would 
 run as follows : Triangle into equilateral, isosceles, scalene (F.D. 
 relative side-length), and into acute-angled, right-angled, obtuse- 
 angled (F.D. side-inchnation). 
 
 (vi.) ' Yorkshire ' into ' North, East, and West Ridings.' 
 This is physical division. 
 
 (vii.) ' Lemonade ' into ' fluid, acid, sweet,' etc. 
 Tliis is an incomplete metaphysical division. 
 
 (viii.) 'Accident ' into ' misadventure or irrelevant predicable.' 
 
 This is verbal division, the discrimination of the possible meanings 
 of an ambiguous term. 
 
 CHAPTER V. 
 
 II. (v.) CLASSIFICATION.* 
 
 The first main object of Classification is to keep control over facts 
 by marshalling them in order ; and the general principle which guides 
 
 * As already indicated in the last chapter (vide p. 42), the term 'Classification' 
 is more comprehensive than the term ' Real Division ' ; for, in the first place, it 
 includes not only the downward movement from summum genus to infimae species 
 to which we are restricted in Real Division, but also the upward movement from 
 the lowest species to the highest genus. In the treatment of Classification here
 
 Chap. V.] CLASSIFICATION 57 
 
 every such endeavour is that of bringing together those things 
 which are most alike and separating those things which are most 
 unhke. Thus, to take the case of animals, we have here an immense 
 and bewildering variety of individual beings. A sufficient know- 
 ledge of Anatomy enables us to detect within this maze of life certain 
 relatively permanent types of structure by the aid of which we form 
 zoological species. When these are compared together, some will 
 be seen to have characters in common by which they resemble 
 one another and differ from all other species. These we group 
 together into what is here technically called a genus. From genera 
 we pass by similar steps to famihes, orders, classes, and finally to 
 sub-kingdoms. 
 
 The words ' class,' ' genus,' ' species ' have here acquired meanings 
 quite different from those involved in stating that the definition of 
 a class or species is given by stating genus and differentia. In this 
 latter statement all the terms are general and relative. From tlie 
 point of view of the predicables, the word ' class ' is used generally 
 for any group of objects resembling each other in certain character- 
 istics.* Thus, a sub-kingdom, or an order, or a class proper, or a 
 genus, or a species is a class in this sense of the word. So, again, 
 if we take any two successive groups in the scheme of classification, 
 the first will stand to the second as genus to species in the predicable 
 sense of these terms. ' Class ' is the genus of which ' Order ' is 
 the species. But in these classifications the words ' class,' ' genus,' 
 ' species ' have fixed specialized meanings. A ' class ' comes 
 between a sub-kingdom and an order, a ' genus ' between a family 
 and a species, a ' species ' between a genus and a variety. 
 
 Types of Classification. 
 
 Classifications are of two kinds : they may be either real or 
 formal. 
 
 When we state that classifications are governed by the para- 
 mount consideration of order, our primary meaning is that classifica- 
 tions arise in response to a dominating subjective purpose, the 
 need for order. But there are two main ways in which tliis subjec- 
 tive purpose realizes itself : it may cither develop in whole-hearted 
 conformity to the nature of the material studied, or it may show a 
 divided adherence, conforming partly to the requirements of the 
 
 given we have used the term almost exclusivelj' in that sense ia which it cannot 
 be mistaken for Real Division — i.e., we have considered the up-building of a 
 classiiication rather than its explication from the most general concepts downwards. 
 But even where the direction of Classification coincides with that of Real Division, 
 the two processes remain distinct. For Classification includes processes of Definition 
 as well as of Division; whereas Division and Definition, as we have defined them, 
 are mutually exclusive. 
 
 * The extension-import of a class is here assumed, as the more convenient for 
 our purpose (vide p. 14G).
 
 58 THE PROBLEM OF LOGIC [11. v. 
 
 material, but partly also to one or other of the specialized demands 
 for order ■which the subject makes in the interests of his own practical 
 life and culture. Li the former case the classification may be called 
 real ; in the latter, formal. In each case the dominating factor is 
 the subjective interest in order, and here, as well as there, the 
 interest may be ' disinterested.' But in the one case this interest 
 is fixed on the discovery of the material's own order imposed upon it 
 by the laws of its own nature ; in the other — whether through choice 
 or necessity — it is bent on arranging the material in a selective spirit 
 by the help of such of its characters as happen to be relevant to the 
 classifier's specific requirements. 
 
 AU the classification-schemes of the Natural Sciences are real in 
 the sense above defined. 
 
 There are two main types of Beal Classification, respectively 
 knovm as natural and diagnostic. But they do not stand on the 
 same level, for the diagnostic type of Classification has its sole raison 
 d'itre in the service of the natural. As the distinction between 
 these two types of Real Classification is particularly important, we 
 proceed to consider it at some length. 
 
 Natural Classification. 
 
 In classifying according to Nature, scientists have been guided 
 by the following important clue, which may be regarded as the 
 guiding-thread of true Natural Classification — to wit, that it is 
 characteristic of the ways of Nature that, when she makes a differ- 
 ence in any single fundamental particular — e.g., possession or lack 
 of a spinal cord — she correlates with this difference a large number 
 of other differences. In the case of the Genetic Sciences, which 
 view their object-matter from the standpoint of its development, 
 this characteristic admits of a ready explanation. Given two 
 species, one with and the other without the rudiment of a spinal 
 cord, it is obvious, from the point of view of Evolution, that they 
 will develop in very different ways, and acquire very different pro- 
 perties. Such classes as are formed of things which agree among 
 themselves and differ from others in a multitude of characters were 
 called by J. S. Mill ' natural kinds.' A classification is natural only 
 in so far as it keeps to natural kinds throughout. 
 
 A natural classification, then, may be defined as one in which, 
 roughly speaking, the divisions are so constituted that the objects 
 included in any one of them resemble each other and differ from all 
 others in many significant respects. 
 
 In Natural Classification the more important characters — i.e., 
 those wliich are accompanied by the larger number of correlated 
 differences — are selected for determining the higher groups, and 
 thus the kinds classified will, on the whole, be arranged, from the 
 primary divisions downwards, according to the principle of ' sub-
 
 Chap. V.] CLASSIFICATION 59 
 
 f 
 
 ordination of characters.' In this arrangement, the higher the place 
 which any class holds in the classification, the more important are 
 the characters which constitute it. This arrangement will prevent 
 any widely dissimilar groups from being brought together in the 
 lower divisions. The ox and the frog will be held apart in the classi- 
 fication, as in Nature. Thus, if we are considering flowering plants, 
 we notice that plants in which the ovules are enclosed in a protective 
 structure resemble one another (and differ from those whose ovules 
 are unprotected) not only in this particular, but in a large number 
 of other points as well, such as the structure of their vascular tissue, 
 the form of the stamens, the germination of the pollen-grain, and 
 the development of tlie endosperm. In classifying flowering 
 plants, we therefore divide them first of all (according as they have 
 protected or unprotected ovules) into Angiospermae or Gymno- 
 sperma?. In subdividing the Angiospermse, we choose the character 
 of the presence of two primary leaves, or of onty one, and thus form 
 the two alternative sub-classes, Dicot3'ledons a,nd Monocotyledons. 
 After this we go on to other characters in descending order of 
 importance, and so form our Orders, Sub-orders, Genera, and 
 Species. 
 
 The characterization of Angiosperms, according as they are 
 dicotyledonous or monocotyledonous. admits of being stated in a 
 relatively untechnical way. Thus : 
 
 (i.) Dicotyledons have the following characters : 
 
 (1) The embryo has two seed-leaves or cotyledons. 
 
 (2) The first or primary root of the embryo branches after 
 
 it leaves the seed. 
 
 (3) The stem branches repeatedly. 
 
 (4) The stem, when perennial, has a distinct pith, con- 
 
 tinuous rings of wood, and separable bark. The 
 stem increases in tliickness by the formation of fresh 
 rings of wood outside those already formed and 
 inside the bark. The hardest wood is insid*?. 
 
 (5) The outer parts of the flower are most commonly in 
 
 fives — i.e., have five members in each whorl. 
 
 (6) The leaves are net-veined. 
 
 (ii.) Monocotyledons have the following characters : 
 
 (1) The embryo has only one seed-leaf. 
 
 (2) The primary root branches before it leaves the 
 
 seed. 
 
 (3) The stem, as a rule, shows little branching, and in the 
 
 monocotyledonous trees (such as Palms) it ma}- be 
 quite unbranched, growing only from a bud at its 
 apex, the buds produced in the axils of the leaves 
 remaining undeveloped.
 
 CO THE PROBLEM OF LOGIC [II. v. 
 
 (4) The stem is without any distinct pith, continuous rings 
 
 of wood, or separable bark. The wood consists of 
 bundles of fibres and vessels, which are separately 
 embedded in cellular tissue. The hardest bundles 
 are outside. 
 
 (5) The outer parts of the flower are in threes. 
 
 (6) With few exceptions, the leaves are straight-veined. 
 
 It is to be noticed that the most important characters are by no 
 means (usually) the most obvious. Our natural groups seem, at 
 first sight, to include extremely heterogeneous kinds. To an 
 unbotanical mind, the yellow cowshp, the scarlet pimpernel, and 
 the purple cyclamen would seem as unlike as flowers could be ; 
 yet these three species are closely related, and we class them all 
 in tlic Natural Order Primulacea. So also the daisy, the goldenrod, 
 and the thistle belong to one Natural Order, the Compositte ; and 
 two flowers so unhke as the blue cornflower and the purple knap- 
 weed belong, not only to the same family, but even to the same 
 genus (Centaurea). We do not, in Classification, give the preference 
 to the most obvious, but to the most significant and the least vari- 
 able characters. Thus, in both Zoological and Botanical Classifica- 
 tion, Analogy (resemblance arising from adaptation to similar 
 functions) is of far less importance than Homology or morphological 
 identity. Hence the paramount necessity, for purposes of Classi- 
 fication, of the study of Development. In classing any organism, 
 we must consider not only its characters at any one moment of 
 observation, but also those exhibited by its past history ; for thus 
 alone is it possible to ascertain the liomologies of structure upon 
 which Comparative Morphology is founded. 
 
 The importance to Classification of a close study of Development 
 has been tenfold increased by the discovery of the connexion 
 between ontogeny and phylogeny, the establishment of the theory 
 that each individual organism (at least among animal forms) 
 ' recapitulates ' in its development the wliole history of its race. If 
 we were to meet, for the first time, a full-grown hen, we might be 
 uncertain of her exact place in the Animal Kingdom ; but when we 
 have watched day by day the development of the chick, in the egg, 
 from the single cell which represents some protozoan ancestor, 
 through the fLsh-hke stage which exhibits a swimming tail and con- 
 spicuous gill-slits, and again through the reptilian form with its 
 four limbs and hands, each with its five digits distinctly shown by 
 the microscope, on to the first emergence of the characteristic bird- 
 like form, then we have no difficulty in relegating our adult fowl to 
 her proper position in our zoological classification. 
 
 Thus, an ideal natural classification of animals or plants would 
 represent, not only the present affinities, but tlie whole ancestral 
 history of the organisms dealt with. It would indicate no mechan-
 
 Chap. V.] CLASSIFICATION 61 
 
 ical arrangement of isolated types, but an organized continuum 
 in which some of the missing Hnks would be supplied by pala^onto- 
 logical research, and others would be ideally reconstructed with 
 more or less of probable exactness. Our scheme of classification 
 would thus become a genealogical tree, showing the vital relation of 
 each kind to all the others, and thus making evident the organized 
 unity of the whole.* 
 
 Definition in Connexion with Natural Classification. — The Problem 
 of Classification involves the necessity of defining the names wliich 
 constitute the Nomenclature. In the case of Natural Classification, 
 Definition takes the form of Characterization — i.e., of giving an in- 
 ventory of the known characteristics common to all the typical 
 members of the class indicated by the term to be defined — a result 
 to be obtained only through those thorough-going analyses and syn- 
 theses which are called for when we study Nature, with reconstruc- 
 tive intention, as a complex and developing system of which all the 
 parts and aspects are intimately interrelated. In a natural classi- 
 fication, as we have seen, every group, from the primary divisions 
 downwards, will possess a number of common characteristics. 
 Thus, the definition of the term ' Dicotyledon ' might be stated 
 somewhat as follows : 
 
 The term ' Dicotyledon ' stands for a plant possessing the distin- 
 guishing characters of the Angiospcrmoe (genus), and further 
 characterized by the following marks : 
 
 Embryo with two cotyledons. 
 
 Stem, when perennial, having a distinct pith, continuous rings of 
 wood, and separable bark, and branching repeatedly. 
 
 Leaves net-veined. 
 
 Parts of the flower usually in fives. 
 
 So, again, the definition of the term ' Vertebrate ' in Zoology 
 would be somewhat as follows : 
 
 The term ' Vertebrate ' stands for a multicellular animal (genus) 
 characterized by the following marks : 
 
 1. The possession (at some stage of the animal's development) of 
 a smooth, elastic, dorsally i)laced rod (the Notochord) lying ventral 
 
 * As a particularly important and impressive instance of Natural Classification 
 in the realm of inorganic Nature, we may mention the classification of the chemical 
 elements according to Mendelecll's Periodic Law. This same instance is also 
 an excellent example of Classification by Series (vide Professor Duncan's ' The 
 New Knowledge,' ch. iii. ; Hodder and Stoughton, 1906). As another, perhaps 
 still more important, instance of Natural Classification — this time in the realm 
 of spiritual values — we would refer to an article on ' The Classification of the 
 Virtues,' by H. W. Wright (The Journal of Philosophy, Psychology, and Scictiiific 
 Methods, vol. iv.. No. G, March 14, 1907). However, Mr. Wright does not so much 
 furnish the classification itself as the principles for making it. ' As the species 
 are classified according to the part they play in the process of organic evolution, 
 so the virtues are classitied according to the office they discharge in the organization 
 of conduct. Thus our ideal of a principle of classification organic to the field of 
 its application is realized ' {ibid., p. 100).
 
 62 THE PROBLEM OP LOGIC [IL v. 
 
 to the norve-cord. This may be replaced by a cartilaginous {i.e., 
 gristly) rod, or by a column of distinct ' vertebra.' These, again, 
 may either remain cartilaginous or be replaced later by vertebrre 
 of bone. (These vertebrae grow round and protect the nerve-cord.) 
 
 2. The possession, at some stage of development, of gill-shts in 
 the anterior part of the alimentary canal, 
 
 3. An unpaired dorsal nerve-cord, which is tubular, having a 
 central canal, and is protected by the notochord or the vertebrae. 
 In the more advanced forms the brain and sense organs are highly 
 developed, the latter being paired. 
 
 4. A highly organized circulation. The heart is always ventral 
 to the alimentary canal. 
 
 5. Symmetrical segmentation. 
 
 Definition by characterization tends, in the case of the develop- 
 mental sciences, to take the form of Definition by Type, a type 
 being defined as ' an example of any class — for instance, the species 
 of a genus — which is considered as eminently possessing the character 
 of the class ' (Whewell). Thus, Dr. P. Chalmers Mitchell says that 
 Morphologists ' are slowly coming to some such conception as that 
 a species is the abstract central point around which a group of 
 variations oscillate, and that the peripheral oscillations of one 
 species may even overlap those of an allied species ' (' Enc. Brit.,' 
 10th edn., vol. xxviii,, article on ' Evolution,' p. 343). Definition 
 by Type is no doubt to this extent logically defective — that it does 
 not provide ideally against ambiguity ; and, in its insistence on a 
 central as distinguished from a peripheral defijiiteness of character- 
 ization, it resembles Description rather than Definition proper ; 
 but it is none the less the Definition natural to elassificatory Science. 
 
 There is good reason why, in Botanical andZoolofrical classification, 
 the reference to organized reality should call for definition by type. 
 Typical structures possessing a complete fitness for existence sur- 
 vive, and the intermediate forms tend to disappear, though there 
 may be many deviations from type that are not important enough 
 to interfere with that fitness to survive upon which the persistence 
 of the type depends. Hence, in the developmental sciences. Real 
 Definition — the definition of a class or concept that is framed to bear 
 the searchlight which Science throws upon Nature — is essentially 
 central in character. The central qualities and tendencies determine 
 the definition ; but in its application tlie definition takes in all varia- 
 tions which show a more marked approximation to the central 
 refjuirements in question than to those of any other definition. 
 This Definition by Type, we may add, forms a transitional link 
 between a rigid peripheral definition, or definition by boundaries, 
 and the more inward and vital definition by ends or ideals. In this 
 latter kind of Definition, the defining marks, far from being pos- 
 sessed in common by all the members of the class defined, may be
 
 Chap. V.] CLASSIFICATION 63 
 
 possessed, in strictness, by none. If ' Man ' is defined by the ideal 
 his nature is capable of reaching, it is not necessary tliat any 
 single individual man should possess the marks in question.* 
 
 Real Definition, finally, is essentially provisional and progressive. 
 The widening of knowledge implies tlie remodelling of scientific 
 principles and scientific classifications, and this implies that the 
 definitions of essential concepts and of natural kinds undergo a 
 sympathetic renewal. Where the aim of Definition is to charac- 
 terize according to Nature, and the knowledge of Nature is continu- 
 ally deepening, the definition must adapt itself if it is not to stultify 
 the very reason for its existence. 
 
 Diagnostic Classification and Definition. 
 
 A natural classification, in order to be really useful, should be 
 accompanied by an analj'tical key. Such a key is a diagnostic 
 classification, its function being to serve as an index or searcher 
 for the corresponding natural classification. The essential dis- 
 tinction between these two types of Classification is that, wliilst 
 the marks which serve to locate a given species witliin the system 
 are, in the ease of Natural Classification, fundamental, in the case 
 of Diagnostic Classification they may be merely superficial. The 
 marks here are external and salient, and easily recognized. A 
 diagnostic classification meets an important practical requirement 
 — that of easy diagnosis — diagnosis being the method of determining 
 the place of a natural kind in a classification, finding the correct 
 name or label for the object by means of its characteristics. 
 
 The botanical system of Linnaeus is essentially of the diagnostic 
 type. It has been called artificial, because its classification-con- 
 nexions do not stand for natural affinities. This is true, but it 
 was not the author's intention that his classification should be 
 natural. He intended that it should serve as a complete practical 
 index or catalogue. ' It is an index to a department of the book of 
 nature, and as such is useful to the student. It does not aspire to 
 any higher character, and although it cannot be looked upon as a 
 scientific and natural arrangement, still, it has a certain facilitj' of 
 application wliich commends it to the tyro. In using it, however, 
 let it ever be remembered that it will not of itself give the student 
 any view of the true relations of plants as regards structure and 
 properties, and that, by leading to the discovery of the name of a 
 plant, it is only a stepping-stone to the natural sj'stem. Linnseus 
 
 * As a suggestive illustration of what is involved in a philosophic definition — 
 a definition, that is, which is framed under the control of such categories as those 
 of dcveloptncjit and pcrsonnlity — see Edward Caird's ' Evolution of Religion,' 
 Lecture II. For the so-called ' pragmatic ' definition, see C. S. Peirce. ' Illustra- 
 tions of the Logic of Science,' in The Popular Science MontWy, vol. xii., under the 
 two headings. The Fixation of Belief ' (November, 1S77), and ' How to Make our 
 Ideas Clear ' (January. 1S78).
 
 64 THE PROBLEM OF LOGIC [II. v. 
 
 himself claimed nothing higher for it. . . . Besides his . . . index, 
 lie also promulgated fragments of a natural method of arrangement ' 
 (' Enc. Brit./ Otii edn., vol. iv., p. 80). 
 
 This distinction between Natural and Diagnostic Classification 
 puts in a clear light the relative character of what is ' essential ' for 
 Definition. The need for a diagnostic classification shows that for 
 scientific purposes the salient mark is often more essential than the 
 structural or the functional mark. If we wish to identify a plant, 
 we don't, as a rule, need to examine its microscopic characters in 
 order to class the specimen by some minute peculiarity of structure. 
 It is, therefore, not true to say that in Diagnostic Definition the 
 specifying mark may be non-essential. For the purposes of 
 Diagnosis and of Diagnostic Definition its saliency is just what is 
 essential. Definition in connexion with Diagnostic Classification is 
 Definition with the purpose of identification. Hence it naturally 
 takes the form of diagnostic characterization. It is a definition 
 giving the salient, easily tested marks. Thus ' Iodine ' may be 
 diagnosticallj^ defined as a substance that colours starch blue. 
 Where the absence of a mark forms the most striking means of 
 identification, the diagnostic definition includes a negative 
 characteristic : 
 
 Manx cats are cats that have no tails. 
 
 An apetalous plant is a flowering-plant in whicli the corolla is 
 absent. 
 
 Formal Classification. 
 
 As we have already stated, formal classifications are characterized 
 by the intimate relation in which they stand to the specific require- 
 ments of the individual classifier. There are two main types of 
 formal Classification to which we may conveniently give the names 
 of Conventional Classification and Index-Classification, the latter 
 existing solely for the sake of, and in the service of, the former. 
 
 Conventional Classification, again, may be either Appropriately 
 Conventional or not. In the latter case, the Classification may 
 suitabl}' be called Artificial. A conventional classification is 
 appropriately carried out when there is no maladjustment between 
 the nature of what is classified and the specific nature of the purpose 
 which directs the classification. The kinds here classified must 
 therefore not be those proper to any natural science, for in that case 
 a subjective, conventional ordering of them would not be proper to 
 their nature. They must be drawn from products of human art 
 and thought, such as statues or books. Here, more especially in 
 the case of books, no valuable end would be gained by attempting 
 to group the types or kinds after the complete manner characteristic 
 of Natural Classification, in which the full resemblances of the 
 types classified are taken into account. It is here more purposive.
 
 Chap. V.] CLASSIFICATION 65 
 
 and therefore more lofrical, to fix on an attribute or group of attri- 
 butes which happens to be of importance for the purpose, and to 
 construct and classify the types in strict relation to it. On the 
 other hand, where the types are natural kinds — e.g., species of plants 
 — and yet are not classified according to their nature, but according 
 to a specifically subjective principle of selection and arrangement 
 chosen without regard to the real nature of the material in question, 
 the classification may conveniently be termed ' Artificial.' This 
 appears to be a right and proper use of the word, though it gives a 
 sense more restricted than that implied in the ordinary contrast 
 between Natural and Artificial Classification. Thus the various 
 kinds of garden plants might be formed and arranged in the light of 
 some subjective interest {e.g., the decorative interest, which may 
 find colour-distinctions essential), and such an arrangement, when 
 contrasted with the true, genetic order in which the various species 
 of plants stand in relation to each other, might suitably be called 
 ' Artificial.' Artificial Classification would then be a kind of 
 Conventional Classification, but would not be identified with 
 Conventional Classification in general. 
 
 Conventional, like Natural, Classification needs the co-operation 
 of a key-classification subordinated to its own special requirements. 
 But this key-classification will not, of course, share the objective 
 character and intention of the analytical keys proper to Diagnostic 
 Classification. As a rule, it will be found to be strictly alphabetical, 
 as in the case of all indexes and catalogues. Thus, a librarian, in 
 constructing and classifying types of books, will do so according to 
 some subjective plan for which his own convenience rather than the 
 nature of the object is the dominating standard. But the librarian's 
 classification of the books is one thing, the cataloguing of the same 
 for the convenience of readers is another thing. The latter classifi- 
 cation is a mere finder to the former as that is represented by the 
 arrangement of the books on the shelves, and stands to it in a rela- 
 tion closely analogous to that in which a diagnostic stands to a 
 natural classification. And yet there are differences. Thus, a 
 diagnostic classification yields in itself a certain superficial know- 
 ledge of the nature of what is classified, and can be translated into 
 other languages, the arrangement not being alphabetical ; but a 
 catalogue, qvxi alphabetical arrangement, yields no knowledge of 
 what it classifies. To be aware that Punch and the ' Principia ' 
 have a common initial letter hardly constitutes a knowledge of the 
 books in question, and any attempt at translation would involve 
 such a complete transformation of the original arrangement as to be 
 equivalent to the construction of a new catalogue. 
 
 Finally, we would draw attention to the fact that spatial grouping, 
 such as that of the books on the shelves of a library, or the arrange- 
 ment of a collection of butterflies in a cabinet, is in no sense a 
 logical classification. It is an arrangement of specimens, and not
 
 GG THE PROBLEIVI OF LOGIC [II. vi. 
 
 a classification of species. Again, the assigning of individuals to 
 their respective classes, though in itself a logical operation, is not a 
 classification of species, but only a classing of objects. We may class 
 specimens, but we cannot classify them 
 
 Classification. 
 
 I I 
 
 Real. Formal. 
 
 I I 
 
 II I I 
 
 Natural. Diagnostic Conventional. Index-Classification 
 
 (Analytical key). | (Alphabetical key). 
 
 I I 
 
 Appropriate. Artificial. 
 
 CHAPTER VI. 
 II. (vi.) SCIENTIFIC TERinNOLOGY AND NOaiENCLATURE. 
 
 Technology, 
 
 Let us first say a few words about Technical Terms. Words, 
 as we have seen, are sensitive, and pass through a process of growth. 
 Various associations cluster round the original nucleus of meaning, 
 making it extremely hard to define sharply even the usual meaning 
 of the word. Now Science, in its anxiety to escape the dangers 
 arising from these clinging associations, often takes the extreme 
 step of inventing symbols which, since they can never be current, 
 can never gain an associative meaning. Hence a great advantage 
 in clearness and precision. On the other hand, we must face the 
 fact that, by using technical terms, we cut Science off from 
 ordinary life. To meet this objection Political Economy has, on 
 the whole, adopted the plan of using popular words — e.g., ' rent,' 
 ' wages,' etc. But it defines these strictly for its own accurate 
 purposes, and is therefore misleading to the uninitiated reader. 
 And yet this plan has the great advantage of keeping the student in 
 close touch with fact. The use of technical terms is, of course, 
 justifiable only on the ground that accurate distinctions are needed. 
 To be technical in one's language and inaccurate in one's thought 
 is to make oneself ridiculous. The two essentials of the technical 
 language of Science are (a) a good descriptive Terminology, (6) a 
 good Nomenclature. 
 
 (a) Terminology. — By descriptive Terminology we mean a collec- 
 tion of word.s wliicli will enabl ; us to describe natural kinds. Of 
 all the sciences, perhaps Botany has the best descriptive terminology.
 
 Chap. VI.] TERMINOLOGY AND NOMENCLATURE 67 
 
 Every part of a plant and every variety of plant structure has been 
 so exhaustively named that the plant can, so to speak, be drawn in 
 words. The flower has its calyx, consisting of sepals, its corolla of 
 petals, its stamens, with their filaments and anthers, its pistil with 
 its carpels, style, and stigma, etc. 
 
 Again, among all the various forms of the leaves of plants tliere 
 is not one which cannot be accurately described. Tlius, when a 
 leaf is long and very narrow, it is said to be linear ; when the length 
 is three or more times as great as the breadth, and the broadest 
 part is below the middle, w^iile the summit is tapering, the leaf is 
 described as ' lanceolate ' ; when the broadest part is above the 
 middle, and the blade tapers towards the base, the leaf is called 
 ' cuneate ' ; and when the blade is broadly cuneate with a rounded 
 top we say that it is ' flabelliform.' A leaf that approaches the 
 form of a spoon or ladle is called ' spathulate ' ; and other forms of 
 leaves are known as ' ovate,' ' obovate,' ' orbicular,' ' oval,' ' oblong,' 
 ' elliptical,' ' rhomboidal,' ' oblate,' and ' falcate.' 
 
 (6) Nomenclature. — jA descriptive terminology must be carefully 
 distinguished from a nomenclature. The nomenclature of any 
 classification consists of the names for the groups or kinds which 
 the classification systematizes ; the words by which these groups 
 are characterized constitute its terminology. 
 
 Nomenclature, like Definition, tends to vary with the point of 
 view from which names arc considered. The purpose of Science 
 being steady — that of naming in accordance with principles of 
 Classification — we have, of course, a corresponding steadiness of 
 nomenclature. With variety of interest comes variety of nomen- 
 clature, as the following extract from Watts's ' Logic ' (quoted b}^ 
 Dr. Gilbart, ' Logic for the Million,' pp. 66, 67) clearly shows : 
 ' ]Most of all [flowering] plants agree in this — that they have a root, a 
 stalk, leaves, buds, blossoms, and seeds ; but the gardener ranges 
 them under very different names, as though they were really 
 different kinds of beings, merely because of the different use and 
 service to which they are applied by men — as, for instance, those 
 plants whose roots are eaten shall appropriate the name of roots 
 to themselves, such as carrots, turnips, radishes, etc. If the leaves 
 are of chief use to us, then we call them herbs, as sage, mint, thyme ; 
 if the leaves are eaten raw, they are termed salad, as lettuce, 
 purslain ; if boiled, they become pot-herbs — as spinach, coleworts ; 
 and some of those plants which are pot-herbs in one family are salad 
 in another. If the buds are made our food they are called heads or 
 tops, so cabbage-heads, heads of asparagus, and artichokes. If the 
 blossom be of most importance we call it a flower, such as daisies, 
 tulips, and carnations, which are the mere blossoms of those plants. 
 If the husks or seeds are eaten, they are called the fruits of the 
 ground, as peas, beans, strawberries, etc. If any part of the plant 
 be of known and common use to us in medicine, we call it a physical 
 
 5—2
 
 68 THE PROBLEM OF LOGIC [11. 
 
 VI. 
 
 herb, as carduus, scurvy-grass ; but if we count no part useful we 
 call it a wood, and throw it out of the garden ; and yet, perhaps, our 
 next neighbour knows some valuable property and use of it ; he 
 plants it in his garden, and gives it the title of an herb or a flower. 
 Now, when things are set in this clear light it appears how ridiculous 
 it would be to contend whether dandelion be an herb or a weed, 
 whether it be a pot-herb or a salad, when, by the custom or fancy of 
 difiFerent families, tliis one plant obtains all these names, according 
 to the several uses of it and the value that is put upon it.' 
 
 In an ideal nomenclature each name would indicate the place 
 occupied by the class named in the classification. This would be 
 done b}' relating the class-names to each other, instead of allowing 
 each group to name itself independently of the rest. Definition 
 per genus et differentiam is thus represented, in a very simple form, 
 in the systems of scientific nomenclature. In Botany and Zoology, 
 for instance, each kind takes the name of the genus of which it is a 
 species, and adds toit adift'erentia giving, usually, some characteristic 
 or salient mark. Thus, the zoological name of the Rabbit is Lepus 
 cuniculus : that of the Common Hare, Lepus timidus. The Red Deer 
 is Cervus elaphus, the Wapiti Deer Cervus canadensis. The Brown 
 Bear is Ursus arctos, the Grizzly Bear Ursus ferox. So also Botanists 
 call the Field Rose Rosa arvensis, the Dog Rose R. canina, the 
 Sweetbriar R. rubiginosa. The Marsh Violet is Viola palustris, the 
 Sweet Violet V. odorata, the Hairy Violet V. hirta, the Sand Violet 
 V. arenaria. The Common and the Creeping Buttercu}:), the Hairy 
 Ranunculus, ' Goldilocks,' and the Lesser Celandine all belong to 
 the genus Ranunculus, and are distinguished respectively by the 
 specific names acris, repens, hirsutus, auricomus, and Ficaria. 
 
 In view of the great efficiency secured by making the name itself 
 a sort of condensed definition, we may feel considerable sympathy 
 with Mr. Garden's protest against what he calls the evil fashion, 
 once so prevalent amongst naturalists, of paying compliments 
 by naming genera and species after each other. ' What am I the 
 better,' he asks, ' for hearing a rare moss called Hedwigia horn- 
 schuchiana, beyond being led to infer that Germany has, or had, 
 two botanists, one called Hedwig and the other Hornschuch ? On 
 the other hand, when I am told that such a moss is called Trichosto- 
 mum lanuginosum, I am, on supposition of previous knowledge 
 of Trichostomum, presented with a definition, lanuginosum 
 (" woolly ") expressing the differentia of this species in the genus 
 Trichostomum, even as Trichostomum does that of the genus when 
 viewed as species of the higher genus which contains it.' 
 
 Chemical Nomenclature is peculiarly efficient. The names of the 
 Elements, indeed, have, for the most part, been arbitrarily chosen, 
 and are of historic interest rather than of scientific value ; but the 
 names of compound substances are assigned on systematic principles. 
 Those of simple binary compounds {i.e., substances composed of
 
 Chap. VI.] TERMINOLOGY AND NOMENCLATURE 69 
 
 two elements only) are formed by combimng the names of their 
 component elements, and as in many cases the same two elements 
 combine in different proportions, tlie different compounds so pro- 
 duced are distinguished by means of tlie addition of terminal 
 syllables or (more usually) of prefixes. Thus, among the compounds 
 of Sulphur and Oxygen we have Sulpliur dioxide, Sulphur trioxide, 
 Sulphur sesquioxide. So, too, we have Potassium monoxide and 
 Potassium dioxide. Lead tetroxide. Arsenic pentoxide, etc. Simi- 
 larly the names of acids indicate their places in the classification by 
 means of significant prefixes and suffixes. Thus Sulphur, in com- 
 bination with Oxygen and Hydrogen, forms a whole series of acids 
 known respectively as hyi:)osulphurous acid, sulphurous acid, sul 
 phuricacid, thiosulphuric acid, pyrosulphuric, and anhvdro-sulphuric 
 acid. In combination with other elements Sulphur forms a series 
 of sulplu'c?es, sulpln7e5, and sulphates, the termination showing in 
 each case the position in the series of the compound indicated. 
 Further, the symbolic nomenclature of Chemistry is even more 
 efficient and precise than the verbal system. Each element is 
 symbolically represented by the initial letter (or two of the letters) 
 of its Latin name, and the symbolic names of compounds are made 
 up of the symbols of their component elements with the addition 
 of numbers which indicate the proportions in which these elements 
 are combined. Thus H is the symbol of Hydrogen, S of Sulphur, 
 and of Oxygen ; and, in the series of acids cited above, hypo- 
 sulphurous acid is symbolically represented as H2SO2, sulphurous 
 acid as HjSOg, and the others in order as H2SO4, H2S2O3, HSgO,, 
 HgSgO,. 
 
 In Astronomy, also, we find both verbal and symbolic names. 
 The verbal name is framed, as a rule, on the ' genus et differentia ' 
 principle. The system is analogous to our method of designating 
 persons by a family name and a Christian name. The family name 
 is represented by that of the constellation, the Christian name by 
 a letter of the Greek or Roman alphabet, or a number. Thus, 
 a Lyra}, Q Pegasi, Z Herculis, T Coronae, 113 Herculis. 
 
 Frequently the number of the star in some given catalogue is 
 used as its designation — e.g., Lac(aille) 7215, Brad(ley) 3077. 
 
 As the constellations, especially those in the southern hemi- 
 sphere, have been variousl}'' mapped out by different astronomers, 
 and as different astronomers, again, use different catalogues, there is 
 still a good deal of uncertainty as to the naming of stars. The same 
 star may thus belong to more than one constellation, and be 
 differently numbered in different catalogues. 
 
 Hence the great advantage in star-naming of using the symbohe 
 name. The symbolic name is a formula or rule for finding the 
 position of a star, and so identifying it. The formula consists in 
 giving what are called the co-ordinates of the star — its latitude and 
 longitude, so to speak (technically its right ascension and declina-
 
 70 , THE PROBLEM OF LOGIC [II. vii. 
 
 tion). Thus, the symbolic name of a Lyrae is : 18 hours, 33', 6" R.A. ; 
 + 38^ 41' Declination. 
 
 It would seem, then, that, since the purpose of naming a star is 
 to be able to identify it by means of its name, and since a star can 
 always be identified by its right ascension and declination, nothing 
 but this statement of the star's position is really necessary. The 
 symbolic name should be completely sufficient. ' Unfortunately, 
 the position constantly changes through the precession of the 
 equinoxes and other causes, so that this designation of a star 
 is a variable quantity.'* The true symbolic name of a star is, 
 tlierefore, given by the formula noting its R.A. and declination + all 
 the rectifications required for precession, refraction, errors of 
 instrument, personal equation, etc. 
 
 CHAPTER VII. 
 
 II. (vii.) CONNOTATION AND DENOTATION. 
 
 DEFI^'ITION and Division are the two fundamental methods for 
 develoi^ing or making exphcit the meaning of a term. The results 
 of these two processes have special names given to them ; — Definition 
 gives the connotation, Division the denotation of a word. Thus, if 
 we define ' Man ' as ' rational animal,' this is its connotation. If 
 we divide ' Man ' into ' Aryan, Semitic, or Turanian,' we are giving 
 its denotation. 
 
 The connotation of a term, then, consists of the defining marks 
 which the name implies ; the denotation, of the alternative class- 
 distinctions into which the meaning of the name can relevantly be 
 di\ided. Let us consider more closely each of these two aspects 
 of a word's meaning. 
 
 1. Connotation may be either formal or real. The connotation of 
 a name will be formal when the name is used in the service of an 
 interest that is more or less subjectiv^e and occasional ; it will be 
 real when it is the connotation of a scientific term. But in either 
 case connotation is essentially definite, being the product of defini- 
 tion. And just in so far as an object possesses the attributes or 
 the characteristics formulated in the connotation does it merit and 
 obtain the name. 
 
 From the two types of logical connotation above referred to, the 
 formal and the real, we must carefully distingush a type of non- 
 logical connotation usually referred to as ' subjective intension.' 
 
 The ' subjective intension ' of a class includes only such marks as 
 
 * Simon Ncwcomb, ' The Stars,' pp. 36, 37.
 
 Chap. VII.] COXXOTATION AND DENOTATION 71 
 
 happen to be suggested on any occasion to any individual when a 
 given word is mentioned. Thus, in the instance given on p, 29, 
 ' crystal ' would have been to the author, as a child, part of the 
 * subjective intension ' of the word ' palace.' The ideas which a 
 word serves to bring before the hearer's mind constitute its indi- 
 vidual or ' subjective ' intension (in the usual sense of these words), 
 but not the logical connotation. The psychological suggestions of a 
 word are one thing, the logical imiDlications of a word quite another.* 
 
 2. The Denotation of a class or class-name depends upon its 
 connotation, and in the same sense in which the division of a class 
 depends upon the definition given to it. As division serves to 
 differentiate and to specify what is indeterminate in a definition, so 
 the denotation differentiates and specifies what is indeterminate in 
 the connotation. The distinction between these two ways of stating 
 the matter is that, in the former case, when we are speaking of 
 Definition and Division, we have logical processes in mind, and, 
 in the latter case, the results of these processes. It is quite custom- 
 ary, however, to use the term ' Definition ' in the sense here reserved 
 for ' connotation.'! Denotation will be formal or real according as 
 the connotation is formal or real. The formal denotation of the 
 term ' horse ' includes, as alternatives, all the different kinds of 
 horses that come logically under the class ' horse ' as formally 
 defined. The real denotation of the term includes, in a similar 
 disjunctive sense, all the kinds of Equus cahallv^ that come witliin 
 the central or ' typical ' definition of that species recognized by 
 classificatory Zoology. 
 
 3. Interpreting ' Denotation ' as a stage in the logical develop- 
 ment of a term's meaning, we manifestly need another term, quite 
 distinct from logical denotation, for specifying the enumerative 
 relation between a class and the individuals to which the class- 
 concept applies. The term ' Extension ' appears to be the most 
 appropriate for this office. This would involve a specialization of 
 the function which Professor Keynes has suggested for this very 
 term. The problem, as it presented itself to Professor Keynes 
 (' Formal Logic,' 4th ed., p. 30), was to fix a type of extension (in 
 our sense of the word) which, in reference to the concept ' horse,' 
 say, should include not only the horses that breathe and eat grass, 
 but the steeds of fiction and imagination — such creatures as 
 Pegasus, tlie Wooden Horse of Troy, the white horse of the fine 
 lady at Banbury Cross, the horses of our dreams and of our desires. 
 
 * Vide Chapter I., p. 17. 
 
 t It would greatly conduce to clearness of nomenclature -were the terms ' Defini- 
 tion ' and ' Division ' exclusively applied to the propositions which represent the 
 defining or the dividing process. Thus, ' Man is a rational animal ' might pass as 
 the definition of ' man,' ' Man is Aryan, Semitic, or Turanian ' as a corresponding 
 d vision of the term. ' Rational animal ' would then be the connotation of ' man,' 
 ' Man is a rational animal ' its definition, ' Aryan, Semitic, or Turanian ' its deno- 
 tation, ' Man is .^van, Semitic, or Turanian ' its division.
 
 72 THE PROBLEM OF LOGIC [II. vii. 
 
 It is for this comprehensive office that Professor Keynes has sug- 
 gested the term ' extension.' ' By the . . . extension of a general 
 name . . .,' he writes, ' we shall understand the whole range of 
 objects, real or imaginary, to which the name can be correctly 
 applied, the only limitation being that of logical conceivability.' 
 But if the term ' extension ' is adopted in the sense here defined by 
 Professor Ke^Ties, it should not be overlooked that in so far as its 
 range is not limited by a corresponding connotation, either in the 
 formal or in the scientific sense, it cannot rank as a logical type of 
 extension. Considered apart from any reference to a definite 
 connotation, there would, for logical purposes, be the same objection 
 to the term ' extension ' as there is to the so-called infinite term 
 ' not-x ' {vide pp. 48-52). It is logically essential that extension 
 should in all cases be determined by connotation. 
 
 We must carefully distinguish between the specification of mean- 
 ing, through Division, and the application of meaning, through 
 Enumeration. We propose to stamp this distinction by a corre- 
 spondinglj' distinct use of the two terms ' denotation ' and ' exten- 
 sion.' Denotation we define as differentiation of meaning, to be 
 interpreted disjunctively through the formula 'G=Si or Sg or S^, 
 etc' Extension we define as application of meaning to individual 
 objects, to be interpreted conjunctively by means of the formula 
 ' G applies to (or indicates) the individuals I^ and I2 and I3, etc' 
 
 Corresponding to this use of the term ' Extension ' we would 
 suggest the term ' Intension.' Let ' Intension ' stand for the full 
 relevant development of the meaning of a concept through 
 Definition and Division. The intension of a term will then be 
 equivalent to its connotation and denotation combined, and we 
 liave the formula : 
 
 Intension — Connotation + Denotation. 
 
 Our use of the term ' Intension ' in the sequel will always be in 
 the inclusive sense here indicated. 
 
 The ' Inverse Relation ' of Connotation and Denotation. 
 
 It has been customary to formulate the relation between Connota- 
 tion and Denotation by pointing out that, as we pass from summum 
 genus to infima species in a classification, at each step increasing the 
 determinate connotation by at least one differentia, we tend to 
 diminish the number of kinds denoted by the concept. Thus, if we 
 specify 'ship' as 'steam-sliip,' the word no longer denotes the 
 mere sailing-vessel. If we further qualify it as ' screw-steamship,' 
 the species paddle-steamer is ruled out. 
 
 But, substantially correct as is this view of the relation in question, 
 it is none the less superficial and misleading in its emphasis. It 
 obscures the fact that the fundamental relation between connotation
 
 Chap. VII.] CONNOTATION AND DENOTATION 73 
 
 and denotation is not ' inverse,' but ' complementary ' — that conno- 
 tation and denotation are, in fact, complementary stages and co- 
 factors in the logical articulation of meaning. Moreover, the 
 relation by no means implies that the total meaning or intension 
 of a genus is poorer than that of a subordinated species. This 
 remains an open question. Let G stand for any given genus, 
 Sp So, S3 for its species (F.D. /) ; let d^ d^ d^ stand for the deter- 
 minate connotation of G, and let f-^, /j, f^ stand for those specifica- 
 tions of / which give the differentiae of S^, S.,, S3 respectively. Then 
 the full meaning or intension of G, as relevant to the present com- 
 parison between G and its species, is given by 
 
 ^- /. L 
 
 1 Ji Ji 
 
 and the corresponding intension of S^ is given by d^ + d^ + d^+f^. 
 The question before us, then, is whether (/j or /g or f^) — for this is 
 what 
 
 / 
 
 amounts to — is richer or poorer in meaning than j^. The difficulty 
 which besets the solution of this conundrum suggests that genus 
 and species are more profitably studied as mutually indispensable 
 links in the development of meaning than as rival claimants for 
 some monopoly of meaning which shall enrich the one at the 
 expense of the other. 
 
 Connotative and Denotative. Names — The Limits of Definition 
 
 and Division. 
 
 A name may appropriately be called ' connotative ' in so far as it 
 possesses a connotation, ' denotative ' in so far as it possesses a 
 denotation. The expressions are useful as helping to give precision 
 to the important inquiry concerning the limits of definability and 
 divisibilit}'. 
 
 If we consider a given conceptual system, such as a natural 
 classification of animals or plants, it is at once clear that all the 
 subaltern genera, the classes between the summum genus and the 
 infimae species, are both connotative and denotative ; they are, we 
 may say, conno-denotative. But it is not so clear that either the 
 summum genus or the infima species is conno-denotative. The 
 summum genus — ' Animal ' in the case of the zoological, ' Plant ' 
 in the case of the botanical classification — cannot, relevantly to the 
 system which it represents, be defijied per genus et difjcrentiam. 
 The summum genus, being the ' highest,' cannot be the species of a
 
 74 THE PROBLEM OF LOGIC [11. vii 
 
 genus higher than itself. It might seem, then, that summa genera 
 were indefinable. So, again, it would seem that the infimae species, 
 being ' lowest ' in the ranks of a division, were not relevantly 
 subdivisible. 
 
 There are, moreover, other difficulties — those, namely, which are 
 associated with the ambiguity of the term ' infima species.' If the 
 infima species is a class-concept, it should be definable in a sense 
 precisely similar to that in which a subaltern genus is definable. 
 If, on the other hand, the proper name, or the singular meaning, is 
 to be identified with the infima species as its hmiting form, we are 
 confronted by a new set of difficulties which centre round the 
 question : ' Can a proper name be defined V 
 
 Having thus briefly stated the difficulties, we pass on to consider 
 how they can best be met. 
 
 (i.) Is the Summum Genus Definable ? 
 
 It is manifestly true that in the case of the summum genus 
 ' Animal ' there is, within the classification, no higher genus by the 
 help of which it could be defined. But this simply points the moral 
 that the zoologist cannot develop the full meaning of his leading 
 concept, ' Animal,' without connecting it with the leading concepts 
 of other sciences — e.g., the botanical concept ' Plant ' — and recog- 
 nizing a superordinate genus, ' Organism,' which dominates both 
 interests ahke. At the same time, even within the limits of the 
 specific classification which it represents, a summum genus must 
 admit of a partial definition through an indeterminate attribute, 
 the primary F.D., of which all subsequent fundamenta di\4sionis 
 are the specifications. Thus, taking as our summum genus one of 
 the two primary groups into which the Animal Kingdom is divided, 
 the Sub-Kingdom ' Metazoon,'* this summum genus is definable 
 as ' an animal organism possessing an anatomical structure of some 
 kind.'' This last-named characteristic, though indeterminate or 
 indefinite, supplies, none the less, a perfectly unambiguous mark, 
 and, as we have seen, an indefiniteness which does not amount to 
 ambiguity is no disquahfication for the purposes of Definition. 
 
 * Protozoa cannot unarabiguoualy be said to possess anatomical structure. 
 If we have shirked the definition of ' Animal,' it is because scientists do not yet 
 seem to have discovered a satisfactory differentia between ' animal ' and ' plant.' 
 But if this should not be obtainable, the logical course would be to absorb the 
 so-called 'Animal' and 'Plant' Classifications within the single Classification of 
 Organic species. The Summum Genus 'Organism' might thus be defined as 
 ' a cellular structure of some kind,' or, better still, ' a protoplasmic structure of 
 Bome kind.' The ess*3ntial point is that no classificatory system can be developed 
 without a primary F.D., and this primary F.D. supplies an adequate differentia 
 of the Summum Genus, distinguishing it unambiguously from all other Summa 
 Genera. The reader who is interested in the attempt to fix the distinction between 
 'plant' and 'f^nimal' will find an excellent treatment of the problem in Prof. 
 Bergson's ' L'Evolution Creatrice' (douxieme Mition, pp. 115-1.30).
 
 Chap. VII.] COKNOTATION AKD DENOTATION 75 
 
 The disjunctive specification of this dominating fundamentum 
 gives the division of ' Vertebrate or Invertebrate' ; and all the subse- 
 quent fundamcnta — e.g., ' Dentition ' — are so many modifications 
 of this original attribute of the summum genus — the possession of 
 some kind of anatomical structure. 
 
 But a fresh difficulty arises when we conceive the process of 
 abstraction, whereby summa genera are reached, carried to its 
 limit, and culminating in a concept like ' Being ' or ' Existence.' 
 Such a concept or meaning can have no more general concept 
 beyond it, since it is posited as ultimate for our thinking. We 
 cannot, therefore, bring it under any superordinate genus, nor can 
 we connect it with any co-ordinate species fulfilling a function 
 logically similar to its own, as we can connect ' Animal ' with 
 ' Plant.' The ultimate summum genus cannot be defined per genus 
 et differentiam. We cannot compare this unique definiendum with 
 any co-species ; we therefore cannot sift agreement from difference, 
 and so distinguish a genus from a differentia. We must look 
 elsewhere for a solution of the problem. 
 
 It might be urged that this arch-concept is self-defining. But 
 if so, in what sense ? It cannot be self-defining in the sense in which 
 connotations are self-defining. The ultimate concept does not 
 tell us its own meaning as do the expressions ' rational animal ' 
 and ' the mother of the two Gracchi ' {vide p. SO). But if not self- 
 defining in this determinate form of self-definition, may it not still 
 be self-evident, and therefore in last resort self -definable ? 
 
 There is no logical justification for supposing this. The ultimate 
 abstraction can make no appeal to immediate experience ; it there- 
 fore does not proclaim its own meaning, in however vaeue and 
 undeveloped a form, by the easy way of unreasoned intuition. 
 
 But it might conceivably be self-evident in another sense. It 
 might proclaim its meaning indisputably to the trained insight of 
 the logical reason, though it failed to impress the exoteric conscious- 
 ness. Can it be self-evident in tliis esoteric sense ? In order to 
 test this point we apply the well-known logical criterion of intuitive 
 certainty ; we ask whether it is impossible to deny the self -evidence 
 of Pure Being without falhng into self-contradiction.* 
 
 Let us first consider the argument in favour of the logical self- 
 evidence of the statement that ' Something, qua pure being, is.' 
 We take as our model Dr. McTaggart's defence of the indubitable 
 certainty of Hegel's dialectical starting-point, the Category of Pure 
 Being stated in the form ' Something is.' Hegel's Pure Being differs 
 in some respects from the summum genus we are here considering, 
 but the differences do not affect the present argument, and our proof 
 of the non-self-evidence of the ' Being ' wliich gives the summum 
 genus tells equally well, in our opinion, against Dr. McTaggart's 
 
 * The ' self-afTirmation ' of Being — namely, the affirmation that it exists — is 
 in a sense, a statement of what it is, and to tliis extent implies its definability.
 
 76 THE PROBLEM OF LOGIC [11. vii. 
 
 defence of the self-evident character of the ' Something is,' which 
 gives the leading category of the Dialectic. 
 
 To deny the self-evidence of ' Being ' — so runs the argument — is 
 to dcnj- the self-evidence of the assertion that ' Something is.'* 
 But this assertion cannot be denied without being at the same time 
 reaffirmed. For the denial at least ' is.' And to doubt the assertion 
 is as conclusive in its favour as to deny it. For onr doubt must be 
 either genuine or not. If it is genuine, then we do not doubt that 
 wo doubt ; we hold that something is — namely, our doubt : and if it 
 is not genuine, then we are all the while admitting the truth that 
 ' Sometliing is,' while we pretend to doubt it. 
 
 Xow if this argument were sound we should have to admit the 
 self-evidence of Pure Being. But the argument is surely fallacious. 
 Suppose I deny the self-evident character of Pure Being. I assert 
 my denial, certainlj', but not in the sense of ' pure being.' I assert 
 it in a much less abstract sense. I may, therefore, without any 
 logical inconsistency, denj'' that ' Something qua pure being is,' for 
 the assertion of my denial is the assertion not that ' Something is ' 
 qua pure being, but that ' Something is ' for me as an immediate 
 experience. In the two propositions ' Pure Being is ' and ' My denial 
 of the existence of Pure Being is ' the word ' is ' has two quite 
 different meanings. 
 
 We therefore cannot admit that the ultimate summum genus is 
 either self-defining or self-evident ; nor, as we have seen, can it be 
 defined per genus et differentiam. 
 
 It would, no doubt, be convenient if at this point we could cut 
 the knot with the short sharp word ' indefmable.'' The stroke would, 
 however, be suicidal, for it would cut at the root of the whole logical 
 theory of Definition. If a term is ' indefinable ' in the strict sense 
 of the word, it must remain permanently infected with ambiguity, 
 should ambiguity ever come to cleave to it ; for, the remedy of 
 Definition being unavailable, the ambiguity must remain to tease 
 logicians to the end of time. But no one will pretend that the term 
 ' Pure Being,' that ' x ' which is the ultimate summum genus, is 
 free from ambiguity. Moreover, if an incurable ambiguity attaches 
 to the summum genus, there is no root of soundness in any classi- 
 ficatory system developed on the genus et differentia principle. 
 For in such a system there is no class-term of which the meaning 
 does not rest ultimately upon the summum genus. ' Man,' we say 
 ' is a rational animal ' ; but both rationality and animality are in 
 last resort specifications of the wholly indeterminate concept from 
 which the developmrnt of all moaning initially flows. If the 
 summum genus is indefinable, our definitions are, one and all, illusory, 
 and we can never ultimately know what we really do mean. Our 
 definitions will all be more or less remote specifications of ' that we 
 know not what.' If x^ be the ultimate concept, and a;„_i a penulti- 
 
 * Vide McTaggart, ' Studies in the Hegelian Dialectic,' § 18, p. 21.
 
 Chap. VII.] COXXOTATION AXD DENOTATION 77 
 
 mate concept — a species of x^ with differentia S,^ — we say that r„_j 
 is x^ quahficd by 5„. But what is a:,^ ? Similarly x^_<y is x^_■^^ quali- 
 fied by S„_i. Thus, a-„_2 inherits the vice of x^_^ which originated 
 in x^ ; and so we might go on to the limit of the infima species. 
 
 At this point we may be met with the impatient reply that the 
 whole difficulty is surely gratuitous, and that the ultimate concept 
 is not only definable, but definable in many ways. When we call 
 ' Being ' the ' summura genus,' are we not defining it ? Or we may 
 define ' Being ' as the ultimate category or concept, the wholly 
 indeterminate meaning. And are not these meanings, it will be 
 added, unidetermining ? Can ' the wholly indeterminate ' be con- 
 fused with any other meaning ? 
 
 To these plausibilities the sufficient response is that, from the 
 point of view of Definition per genus et diijerentiam, they all involve 
 a circulus in definiendo. ' Wholly ' and ' indeterminate ' must 
 themselves be specifications of the wholly indeterminate concept 
 they profess to define, for the ultimate concept is, by hypothesis, 
 ultimate, the ultimate generality whence proceed all the determinate 
 forms of conceptual being. Every term of every definition pre- 
 supposes tliis ultimate summum genus. 
 
 We have, then, still before us the task of showing in what sense, 
 if any, the summum genus is definable. 
 
 Our first step must be to name the summum genus more precisely. 
 From the standpoint of a merely objective view of the nature of 
 ' meaning ' there might be good reason for retaining ' Being ' or 
 ' Pure Being ' as the ultimate concept. But we have already re- 
 pudiated this abstractly realistic conception of ' meaning,' and 
 have adopted an inclusively idealistic conception, which claims that 
 meaning, whatever else it is, must always be the meaning of an 
 object for a subject. Tliis fundamental conviction, wliich has 
 been stated rather than defended, or defended only through its 
 power to reconcile conflicting views, calls for a correspondingly 
 idealistic framing of the ultimate concept. Shall we, then, call it 
 ' Knowable Being ' ? If it were urged that this concept suggests 
 a further-lying concept, 'Being as such,' which may be specified as 
 either knowable or unknowable, we should reply that ' unknowaft^e 
 being ' is, from our point of view, meaningless. We might, indeed, 
 content ourselves here with Professor Ferrier's simple argument 
 that ignorance is relative to knowledge, and that where no know- 
 ledge is possible there can be no possible ignorance. If there be an 
 unknowable, it is out of all relation to Consciousness, and we cannot 
 logically refer to it as ' it.' We cannot say that we are ignorant of 
 it, or that we don't know what it is. It must be nothing for our 
 ignorance as well as for our knowledge. 
 
 Adopting 'Knowable Being ' as our ultimate concept, we abandon, 
 as intrinsically unreasonable, the attempt to define the ultimate 
 concept except in relation to the logical interest through wliich we
 
 78 THE PROBLEM OF LOGIC [II. vii. 
 
 know it, and we begin to realize that the search for the hidden 
 sources of meaning must take us beyond the limits of any con- 
 ceptual tree which does not, throughout its ramifications, involve 
 this reference to interest and to knowledge. Xor is the appeal from 
 * object ' to ' object of interest ' an appeal to the deus ex machind. 
 For the reference to logical interest is involved in the very notion 
 of a concept. A concept is a conccpt-in-relation-to-logical-interest. 
 If the addendum be torn away, the concept itself vanishes. Hence 
 the appeal from the concept per se to the concept as known is simply 
 the demand to have made expUcit what is already implied. But to 
 bring out implications is precisely to do that which renders super- 
 fluous all unintelligible assistance from a deus ex machind. 
 
 The problem now presents a somewhat different aspect. The 
 ultimate genus, the genus that has no superordinate,* is seen to 
 have as its correlative a thorough-going logical interest. It there- 
 fore cannot get its whole meaning from itself. If it is definable at 
 all, it must be definable through the relation in which it stands to 
 such an interest. 
 
 This relation is, in brief, the subject-object relation. Though 
 bereft of all determinate content, the summum genus is still an object, 
 the object of a logical interest. Hence the problem of its definition 
 becomes the problem of deciding what we mean in general by an 
 object of logical inquiry. The attempt to cope with this problem 
 would bring us to the fundamental question of Kant's whole critical 
 inquiry : ' What are the conditions of a possible object of ex- 
 perience V It is not necessary for us to reconsider tliis problem, 
 or to gauge the value of Kant's solution of it. It is not necessary 
 because our aim is not to give the definition of ' Object,' but to 
 consider, and in some sense to answer, the question as to its 
 definability. We, therefore, content ourselves with noting the 
 following points : (1) That there is no meaning more ambiguous or 
 more in need of careful definition than the term ' object of ex- 
 perience ' ; (2) that the problem of its definition — i.e., the question 
 of its function and significance within the unity of exiDcrience — is a 
 fundamental problem of the Theory of Knowledge ; (3) that the 
 ultimate logical postulate which this defining-process presupposes 
 is the postulate of the radical intelligibility of experience. We have 
 only to add that tliis postulate is not optional. We cannot think 
 at all without making it. For to think is the same thing as to think 
 what is not self-contradictory {vide Chapter X. c) ; the self-contra- 
 dictory cannot be thought. But there is nothing ultimately 
 meaningless save the self-contradictory {vide pp. 103, 104). 
 Hence, to think at all is to tln'nk what has meaning and is pro tanto 
 radically intelligible. This ultimate postulate, then, may be 
 accepted as a irov aro). Not only is it true to say that if the universe 
 
 * It makes no difiFerence to the general argument if, with Porphyry and Aristotle, 
 we hold that there are as many ultimate genera as there are categories.
 
 Cu-vp. VIL] COXNOTATION AXD DENOTATION 79 
 
 were in any respect meaningless we could never know it, being 
 unable to think what is meaningless ; we may go further, and say 
 that if there were any such unknowable, we should not be suspecting 
 the possibility of its existence. To be conscious of a limit is to 
 have already transcended it ; the barrier that closes upon our 
 knowledge closes also upon our ignorance. 
 
 (ii.) Is the Infima Species Divisible ? 
 
 The reference to logical interest which has determined the nature 
 of our answer to the question, ' Is the summum genus definable V 
 suffices to give a definite answer to the further question, ' Is the 
 infima species divisible V For, if a given species is ' lowest ' 
 relatively to a stated interest, any further division of the species 
 would be necessarily irrelevant, and would involve that over- 
 distinction and false subtlety which is the inevitable penalty of 
 irrelevance. If we know our purpose and point of view, it must 
 always be possible to fix the critical point beyond wliich refinement 
 and the differentiation of meaning degenerates into irrelevant 
 subtlety. Where a classification is dominated by a scientific 
 interest, the extent to which a division is carried A\dll depend not only 
 on the degree of accuracy required, but on the progress of scientific 
 knowledge. Thus ' giraffe ' {Camelopardis giraffa), wliich used 
 formerly to stand as an infima species, is now known to include 
 eight or ten distinct varieties. We conclude, then, that while the 
 infima species, as such, should be regarded as logically indivisible, 
 yet fresh advances in scientific knowledge are always promoting 
 infimse species to a higher grade, and thus making them divisible. 
 Division, hke Definition, is essentially progressive. 
 
 (iii.) Can Proper Names be Defined? 
 
 Let us once more consider the celebrated conceptual Tree of 
 Porphyry, the tree that grows downwards through a series of 
 successive divisions from the heights of independent Being, through 
 the distinctions between Animate and Inanimate, Sensible and 
 Insensible, Rational and Irrational, to Socrates, Plato, Aristotle, 
 etc. at the terminus {vide p. 47). This tree of concepts is made 
 to include individual elements, and the point which here concerns 
 us is the question whether this inclusion can be logically justified. 
 Can the infima species of a conceptual tree be a proper name ? 
 
 It seems reasonable, and perhaps necessary, to suppose that it 
 can. A division of the term ' Man ' might conceivably be so 
 carried out as to pass from Man to Statesman, thence to Prime 
 Minister, then to the Prime iVIinister of England in May, 1907. 
 This last concept would necessarily be singular in its reference, and 
 would be represented by the proper name, " Henry Campbell-
 
 80 THE PROBLEM OF LOGIC [Il.vii. 
 
 Bannennan.' Similarly we might pass by successive subdivisions 
 through the subaltern genera Greek, Athenian, Philosopher, etc. 
 to the Socrates and Plato of Porphyry's tree. So long as the single 
 individual represented by the proper name is not considered as he 
 is jor himself, but in the same external way as are the many indi- 
 viduals indicated by anj^ ordinary class-name such as ' Man ' or 
 ' Athenian,' there seems to be no ground for differentiating between 
 the status of the general name and that of the proper name in regard 
 to the problem of the development of meaning. The latter becomes 
 simply a limiting case of the former. 
 
 Witli the above-mentioned proviso, we seem to be justified in 
 including individual elements in a conceptual tree and in regarding 
 them as intrinsically general or conceptual in character, though 
 sharpened — intensively — to such a point of conceptual fineness 
 and determinacy as to have — extensively — a unique reference. 
 They are then singular names, the function of a singular name, from 
 the point of view of Extension, being to refer us unambiguously 
 to some single individual object or person. 
 
 Regarded in this light, the singular name or meaning belongs, by 
 natural birthright, to the organized system of logical concepts. 
 There is no opposition between the class-concept and the singular 
 or individual concept, nor is the phrase ' singular concept ' a contra- 
 diction in terms. The individual meaning is simply the concept in 
 its limiting form. Where the only relevant element in this meaning 
 is the individuality of its reference, the name which represents it is 
 known as a 'proper name. 
 
 The singular concept is that limiting form which the infima species 
 of a division tends in last resort to take, thougli whether this 
 terminus is actually reached or not depends entirely on the answer 
 to the question whether it is logically relevant or advantageous 
 to reach it. The singular concept, or the proper name which repre- 
 sents it, we hold to be definable by means of a significant singular 
 name — a name which, by virtue of its very significance, is singular. 
 This appears to be the natural connexion between the proper name 
 and the significant singular term. ' Rational animal,' which defines 
 the general concept ' Man,' is a significant general expression ; 
 ' The mother of the two Gracchi,' which defines the singular concept 
 ' Cornelia,' is a significant singular expression.* Significant singular 
 expressions, precisely alike in this to significant general expressions, 
 arc .self-defining. Thus, ' The highest mountain on the Earth at 
 this present stage of our planet's geological history ' defines itself. 
 
 • Significant singular expressions are sometimes more briefly called ' designa- 
 tions.' Thus Mr. Joseph (' An Introduction to Logic,' p. 21) defines a designation 
 as ' a phrase which by a pronoun or what not serves to indicate an individual 
 otherwi.se than by a name of its own.' This briefer title, however, does not render 
 the longer one superfluous. A singular symbol may either designate or signify. 
 It may designate an individual (Extension), or it may signify a meaning (intea- 
 Bionj.
 
 CuAP. VII.] CONNOTATION AND DENOTATION 81 
 
 Its significance consists in the meaning of the marks that make up 
 the term. The very expression tells us that the single object 
 referred to is a mountain in our ])lan('t, and the highest of all at the 
 present time. As other instances of the .significant singular term 
 we may mention ' The centre of the Earth,' ' The first Emperor of 
 Rome,' ' The town at present situated at the mouth of the Dart.' 
 
 Our attempt to maintain the definability of the individual may 
 be met by certain objections : 
 
 1. It may be urged that proper names do indeed suggest, but 
 do not imply, attributes. The distinction between suggestion, 
 according to psychological laws of association, and logical implica- 
 tion is, indeed, fundamental, and there is a great difference between 
 unregulated ' subjective ' meaning and logical connotation. But 
 once we recognize that such ' implication ' is relative to purpose — 
 that it is only through knowing what our defining purpose is that 
 the implication can be relevantly fixed — the objection loses its 
 point. ' Common ' natures, no less than individual natures, suggest 
 a great deal more than is relevant to the purpose for which their 
 definition may chance to be required. But the defining interest 
 once clearly stated, those attributes or marks can be selected which, 
 in the most direct and economical wav, suffice to render our raeanins: 
 and reference unambiguous. The unselected marks that are not 
 required for the fixation of our meaning will then assume the status 
 of propria, and will remain propria so long as our defining interest 
 remains the same. It may be sufficient, from a given point of view, 
 to identify Mr. Balfour with ' the British Prime Minister in the 
 year 1900,' or with ' the philosopher who wrote " The Foundations 
 of Behef." ' These identifications will give just such meaning as 
 the name relevantly, and therefore logically, implies in respect to 
 the interest which dominates the definition. They will be definitions 
 of a proper name. Proper names, like general names, require as 
 many definitions as there are points of view from which they can 
 serviceably be used. 
 
 2. An objection of quite another kind may be raised against the 
 definition of the proper name. It may be contended that it is just 
 a meaningless mark. ' The only names of objects which connote 
 nothing,' writes Mill, ' are proper names ; and these have, strictly 
 speaking, no signification. ... A proper name is but an unmeaning 
 mark which we connect in our minds with the idea of the object, 
 in order that whenever the mark meets our eyes or occurs to our 
 thoughts, we may think of that individual object.'* 
 
 In the use of the phrase ' unmeaning mark ' it is evident that Mill 
 does not intend us to understand that the name is bereft of all 
 meaning whatsoever. It must at least retain a meaning as a 
 complex of sounds or letters, for if the proper name were bereft of 
 this meaning it could not function as a mark at all. By the phrase 
 
 * J. S. Mill, ' A System of Logic,' Book I. ch. ii., § 5. 
 
 6
 
 82 THE TROBLEM OF LOGIC [II. vii. 
 
 ' unmeaning mark ' he seems to mean a sign which stands for the 
 ' that ' and not for the ' M'hat ' of the object which it indicates. 
 But is tliis possible ? Is the ' that ' intelhgible apart from its 
 development into a ' what ' ? A case might conceivably be made 
 out for the pure thatness of a summum genus, but not of a proper 
 name. Mill compares the imposition of a proper name to the 
 marking of Ali Baba's house by the robber in the ' Arabian Nights.'* 
 But the chalk-mark afifixed to the house in order to distinguish it 
 from the other houses in the row did not represent a mere ' that.' 
 It was a sign which meant ' The house of Ali Baba,' and to this 
 extent it stood for and signified what was the object to which it 
 was attached. It therefore cannot be regarded as ' an unmeaning 
 mark.' 
 
 3. The best (though, in our opinion, still unconvincing) defence for 
 the indefinability of the proper name is based upon the conviction 
 that an individual's princi'pium individuationis is not what the 
 individual is for an outside spectator — or logician — but what it is 
 for itself, the contention being that the individual, qua inwardly 
 individual — qua experient — is strictly indefinable. But is this so? 
 If we cannot here have Definition per genus et differ entiam, may we 
 not have Experience-definition reflecting the point of view of the 
 experient himself ? Such definitions would present in systematic 
 form the meaning of our own immediate experiences stamped with 
 the impress of our own relevant reflection upon them. Concepts 
 like ' Thou ' and ' I,' concepts wliich mark immediate experiences 
 of feeling or sensation, such as ' blue,' ' hot,' ' angry,' ' tired ' (con- 
 sidered not as concepts of Physics or Physiology, but as genuine 
 psychological meanings grounded in immediate experience) — may 
 not all these gain a definite meaning through a reflective reconstruc- 
 tion of what is primarily self-evident ? As experience-concepts 
 they could not belong to any conceptual tree which, in its arrange- 
 ment, failed to recognize either that meaning is ' for a subject ' or 
 that the subject is also an experient. ' Now,' ' here,' ' ultramarine,' 
 ' I,' and all proper names, qvxi personal, belong neither to Porphyry's 
 tree nor to any branch of strictly scientific Cla.ssification. They 
 belong, as perhaps all concepts ultimately do, whether directly or 
 indirectly, to the tree of self-knowledge which is rooted in immediate 
 experience. 
 
 The difficulties which beset the attempt to define experience- 
 concepts apart from their relation to the experience which they 
 signify are excellently illustrated by the following passage from an 
 article by Professor Stout. f 
 
 ' When I refer to the present Lord Chancellor, I determine him as 
 the particular Lord Chancellor existing at the time at which I am 
 speaking ; when I refer to this table, I determine the table as that 
 
 * J. S. Mill, ' A System of Logic,' Book I., ch. ii., § 5. 
 
 t Proceedings of the Aristotelian Society, N.S., vol. vi., pp. 3G0-362.
 
 CnAP. VII] CUXXOTATION AND DEXOTATIOX 83 
 
 wliich is near me or at which I am pointing, or as that of which 1 
 have just been talking. Tlie application of proper names is deter- 
 mined by the particular occasions on which they have been uttered — 
 e.g., in the baptismal service. And so, quite generally, we can never 
 mark off in language the particular existence we mean except by 
 its relation to some other particular existence presupposed as 
 already known. But this process obviously involves a vicious 
 circle unless there is ultimately some direct apprehension of particu- 
 lar existence which supplies a point of departure for thought. If 
 we attempt to reach this ultimate basis by a regressive process, we 
 find ourselves approaching nearer and nearer to our own psychical 
 Hfe as the final centre of reference through which all other particular 
 existence is determined. The limit of this regress is marked by 
 such words as " now " and "I." In such words we indicate a 
 particular existence which is not determined by the thought of 
 relation to some other particular existence, but by the direct 
 apprehension of particular existence just as it is actually existing. 
 For this reason it is impossible without a logical circle to define 
 adequately in language what it is we refer to when we say " now " 
 or " I." This is impossible because we can onl}^ express in language 
 the relatively complex cognition of which immediate apprehension 
 is an element. What is immediately apprehended cannot be so 
 detached as to become b}^ itself a distinct object of knowledge. It 
 is not nameable except as being an element of a relatively complex 
 object. Thus, if I am right, when the application of words to 
 particular existents is directly determined by immediate experience, 
 it ought to be impossible to explain what is meant without a vicious 
 circle. And, as a matter of fact, this is so. Let anyone try to 
 explain what time it is which he refers to when he says '' now." It 
 is not enough to say that " now " means the time at which a person 
 is speaking, for persons speak at different times, constituting a 
 great many nows ; but in saying " now," the reference is to only one 
 particular time. How is this particular time distinguished from 
 the others ? It is circular to say that by " now " I mean the 
 " time at which I am now speaking." Yet anything short of this 
 is inadequate. Again, we cannot define the time meant by assigning 
 its relation to past or future time. For the " now " forms the ulti- 
 mate starting-point from which we determine temporal position in 
 the past or future. The future is what follows the " now," and the 
 past is what precedes it. Thus, any attempt to determine the mean- 
 ing of the now merely by its relation to the past or the future 
 involves a vicious circle. The " now " must be stamped by a peculiar 
 signature of its own — a peculiar character intrinsic to it. What 
 is tliis peculiar character ? W^e may attempt to express it by 
 saying that the " now " is the moment of actual experience. We may 
 say that it is the moment in which sensations, pleasures, pains, etc. 
 are not merely being thought but actually existing. But, again, 
 
 G— 2
 
 84 THE PROBLEM OF LOGIC [II. 
 
 Vll. 
 
 we have to press home the old question. The " now," it is said, is a 
 moment of actual experience. But which moment of actual 
 experience is it ? For there are an indefinite multiphcity of these ; 
 tile mental life of each of us from the cradle to the grave includes an 
 incessant succession of moments of actual experience. How is the 
 particular one which we refer to in saying " now " singled out from 
 the others ? Evidently no general conception of actual experience, 
 and no mere thought of there being particular instances of actual 
 experience, will help us in the least. Mere thought leaves us moving 
 round in the old circle. The moment of actual experience referred 
 to is the present moment ; the "now" is the time of that actual 
 experience which is now existing. If there is any way out of this 
 impasse except the one I propose, I should be exceedingly glad to 
 know what it is. The only escape that I can discover lies in franldy 
 admitting that there is a direct apprehension of particular existence 
 as it is actually existing. The application of the w^ord " now " is 
 determined, not by any mere thought of it, but by our immediate 
 experiences in the way of sensation, sensuous imagery, pleasure, 
 pain, etc., directly cognized in the moment of their existence as 
 they cannot be cognized at any other moment.' 
 
 With the fundamental contention of this passage I should be in 
 cordial agreement. But I would put in a plea against the suggestion 
 that certain ultimate meanings are indefinable. ' It is impossible,' 
 says the writer, ' witliout a logical circle to define adequately in 
 language what it is we refer to when v/e say " now " or " I." ' Is 
 this strictly true ? Has not the passage just quoted shown us 
 rather that it is impossible without a logical circle to define in 
 adequate language what it is we refer to by such terms as ' now ' 
 and ' I ' when they arc used as symbols of immediate personal 
 experience? But is it, we ask, justifiable to attempt a definition of 
 the self-evident qua immediately self-evident ? And if we do not 
 attempt this, deeming the task uncalled for, may we not still live 
 in the hope that the self-evident, when sufficiently reflected on, so 
 as to reveal an inner spiritual structure, may i)rove in the end to 
 be self-defining ? 
 
 The adequate definition of these experience-concepts is, no doubt, 
 a philosophical ideal, but, as such, it is surely a problem of supreme 
 interest and importance. With regard to the nature and con- 
 ditions of such a problem, we cannot do more than throw out one 
 central suggestion. Is not the essential condition this — that the 
 definitions can be grasped only in proportion as the experiences are 
 experienced ? Just as the definitions of ' here ' and ' now ' must be 
 formulation.s of actual experience, so there can be no genuine 
 definitions of religious concepts except such as express genuine 
 rehgious experience ; no genuine definitions of art-concepts which 
 do not express j:"^'nuino artistic experience. The defincr who would 
 understand his definitions will, in such case, be compelled to ex-
 
 Chap. VIII.] CONCRETE AND ABSTRACT TERMS 85 
 
 pericnce and, in some sense, sympathetically to live out what his 
 definitions may subsequently attempt to formulate and express. 
 Experience must, in his definitions, relive in conceptual form, other- 
 wise the definitions are mere deceptive formulae which cannot 
 possibly mean what they say. An adequate definition of the self- 
 evident ' now ' of immediate experience will, at any rate, not be 
 given till our human nature has realized in its own time-experience 
 how past, present, and future stand related not only to each other, 
 but also to the eternal Present which in some way transcends ther-" 
 
 CHAPTER VIII. 
 
 II. (viii.) CONCRETE AND ABSTRACT TERMS. 
 
 A TERM may be said to be concrete when it refers us to concrete 
 objects, abstract when it refers us to an abstract object. Whether 
 a term is concrete or abstract depends, then, on the nature of its 
 objective reference. 
 
 Now, an object is concrete when it is regarded as possessing an 
 immediacy either for sense or for feehng, or individuality in time or 
 space, and is thought of as thus ' immediate ' or ' individual.' 
 
 Abstract objects are derivative. They are derived from concrete 
 objects through a process (1) of discrimination, (2) of analysis. 
 
 Discrimination alone cannot give us an abstract object. It can 
 only differentiate the concrete object. In a yellow orange we may 
 discriminate the yellowness and the roundness. But the yellowness 
 of the orange is not an abstract yellowness ; it is a quality, a dis- 
 criminated quality, of the concrete orange. 
 
 Whether we use the adjectival or the substantival form makes 
 no difference in the nature of the discrimination, and therefore no 
 difference in regard to the question of abstract and concrete. ' A 
 yellow orange ' and ' the yellowness of an orange ' are equally con- 
 crete in their reference, and accordingly are both concrete terms. 
 
 But if, after having discriminated the yellowness as a quality or 
 feature of the orange, we proceed to analyze it out of its concrete 
 context, so that it can no longer be said to be an inherent feature 
 of the orange, but is withdrawal by the abstracting power of thought 
 and brought under the conditions of thought-existence, we have 
 yellowness as an abstract object — an object which has immediacy 
 for thought, but no longer any relevant sense-immediacy, nor yet 
 any immediacy for feeling in the ordinary acceptance of that term. 
 
 An abstract conception or meaning is thus a common quahty as 
 such, a universal, detached from all reference to individual obje<?ts, 
 and considered solely qua universal.
 
 86 THE PROBLEM OF LOGIC [11. viii. 
 
 Terms which indicate or otherwise refer to these abstract objects, 
 these products of mental analysis, are called abstract. They are 
 the names of detached attributes and relations, of attributes and 
 relations mentally isolated, of attributes and relations per se. The 
 merely discriminated attribute or relation which is not mentally 
 severed from the concrete object which it qualifies or relates is, as 
 we saw, still concrete in its reference. (See 1, b below.) 
 
 We obtain, then, the following distinctions : 
 
 1. A term is concrete — 
 
 (a) When it serves to indicate* individual existents (things, 
 
 persons, events, etc.) ; 
 
 [b) When it serves to indicate an attribute presented as actually 
 
 f[ualifying individual existents, or a relation considered 
 in connexion with the individuals related. 
 
 2. A term is abstract — 
 
 (a) When it serves to indicate an attribute considered apart 
 from the individuals (tilings, persons, events, etc.) of 
 which it is the attribute ; or a relation considered in 
 severance from the individuals related ; 
 
 (6) When it serves to indicate an attribute or other qualifica- 
 tion of an abstract object. 
 
 As illustrations of the distinction between Abstract and Concrete 
 Terms regarded from the point of view of objective reference, let 
 us consider the subject-terms of the following propositions : 
 
 ' A tyranVs hate is a thing we need not fear.' 
 
 Here the subject-term is concrete (1, b). 
 
 ' The hate of my fellows is a force I dare not face.' Concrete (1, b). 
 
 ' Hate is old wrathe ' (Chaucer). Abstract (2, a). 
 
 ' The hate of hate is the Poet's dower.' Abstract (2, a). 
 
 ' The hatefulness of sin is an eternal fact.' Abstract (2, b). 
 
 ' The hatefulness of Mr. Hyde is a haunting horror.' Concrete 
 
 (1, i). 
 
 ' A hateful thought is a crime.' Abstract (2, a). 
 
 ' That hateful thought of yours is a disgrace.' Concrete (1, b). 
 
 ' All the Virtues are personified quahties.' Abstract (2, a). 
 
 ' Virtue is a self-rewarding activity.' Abstract (2, a). 
 
 ' A man's virtue is to be truly a man.' Concrete (1, b). 
 
 * Some virtuous people are not pious people.' Concrete (1, a). 
 
 ' All virtuous activities are forms of happiness. Abstract (2, a). 
 
 ' The virtue of suffering is the fostering of sympathy. Abstract 
 (2,6). 
 
 ' This man's thoroitghness is an acquired characteristic. Concrete 
 (1, h). 
 
 * Thus ' Bf-ing ' or ' Person ' is, from this point of view, as concrete as ' Welsh 
 man ' or ' John Jone?.'
 
 Chap. VIII.] CONCRETE AND ABSTRACT TERMS 87 
 
 ' Thoroughness is a test of efficiency.' Abstract (2, a). 
 ' The thoroughness of this work is an admirable quality.' Concrete, 
 (1, b). 
 
 ' Thoroughness of work is a praiseworthy quality.' Abstract (2, b). 
 
 Question. — Examine whether the italicized terms in the following 
 propositions are abstract or concrete : 
 
 1. ' Truth is not mercy. ^ 
 
 2. ' What I ask of you is mercy. ^ 
 
 3. ' Mercy is the sister of Justice.' 
 
 4. ' A light is a characteristic of the glow-worm.' 
 
 5. ' Light is a necessity of plant-hfe.' 
 
 6. ' Light is a mode of motion.' 
 
 7. ' This figure is a square.^ 
 
 8. ' This square is not that squareJ' 
 
 9. ' The square is a rectangle.' 
 
 1. ' Truth is not mercy. ^ 
 
 ' Mercy ' is here used abstractly, apart from any reference to a 
 stated individual possessing it as a quahty of character. Abstract 
 (2, a). 
 
 2. ' What I ask of you is mercy.'' 
 
 ' What I ask of you is your mercy.'' ' Mercy ' is here used as 
 qualifying the person referred to by ' your.' Concrete (1, b). 
 
 3. ' Mercy is the sister of Justice.' 
 
 The personification of ' mercy ' only makes its abstract use more 
 evident. If the proposition had been ' Mercy is the sister of 
 Charity,' with a reference to ' Martin Chuzzlewit,' ' Mercy ' would 
 have been used as a concrete term. The personified \'irtues that arc 
 not of flesh and blood are the sheerest abstractions. Abstract (2, a). 
 
 4. ' A light is a characteristic of the glow-worm.' 
 
 Here ' a light ' is a concrete term. It refers to individual things. 
 Concrete (1, a). 
 
 5. ' Light is a necessity of plant-hfe.' 
 
 ' Light,' as a necessity for plant-life, indicates radiant energy 
 manifested at definite times and places. The term is therefore 
 concretely used. Concrete (1, a). 
 
 6. ' Light is a mode of motion.' 
 
 It is true that ' Light,' as a mode of motion, can be neither sensed 
 nor felt. But the word has reference to undulatory movements 
 passing through definite spaces at definite times. Concrete (1, a). 
 
 7. ' This figure is a square.'' 
 
 Here ' a square ' is used abstractly. Despite the fact that a par- 
 ticular figure is indicated, what is imphed is that this particular 
 figure stands for an ideal square. However badly it may have 
 been drawn, it still represents that ideal form. Abstract (2, a).
 
 88 THE PROBLEM OF LOGIC [11. viii. 
 
 8. ' This square is not Ihat square.^ 
 
 Here both subject and predicate terms are concrete. The 
 meaning is that the given square ABCD, as drawTi or constructed, 
 Avitliout reference to the ideal matliematical square, is not identical 
 with tlie given square EFGH. Of course, if the ' squares ' indi- 
 cated are of the ' Trafalgar Square ' kind, the concrete nature of the 
 terms is still more obvious. Concrete (1, a). 
 
 9. ' The Square is a rectangle.' 
 
 Here the reference is to the ideal square, not to any particular 
 representation of it. Abstract (2, a). 
 
 Since it is the nature of the objective reference which decides 
 whether a term is abstract or concrete, it would seem to follow 
 that the degree of generality which a class-term as such may possess, 
 though it involves a certain kind of abstraction — namely, ' generali- 
 zation ' — has nothing to do with its being ranked as an abstract 
 or concrete term. 
 
 The objection may, indeed, legitimately be raised that class- 
 meanings are as such abstract, and that one class meaning is more 
 abstract than another. ' Animal,' ' mammal,' ' lion,' it may be 
 urged, are abstract in the sense that they are general classes and 
 not individual things ; and, again, ' animal ' is more abstract than 
 ' mammal,' ' mammal ' more abstract than the typical ' lion.' As 
 we pass from lion to carnivore, from carnivore to mammal, from 
 mammal to vertebrate, each higher class — higher in the sense of 
 being more general — is at the same time inore abstract, and the 
 summum genus in any system of classification will be the most 
 general and most abstract of all. 
 
 That the distinctions here referred to are real enough cannot 
 be denied. The presence of degrees of generality among concepts is 
 essential to the existence of a classification. But the word ' general ' 
 is surely good enough to enable us to dispense, in this connexion, 
 with the word ' abstract.' There is, moreover, a more appropriate 
 word still — namely, ' indeterminate.' This word is, in fact, forced 
 upon us by the construction we have put on ' meaning ' and its 
 logical evolution from the indeterminate to the determinate, or 
 involution from the determinate to the indeterminate. From this 
 point of view, every concept has its place within a certain conceptual 
 system. What tliis system precisely is depends on the interest 
 whir-li r-ontrols our use of the concept in question. This system wo 
 may call the logir-al topic, or universe of discourse, and in so far as 
 it i.s allowed to develop itself logically — i.e., in the direction deter- 
 mined by the growth of the logical interest itself — it takes its 
 start from a summum (jfrius as its indeterminate fountain-head, 
 and differentiates itself in an orderly conceptual way, till it reaches 
 the limit of determinacy in the infimaj species. The concept 
 ' Carnivore ' represents, from this standpoint, a stage in the logical
 
 Chap. VIIL] CONCRETE AND ABSTRACT TERMS 89 
 
 development of tlu; Topic of the Animal Kingdom, and it is the aim 
 of Definition and Division, as combined in the single process of 
 Classification, to indicate precisely the nature and position of this 
 stage in relation to the whole Topic. 
 
 Where class distinctions are understood in this way, the use of 
 the term ' abstract ' is unnecessary and uncalled for. It is simpler 
 to say that there are degrees of deterrainacy or indeterminacy 
 among concepts, and to interpret this statement in the way above 
 suggested. 
 
 Logicians are accustomed to raise the question as to whether 
 abstract terms are connotative or not. The question is legitimate, 
 but should include the question whether abstract terms are denota- 
 tive or not. 
 
 There seems to be no reason for dealing differently in this matter 
 with concrete and with abstract terms. For the concreteness or 
 abstractness concerns the objective reference only, and in no way 
 the conceptual interrelations of any system of meanings, whether 
 these be abstract or concrete, whereas it is precisely with these 
 interrelations within a conceptual system that the distinctions of 
 connotation and denotation are concerned. If by ' connotation ' 
 we mean the product of definition 'per geniis et differ entiam, then in a 
 system of abstract concepts the summum genus, relatively to the 
 system, will be non-connotative ; the infimse species — e.g., mathe- 
 matical equality — will be non-denotative ; all the other abstract 
 concepts will be both connotative and denotative. Thus the term 
 'Insanity, 'abstractly used, might have a connotation ' morbid mental 
 state,' and a denotation ' mania, or monomonia, or melanchoha, or 
 dementia, or amentia.' So the connotation of ' Roundness,' ab- 
 stractly used, might be ' spatial form having a curved surface or 
 outline,' and this connotation admits of being differentiated after 
 the ordinary manner of denotation. There are at least two main 
 kinds of roundness : roundness of line and roundness of surface. 
 To the former type belong circularity, the roundness of the oval, of 
 the cycloid, of the catenary, etc. ; to the latter all the varieties of 
 three-dimensional roundness — the roundness of the sphere, the 
 ellipsoid, etc.
 
 III. 
 
 THE LOGICAL PROPOSITION. 
 
 ( i.) The Judgment or Proposition. Introductory Statement (ch. ii.J, 
 (ii.) The Laws of Thought (ch. x.). 
 
 (a) The Law of Logical Identity in its relation to the Proposition. 
 
 (b) The Laws of Non-contradiction and Excluded Middle. 
 
 (c) The Inviolability of the Laws of Thought.
 
 CHAPTER IX. 
 
 III. (i.) THE JUDGMENT OR PROPOSITIOX. 
 
 A JUDGMENT, in the simplest logical sense of the word, is a meaning 
 whicli admits of being characterized as true or false, or at least 
 as self -consistent. Where such characterization is out of place, 
 tlie expression cannot rank as a judgment. Thus, optatives, im- 
 peratives, and ejaculations fall, as such, outside the sphere of logical 
 judgments. An optative expresses a wish, and we cannot say of 
 mere wishes that they are either true or false ; they are merely 
 reasonable or unreasonable. Similarly imperatives call, not for 
 belief, but for obedience ; they announce commands, but do not 
 communicate truths. 
 
 The relation of Proposition to Judgment or Assertion may be 
 defined by saying that the proposition is the judgment in a purpo- 
 sively fixed form. It is the judgment in that form in which it 
 first becomes available for logical purposes. The proposition is not 
 to be understood as a mere drapery of words which the judgment, 
 as a synthetic act of thought, can put on or put off as it pleases. 
 It is quite true that we can think without words. It is not at all 
 essential that the sensory assistance so indispensable to thought 
 should take the form of a conventional verbal sign. But wliatever 
 the sensory symbolism may be, it is only as a purposive fixation of 
 meaning that it has any logical significance. The grammatical, or, 
 to speak more correctly, the philological and phonetic interest in 
 verbal structure as such is non-logical. In Logic we are interested 
 in words only as the visible or audible forms in wliich thought fixes 
 and controls its o\vn meaning. The proposition, qua logically 
 serviceable, is therefore indistinguishable from the judgment. As 
 in Definition the distinction between verbal and conceptual defini- 
 tion was found, on closer inspection, to have no logical raison d'itre, 
 so, in the matter of assertion generally, the distinction between pro- 
 position and judgment, so far as logical interest is concerned, is a 
 distinction without a difference. The terms ' proposition ' and 
 ' judgment ' are logically interchangeable. 
 
 On the other hand, we must distinguish between proposition and 
 sentence. Every proposition is a sentence, but not every sentence a 
 proposition. For the sentence is the unit of speech generally. Any 
 
 93
 
 94 THE PROBLEM OF LOGIC [III. i, 
 
 sjTitactical expression is a sentence, whether it expresses a truth 
 or a mere wisli, a command or a supposition. A proposition is only 
 a certain kind of sentence — a sentence in which an affirmation or 
 denial is appropriately expressed. It has been called an indicative 
 or declaratory sentence. 
 
 We liave referred to the judgment or proposition as the logical 
 unit. The reason for this is that the proposition is the simplest 
 form in which we can state our meaning, the simplest form to which 
 the verdict ' true ' or ' valid ' can be apphed. Ideas in themselves 
 — ' horse,' ' leaf,' ' man ' — are neither true nor false ; nor can we 
 predicate truth and falsity of mere combinations of ideas. We 
 can no more say that such expressions as ' high horse,' ' lingering 
 leaf,' ' miserly man,' are true than we can say that ' hexagonal 
 horse,' ' league-long leaf,' ' microscopic man ' are false. They are 
 not judgments, and therefore do not admit of truth or falsity. Nor 
 can such complex ideas be self-contradictory or invahd. ' Human 
 horse,' ' limping leaf,' ' mineral man ' cannot contradict themselves, 
 because they express no assertions they can proceed to deny. 
 Before I can maintain anytliing to be either true or false, I must 
 express it as an affirmation or a denial. ' This horse is not high,' 
 ' These leaves are lingering,' ' This man is a miser.' 
 
 The statement of meaning which we find in every proposition as 
 such implies an objective reference to fact. For ' meaning,' as we 
 saw, is always the meaning of an object for a subject. But it is by 
 no means essential that we should take this implied objective 
 reference exphcitly into account. Whether we do so or not wiU 
 depend on the nature of our interest. Where the interest in putting 
 forward a proposition hes in ascertaining whether the interpretation 
 of fact, as stated in the proposition, does or does not tally with 
 the more authoritative interpretations based upon experimental 
 analysis of sensible fact, the proposition has a truth-import and 
 calls for proof or disproof. The reference to fact is here of the 
 essence of the proposition. The discussion of such propositions 
 will occupy us at a later stage, when we come to treat of Inductive 
 Method. 
 
 The prehminary inquiries of logical science into the nature of 
 Judgment presuppose a more abstract view of propositional import. 
 We require not only, as in inductive inquiry, to consider the bearing 
 of statement upon fact ; we must also consider the statement itself, 
 the interrelations of its elements, and the relations in which it may 
 stand to other .statements. We have also to consider how the im- 
 plications of a statement, or combination of statements — imphca- 
 tions of quite another kind from that of objective reference — may 
 be made clear and explicit. In a word, we have to consider, not 
 only the truth-import of a proposition, but also, and from the various 
 points of view above specified, its statement-import, its import as a 
 statement of meaning.
 
 Chap. X.] THE LAWS OF THOUGHT 95 
 
 The theory of the proposition, from the relatively abstract stand- 
 point of its statement-import, consists essentially in an elaboration 
 and application of two fundamental principles, respectively known 
 as the Principle of Logical Identity (or, more briefly, the Law of 
 Identity) and the Principle of Non-Contradiction (or, more briefly, 
 the Law of Contradiction). Of these, the Principle of Identity is 
 logically the more fundamental. It is implied in the very stating of 
 a proposition, and is therefore presupposed in the very enunciation 
 of the Principle of Non-Contradiction, which, in a form still to be 
 more closely specified, concerns the relation in which an already 
 stated proposition stands to other propositions. 
 
 CHAPTER X. 
 
 III. (ii). THE LAWS OF THOUGHT. 
 
 (a) The Law of Logical Identity in its Relation to the 
 
 Proposition. 
 
 The fundamental principle of the logical development of meaning 
 is the Principle of Logical Identity.* Its essential purport is that 
 a meaning remains identical with itself through all its manifold 
 developments, that these developments are its developments, and 
 therefore belong to one and the same single system of meaning. 
 A meaning-identity in its most undifferentiated form is known as 
 a topic or subject of discourse. It is the identity of some total 
 interest, qvxi total and undeveloped. We have but to remember 
 that meanings are constantly changing and developing to see that 
 that type of meaning which logicians refer to as the Concept is 
 just some more or less determinate specification of some such topic 
 or universe of discourse. The topic itself is the concept m its 
 most indeterminate and total form, and the first and simplest 
 formulation of the Principle of Logical Identity is the statement 
 that the conceptual system in which all the imphcations of a concept 
 are made systematicaUy explicit is a single, identical unity. A 
 classification-scheme is a conceptual system of tliis kind. As such 
 it is a logical Identity, which finds its incipient expression in that 
 most indeterminate of concepts, the summum genus. 
 
 Such a view of identity has its roots in the very nature of the 
 process according to which our minds habituallj- work. We always 
 think within an interest. What interests us may be a question or 
 
 * The terms ' principle ' and ' law,' as applied to the logical nature of thought. 
 may be treated as equivalent expressions. The Principle or Law of Identtiy is 
 a fundamental Law, or Principle, of Thought.
 
 96 THE PROBLEM OF LOGIC [III. ii. 
 
 a problem, a desire, hope, or aim ; and the intellectual movements 
 of our life, however fleeting and evanescent they may be, gain their 
 identity or organized unity through relation to some such interest, 
 and through working the interest out to some kind of ending or 
 consummation : the answering of a question, the solution of the 
 problem, the reasoned fulfilment of the end we had in view. 
 
 We do not propose at present to discuss in its full bearings the 
 Principle of Logical Identity. Its relation to Inference is as close 
 as its relation to Judgment. We shall see later on in what sense an 
 Inference is an Identity. We restrict ourselves here to considering 
 the principle of Logical Identity in its relation to the unit of all our 
 thinking : the judgment or proposition. 
 
 It has been customary, in tliis connexion, to state the Principle 
 in the unpromising form ' A is A,' or ' A is not non-A.' This is the 
 form of stfttement natural to a view of the significance of ' meaning ' 
 which takes no account of development. It is the Principle of 
 Statical Identity. As such it can only endlessly affirm the separate 
 self -identity of logical concepts. It circles about the concept as 
 a starting-point, impotent to move beyond it. It cannot regulate 
 the development of a concept, or take us from the concept, through 
 the proposition, to inference and a system of inferences. For to 
 say that A is A is precisely equivalent to saying ' A.' If we are 
 told that Logic is Logic, we still hold the undifferentiated concept 
 of Logic before our minds ; ' Logic is Logic ' means ' Logic,' but we 
 have said nothing at all about Logic. This so-called principle of 
 judgment is not itself a judgment at all. ' The prepositional form 
 itself contradicts it : for a proposition always promises a distinction 
 between subject and predicate, while the present one does not fulfil 
 what its form requires ' (' The Logic of Hegel,' tr. Wallace, p. 213). 
 If we were really guided by this so-called principle, we should never 
 make any assertions whatsoever. We should be continually positing 
 concepts, and never developing them ; proposing topics, and then 
 revolving helplessly around the mere idea of them. 
 
 To state a proposition, we must, to put the matter quite generally, 
 specify our meaning. If we wish to make a definite statement of 
 fact, we must first specify that aspect of the total topic which we 
 wish particularly to speak about ; this will give us the subject 
 of our statement. We have then to specify this subject by predi- 
 cating something about it that is other tlian itself. This whole 
 specification of meaning will have taken place within a topic whose 
 unity dominates the whole procedure, so that the specifications will 
 be all specifica'tions within the topic ; and the Principle of Identity 
 will be a Principle of Identity in relation to Differences, a Principle 
 of Identity in L)ifTerence. 
 
 Identity, then, can be fruitfully understood only in the light of 
 difference. Hence, when identity is conceived out of all relation to 
 difference, it is barren, and has no logical value whatsoever. A
 
 Chap. X.] THE LAWS OF THOUGHT 97 
 
 ' principle ' of Identity so conceived is a mere principle of Tautology. 
 Thus, the empty insistence that ' A is A,' or that ' Whatever is, is,' 
 is worse than meaningless. Many who despise truisms in ordinary 
 discourse seem to think that in Logic they are very suitably housed, 
 fulfilling there some kind of mysterious function which must anyhow 
 be very satisfjnng to the initiated. Moreover, the greatest truths, 
 we are told, are the simplest. But the very reverse of all this holds 
 good. A truism is worse in Logic than anywhere else, for it is 
 there paraded, not as a mere truth of fact, but as a truth of the 
 very reason. 
 
 The inanity of the old formula ' Whatever is, is,' or ' A is A ' — 
 distinctions of formal statement are here of no consequence — was 
 first clearly exposed by Hegel in his ' Logic' ' The main thing,' he 
 urges, ' in connexion Avith thought, is not to confuse the true Iden- 
 tity . . . with an abstract identitj'-, identity of bare form. All the 
 charges of narrowness, hardness, meaninglessness, wliich are so often 
 directed against thought from the quarter of feeling and immediate 
 perception rest on the perverse assumption that thought acts only 
 as a faculty of abstract identification ' (' The Logic of Hegel,' tr. 
 Wallace, p. 215). ' It is asserted,' he writes on a previous page 
 (ibid., p. 214), ' that the maxim of Identity, though it cannot be 
 proved, regulates the procedure of every consciousness, and that 
 experience shows it to be accepted as soon as its terms are appre- 
 hended. To tliis alleged experience of the logic-books may be 
 opposed the universal experience that no mind thinks or forms 
 conceptions or speaks in accordance with this law, and that no 
 existence of any kind whatever conforms to it. Utterances after 
 the fashion of this pretended law (A planet is — a planet ; Magnetism 
 is — magnetism ; j\Iind is — mind) are, as they deserve to be, reputed 
 silly. That is certainly matter of general experience. The logic 
 which seriously propounds such laws, and the scholastic world in 
 wliich alone they are valid, have long been discredited with practical 
 common sense, as well as with the pliilosophy of reason.' 
 
 It is extremely important, however, that this lawful contempt 
 for the impotent tautology ' A is A ' should be precisely limited to 
 its lawful object. What is logically inane is an identity that is 
 out of relation to all differences. Identity in relation to difference 
 still remains the basic and the guiding conception of logical Science. 
 
 The logical unit — the Statement or Proposition — is an identity 
 in difference. A statement is a movement of mind — a purposive, 
 ideally directed movement of mind* — and this movement is a de- 
 
 * The purposive character of all assertion whatsoever is implied in itj being a 
 statement of meaning; for ' meaning,' as we have conceived it, ' tells \is what an 
 object is in relation to a specified interest or purpose ' (vide p. 23). Meaning, 
 in so far as it has any logical significance, is always intended meaning. The teleo- 
 logical character of Judgment can, however, be adequately grasped only in 
 relation to the problem of Experience, and the further discussion of this central point 
 is therefore postponed. 
 
 7
 
 98 THE PROBLEM OF LOGIC [III. ii. 
 
 vclopmcnt through which dilYcrent elements are evolved within the 
 unity of a single total meaning. Nor, as we have already said, is 
 the scope of the Law of Identity limited to a single statement. 
 Whatever we tliink together must be knit together tlirough the 
 constitutive activity of the Identity Principle, otherwise it is illusory 
 to suppose that our thoughts really cohere. 
 
 The question now suggests itself : Does the Law of Identity indi- 
 cate, not only a necessary, but also the sufficient condition of sound 
 thinking ? If this were so, the Law of Identity would be the Law 
 of Thought. There is good reason for maintaining that this is really 
 the case, and that, in guarding itself against degenerating into a 
 meaningless formula of difference without identity, the Principle 
 would naturally take the hmiting, or rather the sclf-Hmiting, form 
 of a Law of Self-Consistency. If S and S' are accepted as con- 
 tributing to the development of one and the same idea or topic, 
 S cannot afhrm what li^' denies. There is, however, at tliis point 
 no reason for pressing thus far the claims of a logical Monism. 
 For the present we may conveniently regard the Law of Identity 
 as limited in its application by a second law, wliich insists that differ- 
 ence cannot intelhgibly be carried to the point of inconsistency ; 
 for, in so far as the integrity of our thinking is broken, our statements 
 must cease to mean anything at all. 
 
 (6) The Laws of Non-Contradiction and Excluded Middle. 
 
 The Principle of Non-Contradiction, which supplies the criterion 
 or test of Logical Consistency, states the conditions under which a 
 given statement can be held consistently with the primary requisite 
 of Intelligibility. This involves the relating of a proposition "Z, 
 or, more specifically, ' S is P,' to what is called its contradictory 
 %, or ' SlsT.'* 
 
 The Principle may be formulated as follows : 
 
 If the statement ' S is P ' is accepted {i.e., accepted as an 
 understood statement of meaning), the statement ' S is P ' 
 mu.it be rejected (a). 
 And— 
 
 If the statement ' S is P ' is rejected, the statement ' S is P ' 
 must be accepted (/3).t 
 
 • A convenient symbolic designation for the denial of ' S is P.' The ' bar ' 
 
 placed over the afifirmation is a recognized logical device for expressing the 
 
 corresponfiing denial. The device is a natural one, for we have only to place the 
 bar a little lower down, thus : B ia P . and ' S is P ' is then appropriately cancelled 
 in the orrlinary way. We shall, therefore, use the symbol S is P in place of the 
 more awkward expression ' Not (S is P) ' whenever the subatitutiun proves con- 
 venient. 
 
 t To reject a statement is to accept its contradictory.
 
 Chap. X.] THE LAWS OF THOUGHT 99 
 
 Tliese enunciations, it will be observed, include an explicit 
 reference to the thinker's attitude of acceptance or rejection. This 
 reference is essential. If it were dropped, the Principle would cease 
 to be a Consistency-principle. To insist simply that ' If " S is P " 
 is true, " S is P " must be false ' is, indeed, to postulate the Intelligi- 
 bility of Reality ; but the principle so enunciated has no direct 
 bearing on the Consistency of the Thought which is exercised upon, 
 or witliin, Reality. If % and % could be true together. Reality 
 would not be intelligible ; but there is no inconsistency in our 
 thinking until we attempt to think these two statements together. 
 S and S must be entertained as true together before our thought can 
 be said to be inconsistent and to run into contradiction. 
 
 It must not be supposed, however, that the principle ' If S be 
 true, S~ must be false ' is not, in some sense, a Law of Thought. 
 The Principles which express the necessity we are under of holding 
 Reality as intelligible are of fundamental significance for our 
 thinking. Still, they concern not statement as such, but statement 
 only in relation to fact. They concern the truth-import of the Pro- 
 position, not its statement-import. 
 
 This consideration will serve to explain the omission of the Law 
 of Excluded Middle from the hst of principles upon which the 
 intelligibility and consistency of our statements depend. The Law 
 of Excluded IMiddle may be formulated as below. The statement 
 implies that no third alternative, no mean, no middle course is 
 possible.* 
 
 Either S is false or 5 is false (a). 
 Either S is true or S is true (/?). 
 
 Here, as before, (a) and (/3) are mutually complementary. 
 
 It is plain, moreover, that the attempt to interpolate a reference 
 to the thinker would destroy the self-evidence of the principle. It 
 is by no means necessary that ' Either 5^ or S must be accepted as 
 true,' or that ' Either S or S must be rejected as false.' We may 
 suspend our judgment. % may be a mere unverified hypothesis, in 
 which case it would be an affront to Reality either to accept it as 
 true or to reject it as false. It is, then, only on the supposition that 
 E is actually accepted as true that we are bound, in the interests of 
 consistency, to reject ^ as false. 
 
 The Law of Excluded Middle, then, does not concern the statement- 
 import of the Proposition, and we can have no occasion to apply it 
 until we come to deal with propositions from the point of view of 
 their truth-import. There is, however, an important misconception 
 to which the law itself is liable — a misconception affecting its very 
 statement ; and tliis it will be convenient and suitable to consider 
 at once. 
 
 * For a criticism of the pseudo-form * S is cither P or nou-P,' vide p. 191.
 
 100 . THE PROBLEM OF LOGIC [III. ii. 
 
 The Bdation of the Law of Excluded Middle to Time. 
 
 It is frequently stated that the formulation of the Law of Excluded 
 Middle should include a reference to time. If tliis were so, the Law 
 of Excluded Middle should run as follows : 
 
 ' The two propositions " S is P " and " S is-not P " cannot both 
 be true (or false) simidtancously.^ 
 
 The point is important, but it is not necessary to appeal to its 
 importance to justify the following discussion of it, which I owe 
 to the kindness of Professor Stout. 
 
 ' We sometimes speak as if a proposition and its contradictory 
 could both be true — not, indeed, at the same time, but at 
 ditferent times. Thus we may say. It was true that Jones was 
 a bachelor, but the truth now is that he isn't a bachelor. . . . 
 But closer examination shows that truth and falsity are in no way 
 affected by the flux of. time. Whatever reference there may be 
 to time is alread}'- included in the meaning of the propositions which 
 are true or false. These temporal conditions cannot, therefore, 
 be used over again to qualify the proposition as a whole. " The 
 Battle of Waterloo was fought in 1815 a.d." is a proposition 
 concerning a temporal event and its date. But the truth of 
 the proposition is not an event, and has not a date. The truth 
 of the proposition is not something which occurred in 1815. 
 The occurrence of the battle in 1815 will be a fact a thousand 
 years hence ; and if a thousand years before 1815 some one 
 had assigned 1815 as the date of the battle, he would have been 
 stating a fact. 
 
 ' To bring this out clearly, we may compare with the analogous 
 case of spatial relation. The same argument which may be used to 
 show that truth is subject to time conditions may also be used to 
 show that it is subject to spatial conditions. But in the case of 
 space the fallacy is more obvious. Suppose three persons, A and B 
 and C. A is on one side of the street ; B and C are on the other. A 
 calls to B : " Where is C ?" B answers : " C is here." A rejoins : 
 " C is not here, but there." Thus we have two statements : " C is 
 here," and " C isn't here," and both of these are true. Are we, then, 
 to say that truth is locally variable ? Are we to say that a proposi- 
 tion may be true in one place, whereas it is false and its contradictory 
 true in another only a few j^ards off ? If this were so, the topo- 
 graphj' of truth would be a most interesting study which has been 
 strangely neglected. Of course, the whole supposition is nonsense, 
 and the apparent difficulty is easily removed. The word " here," 
 although it has the same sound in the mouth of A and in the mouth 
 of B, has a different meaning. In the mouth of A it indicates the 
 place where A is at the time, and its immediate neighbourhood ; in 
 the moutli of B it indicates the place where B is at the time, and 
 its immediate neighbourhood. Thus the statements " C isn't here "
 
 CiFAP. X.] THE LAWS OF THOUGHT 101 
 
 and " C is here " are not the same statements in the mouths of A and 
 B respectively. Hence we may admit that when B says " C is here," 
 he is affirming a true proposition ; and also that when A says " C is 
 not here," he is affirming a true proposition ; we can, I say, admit 
 this without admitting citlier that two contradictory statements 
 may both be true, or that truth and falsehood have position in 
 space. 
 
 ' The relation of truth to time is quite similar. Propositions 
 are not true at one time and false at another any more than tliey are 
 true at one place and false at another. If it sometimes looks as if 
 this wore not so, the appearance is illusory, and is due to the fact 
 tliat the same verbal formula may on different occasions express 
 different propositions. Tlie variability in the meaning of the 
 words " here " and " there " attaches also to the tenses of verbs, 
 and to such terms as " now " and " then," " yesterday " and " to- 
 morrow," " past " and " future " and " present." We say, for 
 example, that " to-morrow never comes," but this only means that 
 when it does come we no longer call it " to-morrow," we call it " to- 
 day " ; but when we call it " to-day " we mean the very same day 
 which we had previous^ called " to-morrow." On the other hand, 
 if we said " to-morrow " instead of " to-day," we should not be 
 referring to the same day, but to another. Thus, if a man say on 
 Monday " I shall play golf to-mori'ow," he does not contradict the 
 statement by saying on Tuesday " I shall not play golf to-morrow." 
 He contradicts the proposition that he will play golf on Wednesday ; 
 he does not contradict the statement that he will play golf on 
 Tuesday. The tenses of verbs, and more especially the present 
 tense, fluctuate similarly in meaning with the occasion on which 
 they are used. In the year 1900 the words " Jones is now a 
 bachelor " may express a true proposition ; in the year 1905 the 
 very same words may express a false proposition, for Jones has 
 married in the interval. But we do not give a right account of what 
 has taken place if we say that the proposition which was true in 
 1900 has ceased to be true in 1905. On the contrarj', the truth of 
 this proposition is absolutely unaffected by the lapse of time. 
 What looks like a change from truth to falsity is really only a change 
 in the meaning of the word " now " and of the present tense of the 
 verb. The truth expressed by the verbal formula " Jones is a 
 bachelor " as used in 1900 cannot be expressed b}^ the same formula 
 in 1905, and it camiot be contradicted b}^ saying in 1905 "' Jones is 
 not now a bachelor." If a person speaking in 1905 wishes to con- 
 tradict it, he must say " Jones was not a bachelor in 1900." Simi- 
 larly, if we now say " Jones will be a widower in 1908," we express 
 exactly the same proposition which would be expressed by some one 
 speaking in 1908 in the form " Jones is a widower." In general, 
 what is true does not become false, and what is false does not become 
 true. Wherever this may appear to be so, the appearance is an
 
 102 THE PROBLEM OF LOGIC [111. ii. 
 
 illusion. Wliat lias really happened is that a form of speech has 
 changed its meaning because of a shifting of the point of view from 
 which we determine time-relations.' 
 
 The relation of the Principle of Non-Contradiction to that of 
 Xon- Ambiguity may be briefl}^ expressed as follows : 
 
 The Principle of Xon-Ambiguit}'- logically presupposes that of 
 Non-Contradiction, though the practical application of the latter 
 principle presupposes the former. For what is Ambiguity ? Not 
 all variation in the meaning of terms, but only such as may lead 
 to laconsistenc}'. And Ambiguity is to be avoided just for this 
 reason. Without reference to the Principle of Non-Contradiction 
 we can hardly distinguish Ambiguity from other variation in the 
 meaning of the same term.* 
 
 We have said that in the application of the Principle of Non- 
 Contradiction the Principle of Non-Ambiguity is presupposed. 
 Tliis maj^ be sho\\Ti as follows : If there is not between disputants 
 complete agreement as to the meaning of statements and expressions, 
 it is impossible for the one, with any certainty, to accuse the other 
 of contradicting himself. Similarly, if there is not perfect corre- 
 spondence between what I mean and what I say, two of my state- 
 ments may be apjxirently contradictory — contradictory in form — 
 and j'ct my meaning may not be so. The existence of a genuine 
 thought-contradiction can be clearly pointed out only after we have 
 first seen that the Principle of Non-Ambiguity is adeciuately satisfied. 
 Meanings can be criticized as consistent or inconsistent with each 
 other only in so far as they have been adequately defined. Thus, 
 to borrow Father Clarke'sf illustrations, the two propositions ' This 
 man is wise,' ' Tliis man is not wise ' may be accepted together, in 
 spite of their being verbal contradictions, if by the first we mean that 
 the man is prudent and canny in business matters, and by the second 
 that he holds many foolish opinions on speculative questions. So, 
 again, an event may be said to bo impossible or not impossible ac- 
 cording as we use the word to signify moral or absolute impossibility ; 
 and similarly, in reference to an unwelcome visitor, I may say with- 
 out self-contradiction " I am at home, and yet I am not at home." 
 
 Before we can decide, then, whether two proporitions are contra- 
 dictory or not, we must agree as to their meaning. In this sense the 
 observance of the Law of Non-Ambiguity is presupposed in all dis- 
 cu.ssions as to whether one proposition is or is not consistent with 
 another. But ambiguity does not necessarily in itself imply a 
 contradiction. Two interpretations of one and the same statement 
 may or may not be inconsistent, and therefore contradictory. 
 A priori, they are as likely to prove adjustable and complementary 
 as they are to prove mutually inconsistent. 
 
 * For this whole observation I am indebted to Professor Stout. 
 t ' Logic,' p. 37.
 
 Chap. X.] THE LAWS OF THOUGHT 103 
 
 Tlie foUowing are two main ways in wliich the application of 
 the Principle of Non-Contradiction presupposes the Principle of 
 Non-Ambiguity : 
 
 1. Tlie reference in both the opposed propositions must be to 
 the same standard. Thus, the rate of five miles an hour is both 
 fast and not fast — fast for a man walking;, but not fast for a man 
 running. Hence the two propositions ' This is a quick pace ' and 
 ' This is not a quick pace ' are contradictory only when the standard 
 is kept constant.* 
 
 2. Two apparently contradictory propositions are not really so 
 if the objective reference of subject or predicate differs in the two 
 cases. Thus, the two propositions ' All infants are under two,' 
 * Some infants are not under two ' {e.g., over twenty) are perfectly 
 consistent if ' infants ' is used in the ordinary sense in the first 
 statement, and in its legal sense in the second. So, again, a child 
 may be fair and j^et not be fair — i.e., he may have a fair complexion, 
 but hair that is not fair ; a man may be strong and yet not strong — ■ 
 muscularly strong, but not strong constitutionally. 
 
 Self- Contradict ion. 
 
 There are certain so-called statements that are self-contradictory. 
 These are cancelled by the Principle of Non-Contradiction as 
 logically meaningless ; e.g., ' The statement of Epimenides the 
 Cretan that all Cretans are always liars is perfectly true.' As Pro- 
 fessor Stout says, ' When Epimenides utters the words " All Cretans 
 always lie," then, if he tries to include the statement which he sup- 
 poses himself to be making, he is not making a statement at all. 
 For the meaning of the supposed statement cannot be determined 
 without a vicious circle — i.e., cannot be determined at all. 
 
 AU statements by Cretans, including this one of mine, are false. 
 
 Including what statement of j'ours ? 
 
 Answer : Including the statement that all statements by Cretans, 
 including that one of mine to the effect that . . . etc., are false. 
 
 And so on ad infinitum, with no irov <7Tw.'t 
 
 By way of further illustration we may consider the following 
 example : ' Contradictories may be true together. For, admitting 
 that every rule has an exception, then, since tliis statement is itself 
 a rule, it follows that we must also admit that there is at least one 
 rule without exceptions.' 
 
 Tliis means that if we grant the truth of the rule ' Every rule has 
 an exception,' then the rule which tells us this, being itself a rule, 
 must have an exception. There is an exception to the rule that 
 every rule has an exception as there is to every other rule — i.e., 
 there must at least be one rule that has no exception. The solution 
 
 * Cf. Father Clarke. ' Logic,' ihid. 
 
 t For a further treatment of this ancient puzzle, sco p. 1S2.
 
 104 THE PROBLEM OF LOGIC [III. ii. 
 
 of this puzzle lies in the consideration that the given proposition is 
 self -contradictory. What we have just proved, then, is not that 
 contradictories may be true together, but that if a proposition is 
 implicitly self-contradictory, it is possible by a httle reasoning to 
 make its self -contradictory character exphcit. 
 
 We conclude our treatment of the Laws of Thought by the dis- 
 cussion of a certain fundamental characteristic shared by all the 
 Laws alike in virtue of their common function as Principles of Intel- 
 ligibility. As such, the Laws of Thought are inviolable, and we now 
 propose to discuss the precise significance of this striking property 
 of inviolabilit3\ 
 
 (r) The Inviolability of the Laws of Thought.* 
 
 The Laws of Thought are so constitutive of our rational faculty 
 that it is impossible to violate them. The deliberate attempt at 
 violation results not in falsehood, but in a blank failure to think. 
 What we say is unthinkable, and has no meaning. Thus, the 
 attempt to override the Principle of Identity, in its relation to the 
 proposition, is the attempt to treat an assertion as being void of any 
 principle that holds its various elements together. Subject and 
 Predicate are treated as mutually independent entities, whereas the 
 assertion has no meaning apart from their dciDendence. Hence, an 
 inward contradiction which, qua contradiction, is unthinkable. 
 And if, further, the hypothetical violation of the Law of Con- 
 tradiction reduces the ' proposition ' that attempts the viola- 
 tion to the status of a mere meaningless formula, the attempt at 
 violating the Law of Excluded Middle leads to similar results. The 
 Law of Excluded Middle safeguards the intelhgibihty of fact. If 
 fact is not intelhgible, it cannot be inteUigibly thought of. In so 
 far as the object of our (would-be) thought — e.g., a round square — is 
 meaningless, the ' thought ' is simply non-existent. 
 
 In this fundamental respect of inviolabihty, the Laws of Thought 
 are unlike such other kinds of laws as laws of the land, and laws 
 which regulate the relation between ends and means. Laws of the 
 land are presumably violated by every criminal who finds his way 
 into prison or reformatory. Laws of licalth, and all other laws 
 which prescribe the conditions of success in the attainment of ends, 
 are constantly infringed by all of us. But we cannot so speak of the 
 Laws of Thought : a violation of these can never occur. If we are 
 found accepting two apparently contradictory statements, it is 
 either that they are contradictory in form only and not in substance, 
 or else that, in accepting them, we accept them without thinking, 
 or, at any rate, without recognizing that they are contradictory. 
 Let us consider difTeront classes of cases in which the Law of Con- 
 tradiction is thus apparently violated. 
 
 * The following section owes much to the help and inspiration of Professor Stout.
 
 Chap. X.] THE LAWS OF THOUGHT 105 
 
 The most common instance of a supposed violation of a law of 
 thought is the case in which we are deceived by the ambiguities of 
 the language we make use of — ambiguities which themselves reveal 
 a lack of precision in our own thinking. Thus, to take up an illus- 
 tration already cited, we may accept the two statements ' Tliis 
 man is strong ' and ' This man is not strong ' as contradictories, 
 whereas a little further reflection will show us that we are using the 
 word ' strong ' ambiguously. Until we have made the distinction 
 between ' strong muscularly ' and ' strong constitutionally,' we have 
 not said what we intended to say ; but so soon as our intention is 
 expressed in appropriate language, the contradiction disappears. 
 Distinction is the first remedy to apply to that state of mental 
 disease which we may call the illusion of Contradiction. 
 
 It is a well-known fact that when we set ourselves to think over 
 the fundamental problems of our life — God, Freedom, Immortality, 
 the World — we soon find ourselves reaching apparently contradic- 
 tory conclusions. In their most fundamental form, these conclu- 
 sions are loiown in Philosophy as Antinomies. Kant discusses their 
 significance in his ' Critique of Pure Reason.' Perhaps the most 
 famiUar of these Antinomies is that between Necessity and Free- 
 will. Sincere reflection on the problem may lead to the two 
 opposite conclusions that we are free, and yet that we are not free. 
 But an Antinomy of this kind is not a Contradiction. If it were 
 this, it would not be so mentally stimulating as it is. It is not 
 actual Contradiction that stimulates — what is meaningless cannot 
 fail to be depressing — but the necessity our thinking nature is 
 under to get beyond the illusion of Contradiction. Kant solves the 
 Antinomy by making a distinction. He qualifies the ' we ' differ- 
 ently in the two cases. As members of a phenomenal world, he 
 concludes, we are not free ; we can be free only as members of an 
 ' intclhgible ' or spiritual world. 
 
 In another class of cases we run into apparent contradiction 
 through very failure to think at all. Thus it is conceivable that a 
 student might conclude from the two statements ' No French are 
 not Europeans ' and ' All Parisians are French ' that some Parisians 
 were not Europeans. But it cannot be assumed that the conclusion 
 of an argument is always reached by thinking out the premisses. 
 A mere association or an ill-digested suggestion may very well 
 rcheve the tension of our thought at the critical moment, without 
 our realizing what has happened. But in this case we have not 
 violated a law of thought, but simply the salutary maxim that a 
 result for which the reason is to be held responsible must have been 
 reached under the sole direction of the reason. 
 
 Most people entirely fail to systematize their thinking. They 
 live mentally in a number of isolated compartments, and are quite 
 indifferent if an 'A is B,' which can be justified within its o\va 
 abstract sphere, is irreconcileable with an ' A is not B,' wliich in its
 
 106 THE PROBLEM OF LOGIC [III. ii. 
 
 owii context, again, can be similarly justified. But there is no 
 contradiction here. For if the two spheres in question are labelled 
 S^ and S;, then S^ (A is B) and So (A is not B) are not contradictories. 
 SjA is not the same subject as SoA, nor is S^B the same j)redicate as 
 SoB. There is only the Illusion of Contradiction. 
 
 However we look at this matter, the LaAv of Contradiction is seen 
 to be inviolable. When we imagine that we have violated it, we 
 have simply ceased to think ; and where there is no thought, we 
 cannot violate its laws. 
 
 To think inconsistently, then, is strictly impossible. It would 
 necessitate our thinking the gaps in our thinking. We cannot go 
 on thinldng through a contradiction, though we can think on to 
 the very edge of the gajD and then allow ourselves to be borne 
 asleep on the wings of forgetfulness over the unthinkable itself. 
 
 But the metaphor is misleading. It implies that the unthinkable 
 is something, whereas it is nothing. The only unthinkable is that 
 which breaks a constitutive law of our thinking ; but if these laws 
 are inviolable, the unthinkable cannot exist in any form. It is not 
 Contradiction that exists, but the baffled thought that reahzes the 
 restraining pressure of that which makes Contradiction impossible — 
 the very Reason itself. 
 
 It is, then, impossible, we repeat, to think inconsistently. Where 
 the inconsistency occurs there will be a bare blank of thought, 
 showing up not, indeed, the inconsistency of our thought, but its 
 f ragmen tariness. We shall thus appear to have violated the Prin- 
 ciple of Identity which insists on the meaninglessness of logically 
 connecting two judgments that are logically discontinuous. The 
 Illusion of Identity here consists in the assumed unity and continuity 
 of the reasoning. In treating what is fragmentary as though it 
 were coherent, we have not, however, violated the Principle of 
 Identity, for Ave have not been able to think it as one and continuous. 
 
 The difficulty which besets the conclusion that the Laws of 
 Thought are inviolable is that, on this doctrine, it seems hard to 
 account for the fact of Inconsistency. Inconsistency of Thought 
 can mean no more than Discontinuity of Thought ; in committing 
 the Formal fallacy, we must have ceased to think. But if we are 
 not inconsistent as thinkers, in what capacity is it that we are incon- 
 sistent ? Moreover, even if we replace the conception of Inconsis- 
 tency by that of Discontinuity, and are content to label our fallacies 
 as mere breaches of Continuity, we have only shifted the difficulty 
 from one word to another. We have still to reconcile the fact of 
 discontinuity with the inviolability of the Laws of Thought. If a 
 breach of continuity is not a violation of a law of thought, of what 
 is it a violation ? To hold that nothing is violated is to deny the 
 value of logical distinctions, and reduce the distinction between 
 Consistency and Inconsistency, Continuity and Discontinuity, to a 
 difference that is purely psychological. But if we hold, as we must
 
 Chap. X.] THE LAWS OF THOUGHT 107 
 
 if Logic is to have any distinctive meaning, that our inconsistencies 
 are, in some sense, breaches of rational obligation, we are once more 
 confronted with the old persistent difficulty : How can they be 
 violations of logical requirement if they are not violations of the 
 laws of our thinking ? 
 
 Perhaps the simplest answer to tliis really perplexing question is 
 the best. It may be enough to say that the fallacies which appear 
 to be infringements of the Laws of Thought are, in truth, infringe- 
 ments of the duty to think, of the moral (or religious) obligation we 
 are imder to think faithfully — i.e., to faithfully think whenever we 
 profess to be thinking. 
 
 It may, indeed, be argued that though we cannot violate a law, 
 we may yet break a rule, and that fallacies in Logic are no more 
 than breaches of logical rules. Tliis would no doubt be a happy 
 solution did not the breach of the rule imply the violation of the law. 
 But strictly logical rules embody the requirements of the Laws of 
 Thought. How could they otherwise be qualified for the guidance 
 of our thinking ? A fallacy cannot therefore be a violation of 
 logical rules as such. Again, though the Rules of the Syllogism 
 may no doubt be used to test the logical satisfactoriness of our 
 reasoning, it v.'ould seem that in thus testing its validity we are, at 
 bottom, testing our owti -fidelity to thought. For if we have really 
 been thinking, we must have been thinking vahdly. We are thus 
 brought back to the view that ' logical ' fallacies are breaches of 
 intellectual morality or of the religion of the Intellect, violations of 
 the duty to think faithfully whenever we are pledged to tliink. 
 Logic would thus seem to be rooted in Morality and ReUgion. 
 
 It may seem strange that in a crucial difficulty of this kind there 
 should have been no appeal to the insight of Hegel. The whole 
 perplexity arises from the conviction that Contradiction is meaning- 
 less, whereas Hegel is insistent that Contradiction is not meaning- 
 less. ' Contradiction,' we read (' Logic,' tr. Wallace, p. 223), ' is 
 the very moving princijile of the world : and it is ridiculous to say 
 that Contradiction is unthinkable.' I am, however, obliged to 
 confess — though the fault may vcrj^ well be mine — that I have found 
 Hegel's conception and treatment of Contradiction irrelevant to 
 the solution of the precise difficulty under consideration. In so far 
 as Hegel is insisting that a certain Xegativity, or negative movement, 
 is essential to the vitality of spiritual experience, and that such Nega- 
 tivity, under one aspect or another, is a permanent element of such 
 experience, even in its most harmonious developments, I believe 
 that he is indicating and establishing in its abstract form the most 
 distinctive conv^iction of all genuine Idealism. What I do not see 
 is that he is in any way redeeming or transcending the unintelli- 
 gibility of a contradiction.* 
 
 * For a profound and suggestive treatment of the problem of Negativity, vide 
 Professor Bosanquet's article on ' Contradiction and Reality,' Mind, N.S., No. 57, 
 January, 190G.
 
 IV. 
 
 ANALYSIS OF THE LOGICAL PROPOSITION AS 
 STATEMENT OF MEANING. 
 
 (i.) Kinds of Proposition (ch. xi.). 
 
 (ii.) Analysis of the Categorical Proposition (ch. xii.). 
 (iii.) The Meaning of Possibility (ch. xiii.). 
 (iv.) The Disjunctive Proposition (ch. xiv.). 
 
 (v.) The Hypothetical Proposition (ch. xv.).
 
 CHAPTER XI. 
 IV. (i.) KINDS OF PROPOSITION. 
 There are three main kinds of proposition : 
 
 1. Categorical : ' S is P,' ' S is-not P.' 
 
 2. Disjunctive : ' Either A is B, or C is D, or E is F ' 
 
 3. Hypothetical : ' If A is B, then C is D.' 
 
 If we examine the structure of these three main types of pro- 
 position, we notice that the elements out of which each type is 
 constructed are different in the three cases. 
 
 1. The Categorical Proposition is a synthesis of the two elements 
 which, in their verbal form, are called Terms* — the Subject-Term 
 and the Predicate-Term. The subject of a categorical proposition 
 is that about which something is being said ; the predicate is that 
 which is being said or stated about the subject of the proposition. 
 
 Examples : (The next examination) is (held in June). 
 
 S P 
 
 It is (a tiresome thing) (to fail in an examination). 
 
 P S 
 
 No one need fail. 
 
 = (The possibihty of passing) is (in the hands of 
 S P 
 
 everyone). 
 
 As the proposition is the true logical unit, and the term a mere 
 derivative or abstract from the proposition, it follows that the term 
 has meaning only in relation to the part it plays in a proposition. 
 But this does not mean that we are logically precluded from discuss- 
 ing terms except in so far as tliey actually exist within given pro- 
 positions. A word, or a word-complex such as ' The Science of 
 Logic ' or ' The Cat and Fiddle,' may justly be called ' a term,' 
 provided it can play the part of subject or predicate in a proposition. 
 A term is, in fact, just a potential subject or predicate, so that even 
 in abstraction from the proposition it still remains intrinsically related 
 
 * Oa the etymological significance of the -n-ord ' term,' vide Joseph, ' An Intro- 
 duction to Logic' ch. ii., p. 13, footnote. 
 
 Ill
 
 112 THE PROBLEM OF LOGIC [IV. i 
 
 to it. Though it need not be doing actual service within a proposi- 
 tion, it must at least belong to the reserve. 
 
 2. Li the case of the Disjunctive Proposition — e.g., ' Either the 
 earth moves or Copernicus was mistaken ' — the main elements are 
 clauses, not terms ; and they stand to each other in the relation of 
 alternatives — a relation very different from that of Subject to Predi- 
 cate. 
 
 3. In the case of the Hjqiothetical Proposition, again, it is evident 
 that we cannot call its main elements ' terms.' Take, for instance, 
 the proposition ' If the weather is fine, we shall go for a picnic' 
 Here we have two main elements ; but these do not stand to each 
 other as Subject to Predicate, nor are they terms. They are clauses, 
 and stand to each other as Antecedent to Consequent, the antecedent 
 being represented by the ' if ' clause. 
 
 We conclude that propositional elements are of three kinds ; 
 
 Terms : 1. Subject and Predicate in Categorical propositions. 
 
 [2. Alternative possibiUties in Disjunctive propositions. 
 Clauses : - 3. Antecedent and Consecjuent in Hypothetical proposi- 
 
 { tions. 
 
 The types of proposition that we have been considering may be 
 connected together as different stages in the process through which 
 the human mind passes on its way from question to answer. Every 
 process of reflective activity which is guided, however vaguely, by 
 the logical ideal of clear and consistent thinking is a process of 
 which the essence is to give a more determinate form to what, at the 
 outset, is relatively indeterminate. ' The action of thought,' it has 
 been said, ' is excited by the irritation of doubt, and ceases when 
 behef is attained.'* So, we might add, an act of judgment is excited 
 by the discomfort of a question to which we do not at once foresee 
 the answer, and it crystaUizes into the form of a proposition only 
 when some answer, however partial, has been definitely formulated. 
 As Behef is to Doubt, so is the Answer which a judgment expresses 
 to the Question out of which it arises. 
 
 Now, a question, if it is to be more than a vague unprogressive 
 state of wonder or curiosity, is already, by the very fact of its being 
 a question, partially determinate. It takes its start from a more or 
 less vague conception, and the motive wliich prompts a more deter- 
 minate answer is the pressure which the idea of the determinate 
 necessarily exercises upon the undetermined whenever the logical 
 interest is in any degree awake. This pressure exerts itself most 
 naturally in the work of transforming the vague possibility of a 
 solution which is implied in the existence of the C{uestion into a 
 set of definite alternative possibilities. If I am told that a happy 
 
 * C. S. Peirce, ' Illustrations of the Logic of Science,' Popular Science Monthly, 
 xii.. p. 289.
 
 Chap. XL] KINDS OF PROPOSITION 113 
 
 event has raised nic to the dignity of an uncle, but am not furtiier 
 informed, the knowledge I have is, in its indeterminateness, itself 
 a question wliicli tends to determine itself forthwith in the form 
 of tlie disjunctive proposition : ' Either it is a nephew or it is a 
 niece.' 
 
 A further stage of determination is reached when I make an 
 assumption as to the sex, and proceed to define my mental attitude, 
 prospectively, on that assumption. ' If it is a boy, my brother-in- 
 law will perhaps be sorry ; but my sister, I know, will be pleased. 
 If it is a girl, there is no reasonable possibility of its being called after 
 me.' And so forth. The second stage is thus that of the Hypo- 
 thetical Proposition. 
 
 The third stage in the process is reached when the news arrives 
 that my sister is pleased, and that the infant is to be called after 
 me. The question ' What sex V with which the whole thought- 
 process started now receives the definite answer ' The child is 
 a boy.' Tiie disjunctive proposition passes into a categorical pro- 
 position which asserts the one alternative to the exclusion of the 
 other. 
 
 It must not, however, be supposed that the Disjunctive Proposi- 
 tion, which is logically prior to the Hypotlietical, is logically prior 
 to tlie Categorical as well. It is certainly prior to the more deter- 
 minate form of categorical which asserts one out of many alterna- 
 tive possibihties ; but it is posterior to the more indefinite categorical 
 out of which the question and the disjunction sprang. Thus, the 
 proposition that the being is a human child is the necessary preface 
 to the question ' What sex V and to all that follows between the 
 question and the determinate categorical answer, ' The cliild is a 
 boy.'* 
 
 In many cases, indeed, a categorical statement possesses no in- 
 determinateness at all, hence suggests no question, and therefore no 
 disjunction as the first step towards a more determinate answer. 
 Moreover, there is another class of cases in which we have a cate- 
 gorical proposition which, though determinate, cannot have been 
 reached through the cancelling of alternatives in a disjunctive pro- 
 position. The statement ' The part is not greater than the whole ' 
 cannot have been preceded by a disjunctive proposition stating an 
 alternative to the assertion in question. The proposition is self- 
 evident, and in the case of all self-evident propositions the cate- 
 gorical statement cannot be considered in the light of an answer at 
 all. Where there can be no doubt there can be no question, and 
 therefore no answer. 
 
 * The Categorical Proposition is, in fact, implied all through. It mediates the 
 transitions from the question to the disjunctive, and from the disjunctive to tho 
 hypothetical. Thus the statement ' Either it is a nephew or it is a niece ' pre- 
 supposes the categorical ' A baby is either a boy or a giil ' ; and the statement 
 ' If it is a boy . . . my sister . . . will bo pleased ' presupposes the categorical ' ^ly 
 sister has a preference for boys.'
 
 114 THE PROBLEM OF LOGIC [IV. i. 
 
 We conclude, then, tliat the logical order of the propositional 
 forms is : 
 
 1. Categorical ; 
 
 2. Disjunctive ; 
 
 3. H^-pothetical ; 
 
 and we propose to adopt this order in the following discussions on 
 the nature of Proposition and Inference. 
 
 But before we proceed to the detailed discussion of the Categorical 
 Proposition, it is essential that we should emphasize a distinction of 
 fundamental importance arising out of the relation in wliich a ques- 
 tion stands to the reahty in reference to which the answer is sought. 
 
 Tlie logical interest that inspires a question may be either 
 occasional or systematic. It may be conventionally restricted 
 within the limits of some definite topic, or it may aim at an answer 
 that shall respect systematic connexion of facts apart from any such 
 conventional restriction. In the former case the answer has a 
 ' formal ' value ; in the latter case it has a ' real ' value. Where it 
 has only a formal value, the judgment or proposition in which it is 
 expressed may be designated as ' formal ' ; where it has a real value, 
 as ' real.' 
 
 Thus, if I inciuire ' Is my friend in a good temper this morning V 
 my question may express the merely occasional interest of ascertain- 
 ing whether our meeting is likely to be pleasant or not. On the 
 other hand, if I should chance to be a doctor, I may be scientifically 
 interested in the efifects on his nervous organization of some special 
 condition such as neuralgia or a sleepless night. In the former case 
 the question refers to formal reality, and expects a formal answer ; 
 in the latter both question and answer have a real or systematic 
 character.* 
 
 The distinction between a formal and a real logical interest may 
 be aptly illustrated by the difference in meaning acquired by the 
 Singular Proposition (the proposition wliich has for its subject a 
 singular term) according as it is considered in the light of the one 
 interest or of the other. 
 
 In the service of our unorganized everyday experience as well as 
 in the service of Science the singular proposition plays an important 
 part. In all our talk about individuals, be they pcr,sons, tilings or 
 events, as well as in scientific observations and verifications, our 
 direct, expUcit reference is to the individual fact. In the case of 
 Science, however, the sense-individuaUty of observed fact derives 
 all its significance from the scientist's belief in natural law. A fact, 
 for natural Science is a fact under law, and a species, instance, or 
 example of a common nature. The reverence for fact which is so 
 
 * The reality of Nature, as interpreted by Science, is not the reality of Spiritual 
 Experience. The same question might express a spiritual interest — e.g., in my 
 frieii'J's power vo be cheerful in difficult circumstances. The logical discussion 
 of this more personal point of view belongs, however, to a Philosophical Logic.
 
 Chap. XL] KINDS OF PROPOSITION 115 
 
 characteristic of the scientific attitude is ultimately rooted in 
 reverence for this common nature. This attitude towards the 
 facts of observation gives to the singular proposition as understood 
 by Science — i.e., to the ' real ' singular proposition — a pecuUar 
 import. The reference to individual fact is not to the individual qua 
 individual, but to the individual as symbolic of a universal or 
 common nature. In the case of the formal singular proposition, 
 thougli this reference to a common nature must always be implied — 
 the isolated individual being at bottom unthinkable — it is not 
 implied in the same systematic sense. The fact is not conceived as a 
 fact under law, but as having an individual importance relative to a 
 certain topic. If I say ' This chair has only three legs,' my topic is 
 the possible uses of a chair, and my meaning is that, whatever chair 
 I may choose to sit on, I must avoid the chair in question. 
 
 It is convenient to distinguish from formal and real propositions 
 ahke the strictly verbal proposition. The Verbal Proposition states 
 the meaning of a word or verbal sign qua word. ' Man means 
 rational animal ' and ' Man is a verbal symbol which stands for 
 " rational animal " ' illustrate the two forms in wliich a verbal 
 proposition may be expressed. In either ease the meaning we define 
 is that of the word as such, and not that of the object signified by 
 the word. Thus, the proposition wliich states a definition is not 
 usually verbal, since what we wish to define is, as a rule, not the 
 meaning of the word qiui purposive combination of letters or sounds, 
 but the meaning of that significate of which the word is but the 
 verbal sign. The proposition ' Man is a rational animal ' is a defini- 
 tion of the common nature conventionally symbolized by the sign 
 ' Man,' and not a definition of the sign itself. It is therefore mate- 
 rial (formal or real), and not verbal. 
 
 CHAPTER XII. 
 IV. (ii.) ANALYSIS OF THE CATEGORICAL PROPOSITION. 
 
 1. Subject aiid Predicate in the Categorical Proposition. 
 
 We have defined — or, rather, distinguished — the Categorical Proposi- 
 tion as one which states something about sometliing else, as a 
 proposition, therefore, including a subject as well as a predicate term.* 
 
 * Whether these two terms stand for existents or not, and if so, in what sense, 
 are further questions which concern what is called the Theory of the Existential 
 Import of propositions. Prof. Keynes, in particular, has given an elaborate 
 discussion of this problem in his ' Studies and Exercises in Formal Logic,' 
 4th edition, Part ii., ch. viii. But on this point see p. 189. 
 
 8—2
 
 116 THE PROBLEM OF LOGIC [IV. ii. 
 
 There are certain embryonic forms of the Categorical Judgment — 
 viz., the Exclamatory and the Impersonal Judgments — in which the 
 Subject-Predicate relation, though present, is still not obviously 
 expressed. By Exclamatory Judgments we mean such ejaculations 
 as ' Fire !' ' Man overboard !' where ' Fire !' is equivalent to 
 ' There is a fire,' or ' A fire has broken out ' ; Man overboard !' 
 to ' There is a man overboard,' or ' A man has fallen over- 
 board.' 
 
 Impersonal Judgments are, from this point of view, abbreviations 
 of a more perplexing kind. They are sometimes instanced as cate- 
 gorical propositions without a subject, but there seems to be no 
 adequate reason for tliis view. In such cases as ' It is too bad,' ' It 
 isn't fair,' the ' it ' is quite properly regarded as an indeterminate 
 subject-concept, which through its very indeterminateness truly 
 represents some unnameable and unlabellable subject-thing, some 
 total impression in its natural vagueness and indeterminacy. Such 
 propositions as ' It snows,' ' It is foggy ' hardly seem to be true 
 impersonals. Grammatically, indeed, they have an indefinite 
 ' provisional ' subject (c/. Fr. Il-y-a) ; but as soon as they are 
 reduced to the form of logical propositions — ' Snow is falling,' 
 ' Fog is prevaihng ' — we see that the subject-terms are really 
 definite. 
 
 By the ' subject ' of a categorical proposition we mean, of course, 
 the ' logical subject.' The question which term is subject and wliich 
 predicate must be decided, not by grammatical structure or by the 
 respective positions of the words, but by the meaning of the sentence. 
 At the same time, the subject is usually the less emphatic, the predi- 
 cate — that which is asserted — usually and naturally the more 
 emphatic term. Hence in ordinary conversation we have the help 
 of the speaker's voice and intonation ; and in written work, where 
 the right stress can be given through our knowledge of tlio meaning 
 of the whole context in which the proposition occurs, we may fre- 
 quently be helped out by emphasis. Thus, in reading the ancient 
 proverb ' In the multitude of counsellors there is safety,'* we feel 
 that the emphatic part of the proposition is ' in the multitude of 
 counsellors.' ' Saiety ' is not what we are asserting about ' in the 
 multitude of counsellors,' but ' in the multitude of counsellors ' 
 is what we are asserting about ' safety.' 
 
 But if the .sentence occurs quite alone, and we have not the 
 requisite data for unambiguously specifying the logical subject, 
 we are reduced to stating the various alternatives which the 
 logical .«>ubject may assume. Take, for instance, the isolated 
 sentence : 
 
 ' The Novum Organum of Bacon was not intended to supersede 
 the Organon of Aristotle.' 
 
 * Proverbs xxiv. 6.
 
 CflAP. XII.] THE CATEGORICAL PROPOSITION 117 
 
 Here the meaning may be — 
 
 1. The Novum Organum of Bacon was not intended to supersede 
 
 the Organon of Aristotle. 
 I.e. (The work of Bacon which was intended to supersede 
 
 the Organon of Aristotle) — Subject — 
 is-not 
 (his Novum Organum) — Predicate. 
 
 2. The Novum Organum of Bacon was not intended to supersede 
 
 the Organon of Aristotle. 
 I.e. (The Novum Organum that was intended to supersede 
 
 the Organon of Aristotle) — Subject — 
 is-not 
 (Bacon's work of that name) — Predicate. 
 
 3. The Novum Organum of Bacon was not intended to super- 
 
 sede the Organon of Aristotle. 
 I.e. (The statement that the Novum Organum of Bacon was 
 
 intended to supersede the Organon of Aristotle) — 
 
 Subject — 
 is-not 
 (true) — Predicate. 
 
 4. The Novum Organum of Bacon was not intended to supersede 
 
 the Organon of Aristotle. 
 I.e. (The superseding of Aristotle's Organon by Bacon's 
 
 Novum Organum) — Subject — 
 is-not 
 (a result that was intended by Bacon) — Predicate. 
 
 5. The Novum Organum of Bacon was not intended to supersede 
 
 the Organon of Aristotle. 
 
 I.e. (The relation which Bacon intended his Novum Organum 
 to bear to the Organon of Aristotle) — Subject — 
 
 is-not 
 
 (the relation of superseding) — Predicate. 
 G. The Novum Organum of Bacon was not intended to super- 
 sede the Organon of Aristotle. 
 
 I.e. (The work of Aristotle which Bacon intended that his 
 own Novum Organum should supersede) — Subject — 
 
 is-not 
 
 (the Organon) — Predicate. 
 7. The Novum Organum of Bacon was not intended to super- 
 sede the Organon of Aristotle. 
 
 I.e. (The Organon which Bacon intended to supersede by 
 his own Novum Organum) — Subject — 
 
 is-not 
 
 (the work of Aristotle) — Predicate. 
 
 The main conclusion to which we are driven by the foregoing 
 analysis is that a proj)osition depcn.Is as much on context for its
 
 US THE PROBLEM OF LOGIC [IV. ii. 
 
 true meaiiino; as do the terms which are its elements. Everywhere 
 we find the part incomplete and pointing beyond itself. 
 
 Now. it is of fundamental importance that we should not interpret 
 tliis reference of part to whole in abstraction from the limiting or 
 defining reference to purpose or interest ; for, apart from this 
 defining reference, we cannot hoj)e to fulfil the logical requirement 
 of relevance. A logical whole is objective reality as defined and, 
 as it were, individualized through the selective agency of some 
 specific interest. What Logic here demands as a requirement of 
 right tliinking is recognized in Art as a canon of right feeling. 
 If, for instance, in looking at a picture we are to feel a}sthetically, 
 we must be able to feel the full appeal of the picture within the 
 frame of the picture itself. As Professor Stout has somewhere said,* 
 ■ Whatever content enters into the work of art must be so connected 
 with the whole as never to divert attention from the whole : picture 
 or poem must be apprehended as a world in itself, the whole interest 
 being gratified witliin it ; hence to tliink of the real landscape 
 whilst looking at its picture is to slip away from the artistic unity 
 of the picture, and the enjoyment is no longer sesthetic' So it is 
 with a logical whole, the object that can satisfy a given logical 
 interest. Its natural framework is defhied by the Hmitations of the 
 interest. What in respect of that interest is extra-marginal is 
 logically irrelevant : it cannot enter into the whole TV'ithin which 
 the definite interest finds its definite satisfaction. 
 
 It follows that, when we say of a proposition that its true, or 
 logically ultimate, meaning is given to it by its logical context, 
 we understand by ' logical context ' that limited topic or ' universe 
 of discourse ' within which the interest which inspires the statement 
 in question lives and moves and has its being. 
 
 The relation in which the subject of a proposition stands to the 
 relevant universe of discourse may, perhaps, be made clearer by the 
 help of an illustration. Let the subject (or ' universe ') of discourse 
 (S) be the wanderings of Odysseus, and the proposition in question 
 ' Ulysses (S) bends the bow that no other could bend.' Here the 
 true and logically ultimate subject of the sentence is 8 as interpreted 
 in the light of S- It is not S itself, but S as qualified by "%. We are 
 .speaking of Ulysses, the hero of a hundred adventures already 
 detailed, but now in Ithaca once again, and just about to reassert 
 himself as lord of his own house and country. The true and ulti- 
 mate subject of the proposition is therefore ' Ulysses ' — Ulysses as 
 we have come to know him through the story of his past adventures, 
 not the bare ' Ulysses ' severed from all reference to a past which 
 alone gives to the present action, the bending of the bow, its critical 
 significance. As for the predication ' bends the bow that no other 
 could bend,' its relation to the subject ' Ulysses ' may be defined by 
 
 * I quote from notes, but the atatement substantially reproduces Professor 
 StouVa point.
 
 CnAP. XII.] THE CATEGORICAL PROPOSITION 119 
 
 saying that it serves to develop, through its additional item of in- 
 formation, the meaning which the subject has already acquired 
 througli the previous development of the topic. 
 
 The universe or subject of discourse imphes, as we have seen, a 
 reference to reality, of which the character will vary according as 
 the judgment is formal or real. Where the judgment is formal, the 
 subject of discourse is conceived in isolation from the objective 
 systems of Nature and History.* Thus the wanderings of Odysseus 
 constitute, in the connexion already cited, a formal subject of dis- 
 course. There the bending of the bow is considered as an incident 
 in the old Homeric story, which is itself uncritically accepted. We 
 have in mind the Odysseus of the legend, and are quite unconcerned 
 with what historical science may have to say about his existence 
 or his wanderings. We say ' Ulysses bends the bow that no other 
 could bend ' under the implicit reservation that the relevant context 
 and topic of interest is the legendary and not the historical context. 
 On the other hand, the S, as the ultimate context of a real proposi- 
 tion, is Nature as interpretable by that department of Science within 
 which the assertion falls. A real proposition is synonymous with 
 a scientific proposition — a proposition stated, as it were, in the 
 hearing of Nature, and open at every point to Nature's own correc- 
 tion. 
 
 This, of course, does not imply that the real or scientific proposi- 
 tion is a statement unhmited by any reference to purpose or direc- 
 tion of interest. A scientific proposition about a plant may be stated 
 from the mere chemist's point of view, and imply no vitality what- 
 soever in the organism which it considers. Thus limited, the state- 
 ment would be abstract enough, and the more faithfully it fulfilled 
 the logical requisite of relevancy to purpose, the more abstract would 
 it be. At the same time, it would be none the less real as 
 measured by its significance for a scientific reconstruction of Nature. 
 Reference to purpose, as we have seen [vide p. 5), is as essential 
 to the real proposition as reference to reality is to the formal pro- 
 position. 
 
 Hence, when Dr. Bosanquet seeks to give a deepened significance 
 to the meaning of Judgment by pressing ' subject of discourse ' back 
 upon ' reality as a whole ' or ' the real world as a whole,' and the 
 real world as a sj'stematic whole is taken to be the ultimate or 
 absolute Subject, the same for every judgment, there is involved 
 in these interpretations a clear disregard of reference to purpose ; 
 and this essentially differentiates his theory of propositional import 
 from that which is here defended. Students interested in the 
 question will find Dr. Bosanquet's view developed in a masterly- 
 way in the first three lectures of the ' Essentials of Logic' We 
 content ourselves here with the remark that, from the standpoint 
 
 * By History I hero understand the ordered connoxioa of events in time and 
 space, as treated by methods of scientific criticism.
 
 120 THE PROBLEM OF LOGIC [IV. ii. 
 
 adopted in the present work, reference to context, whether that 
 context be formal or real, is alwaj's conceived as limited by an 
 involved reference to purpose or interest ; and that we take con- 
 siderations of relevancy to be logically ultimate. 
 
 The relation of Predicate to Subject witliin the unity of the 
 Judgment suggests, on closer view, a distinction which, though 
 based on a confusion of thought, is sufficiently widespread and 
 popular to require special notice. 
 
 When the information supplied by the predicate is compared with 
 the definition of the subject-term, the result — so the ordinarj^ state- 
 ment runs — will show either that the predicate gives no information 
 not supplied by that definition, or that it takes us beyond the con- 
 tent of the defuiition. In the former case the judgment is said 
 to be analytic or explicative ; in the second case synthetic or amplia- 
 live. The words ' analytic ' and ' synthetic ' are usually adopted 
 to express the distinction in question ; but Kant, who is responsible 
 for giving to the distinction between Analytic and S^-nthetic Judg- 
 ment its great historical importance, himself suggested as alterna- 
 tive expressions the words ' exphcative ' and ' augmentative or 
 ampliative.' 
 
 An Exphcative Judgment (' analytic,' in Kant's sense of the 
 word) is, then, a judgment which, on reference to the definition of the 
 subject-term, turns out to be a mere verbal truism, telhng us nothing 
 about the subject that was not already given in its definition. Thus, 
 if ' man ' be defined as a ' rational animal,' it is a mere verbal 
 truism or tautology to assert that ' All men are rational,' for no 
 information is thereby given to anj^one who knows the meaning 
 of tlie terms wliich are being used. 
 
 An Amphative Judgment (' synthetic ', in Kant's sense) is a judg- 
 ment which predicates of the subject something which is not already 
 stated in the definition of the subject-term. Thus, if ' man ' is 
 defined as a rational animal, it is an ami)lification of this original 
 meaning to say that man has adopted the habit of wearing clothes. 
 So, again, if we take the proposition ' Man is man ' {i.e., Man is 
 truly human in the sense of being master of his fate), we see that the 
 predicate amplifies the meaning of the subject-term. The meaning 
 of the statement seems to be this : ' Man, a natural being, is also a 
 free agent.' The proposition, through its subject-term, introduces 
 man as a natural being, and the predicate informs us that this 
 * natural ' being is also ' spiritual.' 
 
 This distinction between explicative and amphative judgments 
 is often stated in a more precise form by reference to the system of 
 predicables. It is argued that if, in the judgment ' S is P,' P is 
 the genus to which S belongs or the differentia of S, or at once both 
 the genus and the difTerentia, it simply repeats, in whole or in part, 
 the meaning which the species S bears in virtue of its definition.
 
 Chap. XII.] THE CATEGORICAL PROPOSITION 121 
 
 and thus the judgment is explicative. But if P gives a property or 
 'accident' of S, the judgment is ampliative ; for these predicables 
 can be assigned only after the meaning has been unambiguously 
 fixed through definition. The statement that an equilateral triangle 
 is equiangular — an equilateral triangle being defined as a plane 
 rectilinear figure having three equal sides — is neither a truism nor a 
 tautology. 
 
 Having stated this familiar distinction in its usual form, we now 
 proceed to consider the question of its logical value. 
 
 As we have already pointed out, the distinction is illogical. It 
 is, indeed, open to a fatal objection which deprives it of all value. 
 We state this objection when we maintain that the expUcative 
 proposition, as above defined, is no proposition at all. If, in the 
 statement ' Man is a rational animal,' ' man ' is already understood 
 to mean ' rational animal,' we have fallen back upon the tautology 
 ' Rational animal is rational animal,' which, as we saw, takes us no 
 further than the bare concept ' rational animal,' and is therefore no 
 statement at all. If the principle of Identity-in-Difference is to 
 rule the logical Proposition, the predicate must not repeat the 
 subject in whole or in part. Otherwise we have a circulus in pro- 
 ponendo. Every proposition is therefore essentially ampliative. 
 If I define ' man ' through the statement ' Man is a rational animal,' 
 the meaning of ' man ' which I undertake to define is the as yet 
 unspecified common nature which, when analysed and reconstructed, 
 is specified as ' rational animal.' A proposition wliich states a defini- 
 tion involves as genuine a development of meaning as does any other 
 proposition, the meaning growing in clearness as we pass from the 
 as yet indeterminate meaning of the subject-term to the determinate 
 meaning of the predicate which suppfies the defining marks. The 
 term to be defired, the subject-term or definiendum, is a term which 
 threatens to give ambiguity, and calls for the remedy of definition : 
 it therefore cannot be identical in meaning with the predicate which 
 supplies the definition. 
 
 Similarly, to predicate genus alone, or differentia alone, of an as 
 yet undefined concept is in no sense to repeat a part of that concept. 
 The subject-term is here a relatively indeterminate meaning wliich 
 gains through the predication of genus or of differentia some further 
 though partial development. We conclude, then, that whatever 
 predicable be predicated of the undefined subject-term, the predica- 
 tion is ampliative. When once the meaning of the term is fixed 
 through definition, the term, qua defined, may, of course, become 
 the subject of further predication. Hence, when we predicate 
 propria of a given term, we are ampUfying that meaning of the term 
 wliich it has secured through its definition. But until the definition 
 of a term is fixed, the term, qua definiendum, cannot be treated as 
 already defined. 
 
 With the recognition that the Proposition is intrinsically ampUa-
 
 3 22 THE PROBLEM OF LOGIC [IV. ii. 
 
 tive in character, the distinction between amj)hative and explicative 
 — and with it the synonj'mous distinction between analytic and 
 s^Tithetic propositions — is necessarily abandoned. 
 
 The distinction between Analytic and Synthetic has been re- 
 interpreted by some logicians {e.g., Mr. Joseph) so as to correspond 
 to the distinction between the dififercntiation of S from P and the 
 identification of S and P in the Judgment considered as an Identity- 
 in-Difiference. ' Every judgement,' writes Mr. Joseph, ' is at once 
 analytic and sjTithetic ; for the act of judgement at once holds dif- 
 ferent elements apart and recognizes them as elements in a single 
 whole. As held apart, it requires an act of synthesis to see that they 
 make one whole : as recognized to make one whole, it requires an 
 act of analysis to find and hold them apart.'* 
 
 If we do not endorse tliis application of the words ' analytic ' 
 and ' sjTithetic,' it is partly because we require the word ' synthetic ' 
 for another office in connexion with the distinction between Affirma- 
 tion and Negation, and partly because the function of holding S 
 apart from P, so that their identity does not degenerate into a 
 fusion of one with the other, appears to be more specifically and 
 accurately referred to as a ' differencing ' or ' differentiation ' than 
 as an analysis. We reserve the use of the term ' synthesis,' as 
 contradistinguished from ' aiialysis,'' for the jpurposes of Explanation 
 and Inductive Method, f 
 
 2. The Quality of a Categorical Proposition. 
 
 The psychological analysis of a complete act of judgment, as we 
 understand it, shows (1) that it is an assertion or statement of 
 intended meaning, whether affirmative or negative ; (2) that, qua 
 assertion, it is an identity-in-difference develoi^ing its meaning ac- 
 cording to the principle of logical Identity ; (3) that it involves an 
 accompanying attitude of behef. 
 
 Some psychologists make no distinction between Assertion and 
 Behef. But the distinction is surely necessary. I may assert a 
 statement as stating what I mean to say, and not at all what I 
 beheve to be true or hold to be false. Such a proposition as ' All 
 donkeys are daffodils ' may be taken in this sense as a mere state- 
 ment of meaning. J Moreover, in so far as we are studying proposi- 
 
 * ' An Introduction to Logic,' ch. viii., p. 187. t Vide Chapter XXXIX. 
 
 X It has been urged that propositions of this kind are meaningless formulas, 
 and therefore no propositions at all. But is this really the case ? A reductio 
 ad absurdum might very well conclude with the statement in question, whereas, 
 if this statement were in reality no statement at all, the reductio ad absurdum 
 would at once be vitiated. Further, the writer of a fairy-talc, or a defender of 
 transmigration, or a believer in the absolute identity of all being might very 
 well maintain that donkeys were daffodils in posse, or that, in so far as donkeys 
 had any reality at all, they were essentially identical with daffodils, and that the 
 apparent difference between the two species was a mere illusion. The proposition, 
 in fact, is not necessarily self-contradictory, and only self-contradictory statements 
 are meiningless.
 
 Chap. XII.] THE CATEGORICAL PROPOSITION 123 
 
 tions in themselves, their logical relations with other propositions, 
 or their logical implications, we cease to interest ourselves in their 
 reference to reality or their significance for knowledge. The question 
 whether we beheve or disbelieve them is irrelevant to the purpose 
 we have in considering them. Judgments, when thus abstractly 
 studied, are asserted without being either beUeved or disbeheved. 
 
 Some difficulty may, perhaps, be felt in accepting Affirmation 
 and Negation as co-ordinate forms of Assertion. A consideration 
 of the parallel relation between beUef and disbehef may help to 
 remove it. It is a psychological commonplace that the true oppo- 
 site of belief is not disbelief, but doubt. Disbelief is a form of 
 beUef ; or, at any rate, behef and disbelief ahke are forms of con- 
 viction. Again, the real opposite of a given extreme is not the 
 counter-extreme, but the mean between them. If two men are 
 fighting, they are, qua fighters, akin, and their common foil is the 
 man who stands by and watches. These illustrations suggest a 
 corresponding relation between Affirmation and Negation. They 
 suggest that, just as behef and disbehef agree in being forms of con- 
 viction, so Affirmation and Negation agree in being forms of Asser- 
 tion ; and, further, that just as beHef and disbehef agree in a common 
 opposition to doubt, so the true opposite of Affirmation and Negation 
 ahke is a suspense of judgment. If there is still a difficulty felt in 
 apprehending the view that Affirmation and Negation alike are 
 forms of Assertion, or if it be set down as a scholastic subtlety, the 
 reason is probably tliis : that there is no word in ordinary use to 
 set off against the term ' synthesis ' (with its distinctively affirmative 
 imphcation) to represent the operation involved in denial. But the 
 term ' dialysis,' which means a ' sundering ' or ' separation,' and is 
 a respectable dictionary word, might very well be adopted, and we 
 should then be left free to define a negation or denial as the assertion 
 of a dialysis, and an affirmation or ' position ' as the assertion of a 
 synthesis.* 
 
 Affirmation and Negation, then, are differentiations of the more 
 fundamental activity of Assertion. All Judgment or Proposition 
 is Assertion, or statement of intended meaning, and the quality- 
 difference between Affirmation and Negation is not ultimate : they 
 are differentiations of a common nature, the act of Assertion. 
 ^Yhether affirmative or negative in quality, propositions are assertive 
 in character. Hence, though a negation must be an assertion, it 
 cannot, qua negation, involve any affirmative synthesis. Pure nega- 
 tion is denial pure from all affirmative or sj'-nthetic intention. 
 
 * The restriction here put upon the use of the term ' synthesis ' should bo 
 noted. The term is frequently used to signify that application of the principle 
 of logical Identity -which first constitutes a judgment as such. According to this 
 use (which is that of Mr. Joseph, referred to above) every judgment is synthetic 
 in virtue of its expressing an identity in difference. It seems better, however, to 
 reserve the term ' identity ' to denote this fundamental relation, and to use the 
 term ' synthesis ' in that restricted sense in which it is opposed to ' dialysis.'
 
 124 THE PROBLEM OF LOGIC [TV. ii. 
 
 Our subordination of the distinction between affirmative and 
 negative quality to the common assertive character proper to all 
 propositions does not in any way affect its genuineness. This 
 genuineness of the qualitative distinction has, however, been dis- 
 puted. Some logicians have attempted to state all propositions in 
 affirmative form, insisting that ' S is-not P,' the tj^ical form of the 
 negative proposition, is logically equivalent to ' S is not-P.' The 
 attempt, however, ignores the primary significance of negation in 
 the act of judgment. We often require to negate simply, to give to 
 a statement a bare, blank denial, without any positive implication 
 in the background. If we say that a thing is not white, we merely 
 mean that the term ' white' is not applicable to it. We just contradict 
 quite barely the statement that the thing in question is white. 
 We destro}' the predication ' white ' without making the shghtest 
 mental cfifort to replace it by another. Genuine logical denial can, 
 in fact, do nothing more than deny : it fulfils itself in negating. 
 We conclude, then, that ' S is not-P,' the meaning of which we have 
 discussed in connexion with our treatment of negative terms, is not 
 synonymous with ' S is-not P,' and that the two can be equated only 
 by weakening the natural distinction between the affirmative and 
 the negative. 
 
 3. The Copula. 
 
 The Copula may be briefly defined as the identity-pruiciple* as 
 operative within a categorical judgment. This principle is the 
 rea.son itself as inspired by a logical ideal. Present in subject and 
 predicate alike, it claims both, in their interrelatedness, as specifi- 
 cations of its own meaning. It is in virtue of the unity and con- 
 tinuity of the logical interest that our meaning developes all of a 
 piece. Oneness and identity of logical interest means oneness and 
 identity of the judgment, or the system of judgments, through 
 which that interest expresses itself. We have here to do with the 
 very nature of the logical interest itself, the fundamental and authori- 
 tative factor in all logical inquiry, and we cannot get behind it. 
 What makes an identity of an affirmation (or a denial) is that a 
 reasoned interest specifies, and to that extent fulfils, itself in the 
 assertion of it. The general form in which such an interest fulfils 
 itself is in specifying the meaning of a subject through a predicate. 
 The copula is, strictly speaking, the judgment-activity itself in 
 process of self-fulfilment in the form of a judgment : this active 
 logical interest is, in fact, constitutive of the identity-in-dillerence 
 wliich pervades and characterizes the Judgment. 
 
 * The copula Hhould not be confused with the copula-warZ;. The copula-mark 
 takes two typical forms : ' is ' when the assertion i.s affirmative in character, ' is- 
 not ' when the assertion is negative. ' Is ' may conveniently be called the synthetic, 
 ? id-not ' the dialytic copula-mark
 
 Chap. XII.] THE CATEGORICAL PROPOSITION 125 
 
 The Copula, then, is omnipresent in the Judgment. It cannot be 
 identified witli any part or aspect of it, for in that case whatever lay 
 outside could have no part or lot in the judgment. An clement that 
 had no share in that which unifies a proposition could not be an 
 element of that proposition. This point is important, and we pro- 
 ceed to amplify and to emphasize it. 
 
 The copula is frequently called a coupling-link between the subject 
 and the predicate. This metaphor is misleading if it be interpreted 
 as meaning tliat the terms of a judgment can be given independently 
 of the relation between them. This is, in the nature of things, im- 
 possible. For the words or concepts, set down independently of 
 the relation between them, are not ' terms in the judgment,' but 
 terms out of the judgment. The act of synthesis or dialysis which 
 defines the relation between S and P first bri»igs S and P into the 
 judgment. They are not there prior to the relating act. But a 
 coupling-link cannot be considered as first constituting railway- 
 carriages into railway-carriages. These remain the same before 
 and after the coupling. It is true that the car first becomes part of 
 a train by being thus coupled with another ; and if tliis is insisted on, 
 the coupling metaphor might serve the purpose, though still rather 
 lamely. The essential point to recognize, however, is that the term 
 of a proposition exists only in the proposition itself as organically 
 one with it, so that the Copula, as relating activity, cannot be identi- 
 fied with any single partial element in the Judgment — with a relation, 
 for instance, that is outside the terms. It must be the activity 
 which brings terms and relation, content and form, not together — 
 for this implies a previous separate existence in mutual isolation — 
 but into-birth-together. 
 
 We may illustrate this important point by what is really much 
 more than a mere analogy. It is sometimes stated that a poet 
 works upon a certain content, moulding it into poetic form ; and 
 we are left with the impression that what is intrinsically poetic is 
 the form. But the truth is that the poet works upon a certain 
 subject-matter, wliich, as such, is certainly not the content of his 
 poem. In bringing poetic unity into tliis subject-matter, he brings 
 into birth, in intimate unison, content and form together. With 
 the content comes the form, and with the form the content. The 
 content is the subject-matter poetically formed ; the form is the 
 form of the content. The poetic end is not to superinduce form 
 upon content, but to transform subject-matter into a formed 
 content.* 
 
 So with the Categorical Judgment. The matter of the judgment, 
 that about which we judge, exists prior to the judgment. But the 
 act of judgment consists not in superinducing a relation upon given 
 terms, but in transforming the given matter, through the selective 
 and unifying agency of a dominating interest, into terms-in-rclation 
 
 * Cf. Professor A. C. Bradley, ' Poetry for Poetry's Sake.'
 
 126 THE TROBLEM OF LOGIC [IV. ii. 
 
 (the relation of subject to predicate), or into that formed content 
 whicli we call a proposition. 
 
 The systematic intimacy of subject and predicate within a proposi- 
 tion is customarily indicated through the use of copula-marks — of 
 the expressions ' is ' and ' is-not.' The function of the word ' is ' 
 in the judgment ' S is P ' is not that of serving as a coupling-link 
 between subject and predicate, but that of indicating the identity- 
 relation between the two. It tells us that the term ' P ' which it 
 precedes and introduces must be used as predicate — i.e., as the 
 predicate of the subject-term S.* 
 
 The foregoing discussion will serve to meet Dr. Sigwart's objection 
 against a ' negative copula.' A copula, he argues, has by its very 
 nature a s^mthetic function. Hence a copula that divides is a self- 
 contradictory absurdity. ' There is no such thing,' he writes, ' as 
 a negative, but onh" a negated copula.' f If the function of the 
 copula were ' synthetic ' in that sense of the word in whicli it is 
 opposed to ' dialytic,' this argument would be pertinent enough. 
 But, as we interpret the copula, its function is to operate as the 
 identity which the assertion expresses, whether that assertion take 
 the form of a s^mthesis or that of a dialysis. Concepts do not cease 
 to be related to each other because tlie relation between them 
 happens to be an opposition or a severance. We show our depen- 
 dence upon society most energetically, it has been said, when we 
 assert our independence against it. The ascetic who renounces the 
 world has been made an ascetic by the very evil wliich he shuns. 
 The young lecturer who lectures in direct opposition to the tenets 
 he has imbibed from his late University teachers thereby proclaims 
 the potent eflfect which the lectures have produced upon his mind. 
 
 The assertion of a relation between^ S and P which the copula- 
 mark expresses is independent of the nature of that relation. Hence, 
 if by ' negative ' copula we mean the dialytic relation between S and 
 P, the expression is perfectly reasonable. The ' negated copula,' 
 on the other hand, as we understand the term ' copula,' is a logical 
 fiction. It means nothing. It indicates an operation that cannot 
 be carried out. We cannot deny an assertion without ourselves 
 asserting. For denial is, as we have seen, a form of assertion. 
 Hence the negated copula is itself a copula — a self-contradictory 
 conclusion which we could never have reached had not the original 
 conception itself been seLf-contradictory and therefore meaningless. 
 
 * Cf. Dr. Christoph Sigwart, ' Logic,' vol. i., ch. ii., § 17, English translation 
 by Helen Dendy (.Mrs. Bosanquet), p. 94. 
 
 t Ibid., ch. iv., § 20, Eng. tr., p. 122. 
 
 X The word ' between ' must not be understood in the coupling-link sense. 
 What the copula-mark expresses is the S and P relation, a relation which cannot 
 be understood as distinct from the terms related.
 
 Chap. XIIL] THE MEANING OF POSSIBILITY 127 
 
 CHAPTER XIII . 
 IV. (iii.) THE MEANING OF POSSIBILITY. 
 
 Real Possibility. 
 
 The ultimate source of the idea of possibility is to be found in the 
 ' I can ' of the free agent — the ' I can ' itself implying the ' I ought.' 
 As personalities, we have ideals which we ought to cherish, duties 
 which we ought to perform, spiritual imperatives wliich we ought 
 to obey ; and, as personalities, it is also possible for us to fulfil 
 the essential obUgations of our free-born nature. It is as an 
 agent capable of free choice that a man is justified in saying 
 ' I can be patient (or sincere),' ' I can obey my conscience (or 
 my reason).' 
 
 Similarly, the ultimate source of the idea of necessity is to be 
 found in the ' I cannot ' of the free agent, in the compulsion which 
 limits his freedom. ' I must ' because ' I cannot help it.' It is true 
 that we frequently use the phrase ' I must ' in a more j)03itive sense. 
 But such usage is misleading. It is misleading to say ' I must ' 
 when what we really mean to say is ' I ought to,' or, more decisively, 
 ' I ought to, and I will.' ' I must obey my conscience,' we say, 
 whereas what we really mean to say is, ' I ought to obey my con- 
 science.' So, when Regulus insists that he must return to Carthage, 
 his true meaning is given in the words, ' I ought to return to Car- 
 thage, and I wiU.' 
 
 There are, however, many derivative uses of the words ' can ' 
 and ' must.' Thus, restricting our attention to the former word 
 and its uses, we may allow that there are many aspects under wliich 
 even that wliich is not free admits of being treated as free ; and 
 consequently a corresponding set of senses in which the word ' pos- 
 sible ' can be used. We may conceive the universe, for instance, as 
 containing the whole ground of everything wliich the future wall 
 bring forth ; and we should then say : ' The universe can be what in 
 the fullness of time it will be.' So, again, we may appropriately 
 say, ' Water can freeze and evaporate,' for we here mentally isolate 
 the idea of water, and, abstracting it from the conditions of actual 
 existence, consider its changes of composition as though they eman- 
 ated from itself alone, as the changeless, persistent ground of all 
 these various transformations. Thus it comes to pass that we say 
 ' Water can freeze and evaporate,' much as we should say ' A cat 
 can mew and purr.' Again, the word ' can ' may suitably be used 
 to denote the power of further determination that a general idea 
 possesses. Thus we say ' A triangle can be obtuse-angled ; it can 
 also be right-angled ;' or, again, ' A horse can be black ; it can also 
 be grey ;' and so forth.
 
 12S THE PROBLEM OF LOGIC [IV. iii. 
 
 The Scheme of Opposition* proper to these relations of intrinsic 
 necessity or possibihty ma37^ be set down as follows : 
 
 Assertion of Intrinsic Necessity : S mustf be P. 
 
 (E.g., ' A proposition must be an identity-in-di£ference ' ; 
 
 i.e., the nature of a proposition is such that it must be an 
 
 identity-in-dififcrcnce.) 
 Contradictory : S need not be P. 
 
 {E.g., ' A proposition need not be affirmative.') 
 Contrary : S cannot be P. 
 
 (E.g., ' A proposition cannot be self-contradictory.') 
 SuhaltQrnate [Contradictory of Contrary) : S can be P. 
 
 {E.g., ' A proposition can be indesignate.') 
 
 Where the necessity — or possibility — is explicitly teleological, 
 bearing on the relation between means and end, the Scheme of 
 Opposition needs a certain readjustment, giving what we may call 
 the Teleological Scheme : 
 
 'O' 
 
 Assertion of Teleological Necessity : If X is accepted as end, 
 then Q must} be accepted as means ; or : Teleologically, 
 S must be P. 
 
 Contradictory : If X is accepted as end, then S need not be P. 
 
 Contrary : If X is accepted as end, then S must not be P. 
 
 Subalternate : If X is accepted as end, then S may§ be P. 
 
 Example of Teleological Opposition. 
 
 Assertion of Teleological necessity : If a man's aim is to keep 
 well, he must take regular exercise. 
 
 Contradictory : If a man's aim is to keep W'ell, he need not 
 take regular exercise {i.e., the failure to take regular exer- 
 cise will not be fatal to the attaining of his end). 
 
 Contrary : If a man's aim is to keep well, he must not take 
 regular exercise. 
 
 Subalternate : If a man's aim is to keep well, he may§ take 
 regular exercise. 
 
 {I.e., taking regular exercise will not be fatal to the 
 attainment of his end.) 
 
 The various uses of ' possibihty ' we have so far discussed have 
 ])ointed to a positive capacity in the subject considered, whether 
 that subject be a personal agent, a spatial oVjject, a proposition, 
 means to an end, or what not. Thus, when I say ' Tliis acorn can 
 
 * This section presupposes an acquaintance with ch. xix. 
 
 t Note that ' must ' is here not exclusive of ' can.' The necessary is also the 
 possible. 
 
 X Here, again, ' must ' is not exclusive of ' may.' 
 
 § The ' may ' is here permissive, and by no means implies limitation of know- 
 lodge (vide infra).
 
 Chap. XIII.] THE MEAXIXG OF POSSIBILITY I2fl 
 
 become an oak,' or ' Water can freeze and evaporate,' or ' The as- 
 pirant after health may forgo regular exercise,' I am abstractly 
 considering the changes which it lies in the nature of these subjects 
 to undergo. 
 
 Possibilities of this kind are potentialities, capacities, real dis- 
 positions, and are frequently referred to as real possibilities. They 
 presuppose, in each case, some positive nature of which they are the 
 capacities or dispositions. We might refer to them as disposi- 
 tional POSSIBILITIES, and in designating them we would propose, 
 as far as possible, to adopt the word ' can.' To say that ' S can be 
 P ' is to say that S is potentially or dispositionally P. It is only in 
 the case of teleological connexions that it seems necessary to sub- 
 stitute ' may ' for ' can ' in order to express tliis dispositional 
 quality. But the teleological ' may,' as in the instance already 
 cited, is closely alhed to the potential ' can.' It presupposes, in 
 particular, a positive nature (the agent inspired by an end or motive), 
 to which it adds a quaUfication in the form of a permissive and pur- 
 posive possibility. 
 
 Modal Possibility. 
 
 The distinctive use for wliicli we reserve the term ' may ' is that 
 of expressing modal possibility. 
 
 A modal possibihty is a possibility of wliicli the problematic 
 character expresses, not a potentiahty on the part of the object 
 concerning wliicli some predication is made, but imperfection of 
 knowledge on the part of the predicating subject. The ' may ' 
 which expresses modal possibility is known as the problematic 
 ' may.' 
 
 Let us consider the case of a particular object whose development 
 from witliin is circumscribed by its environment. Here, in so far 
 as the object contains in itself only the partial ground of its future 
 evolution, the possibihty of its reaching certain subsequent states 
 is not only a question of time, but also of external influences beyond 
 our power to determine or foresee. The plant-embryo in the seed 
 may become a full-gro^vn tree ; the child mxiy become a man. When 
 the nature of the environment is thoroughly understood, as also the 
 connexion between it and the object, the possibilitj^ gains a real, 
 objective value, and is no longer modal. In that case (as when we 
 say, ' The moon can be echpsed '), we no longer use the problematic 
 ' may.' 
 
 Consider, again, the specific statement ' It may freeze to-night.' 
 In making tliis statement, I ground my judgment on a very partial 
 knowledge of the conditions wliich occasion frost. The consequence 
 is that I can draw no decisive conclusion, and must content myself 
 with predicating possibility. But tliis possibility which I predicate 
 is simply a confession of ignorance — i.e., of inadequate knowledge. 
 
 9
 
 130 THE PROBLEM OF LOGIC [IV. iii. 
 
 ;My judgment, wliilc it appears to be busy with the frost, treats 
 really of my o^%^l scant knowledge of the frost-bearing conditions. 
 Should it happen that I am conscious of none of the conditions on 
 whose agency the event depends, ' the judgment,' as Dr. Sigwart 
 says, ' passes into the subjective possibility of conjecture, and thus 
 into the expression of uncertainty.'* It is in this purely subjective 
 sphere that possibility becomes synonymous with mere conceiva- 
 bihty or ' absence of contradiction.' 
 
 The. Modal Square of Opposition.-f 
 
 The attitude wliich contradictorily opposes the suggestion of a 
 limitation of one's knowledge on a certain point may be said to find 
 suitable expression in the proposition ' S is assuredly P.' The 
 contradictor}^ of this is ' S is-not assuredly P '; the contrary, ' S is 
 assuredly not P '; and the subaltern (or the contradictory of the 
 contrary) ' S is-not assuredly not P.' If we construe these propo- 
 sitions into equivalents expressive of the subjective attitude toward 
 the evidence which we may conceivably adopt, we obtain the 
 following Modal Square : 
 
 Mj : The evidence for the statement S is P is decisive. 
 ]\Iq : The evidence for the statement S is P is not decisive. 
 Mg : The evidence against the statement S is P is decisive. 
 ]\Ii : The evidence against the statement S is P is not decisive. 
 E.g., The evidence in support of a whale being a fish is decisive. 
 Tlie evidence in support of a whale being a fish is not decisive. 
 The evidence against a whale being a fish is decisive. 
 The evidence against a whale being a fish is not decisive. 
 
 The So-called Inferential Possibility. 
 
 Where, as in Formal Inference, the reference to knowledge is 
 excluded, and with it the modal use of ' may,' the idea of a possi- 
 bility ceases to have meaning. If by a logical possibihty we mean 
 an ' inferential ' possibihty, there can be no such thing as a logical 
 possibility. All Inference is necessary. If the premiss is ' All S 
 is P,' then all we can infcrentially say concerning the proposition 
 ' All P is S ' is that our a-ccepted premiss affords no ground either 
 for accepting it or for rejecting it. But when there is no ground for 
 inference, we cannot have an inferential possibihty. 
 
 * ' Logic,' vol. i., English translation, p. 204. The analysis given above of 
 the uflcfl of the term ' possibility ' owes much to Dr. Sigwart's treatment, 
 t Thw section presupposes an acquaintance with Chapter XIX.
 
 Chap. XIV.] THE DISJUNCTIVE PROPOSITION 131 
 
 CHAPTER XIV. 
 
 IV. (iv.) THE DISJUNCTIVE PROPOSITION. 
 
 The essential function of the Disjunctive Proposition is to develop 
 a given Ccategorical basis, of a more or less general and indeter- 
 minate character, by specifying the alternative possibilities* which 
 the predicate of the given categorical presents. Thus, the Disjunc- 
 tive Proposition comes to the aid of the Categorical ' All plane 
 rectilinear triangles are three-angled plane rectihnear figures ' by 
 adding that every tliree-anglcd plane rectilinear figure has either 
 one obtuse angle, or one right angle, or three acute angles. The 
 relation in which the Disjunctive Proposition stands to the Cate- 
 gorical is analogous to that in which Division stands to Definition 
 {vide p. 41). In so far as weins ist on the mutual exclusiveness 
 of the alternatives, the correspondence is complete. 
 
 ' The disjunctive expresses the relation of possibiUties to their 
 categorical basis — ultimately to the nature of actual existence — 
 which determines them and limits their range. There is no actual 
 existence wliich has not a common nature which it shares with other 
 actual existences, and wliich constitutes it a member of a class — an 
 instance or example. But the common nature, when scrutinized, 
 is seen to be capable of certain alternative determinations, and not 
 of others. To enumerate a series of such determinations which 
 omits no possible alternative is to state a disjmictive proposition. 
 . . . Among the specific determinations which a common nature 
 admits of, there may be many wliich are not actuaUzed, and per- 
 haps never have been nor will be actualized. Nevertheless, they 
 possess the kind of being which we call possibility. The common 
 nature which we are dealing with is intrinsically capable of them. 
 Thus, a fluid is either " partially viscous or a perfect fluid." A 
 colour is either red, green, blue, yellow, white, black, or of some 
 intermediate quahty — e.g., blue-green, blue-yellow, etc. Here blue- 
 yellow is not, as a matter of fact, an actual colour ; and the actual 
 physiological conditions of human vision seem to exclude its occur- 
 rence. None the less, we cannot say that a " blue-yeflow " is in- 
 trinsically impossible, any more than a " blue-green." If we con- 
 sider only the intrinsic nature of colour-mixture and of blue and of 
 yellow, there is nothing to exclude the possibiUty of a blue-yellow. 
 On the other hand, a " four-sided triangle "is an impossibility. It 
 is not a specific determination of " triangle " as such ' (Professor 
 Stout). 
 
 The essential requisite of a disjunctive proposition, if it is to have 
 any logical value, is that the alternatives wliich it enumerates shall 
 
 * The possibilities with which the disjunctive proposition is concerned are real, 
 dispositional possibilities. They are not modal possibilities.
 
 132 THE PROBLEM OF LOGIC [IV. iv. 
 
 be exhaustive. If I say that men are either wliite-skinned or black- 
 skinned, the statement is positively misleading, as the assumption 
 is always tacitly made that the alternatives of a disjunction are 
 exhaustive, and the conclusion would naturally be drawni that if a 
 man's skin is not white, it must be black. But if I say that men are 
 wliite-skinned or black-skinned, or have their skins otherwise 
 coloured, the statement is sound enough, and is in principle exhaus- 
 tive. We learn that white and black are two definite possibiUties 
 of skin-colour in the case of human beings, and the disjunction ex- 
 hausts the possibihties. 
 
 For brevity's sake, we have in the foregoing assumed that the 
 tj'pical disjunctive proposition takes the form ' A is either B or C 
 This, however, is not the case. No doubt the meaning of this formula 
 is genuinely disjunctive, and the proposition may be expressed in the 
 genuine disjunctive form ' Either A is B or A is C '; but the form as it 
 stands is categorical, A being the subject-term, and 'either B or C the 
 predicate-term. No proposition is truly disjunctive that cannot be 
 api:)ropriately expressed in the form ' Either P or Q,' where P and Q 
 are statements. Thus, the proposition ' All A's are either B or C ' 
 is disjunctive only because it may be read in the sense ' Either an A 
 is B or it is C The i:)roposition ' No A is either B or C ' is 
 equivalent to ' A is neither B nor C ' — i.e., to ' A is not B and A is 
 not C ' — a compound categorical proposition. With regard to the 
 particular propositions ' Some A's are either B or C ' and ' Some A's 
 are neither B nor C,' it seems difficult to consider tliem in any other 
 hght than that of categoricals with ' either B or C ' and ' neither B 
 nor C ' as their respective predicate-terms. 
 
 Of the many problems connected wath the import of the Disjunc- 
 tive Proposition none has excited so much interest as the question 
 whether the alternatives of a disjunction should be treated as mutu- 
 ally exclusive or not. Does the expression ' Either ... or ' 
 include or exclude the possibihty of both alternatives being true 1 
 When we say ' Either tliis man is a fool or he is a knave,' do we 
 imply that he may be both, or that he must be one and not the 
 other ? In a word, is ' either ... or ' to imply ' not both ' or ' it 
 may be both ' ?* 
 
 Before we pledge ourselves to either of these uses to the exclusion 
 of the other, it would be well to ask whether the option is a forced 
 one. Is it necessary, is it even relevant, to decide either in one way 
 or the other ? 
 
 * The distinction between these two alternative uses may perhaps be expressed 
 more clearly as that Vxjtween ' assuredly (or knowedly) not both ' and ' not 
 assuredly both.' The truth-view of import is here presupposed. From the stand- 
 point of the mere statement-view of import the distinction would be best expressed 
 as that between ' statedly not both ' and ' not statedly both ' (c/. pp. 148, 157). 
 This interpretation of ' Either ... or ' is exactly parallel to the interpretation of 
 ' some ' adopted in the case of the particular categorical. In the non-exclusive dis- 
 junctive ' Either P or Q,' ' Both P and Q ' is an unstated possibility {vide pp. 156, 157).
 
 Chap. XIV.] THE DISJUNCTIVE PROPOSITION 133 
 
 This query does not imply any dislike of giving a fixed meaning 
 to the expression ' either . . . or.' In the interests of the Principle 
 of Non- Ambiguity, it is essential tliat we should know what we mean 
 when we use the expression, so far as such knowledge is relevant to 
 the logical purpose we have in using the disjunctive form. But, 
 in ordinary cases, the only meaning which it is relevant to give to 
 ' cither ... or ' is that of ' not neither.' This meaning satisfies 
 the requirement of exhaustiveness, and it enables us to waive alto- 
 gether the issue as to whether ' either . . . or ' is to imply ' not 
 both ' or ' it may be both.' 
 
 Let us consider the types of inference that we are naturally 
 inclined to draw from the disjunctive proposition ' Either P or Q.' 
 They are two in number : (1) The hypothetical i)roposition — ' If 
 not P, then Q '; and (2) the hypothetical proposition — ' If P, then 
 not Q.' The first of these is the inference from the exhaustiveness 
 of the disjunction ; the second is the inference from the mutual 
 exclusiveness of the alternatives. If there are more than two alter- 
 natives, as in the proposition ' Either P or Q or R,' the two infer- 
 ences will take the forms : (1) If not P, then either Q or R ; and (2) if 
 P, then neither Q nor R. 
 
 Now, in so far as it is the first of these two inferences that alone 
 concerns our logical purjDose in the framing of a disjunction, the 
 question as to whether we are to side with the exclusivists or tlie 
 non-exclusivists is wholly irrelevant. We are ' exhaustivists ' — 
 that is enough. ' Either ... or ' means for us ' not neither,' and 
 nothing further. 
 
 In an able article in Mind, October, 1903, Mr. G. R. T. Ross has 
 endeavoured to show ' that the function of the disjunctive judg- 
 ment both in science and in practical reasonings is to be exhaustive 
 and not necessarily exclusive.' The contention appears, as a primary 
 contention, at any rate, to be essentially sound. When we develop 
 an indeterminate categorical basis disjunctively, our primary aim 
 is attained if we can substitute for the comparatively indeterminate 
 categorical statement a list of specific possibiUtics wliicli collectively 
 cover precisely the same range as the vaguer generality which they 
 disjunctively specify. For, since the same range is covered in both 
 cases, the choice between a number of specific possibilities is a fresh 
 advantage that the disjunctive oft'ers over and above those oft'cred 
 by the indeterminate categorical. This particularization without 
 loss of scope is in itself something to the good. But once we Imve 
 the specific possibilities exhaustively before us, the question as to 
 whether they mutually exclude each other may remain a matter of 
 complete indifference to us. 
 
 Let us suppose that the assertion that X is an objectionable person 
 is adequately represented by the disjunctive statement that ' Either 
 X is a knave or he is a fool.' Without inquiring wliether X's 
 knavery is or is not compatible with his fooUslmess, we can infer at
 
 13i THE PROBLEM OF LOGIC [IV. iv. 
 
 once, from this statement, that if X is not a knave, he must be a 
 fool, and that if he is not a fool, he must be a knave, and thereby 
 emphasize in a fresh and specific way our original contention that X 
 is an objectionable character. The inference that ' If X is a knave, 
 he is not a fool ' would be beside the point. It would not emphasize 
 his objectionableness, but qualify and limit it. 
 
 Suppose, apfain, tliat X is in a perilous situation at the edge of a 
 chasm, and that the categorical requirement of immediate action 
 is presented in the disjunctive form : ' Either jump or starve.' 
 Here the essential inference is that ' If X does not jump, he will 
 starve,' for it is this consideration that supphes the incentive to 
 action. Little can be gained by inferring that ' If X does jump, 
 he wiU not starve,' and that ' If X starves, he will not have 
 jumped.' 
 
 I quote from the article by Mr. Ross the following illustrations 
 in further suj^port of the contention that, under ordinary cir- 
 cumstances, an exhaustivist need not pledge himself either to the 
 cxclusivist or to the non-exclusivist theory. 
 
 L ' Planetary orbits* fall either wholly inside or wholly outside 
 the Earth's orbit.' 
 
 Here the disjunction admits of being read exclusively. Vv'e can 
 infer, if we hke, that ' Jupiter's orbit, lying v.ithout that of the 
 Earth, cannot lie wholly nearer to the sun than it.' But this is 
 futile. The real force of the disjunction lies, as Mr. Ross points out, 
 in its cxhaustiveness, in the implied denial that there are any planets 
 with orbits intersecting that of the Earth. It lies in the possibility 
 of the inference that if a planetary orbit does not fall wholly inside 
 the Earth's orbit, then it must lie wholly outside it. 
 
 2. ' Planets whose orbits lie between the Earth and the Sun are, 
 when visible, to be seen either in the morning or in the evening.' 
 
 Here we have a statement which we may very profitably utilize 
 without possessing any proof that Venus, for instance, when visible 
 in the evenings, must in the morning rise after the Sun, and so be 
 lost in his light. The force of the disjunction Hes, again, in its cx- 
 haustiveness, in ,^the certainty which it professes to give us that 
 planets of the kind specified are never to be seen during the middle 
 of the night. It assures us that if we see a starry object of peculiar 
 briUiance at midnight, though it may be Jupiter, it cannot be 
 Venus. 
 
 We have so far been considering the Disjunctive Proposition from 
 a point of view that compels no reference to the Exclusivist contro- 
 versy. But a point is necessarily reached where a decision on this 
 question becomes imperative. The point is reached so soon as the 
 distinction of the alternatives from each other becomes a matter of 
 logical interest. It may become important to decide whether an 
 
 * In this statement the Earth is not included among the planets.
 
 Chap. XIV.J THE DISJUNCTIVE PROPOSITION 135 
 
 objectionable person is a pure knave or a pure fool, the knavery 
 being unadulterated with fooHshness, and vice versa, or whether he 
 is at once a knave and a fool. In that case, when we make the state- 
 ment that ' Either X is a knave or he is a fool,' we are directly in- 
 terested in knowing whether the alternatives are intended or are not 
 intended to be mutually exclusive. 
 
 We must distinguish here between two quite different interests — 
 the interest in non-ambiguity and the interest in scientific precision. 
 
 1. The statement in question may not be our own, but may bo 
 made by some other person, and we may be interested in understand- 
 ing what that same person precisely means by his statement. In 
 this case, however, it does not lie with us to decide on the sense to 
 be put upon the ' either . . . or.' We must leave that to the 
 
 * other person.' The most we can do in defence of logical interests 
 is to insist that he shall make his meaning exphcit, and that a 
 rational convention shall be reached wiiich, in respect of the asser- 
 tion in question, shall secure complete freedom from ambiguity. 
 Thus, we might suggest that, if the exclusive reading be intended, 
 the statement should take the form ' Either he is a non-foolish knave 
 or a non-knavish fool,' or perhaps, more simply, ' Either he is a fool 
 or else he is a knave ' ; the ordinary form, ' Either he is a knave or he 
 is a fool ' being reserved for the non-exclusive reading. 
 
 If our interlocutor is stating his disjunctive proposition in sym- 
 bolic form, it seems simpler to reserve the ordinary form ' Either 
 P or Q ' for the non-exclusive reading. The attempt to expand tliis 
 into ' Either P or Q or PQ ' (the three alternatives being taken as 
 mutually exclusive) results in a logical disaster or else in a confusion 
 of symbols. For ' P ' here means ' P and not Q,' and ' Q ' means 
 
 * Q and not P.' Hence PQ should mean ' P and not Q combined 
 with Q and not P,' which leaves httle distance between PQ and logical 
 nonentity. In symbolic language, then, ' Either P or Q ' represents 
 most suitably the non-exclusive reading, and ' Either FQ or QP ' — 
 or, more simply, ' Either P or else Q ' — the exclusive reading. The 
 form ' Either P only or Q only ' is obviously unsatisfactory as an in- 
 terpretation of the exclusive meaning, for ' P only ' impHes a much 
 greater restriction than ' P but not Q.' There is, indeed, another 
 alternative. The exclusive form may be uniformly adopted. In 
 this case the proposition ' Either P or Q ' would be transformed 
 into ' Either PQ or QP or PQ ' when ' Either ... or ' implies ' it 
 may be both,' and into ' Either PQ or QP,' when ' Either ... or ' 
 implies ' not both.' But it is very doubtful whether tliis gain in 
 uniformity would sufliciently compensate for the loss of the simplo 
 form ' Either P or Q.' 
 
 2. We come now to the second of the two interests in connexion 
 with which a decision on the Exclusivist question becomes essential. 
 The distinction of alternative possibilities from each other may have
 
 136 THE PROBLEM OF LOGIC [IV. iv. 
 
 a real value for Science, and the interest in exclusiveness may ally 
 itself with the interest in a logical ideal of disjunction. Assuming 
 that this distinction of alternative possibilities is desirable — as it is, 
 for instance, in the classification of species under a genus — we have 
 to ask what the logical ideal of disjunction demands of the disjunc- 
 tive proposition in view of tliis requirement. 
 
 The natural answer would seem to speak wholly in favour of the 
 Exclusivists. For surely it is only when the disjunctive proposition 
 is read exclusively that we obtain tliis desired distinctness between 
 the various alternatives. 
 
 But in this view tliere is involved an assumption wliich we must 
 try to make clear. It is this : We are assuming that the way in 
 wliich the disjunctive proposition can always best further the 
 logical idea of precision is by being itself ideally precise. But this 
 is a fundamental misconception. In the service of logical ideals 
 the various forms of judgment must co-operate. In some cases 
 the ideal — that of precise characterization, for instance — will be 
 reached by means of a categorical proposition. As an example of 
 this, we may cite the record of some delicate scientific observation. 
 In other cases, as in the development of a supposition into its con- 
 sequences, it is the hj^pothetical projDosition that must embody the 
 logical ideal — in this case that of necessary connexion between the 
 parts of the proposition. In other circumstances, again, as in scien- 
 tific Classification, the ideal, that of the mutual exclusiveness of a 
 number of co-ordinate possibiUties, requires for its embodiment a 
 disjunctive proposition. 
 
 Now, where the function of embodying the logical ideal falls to 
 the lot of the disjunctive proposition, there can be no doubt that 
 the alternative iDossibilities wliich it enumerates must be mutually 
 exclusive as well as exhaustive. Here the junction of the disjunctive 
 logically requires that its form shall be perfect. Hence, when scien- 
 tific results are tabulated in disjunctive form, the disjunctive should 
 be of the exclusive tjqoe, or should approximate as closely as possible 
 to that type. In Mathematics these perfect disjunctions are always 
 obtainable, but in the more concrete sciences this is not the case. 
 In that classification of species in which the species are defined ' hy 
 tj'pe ' {vide p. 62) the boundaries between two co-ordinate species 
 will often be somewliat uncertain, though the various types, as 
 centrally defined, are distinct and mutually exclusive. 
 
 But where the function of embodying the logical ideal falls, shall 
 we say, to the lot of the categorical proposition, and the disjunc- 
 tive proposition fulfils its service as a mere means towards ensuring 
 to the categorical the maximum precision of statement, that service 
 may frequently be best fulfilled when the disjunctive is non- 
 exclusive. The ideal functioning of the disjunctive requires here 
 the imperfection of its form. Suppose that, starting from a given 
 categorical basis, we are able to state our alternatives exhaustively
 
 Chai'. XIV.] THE DISJUNCTIVE PROPOSITION 137 
 
 in the form ' Either P or Q or R.' A methodical scientific inquiry 
 may cancel P as a possibility wliich in the given circumstances 
 cannot be actualized, and we are left with the conclusion ' Eitlier 
 Q or R.' The possibihty Q, we suppose, is similarly cancelled, and 
 we are left with the categorical assertion ' R.' In a case of this kind 
 the imperfect, non-exclusive t3rpe of disjunction is the most service- 
 able, and for that reason the most ideal. Where our aim is to reach 
 the truth of one alternative by the elimination of the others, it is 
 otiose to insist on the various alternatives being ab initio mutually 
 exclusive. The successive canceUing of P and Q is a process that is 
 quite independent of the question of exclusiveness or non-exclusive- 
 ness. Suppose that the disjunction is given in exclusive form. We 
 are then told that either PQR is true, or QPR is true, or RPQ is true. 
 P and Q are cancelled, and we are left with RPQ. But this is really 
 the same result as that which we reached abov^e, when we started 
 with the imperfect, non-exclusive disjunctive ' Either P or Q or R.' 
 For the result R there obtained might equally well have been 
 expressed ' RPQ,' since it was certainly exclusive of P and Q. 
 
 The inconvenience of refusing to utilize a disjunctive proposition 
 at all except when expressed in its most perfect form may best be 
 brought out by a comparison which is much more than a mere 
 analogy. It would be like refusing to use a hypothetical proposition 
 until it had been rendered so precise in both its parts that not only 
 should the affirmation of the antecedent involve that of the conse- 
 quent, but also the affirmation of the consequent should involve 
 that of the antecedent. 
 
 We conclude, then, that the disjunctive, in the service of the 
 categorical, may profitably be left in non-exclusive form, and that 
 this imperfect, non-exclusive form is the form required in the in- 
 terests of Scientific Explanation. Formal Classification, on the other 
 hand, and the laying out of alternative laossibilities in the mathe- 
 matical sciences, require the service of the disjunctive judgment in 
 its perfect and exclusive form. 
 
 A disjunctive proposition, then, may be defined as a statement to 
 the effect that, of a closed number of alternative possibilities, one 
 is taken to be actuahzed. The one obligatory rule of disjunction — 
 Rule I. — is that the alternatives shall exhaust the possibihties. 
 Where it falls to the lot of the disjunctive to uphold the ideal of 
 scientific precision, we have further to observe Rule II., that the 
 alternatives shall be reciprocally exclusive. In mathematical 
 inquiry, where scientific precision is as imperative at the beginning 
 of the inquiry as it is at its close, both rules must necessarily hold 
 good for all disjunctive propositions. 
 
 Example. — Criticize the following disjunction : 
 
 ' Either triangles are equilateral, or they are isosceles, or they 
 are rii^ht-a igled.'
 
 138 THE PROBLEM OF LOGIC [IV. v. 
 
 Tliis breaks Rule II., for an isosceles triangle may be right-angled. 
 Whether it can be equilateral depends on the precise definition of 
 the isosceles triangle. 
 
 It also breaks Rule I. There are triangles which are neither 
 equilateral, isosceles, nor right-angled — namely, the scalene tri- 
 angles which are not right-angled. 
 
 CHAPTER XV. 
 
 IV. (v.) THE HYPOTHETICAL PROPOSITION. 
 
 The Categorical Proposition is a proposition which purports to 
 be a statement of fact. The fact to which the statement refers 
 need not be a ' real ' fact, or fact of Nature, a fact under Causal 
 Law. It may be a ' formal ' fact, a fact that can have actuaUty 
 only in reference to a certain limited universe of discourse. 
 
 The Hypothetical Proi)osition, on the other hand, may be defined 
 as the statement of a connexion between two possibihties. It 
 contains two clauses, of which the first is called the Antecedent, 
 and the second the Consequent. The Disjunctive Proposition also, 
 as we have seen, states a connexion between possibilities, but in 
 that case the possibihties are regarded as alternative possibihties. 
 
 The connexion between the Hj^Dothetical Proposition and the 
 Disjunctive is, from the point of view of the logical development of 
 thought, a very close one. The Hypothetical Proposition, as we 
 have already seen {vide p. 113), takes one of the possibihties wliich 
 the Disjunctive Proposition specifies, and develops by connecting 
 it with another possibihty. Thus the hypothetical selects for its 
 antecedent one of a number of possibihties cUsjunctivel}' presented. 
 The consequent also may be regarded as having been drawTi from 
 the alternatives of another disjunctive series. Thus both the 
 possibihties with wliich the Hypothetical Proposition is concerned 
 are of the disjunctive type, and therefore real, and not modal, 
 possibihties. 
 
 Suppose that we have before us the disjunctive proposition 
 ' Either a triangle is obtuse-angled, or it is right-angled, or it has 
 three acute angles.' Selecting one of these alternative possi- 
 bihties, we say ' If a triangle is right-angled, a semicircle may be 
 circumscribed to it having its hypotenuse as diameter.' 
 
 As the Hypothetical Proposition is concerned with possibihties, 
 and the Categorical with actuahties, or with what purport to be 
 actuahties, whether formal or real, it is logically impossible to 
 express a hypothetical proposition as a genuine categorical, though 
 it may be equivalently expressed in categorical form. Consider
 
 Chap. XV.] THE HYPOTHETICAL PROPOSITION 139 
 
 the statement : ' If anyone trespasses on this property, he will be 
 prosecuted.' We may transvcrt this into categorical form, but we 
 cannot give it a categorical meaning. We may state it in the 
 form : ' AU trespassers will be prosecuted.' But this is not a propo- 
 sition wliicli purports to be a statement of fact. It docs not imply 
 that anyone has trespassed in the past, nor even, indeed, that 
 anyone will do so in the future ; it does not imply that any 
 prosecution has taken place, or is certain to take place. On 
 the other hand, the statement : ' All those trespassers are being 
 prosecuted ' is a categorical proposition. It claims to state an 
 actual fact, and it cannot be equivalently expressed as a genuine 
 h3rpothetical. 
 
 We conclude, then, that, since it is a fundamental requisite of 
 logical tliinking to be guided by meaning and not by form, and 
 since the meaning of a hypothetical statement is entirely different 
 from the meaning of a categorical statement, the two types of 
 statement are irreducible the one to the other. Further, since 
 hypothetical propositions are frequently given in categorical form, 
 and categorical propositions in hypothetical form, it is essential, 
 when reducing a proposition to strict logical shape, to look to the 
 meaning and adjust the form accordingly. 
 
 Consider, for instance, the proposition : 
 
 ' If air is liquefied, it is in that state dangerous to handle.' 
 
 Tliis is a pseudo-hypothetical. Its meaning is precisely equiva- 
 lent to that of the genuine categorical ' Liquefied air is dangerous 
 to handle.' On the other hand, the proposition ' If this fiuid is 
 liquefied air, it is dangerous to handle ' is a true hypothetical. It 
 cannot be reduced to a genuinely categorical equivalent. 
 
 In speaking of the Hypothetical Proposition as a connexion of 
 possibihties we may be using the word ' connexion ' in one or the 
 other of two senses : either (1) as indicating an assertorial — i.e., an 
 assertorially intended connexion ; or (2) as indicating an apodeictic 
 — i.e., an apodeicticaUy intended connexion. The connexion is 
 assertorial so far as it is merely asserted and notliing is imphed 
 with regard to its nature. It is apodeictic when it is intended to 
 be a logically necessary connexion, a connexion that cannot be 
 denied without either rendering Nature unintelligible or our oa^ti 
 thought inconsistent. 
 
 As instances of assertorial connexion we may cite the following 
 propositions : 
 
 ' If you go out- wdthout an umbrella to-day, you will come 
 
 home wet.' 
 * If sun-spots are numerous, magnetic storms on tliis planet 
 
 will be correspondingly numerous.'
 
 UO THE PROBLEM OF LOGIC [IV. v. 
 
 The following are instances of apodeictic connexion : 
 
 1. ' If there is a way up, (it logically follows that) there must 
 
 be a way down.' 
 
 2. ' If all men are mortals, (it logically follows that) some 
 
 mortals are men.' 
 
 3. ' If all men are mortals, (it logically follows that) all mortals 
 
 are men.' 
 
 4. ' If you accept the statement that all persons are selfish, 
 
 vou are logically compelled to admit that you are yourself 
 selfish.' 
 
 In the first and second of these last examples we see that denial 
 of the validity of the connexion would render Nature unintelligible. 
 In the third example the apodeictic intention has miscarried. In 
 the fourth, denial of the validity of the connexion would render 
 thought inconsistent. 
 
 When the connexion of possibihties depends on the human will 
 as a synthetic i)rinciple, and the reference to Reality is still 
 restrictedly conceived as either ' formal ' or ' real,' the free con- 
 nexion which the statement imphes must be interpreted as equiva- 
 lent to a material connexion in this essential respect — that it can be 
 justified through observation or scientific investigation. So inter- 
 preted, the statement ranks as an assertorial hypothetical. Let us 
 from this point of view consider the proposition : ' If anj'one 
 trespasses on tliis property, he will be prosecuted.' Here the 
 prosecution must not be supposed to depend on the caprice of the 
 ouTier. It must dejjend on such facts as the apparatus of the law 
 re. trespassing and the policing of the property, which can be 
 investigated as actual facts can be investigated. Otherwise the 
 validity of the proposition does not admit of being scientifically 
 tested. 
 
 We have, then, two main types of Hypothetical proposition — the 
 Assertorial (formal or real) and the Apodeictic. There are also 
 two main varieties of the Apodeictic, respectively regulated by the 
 Law of Xon-Contradiction and the Law of Excluded IMiddle. The 
 first of these two varieties may be called the Formal Apodeictic 
 H\T)othetical, the second the IMaterial Apodeictic Hypothetical. 
 The three tj-pes of Hypothetical Proposition may suitably be symbo- 
 lized as follows : 
 
 Assertorial Hypothetical. If P, then ascertainably* Q. 
 
 rif P is accepted, then by implica- 
 
 A 1 • i.- TT XI- i.- 1 tion Q must be accepted (Formal). 
 
 Apodeictic Hypothetical. ^ Tr r) • ^ ^^i k ■ v *.- 
 
 '■ •'^ If P IS true, then by implication 
 
 i 
 
 Q is true (Material). 
 
 * The ascertainability of Q is, of course, only a claim made by the judger 
 and the form of verification may vary with the Universe of Discourse.
 
 Chap. XV.] THE HYPOTHETICAL PROPOSITION 141 
 
 Illustrations : 
 
 If water is heated at standard pressure, it will boil at 100^ C. 
 
 (real assertorial). 
 If Black mates White in three moves, Black will have won the 
 
 tournament (formal assertorial). 
 If the moon is made of green cheese, (it logically follows that) 
 
 it must at least be subject to the law of gravitation 
 
 (material apodeictic). 
 If the moon is made of green cheese, (it logically follows that) 
 
 some heavenly bodies are not incorruptible (material 
 
 apodeictic). 
 If the statement that the moon is made of green cheese is 
 
 accepted, (it logically follows that) we must accept the 
 
 further statement that all bodies not made of green cheese 
 
 are-not the moon (Formal apodeictic). 
 If the statements that all green cheese is corruptible and that 
 
 the moon is made of green cheese are accepted, (it is 
 
 logically necessary that) the further statement that some 
 
 heavenly bodies are not incorruptible must also be accepted 
 
 (Formal apodeictic). 
 
 Should the Hypothetical be presented in the general form ' If P, 
 then Q,' or in the form ' If P is true, then Q is true,' the more specific 
 nature of the given hjrpothetical will depend on the meaning of 
 the connecting- word ' then.' If ' then ' means ' in that case, as a 
 matter of ascertainable fact,' the proposition is assertorial ; if 
 ' then ' means ' it follows with logical necessity tliat,' the proposi- 
 tion is an apodeictic hypothetical of one kind or the other. 
 
 With this distinction between assertorial and apodeictic hypo- 
 theticals before us, we are able to give more precise expression to 
 the distinction between the categorical and hypothetical forms of 
 statement. A categorical proposition must admit of being pre- 
 sented, whether in a formal or in a real sense, as true or false. The 
 apodeictic hypothetical can only be valid or invahd. It can be 
 * vaUdated ' as a sound inference, or ' invahdated ' as an unsomid 
 or illegitimate inference. An assertorial hypothetical, on the other 
 hand, may either be justified as a correct prediction or be dis- 
 credited as an incorrect prediction. Moreover, it is only categorical 
 statements that can be verified. When we verify the truth of the 
 alleged connexion between antecedent and consequent, we are 
 verifying the categorical proposition ' Q is connected ■with P.' 
 Similarly, as we shall see, when we come to Induction, to verify 
 a hypothesis is to verify not a hypothetical proposition, but its 
 (categorical) antecedent. 
 
 The logical connexion of possibilities asserted by the Apodeictic 
 Hypothetical, when not invalid, and therefore, in first or last resort, 
 meaningless, may be cither conditionally or unconditionally vahd
 
 142 THE PROBLEM OF LOGIC [IV. v. 
 
 It is unconditionally valid wlien the antecedent cannot be affirmed 
 ■vnthout necessarily implying the consequent. Thus the x^ropo- 
 sitions : 
 
 ' If all men are mortal, then some mortals are men ' (material), 
 and 
 
 ' If it be granted tliat all men are mortal, then it must be 
 granted that some mortals are men ' (Formal) 
 
 are unconditionally vahd hypotheticals. But the propositions : 
 
 ' If all men are mortal, then Socrates is mortal ' (material), 
 and 
 
 ' If it be granted that all men are mortal, then it must be 
 granted that Socrates is mortal ' (Formal) 
 
 are conditionally vaHd hj^jotheticals. They are conditionally 
 valid because in each case the consequent can be inferred from the 
 antecedent only in virtue of a further assumption — namely, the 
 assumption that Socrates is a man, or is granted to be a man. The 
 strictly logical forms of these proj)ositions would be : 
 
 ' If aU men are mortal, and Socrates is a man, then Socrates 
 
 is mortal ' (material), 
 and 
 
 ' If it be granted that all men are mortal, and that Socrates is 
 
 a man, then it must also be granted that Socrates is mortal ' 
 
 (Formal) 
 
 All hjqoothetical validity-connexions, in fact, when expressed 
 in strict logical form, are of the unconditionally vahd type. The 
 hypothetical assertion cannot justify itself except by, as it were, 
 elongating into an explicit logical inference. We have here the 
 point of departure for the study of Inference.
 
 V. 
 
 THE FORMAL TREATMENT OF THE LOGICAL 
 
 PROPOSITION. 
 
 (i.) Transition to the Formal Treatment of Logic (ch. xvi.)- 
 
 (ii.) The Formal Import of the Categorical Proposition (ch. xvii.). 
 
 (iii.) The Reduction of Categorical Propositions to strict logical form(ch. sviil.). 
 
 (iv.) The Opposition of Propositions (ch. xix.).
 
 CHAPTER XVI. 
 
 V. (i.) TRANSITION TO THE FORMAL TREATMENT OF LOGIC. 
 
 The essential significance of the connexion between Inference and 
 the apodeictic hypothetical form of statement is concentrated in 
 the meaning of the word ' if.' Whether we say ' If P is true, then 
 Q is true,' or ' If P is accepted, then Q must be accepted,' ' if ' 
 means no more than ' assuming that.' If the word is understood 
 as equivalent to ' given that,' then by ' given ' we must mean 
 * given as an assumption.' ' If ' therefore plays the important 
 part of introducing truth and falsity relations in a purely hypo- 
 thetical way. Wlien a person states an inference in the form ' If 
 P, then Q,' no call is made on him to substantiate the truth of P. 
 He is not expected even to have an opinion as to the truth or un- 
 truth of P. When respectively introduced in the form of ante- 
 cedent and consequent of a hypothetical proposition, the premisses 
 and conclusion of an argument are shielded from all direct contact 
 with the world of Knowledge and of Reality. The shelter of the 
 word ' if ' completely protects them from the fortunes and per- 
 plexities of the truth-interest. 
 
 Now it is precisely in tliis abstract aspect that the problem of 
 Inference can best be studied. In order to understand the object 
 of our study we must isolate it, after tlie fashion of all successful 
 experimentation ; and in the present instance the hypothetical 
 form of statement is the isolating apparatus. This provisional 
 isolation from the larger interests of Truth has very great advan- 
 tages. It enables us, more particularly, to present our object in 
 such artificially devised forms as will best serve our purpose, and 
 thus makes possible a degree of accuracy and precision wliieh would 
 not be possible were the object studied in the vast, uncontrolled 
 context of its natural surroundings. 
 
 We propose, then, to study Inference as Inference, to substitute 
 a pure validity-interest for the more comprehensive truth-interest, 
 and — as an important provisional step — to adapt the statement- 
 import of the proposition in such wise as to facilitate still further 
 our study of the conditions of valid Inference. Moreover, of the 
 two t\iies of apodeictic h^'pothetical we take as our model the 
 Formal type, of which the purport is to uphold the necessity for 
 consistent statement in the name of the Formal principles of Identity 
 
 145 10
 
 UQ THE PROBLEM OF LOGIC [V. i. 
 
 and Xon-contradiction. Adopting, as we propose to do, the 
 statement-import of the Proposition, the material type ' If P is true, 
 Q is imphcitly true,' with its ultimate appeal to the Law of Material 
 Identity and the Law of Excluded Middle {vide p. 99), would be 
 a wholly inconfrruous model. 
 
 The study of the Inference problem under the restrictions above 
 set forth is the central, and perhaps the exclusive, topic of a ' Formal' 
 Logic. Where Proposition and Inference are studied not only as 
 more or less complicated statements of meaning, but also on the 
 basis of an artificial modification of statement-import, the treat- 
 ment is at once abstract and conventional, and therefore Formal 
 in a very genuine sense of the word ; and what we have here called 
 a Formal treatment of Logic is usually more briefly referred to as 
 Formal Logic. It is with this Formal aspect of Logic that we shall 
 now be exclusively concerned until we pass from tlie problem of 
 Liference to the wider and deeper problem of Scientific Explanation. 
 
 CHAPTER XVII. 
 
 V. (ii.) THE FORMAL DIPORT OF THE CATEGORICAL PROPOSITION. 
 
 The import, or structural significance, of a proposition depends on 
 the character of the identity relation which it affirms. Identity 
 determines Import : as the identity is conceived, so will the import 
 be conceived. Where this identity is conceived in the form which, 
 being the simplest and most manageable, is therefore, regarded from 
 the point of view of the Formal treatment of Logic, the most rele- 
 vant, we have wliat is known as the Formal import of a proposition. 
 
 The most simple interpretation of the meaning of identity is 
 that of numerical coincidence. When I say ' All lions are carni- 
 vores,' I am considering Uons and carnivores alike simply as count- 
 able objects. Qua countable objects they are identical, and the 
 difference is a mere difference of quantity : the hons as countable 
 objects are numerically coincident with the same number of carni- 
 vores. Each of all the lions is a carnivore. 
 
 The form of propositional import which interprets the Propo- 
 sition in this simple way, and reads both subject and jiredicate 
 terms in extension, may suitably be called the Extensive Import 
 of a proposition. The extension of a term, as we have seen, is not 
 to be confused with its conno-denotation or intension, the joint 
 product of those processes of Definition and Di\asion through 
 which the meaning of a term is developed. The extension includes 
 the objects to wliicli the meaning of the term applies. But in 
 r'Jirring to the extension of a term we need not refer to the whole
 
 Chap. XVI I.] THE FORMAL CATEGORICAL 147 
 
 of it : the reference may be either partial or totah Hence a term 
 read ' extensivelj'' ' is a term considered from the point of view of 
 its extent, but dependent on a defining quantity-mark (or its logical 
 equivalent) to make clear whether this reference to extension is 
 partial or total. 
 
 The ' extensive ' interpretation or reading of a proposition is 
 customarily called its ' denotative ' reading. But this usage 
 involves a confusion between extension and denotation, and must 
 carefully be avoided by those who hold, as we do, that tlie meanings 
 of these words should be differentiated. The proposition ' All men 
 are mortal,' when taken as equivalent to the true extensive meaning, 
 ' The objects indicated by the term " man " are numerically co- 
 incident with objects indicated by the term " mortal," ' is commonly 
 said to have its terms read in denotation, and the word ' denoted ' 
 is substituted for ' indicated.' The same misuse (as we hold it to 
 be) of the word ' denoted ' occurs in the enunciation of what is 
 called the predicative view of Import. Here the subject-term is 
 said to be read in denotation, the predicate-term in connotation, and 
 the proposition '^^AU men are mortal ' is then rendered as follows : 
 'The objects denoted by "man" possess the attributes connoted 
 by " mortal." ' We should prefer to say that in this case the 
 subject-term is read in extension and the predicate-term in 
 intension.* 
 
 We are not, however, directly concerned Avith the various ways 
 of interpreting a proposition, but rather with explaining and 
 developing that one form of propositional import which we have 
 selected as most adequately meeting thft requirements of a Formal 
 logical treatment. 
 
 The extensive proposition is most conveniently stated in one of 
 four ultimate forms, which are traditionally known as the A, E, I, 
 and propositions. The scheme is as follows : 
 
 fUniversal Affirmative : A : 'All S's are P's,'^ 
 
 symbolized by SaP. | , ^gjrmo ) 
 
 Particular AfKirmative : I : ' Some S's are P's,' j ^' 
 symbohzed by SiP. J 
 
 fUniversal Negative : E : 'All S's are-not P's,' \ 
 
 symbolized by SeP. \ ,p^Qs 
 
 Particular Negative : : ' Some S's are-not P's,' ' ^^^^'^^ 
 symbolized by SoP. 
 
 1 
 
 * A third variety in the rendering of a proposition may here conveniently be 
 noticed. It is called the attributive view. On this view both terms are read in 
 connotation. Thus, the proposition ' All men are mortal ' would run as follows : 
 ' The attributes connoted by the term " man " are accompanied by the attributes 
 connoted by the term "mortal";' or else 'The attributes connoted by the 
 term " man " are accompanied by the attribute " mortal." ' 
 
 t A distinction is commonly drawn between the generic imiversal judgment 
 and the ' general ' or enumerative universal judgment. The A and tlie E pro- 
 
 10—2
 
 us THE PROBLEM OF LOGIC [V. ii. 
 
 lUusfralions 
 
 A : All dukes are members of the House of Lords. 
 I : Some lords arc members of the House of Commons. 
 E : All dukes are-not members of the House of Commons. 
 : Some lords are-not members of the House of Commons. 
 
 The fact that according to this scheme there are four proposi- 
 tions, and only four, follows from the nature of a proposition as 
 Formally conceived. A proposition must be either affirmative or 
 not ; and in either case its subject-term must either be used in its 
 whole extent or not so used.* Moreover, the choice of the two 
 fundamenta of Quality and Quantity, ujion which this fourfold 
 division depends, is necessitated by the adoption of the extensive 
 view of Imjiort. The adoption of this specific view makes it im- 
 perative to accept the basis of Quantity or extensive reference as 
 one fundamcntum ; and the fact that the distinction in quality 
 is a distinction which is logically prior to any specification of 
 identity-import whatsoever obliges us to include the quality-basis 
 as a second fundamcntum. 
 
 On the Distribution of Terms in a Proposition. 
 
 In a previous paragraph we pointed out the fact that our refer- 
 ence to the extent of a term may be either total or partial, or, to use 
 the more technical language of Formal Logic, either universal or 
 particular. We have now to give to this distinction the preciser 
 :*orm required by the abstract validity-interest wliich is at present 
 dominating our whole inquiry. From tliis restricted standpoint 
 our sole concern is with unambiguous statements of meaning, and 
 inquiry into the implications of these same statements. It is with 
 the aim of clearly expressing this limitation of standpoint that, 
 when speaking of the extension of terms, we shall use the words 
 ' distributed ' and ' undistributed,' as defined below, in the place 
 of the words ' universal ' and ' particular.' 
 
 A term is said to be distributed (within a proposition) when it is 
 used in its whole extent — that is, when there is either an explicitly 
 stated or a logically implied reference to all the individuals con- 
 tained in the class for wliich it stands. 
 
 A term is said to be undistributed when it is not used in its whole 
 extent — that is, when no reference to the whole extent of the class 
 is either explicitly stated or logically imphed. 
 
 Let us consider A, E, I, from tliis point of view. 
 
 j,jT..>ii3 are, in our vn;.;, ouuiuerative universals. The difference between 
 ' generic ' and ' enumerative ' corresponds, in fact, to the distinction between 
 ' intensive ' and ' extensive.' 
 
 ♦ For further substantiation, see the discussion on the Quantified Predicate, 
 pp. 159-161.
 
 CiiAP. XVII.] THE FORMAL CATEGORICAL 149 
 
 i. Distribution of S : 
 
 A, I, and present no difficulty. S is distributed in A, 
 undistributed in I and 0. 
 
 With regard to E a difficulty may present itself, arising from the 
 familiar, though misleading, form ' No S is P.' In objection to the 
 statement that S is distributed in the E proposition it may be 
 urged tliat, since the proposition states that no S's are identical 
 with P's, therefore no individuals belonging to the class S are being 
 referred to. Now, it is quite true that no such individual objects 
 are being referred to as being identical with P's. But ' No S is P ' 
 is not an affirmative projiosition with a subject of zero extent. 
 It is a negative proposition. It is therefore equivalent to ' All the 
 S's are-not P's.' Thus E, the universal negative proposition, 
 distributes its subject. It is obvious that, so long as the E propo- 
 sition is consistently presented in the stricter form ' All S's are-not 
 P's,' the misapprehension cannot arise. 
 
 ii. Distribution of P : 
 
 A. Does the statement ' All S's are P's ' imply the statement 
 'AU S's are a// P's'?* 
 
 Obviously it does not. For the statement that all the S's 
 are severally identical with the same number of P's does 
 not imply that the two classes S and P are coextensive, 
 coinciding point for point. What the given statement 
 necessitates is simply the identification of eacli of all the 
 S's with each of some of the P's. It is equivalent to the 
 statement that ' All S's are some P's.' 
 Therefore P is undistributed. 
 
 E. Does the statement ' All S's are-not P's ' necessarily imply 
 the statement ' All S's are-not any P's '? It must be so. 
 If it did not imply this, its acceptance would not preclude 
 the acceptance of the statement ' Some P's are S's,' 
 wliich, if accepted, would necessitate the acceptance of 
 ' Some S's are P's.' But if ' All S's are-not P's ' does not 
 preclude the acceptance of ' Some S's are P's ' it is 
 meaningless.! 
 Therefore P is distributed. 
 
 I. Does the statement ' Some S's are P's ' imply the statement 
 ' Some S's are all P's ' ? 
 
 No, certainly not. The assertion ' Some cats are black objects ' 
 does not imply tliat, when each of the ' some cats ' re- 
 ferred to has been identified with a black object, there are 
 no black objects remaining over. It docs not imply that 
 the ' some cats ' arc nunierous enough to account for all 
 the individuals contained in the class Black object 
 * On the quantified predicate, see p. 159. t ^Z- PP- 98, 104.
 
 150 THE PROBLEM OF LOGIC [V. ii. 
 
 Therefore P is undistributed. 
 
 0. Does the statement ' Some S's are-not P's ' imply the 
 statement ' Some S's are-not any P's ' ? 
 
 Yes, for if not, its acceptance would not preclude the accept- 
 ance of the statement that Some P's are identical with 
 tliese same ' some S's ' of which we are stating that they 
 are not identical with. P's. But if we were to state that 
 P's are identical with them, we should also be stating that 
 they are identical with P's, which is contrary to the given 
 statement. Now an accepted statement which docs not 
 preclude the acceptance of a statement inconsistent with 
 itself is self-contradictory, and therefore meaningless. 
 Therefore P is distributed. 
 
 From the consideration of E and we see that negative propo- 
 sitions, as such, must have their predicates distributed. 
 
 We have, then, the four following rules for the distribution of 
 terms : 
 
 L All universal propositions distribute their subjects. 
 
 2. Xo particular propositions distribute their subjects. 
 
 3. All negative propositions distribute their predicates. 
 
 4. No affirmative propositions distribute their predicates. 
 
 Rule No. 3 in particular is worth remembering. 
 
 An A proposition, then, is technically defined as a logical propo- 
 sition having its subject distributed and its predicate undistributed ; 
 and E, I, and are similarly definable. These definitions are funda- 
 mental in a Formal treatment of Logic, and it is important, when 
 thinking of these propositions, to think of them in this way. 
 
 The Diagrammatic Representation of A, E, I, 0. 
 
 ' Diagrams,' as Professor Welton says, ' are intended to make 
 obvious at a glance the relations between the terms expressed in a 
 proposition.'* The relations thus diagrammatically exjiressed are 
 necessarily extensity-relations, so that it is only when the extensive 
 view of Import is adopted, and both subject and predicate terras 
 are read in extension, that we can express diagrammatically the 
 relations wliich we state to exist between the two classes S and P. 
 Esich term is treated as a class-term and diagrammatically repre- 
 sented by a circle. The relation stated to hold between the two 
 term.s is thus represented diagrammatically by a mutual coinci- 
 dence or non-coincidence, partial or total, of two circles. 
 
 We proceed to point out the precise diagrammatic equivalents of 
 the A, E, I, and propositions. 
 
 In the following diagrams the shaded part of a circle stands in 
 
 * ' A Manual of Logic,' vol. i., ch. iii., p. 215.
 
 Chap. XV'II.J 
 
 THE FORMAL CATEGORICAL 
 
 151 
 
 each case for that part of the class-extension to which reference is 
 made either exphcitly or by logical implication, so that distributed 
 terms are represented by wlioUy shaded circles, undistributed by 
 partially shaded circles. The horizontal shading in each case 
 belongs to S, the jierpendicular to P. 
 
 The A proposition must be represented thus 
 
 It is, indeed, frequently stated that A requires two diagrams 
 to represent it adequately — the one already given, and a second 
 
 representing the coextensiveness of S and P : 
 
 ./ 
 
 But since, when asserting the A proposition, we do not state (or 
 imply) that we are referring to the whole extension of P, we are not 
 at liberty to make the statement diagrammatically. We cannot 
 logically draw what it is illogical to state. 
 
 The I proposition gives 
 
 The alternative 
 
 is ruled out, since we do not state 
 
 that some S's are all the P's, and, as we do not state this, we are 
 not at liberty to indicate it diagrammatically. 
 
 The proposition gives : 
 
 Finally, the E proposition gives
 
 152 
 
 THE PROBLEM OF LOGIC 
 
 [V. ii. 
 
 It follows from the above that an A or I proposition must invari- 
 ahJij be taken as possessing an undistributed predicate, even though 
 we happen to know that the statement made b}'' the proposition 
 would still be true though taken to involve a reference to all 
 the individuals included in the extension of the predicate term. 
 Thus the predicate of the pro]')osition ' All equilateral triangles are 
 ecpiiangular triangles ' is undistributed, and so is the predicate of 
 the proposition ' Some animals are horses.' If our intention is to 
 distribute both subject and predicate — if we wish, for instance, to 
 assert that all the objects contained in the class Equilateral triangle 
 are severally identical with all those contained in the class Equi- 
 angular triangle, we must make use of two propositions, and say 
 ' All equilateral triangles are equiangular triangles,' and ' All equi- 
 angular triangles are equilateral triangles.' 
 
 It is most important, in connexion with logical diagrams, not 
 to confuse the diagrammatic representation of statements con- 
 cerning the relations of classes to each other with the repre- 
 sentation, in diagrammatic form, of the possible class-relations 
 themselves. 
 
 Thus, as Dr. Venn and Professor Welton insist, there are only 
 five ways in which two classes can partially or wholly coincide with 
 one another — namely, those represented by the five diagrams : 
 
 For, as Mr. Stock points out (' Logic,' 2nd edit., p. 85), two 
 classes, S and P, must either — 
 
 Entirely coincide 
 
 
 
 or not 
 
 partially coincide 
 
 by total inclusion 
 
 or not
 
 Chap. XVII.] THE FORMAL CATEGORICAL 153 
 
 But this docs not justify us in concluding that, on the extensive 
 view of import, there are five, and only five, elementary propooilions : 
 
 1. S coincides with P : 
 
 2. S is wholly included in P : 
 
 3. S Avholly includes P : 
 
 4. S partially includes and partially excludes P : 
 
 5. S entirely excludes P : 
 
 Qua statement of meaning, the first of these is, in fact, not an 
 elementary type. It is equivalent to the second and the third 
 conjointly. 
 
 More generally we may say that, whilst we admit that classes 
 can be related to each other (in respect of mutual coincidence and 
 non-coincidence) in five ways and five only, we do not hold that 
 there are only five ways in which such relations of classes can be 
 stated. We may, as we have just seen, reduce them to four ; or 
 we may extend them, though perhaps without sufficient reason,* 
 to eight. And it is particularly to be noted that in Formal Logic 
 we are concerned not with the relations of classes, but with the 
 statement of their relations. Logical diagrams should represent 
 statements only, sc that two ways of saying the same tiling may 
 appropriately be represented by different diagrams. Thus the 
 proposition ' All the S's are all the P's ' is represented by the 
 
 diagram p^ 'li;' p \ which directly expresses the coextensiveness 
 
 of the classes S and P. But when our statement takes the com- 
 pound form ' All S's are P's and all P's are S's,' the corresponding 
 
 diagrammatic rendering will be : I |:g i^ ) 4" 
 
 ' All ' and ' Some.* 
 
 The meanings which we ascribe to the words ' universal ' and 
 ' particular ' as applied to propositions, and to ' distributed ' and 
 ' undistributed ' as apphed to terms, need to be further defined by 
 a discussion of the precise meanings of the two words ' All ' and 
 
 * Vide p. 159.
 
 15i THE PROBLEM OF LOGIC [V. ii. 
 
 ' Some ' as used to introduce, respectively, the universal and the 
 particular proposition. 
 
 The extension of a term, as we have seen, must be conjunctively 
 interpreted (the intension being disjunctively interpreted). Thus, 
 the extension of ' Man ' is ' tliis man and that man a7id the other,' 
 etc. ' All ' and ' Some,' then, in Formal Logic are used con- 
 junctively. Tliis conjunctive use must not be identified with the 
 collective use of these words. In ordinary discourse ' All ' and 
 ' Some ' mfiy be used collectively. In speaking of ' all the men ' 
 we may be referring to them as a single group, as in the statement 
 ' All tl)e men in the room were but a tithe of those who were 
 invited ' ; and ' Some ' is collective^ used in such statements as 
 ' Some days of rest are all that he needs,' ' Some coppers will make 
 these children happy.' But tlie two words may also be dis- 
 tributively understood, and this is the case in their conjunctive use. 
 The expression ' A and B and C,' when conjunctively understood, 
 means ' A, also B, also C,' and in tliis case whatever is predicated of 
 * A and B and C ' is predicated of each individually. This is the 
 case when All and Some are used to introduce the universal and 
 particular propositions respectively. 
 
 L Oil the Logical Meaning of ' All.'' — We have seen that in 
 ordinary discourse the word ' All ' (as also the word ' Some ') is 
 ambiguous. It bears two distinct meanings of wliich one only is 
 accepted as appropriate to the purposes of Formal Logic. Thus, 
 when I sa}' ' All these articles cost sixpence,' I may mean either 
 that I paid sixpence for each article, or that I gave sixpence for the 
 lot. In the former case, ' All ' is the universal quantity-mark. 
 Both it and the statement that it introduces are distributive in 
 meaning. The sentence is an A proposition in strict logical form. 
 In the latter case, the meaning is collective, and ' All ' is not a 
 quantity-mark in the strictly logical sense. For the purposes of 
 Forma) Logic the statement needs to be reduced to strict logical 
 form, and will then appear as a Singular Proposition : ' This 
 collection of articles is a lot that costs sixpence.' 
 
 The simple test for discriminating between the distributive and 
 the collective ' all ' is to substitute ' each of ' for ' all,' and see 
 whether the sense of the statement is thereby affected. If the 
 substitution does not alter the sense, the proposition is distributive ; 
 if it does, we are deahng with a collective statement wliich is not in 
 strict logical form. For instance, in the sentence ' All these trees 
 here hide the view ' we cannot read ' each of ' for ' all ' without 
 destroying the sense. Therefore the statement is collective. It 
 is a Singular Proposition in disguise. 
 
 2. On the Logical Meaning of ' Some,^ and the Import of the 
 Particular Proposition qua Particular. — The distinction between 
 the universal and the particular proposition coincides, as we 
 have seen, with the distinction between a proposition with a
 
 Chap. XVIL] THE FORMAL CATEGORICAL 155 
 
 distributed subject and a proposition with an undistributed 
 subject. The logical meaning which ' Some ' bears in the par- 
 ticular propositions I and must, therefore, be that which appro- 
 priately expresses the characteristic of undistributedness. As we 
 have seen, a term is undistributed when there is no reference, 
 either expressed or implied, to the whole extent of the class for 
 which it stands. This definition of an undistributed term is not 
 arbitrary. It is the meaning dictated by the true interests of 
 Inference. A term is of interest for purposes of inference just in 
 so far as we definitely state or else logically imply what we take its 
 extensity-reference to be. The primary essential here is to be able, 
 in any given case, to distinguish mth certainty between what is 
 stated and what is not, and it is this very distinction wliich the use 
 of the words ' distributed ' and ' undistributed,' as above defined, 
 enables us to express. 
 
 The true logical meaning of ' Some ' in its relation to ' All,' a 
 meaning which we may call the undistributed meaning of tlie word, 
 is therefore correctly expressed in the phrase ' not statedly all.' 
 As regards the inferior hmit, the appropriate phrase is ' one at 
 least.' If we are told that Socrates was wise, and that he was a 
 citizen of Athens, we are entitled to conclude that at least one 
 Athenian citizen was wise. But this is essentially a proposition 
 with undistributed subject, for we do not state that all Athenian 
 citizens were wise. Hence, in order to include such cases as this, 
 ' Some ' must be able to stand for ' one at least,' and we see that 
 ' one at least, but not statedly all ' is the proper logical equivalent 
 of the word ' Some.' 
 
 From tliis strictly logical meaning of ' Some ' we must carefully 
 distinguish otlier meanings that the word is capable of bearing : 
 
 (i.) There is the popular use of ' Some ' in the sense of ' a few at 
 any rate, yet considerably less than all.' The logical disadvantage 
 of tliis use of the word is that, if the particular quantity-mark is 
 thus interpreted, there are many propositions wliich can then be 
 neither particular nor universal. Thus, ' One S at least is a P,' 
 and ' Very nearly all the S's are P's ' would not be particular 
 propositions if tliis popular meaning of ' Some ' were adopted, and 
 they certainly are not universal. Tliis, of course, presents no 
 difficulty to the practical consciousness. If it wishes to talk about 
 ' one at least,' it says ' one at least ' ; if it desires to refer to ' nearly 
 all,' it says ' nearly all.' It is only when we wish to generalize and 
 to regulate that we aim at minimizing distinctions and making them 
 strictly relevant to our purpose — to the interests, for instance, of 
 logical inference. 
 
 (ii.) Secondly, there is the exclusive use of 'Some.' That which 
 we have called the strictly logical meaning of ' Some ' is open, in 
 the opinion of a certain school, to a fimdamental objection. It 
 shares with other interpretations the defect of not allowing our
 
 156 THE PROBLEM OF LOGIC [V. ii. 
 
 statements to be sufficiently explicit. A man may reasonably be 
 challenged to explain why he prefers to point out what he does not 
 state instead of stating clearly and exclusively what he does intend 
 to say. Granting that a proposition, as Formally used, expresses 
 nothing beyond a statement of meaning, would it not be better to 
 render ' Some S's are P's ' as follows : ' What I mean to state is 
 that one S at least is a P, but that not all of the S's are P's.' Here 
 there is no reticence, no inscrutable reference to what is 7iot stated. 
 Whv not define ' Some' as exclusive of ' All,' and ' All ' as exclusive 
 of ''Some ' ? 
 
 Tliis proposed interpretation of ' Some ' is known as the ' ex- 
 clusive ' meaning. ' Some ' is here taken as equivalent to ' one at 
 least, but Jiot all,' or, if we prefer to keep more explicitly to the 
 statement-view of Import, ' one at least, but statedly not all.' 
 
 The essential objection to this exclusive interpretation of ' Some ' 
 is that it reduces the distinction between the I and propositions 
 to a mere difference of emphasis. If we state that one S is a P, but 
 that not all S's are P's, we are also stating that some S's are-not P's. 
 The fourfold scheme reduces to a threefold scheme, including three 
 types of proposition — A, E, and <J>. This would confuse the 
 whole scheme of logical opposition {vide Chapter XIX.). 
 
 There is also the further objection that this use of ' Some ' 
 cannot cover all cases. If I say ' Some cats are fond of fish,' I do 
 not, as a rule, wish to imply that some cats are-not fond of fish. 
 What the statement does invariably imply is that one cat at least 
 is fond of fish. As to the Ciuestion whether all cats are fond of fish 
 no statement is made. 
 
 (iii.) There is, further, an indefinite or semi-indefinite use of the 
 word still suggested as the correct use in many logical treatises. 
 According to this interpretation ' Some ' is defined as meaning 
 ' one at least, possibly all,' or ' one at least, it may be all.' ' Some 
 S's are P ' could then be paraphrased thus : ' I state the predicate 
 P of at least one S, but I do not state of how many S's it holds good : 
 it mmj hold good of all of them.^ Thus, when I say ' I saw some of your 
 friends at the gathering yesterday,' I may mean to include the possi- 
 bihty of my having seen all the friends of the person to whom I am 
 speaking. 
 
 This interpretation of ' some ' has a plausible appearance. But 
 this plausibility is really derived from a confusion of the positive 
 expression ' it may be all ' with the negative limitation ' not 
 statedly all ' ; and it shares with other renderings the inconvenience 
 of leaving certain tj^es of proposition not accounted for either as 
 particular or as universal. Thus, ' Some human beings are 
 children ' cannot be interpreted as meaning ' Some human beings, 
 possibly all, are cliildren.' ' Some ... it may be all ' is incom- 
 patible with ' Some . . . but not all.' On the other hand, ' Some 
 . . . but not statedly all ' is compatible with ' Some . . . but not all.'
 
 Chap. XVIL] THE FORMAL CATEGORICAL 157 
 
 If I say ' I don't state that all human beings are children,' it is open 
 to some one else to complete my statement by saying : ' But I do 
 statft that some human beings are not children.' The rcnderine: 
 ' one at least ... it may be all ' lacks the comprehensiveness of the 
 rendering ' one at least . . . but not statedly all.' 
 
 But there are other objections of a more fundamental kind. The 
 reading ' possibly all ' does not truly interpret the meaning of 
 the undistributed term. By unduly emphasizing the indefinite 
 import of the word ' some ' it fails to grasp the true logical character 
 of the particular proposition. It is quite true that ' indefmite ' is 
 a better label for the I and propositions than is the word ' par- 
 ticular.' For ' particular,' as popularly used, implies a definiteness 
 of a positive kind, as when we speak of ' this particular individual.' 
 So, again, we talk of reasoning from particulars to particulars, when 
 we mean that in each case we argue from one given instance to 
 another. But we really need a still better label which should convey 
 the negatively definite idea that the proposition makes no universal 
 statement. For there is certainly no logical value in ' indcfinite- 
 ness.' Merely to suggest the possibiUty that all of the S's are P's 
 is to make a statement that has no inferential value whatsoever. 
 There is no logical vitality in a mere possibility. 
 
 The reasons given above are perhaps sufficiently decisive against 
 the use of ' Some ' in its indefinite sense. But the crucial objection 
 arises out of the imphcations of the word ' possibly.' The phrase 
 ' Some, possibly all ' may indicate either a limitation of subjective 
 c?rtainty, in wliicli case it means ' One at least, all for aught I 
 know,' or else a more definite limitation of knowledge, its meaning 
 then being ' One at least, not assuredly all.'* But both these read- 
 ings of the word ' Some ' are entirely inadmissible on the view we 
 have adopted as to the import not only of the undistributed term, 
 but of a Formal Logic generally. It cannot be too emphatically 
 stated that in Formal Logic — i.e., in a Formal treatment of Logic — 
 we are not directly concerned with any questions of truth or 
 knowledge or ignorance, or even of opinion, or with the hmitations 
 of any of these. No doubt a more intimate aim of Logic is to analyse 
 the scientific reasoning which is concerned with knowledge and 
 with truth ; but Formal Logic is a proprodeutic which is abstractly 
 concerned with consistency of reasoning without any reference to 
 the truth or the falsehood of the accepted premisses, or to the 
 knowledge or the ignorance of the reasoner. In Formal Logic we 
 are concerned not with what is, or with what is known, but with 
 what is stated either explicitly or by logical implication. Our 
 business is to develop not knowledge or opinion, but significant 
 statement. Doubtless the Formal logician may have to deal with 
 
 * The word ' Imowcdly ' or ' knownly ' would perliaps hare expressed the 
 meAning more directly than the word ' assuredly,' but might not have sounded so 
 well.
 
 158 THE PROBLEM OF LOGIC [V. iL 
 
 stafcrnoifs about truth, as also ■nnth statements concerning know- 
 ledge, certainty, and possibility ; but with the truth of statements 
 or viith. the knowledge, ignorance, certainty, or possibility from 
 wliicli those statements spring he has nothing at all to do. He 
 develops t!ie meaning of statements, and notes their limitations of 
 meaning. His business is not with what is known or not kno"\\ii, 
 but simply with what is stated. Thus, for liim the particular as 
 opposed to the universal quantity-mark indicates never the state- 
 ment of a limitation, but always the limitation of a statement. 
 
 Justification of the ' Extensive ' View of Import. 
 
 Mr. Joseph has brought against the conception of extensive 
 import the crucial objection that ' we cannot predicate of the exten- 
 sion of one term the extension of another.'* But what does this 
 precisely mean ? It is true that we cannot predicate of all the 
 S's, taken distributively, that they are all the P's, or some of them, 
 if the word ' all ' or ' some ' in the predicate is used collectively. 
 We cannot say ' All S's (distr.) are all, or some, P's (coD.).' Nor 
 can we say ' AH S's (distr.) are all, or some, P's (distr.) ' if by tliis 
 is meant ' Every S is every P (or each of some P's).' But we can 
 say ' Every S is a P,' and the predicate of this proposition may be 
 regarded as a genuinely ' undistributed ' term. The meaning is 
 ' Si and So and S3 . . . — each of all the S's — is a P.' Evidently the 
 P's must be at least as numerous as the S's, but whether they out- 
 number them or not we do not state. If we wish to state that they 
 do, we may do so unambiguously by means of the compound propo- 
 sition ' Every S is a P, and each of some P's is-not any S.' And if 
 we Avish to state that the two groups exactly coincide point for 
 point, we can say ' Every S is a P, and every P is an S.' When we 
 employ the simple form ' Every S is a P ' we are not identifying 
 each of all the P's with an S. The reference, therefore, is neither 
 statedly nor implicitly to each of all the P's, and hence the predicate 
 is undistributed. Thus we see that the mark ' a ' in ' Every S is 
 a P ' may logically represent the undistributedness of the predicate- 
 term. It may do so with equal appropriateness in the particular 
 proposition ' Each of some S's is a P.' Here again the proposition 
 does not imply that each of all the P's is an S. If we desire to dis- 
 tribute the predicate term, we can use the compound proposition 
 ' Each of some S's is a P, and each of all P's is an S.' -In the case 
 of negative propositions, for E we have ' Every S is-not any P,' 
 and is equivalent to ' Each of some S's is-not any P.' Here the 
 ' any ' is a mark of distributedness. We state that no one of the 
 S's referred to is any single one of all tlie P's. 
 
 It follows from the above that there is a sense in wliich we can 
 
 * ' An Introduction to Logic,' ch. ix., p. 202.
 
 CuAP. XVII.] THE FORJIAL CATEGORICAL 
 
 159 
 
 perfectly well ' predicate of the extension of one term the extension 
 of another.' What we have already said may perhaps he sufficient 
 to establish the point. But it may be useful, further, to draw atten- 
 tion to the ambiguity of the verb ' to predicate.' It is possible to 
 use tliis verb in such a way as to presuppose the predicative view of 
 import, and so exclude ah initio the possibihty of adopting an ex- 
 tensive reading. In the proposition ' All S's are P's ' the predicate 
 is not predicated of the subject as its attribute. P cannot be a 
 differentia or a proprium of S. But it may still be a predicable, 
 for it may stand to S as genus to species, or as a class-term in one 
 division may stand to a class-term in another division. What is 
 predicated of the extension-of-S is that it is related in the way of 
 at least partial coincidence to the extension of a certain correlate P. 
 Thus P is predicated of S, not as its attribute, but as its extensional 
 correlate. 
 
 Our conclusion, then, is that we are justified in retaining the 
 fourfold scheme in its extensional form, and that we may posit the 
 following equivalences :* 
 
 AU S's are P's = Each of all S's is a P. 
 
 Some S's are P's = Each of some S's is a P. 
 
 All S's arc-not P's = Each of all S's is-not any P. 
 
 Some S's are-not P's = Each of some S's is-not any P. 
 
 The Quantification of the Predicate. 
 
 Sir William Hamilton proposed to develop the fourfold scheme of 
 Categorical Propositions by adding the quantity-mark ' all ' or 
 ' some ' to the predicate of A and I, and ' any ' or ' some ' to that 
 of E and 0, thus obtaining an eight-fold scheme. 
 
 T. ./All Sis all P.. 
 
 From A{ .„ o • r) 
 
 lAli o IS some P 
 
 -ni T f Some S is all P 
 From I -^ o c • -r> 
 
 I Some b IS some P . . 
 
 -r, -c^ f No S is anv P 
 From E{,x o • " -d 
 liNo b is some P 
 
 -ri r\ I Some S is not any P 
 From 0{ o o • 4. t> 
 
 Ibome b IS not some P 
 
 The interpretation and discussion of this scheme still occupies a 
 considerable place in modern treatises on Logic. The reader will 
 find excellent critical appreciations in Professor Welton's ' ]\Lxnual 
 of Logic' (vol. i., bk. ii., ch. ii., pp. 200 fj.) and Mr. Joseph's "Intro- 
 duction to Logic' (ch. ix., pp. 198, ff.). The ambiguities of the 
 
 * For the relation of this ' coincidence ' or ' identity ' form of extensional 
 import to the class-inclusion view, vide p. 239. 
 
 
 . u. 
 
 
 . A. 
 
 
 . Y. 
 
 
 . I. 
 
 
 . E. 
 
 
 . v 
 
 
 . 0. 
 
 
 U).
 
 160 THE PROBLEM OF LOGIC [V. ii. 
 
 words ' All ' and ' Some ' and the confusion between the distributive 
 and collective uses of those marks supply ample opportunity for 
 criticism and reconstruction. 
 
 Into the complexities of tliis discussion we do not propose to 
 enter. We content ourselves with connecting the doctrine of a 
 quantified predicate with the fourfold scheme as we have adopted it, 
 and considering the logical significance and imjaortance of the eight- 
 fold scheme from tliis single point of view. 
 
 In the first place, the conception of a quantified predicate appears 
 to us to be perfectly reasonable. The quantified predicate is 
 indeed already present in the fourfold scheme under the guise of 
 the distributed and undistributed predicate-term. Hence, assuming 
 as we do the undistributed meaning of the word ' Some,' and reading 
 botli subject and predicate terms in extension, we hold that there 
 is obviously no difference between the A, I, E, and of the quanti- 
 fied scheme and the corresponding propositions of the fourfold 
 scheme. 
 
 Our sole criticism of the Hamiltonian scheme when interpreted 
 in this icay is that the four additional propositions, U, Y, 77, m are 
 superfluous. 
 
 The U proposition, as we have already seen, is equivalent to the 
 compound proposition ' All S is P and All P is S,' of which the 
 elements are A propositions. 
 
 Again, the Y proj^osition is equivalent to the A proposition 
 ' All'P is S.' 
 
 The 7) proposition may be disposed of in a similar way. We may 
 diagrammatically represent the 77 proposition — ' No S is some P ' — 
 as follows : 
 
 or 
 
 But if we excliange the S and P in these diagrams, we obtain the 
 diagrams of the proposition : 
 
 or 
 
 ' Xo S is some P ' is, in fact, identical with ' Some P is no S.' 
 As for the co projKjsition, it cannot be denied wJuitever we intend 
 to state. Even if wo intend to state that the extension of the
 
 Chap. XVIII.] REDUCTION OP CATEGORICALS 161 
 
 class S exactly coincides with that of P, still we evidently cannot 
 deny the statement tliat a certain number of S's are not coincident 
 with a certain other number of P's. 
 
 Thus, of tlie four propositions U, Y, rj, w, 
 
 U is reducible to two A propositions, 
 Y is reducible to an A proposition, 
 77 is reducible to 0, 
 0) is truistic, and therefore useless.* 
 
 CHAPTER XVIII 
 
 V. (iii.) THE REDUCTION OF CATEGORICAL PROPOSITIONS TO 
 STRICT LOGICAL FORM. 
 
 In reducing a proposition to strict logical form, our first care must 
 be to interpret the given statement as an ' extensive ' proposition. 
 Thus, if ' S is P ' is naturally read on the predicative view, to the 
 effect that the objects indicated by S possess the attribute P, it 
 must be reinterpreted, so as to read as follows : 'The objects indicated 
 by S are objects wliich possess the attribute P.' In the case of 
 abstract propositions (statements, that is, with an abstract term 
 as subject) the object indicated by the subject-term will be abstract, 
 and the correct interpretation will be somewhat as follows : ' The 
 abstract object indicated by the term S is identified with an abstract 
 object indicated by the term P.' Thus, ' Mercy is twice blest ' may 
 be interpreted as ' Mercy is a twice-blest virtue.' It will then rank 
 as a universal affirmative with distributed subject and undis- 
 tributed predicate. 
 
 The singular proposition need cause no difficulty from the point 
 of view we are here considering. For, as we have seen, it is but the 
 limiting form of the universal proposition. ' This S is P ' means 
 ' All objects indicated by the term " this S " are objects indicated by 
 the term P.' As a matter of fact, ' this S,' like a proper name, 
 restricts the extension to one object, so tliat, while the proposition 
 is singular, its subject-term is distributed. The singular proposi- 
 tion ranks, therefore, as a universal. It could not rank as a par- 
 ticular proposition for another reason. For ' some ' means ' one 
 at least,' and ' one at least ' is not the same as ' one.' Hence ' This 
 S is P ' could not be brought under the form ' Some S's are P's,' for 
 
 * In the one case in which it might seem possible not to accept w (namely, 
 where S and P extensionally coincide, and both refer to one and the same indi- 
 vidual) the use of the proposition would be inappropriate. If we wished to state- 
 such coincidence we should employ a singular proposition. 
 
 11
 
 1G2 , THE PROBLEM OF LOGIC [V. iii. 
 
 the latter form implies that one S at least is P, and this rendering 
 goes beyond the meaning of the singular proposition. 
 
 Once a 2)roposition is understood extensively, the main rules for 
 reduction to strict logical form may be briefly formulated. 
 
 Putting a categorical proposition into strict logical form means, 
 we maj' say, expressing it in one of the four typical forms, A, E, I, 0. 
 This involves : 
 
 1. Finding which is the true logical subject and which the true 
 
 logical predicate. 
 
 2. Giving the subject its correct quantity-mark, either ' All ' 
 
 or ' Some.' 
 
 3. Giving the proposition its correct quality-mark, either ' is ' 
 
 or ' is-not.' 
 
 The second of these requisites, however, is subject to an important 
 modification. We have seen that, for the purposes of a Formal 
 treatment, the Singular Proposition ranks as a universal. Con- 
 sequently we have a ' singular ' form of the A proposition, the form 
 typically represented by ' Tliis S is a P.' 
 
 We have, then, as tlie two recognized forms of the A propo- 
 sition — 
 
 1. ' All the S's are P's,' where 'All' is understood distributively. 
 
 2. ' S (singular) is a P.' 
 
 A difficulty frequently arises from the fact that propositions 
 collective in meaning are presented in the ordinary distributive 
 form. Tlius, a sentence given to us in the form ' All the S's are 
 P's ' may be incorrectly expressed. A proposition, we know, is 
 distributive if, on putting ' each of ' in the place of ' all,' we find 
 that the sense remains unaffected. The distributive use of ' all ' 
 being accepted as its correct Formal use, the word ' all ' should be 
 retained, in the logically stated proposition, wlienever this substi- 
 tution does not alter the sense. But if the substitution changes 
 the sense, tlien ' all ' is used collectively, and therefore must not 
 be retained in the proposition when this is stated in logical form. 
 In its place we must use a collective expression with a singular 
 import,* so that the elaborated proposition will take the form 
 ' S (singular) is a P ' instead of ' All the S's are P's '; the essential 
 defect of the latter expression in such a case being that it has 
 
 * Collective terms, like ' family ' or ' regiment,' which refer to a collection of 
 objects 'jua aggregate, may be either singular or general. Such terms as ' This 
 regiment ' or ' The Light Brigade ' are singular collectives. ' Regiment ' is a 
 general collective. Collective terms, whether singular or general, are always used 
 collectively with regard to the individuals of the group or kind of group specified. 
 But general collectives are used disjunctively with regard to the various kinds 
 or classes which constitute their denotation. ' Family ' indicates the members 
 collectively, but it denotes the various kinds of family disjunctively. A family 
 is large or small, rich or poor. The term ' family ' may correctly be predicated 
 of each of these types taken apart from the rest.
 
 Chap. XVIII.l REDUCTION OF CATEGORICALS 1G3 
 
 distributive form but collective meaning. This distinction applies 
 also to some propositions which are apparently particular. * Some ' 
 may be used in a collective sense, as meaning, for instance, ' a 
 handful of '; and, when so used, it should not appear as mark of 
 quantity in the proposition logically expressed, but should be 
 superseded by a collective expression. Thus, 
 
 ' All these weeds choke the flower-bed \ 
 
 should be transformed intc 
 
 ' This mass of weeds is a mass that chokes the flower-bed.' 
 
 So also ' Some soothing words appeased him ' is only apparently 
 an I proposition, for the ' some ' is here used collectively. Reduced 
 to proper logical form, the proposition would run : 
 
 ' A string of soothing words is a thing that appeased him.' 
 
 Further difficulty is caused by propositions which refer to a 
 single individual, but do not specify that individual, and therefore 
 are not singular in the sense required to justify the use of the 
 singular form. Examples of such propositions are : 
 
 ' A friend of mine has gone abroad.' 
 
 ' A ship was on fire.' 
 
 ' An earthquake had occurred in Jamaica.' 
 
 A legitimate method of reducing sentences of this type would be 
 to utilize X, the usual symbol for the unknown quantity, as follows : 
 
 ' The friend whom I call X is a person who has gone abroad.' 
 ' The ship X is a ship that was on fire.' 
 
 ' The earthquake specified as X is an event that had occurred 
 in Jamaica.' 
 
 In arguments this substitution would perhaps be convenient. 
 
 The third of the requisites of strict reduction also needs some 
 words of comment. The quality-mark may take any one of several 
 forms. If affirmative, it may be ' am,' ' art,' ' is,' or ' are.' If 
 negative, ' am-not,' ' art-not,' ' is-not,' or ' are-not.' Thus, ' I am 
 a man,' ' Thou art a woman,' ' We are human beings,' may be 
 regarded as reduced propositions. They are aU universal affirmatives. 
 
 It is important that the quality-mark should not be confused 
 with the tense-mark. The quahty-mark is the copula-mark, and 
 has a strictly logical significance. Distinctions between present, 
 past, and future belong to the predicate. Thus ' were ' and ' were 
 not,' ' will be ' and ' will not be ' are not strictly permissible as 
 copula- marks. 
 
 Example. — ' The Drake was in harbour. She is a splendid vessel, 
 and will be the Admiral's flagship.' 
 
 11—2
 
 164 THE PROBLEM OF LOGIC [V. Ui. 
 
 Strict Logical Form : ' The Drake is a vessel wliich at the time we 
 are speaking about was in harbour. The Drake is a splendid 
 vessel. Tlie Drake is a vessel destined to be the Admiral's flagship.' 
 
 Where the argument presupposes throughout either the past or 
 the future tense, so that there is no danger of confusion, disregard 
 of these logical requirements is apt to pass without protest, even 
 from the flintiest logician. Logic owes much to the genius of 
 language, and may waive its claims upon form not only graciously, 
 but \\ithout inconsistency, so long as the yielding does not involve 
 or lead up to any ambiguity or contradiction. 
 
 071 the Reduction of Certain Ambiguous Expressions. 
 
 Any is an ambiguous expression. The following are characteristic 
 meanings : 
 
 Affirmatively ' Any ' = ' every.' 
 
 ' Any man will tell you that ' reduces, therefore, to ' All 
 men are persons competent to tell you that.' 
 Negatively ' Not any ' = ' no '; or else, ' not any '= ' not every ' 
 = ' some . . . not.' 
 Thus 
 
 And 
 
 ' I have not any money ' reduces to ' I am a person with 
 no money.' 
 
 ' Any excuse will not suffice ' reduces to ' Xot every excuse 
 will suffice ' — i.e., ' Some excuses are-not adequate 
 excuses.' 
 N.B. — In questions, ' Any '= ' Some.' 
 E.g., ' Have you any coppers about you V 
 
 ' A few are ' = ' Some are.' 
 
 ' A few are not ' = ' Some are-not.' 
 
 Thus ' A few were present ' reduces to ' Some persons are persons 
 who were present.' 
 
 ' All . . . arc not ' is ambiguous. It may mean either ' All . . . 
 are-not ' or ' Some . . . are-not.' The latter gives the natural 
 meaning ; for ' All . . . are not '= ' Not all . . . are '= ' Some . . . are- 
 not.' It is with the express object of avoiding this ambiguity that 
 we have used the hyphen in the case of the E proposition, ' All S's 
 are-not P's.' 
 
 E.g., ' All men are not honest who say that they are.' 
 = ' Not all men are honest who say that they are.' 
 = ' Some men who say they are honest are-not honest men.' 
 
 So again, ' Every pun is not a joke ' reduces to ' Some puns are- 
 not jokes.'
 
 Chap XVIII.] REDUCTION OF CATEGORICALS 165 
 
 In man}^ cases the quantity-mark is lacking altogether where it 
 is needed, and we have then to insert it. Propositions of this 
 kind arc known as Indcsignate or Pre-indesignate Propositions ; 
 and in reducing them we may guide ourselves by the following 
 rules : 
 
 1. An Indesignate Proposition is universal if, when it is read 
 according to the predicative view, the predicate is found to be 
 furnishing some element of the connotation, or else a property 
 implied in the definition of the subject ; for then P belongs to S as 
 such.* 
 
 Examples : 
 
 ' Angles in a semicircle are right angles.' 
 
 ' Cows are ruminants.' (The connotation of ' cow ' is here 
 
 assumed to include, exphcitly or imphcitly, the marlr 
 
 ' ruminant.' )f 
 
 2. If P is a problematic property of S, the proposition is obviously 
 particular. 
 
 Examples : 
 
 ' Frenchmen are vivacious.' 
 ' Itahans are musical.' 
 
 Reduced to logical form, these will be : 
 
 ' Some Frenchmen are vivacious individuals.' 
 ' Some Itahans are musical individuals.' 
 
 Here the ' Some ' is, of course, equal to ' most '; otherwise the 
 indesignate form would be positively misleading. Still, ' most ' 
 is a mark of particularity. 
 
 3. If P is a characteristic property, the proposition is most 
 naturally treated as universal. There is no logical necessity for 
 our supposing tliat all ruminants have divided feet ; but, so far as 
 our experience goes, the chewing of the cud is invariably accom- 
 panied by the divided foot. Hence the proposition 
 
 ' Ruminants have divided feet ' 
 
 would appropriately reduce into 
 
 ' All ruminants are creatures with divided feet.' 
 
 * Cf. Welton, ' A Manual of Logic,' vol. i.. pp. 1G9-171. Mr. Joseph uses the 
 terra ' indefinite judgement ' ('An Introduction to Logic,' ch. viii., p. 156). 
 
 t The connotation of the subject-term in this proposition is, of course, not 
 the meaning which it actually bears in the proposition. This meaning is a rela- 
 tively indeterminate meaning which the predication of the term ' ruminant ' 
 serves to render more determinate {vide p. 121).
 
 160 THE PROBLEM OF LOGIC [V. iii. 
 
 Exclusive and Exceptive Propositions. 
 
 An exclusive proposition denies a predicate of all save the members 
 of a specified class. 
 
 Example. — ' Xone but the free can obey.' 
 
 = ' All not-free persons are-not persons who can obey.' 
 
 Thus the logical equivalent of the exclusive is an E proposition 
 with a negative term as subject. 
 
 An exceptive proposition afifirms the predicate of the members of 
 a whole class with the exception of those constituting a certain part 
 of it. 
 
 Example : ' All is lost but honour ' 
 
 = ' All possessions other than honour are lost posses- 
 sions ' — i.e., an A proposition with a negative term 
 as subject. 
 
 With regard to the reduction of exclusive propositions to strict 
 logical form, we may provisionally adopt the following rule ; 
 
 Express ' Only S's are P's ' or ' S's alone are P's ' 
 as E : ' All not-S's are-not P's ' (preferably), 
 or as A : ' All P's are S's.' 
 
 Example. — ' Only drakes are curly-tailed ' 
 
 ^ ' All not-drakes are-not curly-tailed creatures,' E. 
 or ' All curly-tailed creatures are drakes.' A. 
 
 It is important to note that the statement ' Only S's are P's ' 
 does not imply that ' all S's are P's.' Granted that ' only heroes 
 are wearers of the Victoria Cross,' it does not follow that all heroes 
 wear it. 
 
 The particular proposition in which ' Some ' is used in its exclusive 
 sense is not strictly exclusive according to the definition given above, 
 and its reduction is different from that of the exclusive universal 
 to which the given definition alone, in strictness, apphes. It 
 resolves itself, in fact, into two independent propositions. Take, 
 for instance, the proposition : 
 
 ' Some only who promise keep their word.' 
 
 Here the logical function of the word ' only ' precisely coincides 
 with that of the expression ' not all ' — i.e., ' some . . . not.' The 
 proposition therefore means ' Some who promise keep their word, 
 but some do not ' — i.e. : 
 
 /I. ' Some makers of promises are breakers of promises.' I. 
 1 2. ' Some makers of promises are-not breakers of promises.' 0. 
 
 Propositions which are thus analysable into two or more inde- 
 pendent propositions are technically called Exponible Propositions.
 
 Chap. XVIII.] REDUCTION OF CATEGORICALS 1G7 
 
 With regard to exceptive propositions, some diflficulty is pre- 
 sented by those in which the excepted class is numerically chara/C- 
 terized. Such Numerical Exceptives may be divided into two classes : 
 
 1. The excepted part may be individually indefinite. 
 
 Examples : 
 
 ' All except a few were lost.' 
 = ' The proportion of persons lost to those not lost is a 
 
 very large proportion.' A. 
 
 Again : 
 
 ' All except seven were saved ' 
 = ' The number of persons who were not saved is seven.' A. 
 
 2. The excepted part may be individually definite. 
 
 Example. — ' All except A and B were lost.' 
 
 This should be treated as an A proposition, since in an argument 
 it would have the force of a universal statement. When it is further 
 stated that C was on board, we can definitely infer that he was lost. 
 As regards logical form, the simplest rendering is reduction to the 
 A proposition : 
 
 ' All persons other than A and B were lost.' 
 
 Propositions in which an exceptive or exclusive phrase modifies 
 some part of the predicate are not exceptive or exclusive propo- 
 sitions, and in reducing them there is no necessity for changing the 
 phrase in question. 
 
 Examples : 
 
 ' Except when they are naughty, children are invariably 
 good.' 
 = ' All children are beings who are invariably good except 
 
 when naughty.' A. 
 
 ' Cliildren are only good when they are asleep.' 
 = ' All cliildren are beings that are good only when asleep.' A. 
 
 ' It concerns no one but myself.' 
 = ' Tliis is a matter that concerns no one but myself.' A. 
 
 ' No human beings experience nothing but trouble.' 
 = ' All human beings are-not beings that experience nothing 
 
 but trouble.' E. 
 
 On the Bcduction of Kon-propositional Sentences to Propositional and 
 
 Strictly Logical Form. 
 
 In so far as optatives, interrogatives, etc. imply assertions, tliis 
 implication can be brought out, and the sentence thus transformed 
 into a sentence indicative, or proposition.
 
 168 THE PROBLEM OF LOGIC [V. iii. 
 
 Optative. — ' Would I were a cassowary !'= ' To be a cassowary is 
 an object of my desire' 
 
 Imperative. — "Beware the Jabberwock !'= ' The Jabberwock is 
 a creature to be avoided.' 
 
 (Rhetorical) Question. — 'Is thy servant a dog ?'='Thy servant 
 
 is-not a dog.' 
 ' Am not I tliine ass ?'= ' I am thine ass.' 
 
 Interrogatives with indicative implications have negative force 
 if they are in positive form, positive force if in negative form. 
 
 Exdajnation. — 'Just tlie place for a Snark !' = ' This place is a 
 
 promising spot for snarks.' To the Bell- 
 man's exclamation the crew might have re- 
 phed : ' We don't believe it !' In that case 
 they would have taken the remark as a 
 statement that had no sufficient ground. 
 This would have stamped it as a Proposition. 
 
 On the Reduction of Given Sentences to Strict Logical Form. 
 
 Practical Hints on what is impHed by ' strict logical form ' : 
 
 (a) As regards quantity and quality. 
 
 L^se ' all,' and not ' every,' ' each ' or ' any.' 
 
 Use ' all . . . are-not ' for the negative universal, and not 
 
 ' no . . . are ' or ' all . . . are not.' 
 Use ' some,' and not ' several ' or ' many,' etc. 
 
 (b) As regards the constitution of subject and predicate 
 
 terms. 
 These should be expressions that have a meaning in them- 
 selves, and they should also admit naturally of being 
 quahfied by ' all ' or ' some ' (except in the case of 
 Singular Propositions) — e.g., all ' glittering things,' 
 not all ' that glitters.' 
 
 Examples. — Express each of the following sentences in such 
 (strict) logical form as reproduces most nearly the natural meaning 
 of the sentence : 
 
 1. Not all who are called are chosen. 
 
 = Some persons called are-not persons chosen. 0. 
 
 2. Xot all your efforts can savx him. 
 
 = The sum-total of your efforts is-not a force that can 
 save him. E. 
 
 3. All kings are not wise. 
 
 = Some kings are-not wise persons. O. 
 
 4. Every bullet does not kill. 
 
 = Some bullets are-not fatal missiles. 0,
 
 p. XVIII.] REDUCTION OF CATEGORICALS 169 
 
 5. Few men succeed in life. 
 
 =: The number of men who succeed in life is-not a large 
 
 number.* E. 
 
 6. All my guesses but two were correct. 
 
 = The number of guesses which I correctly made is 
 
 the total number diminished by two. A. 
 
 7. Great is Diana of the Ephesians. 
 
 = Diana of the Ephesians is a great goddess. A. 
 
 8. An honest man's the noblest work of God. 
 
 = Honest manhood is the noblest work of God A. 
 
 9. He envies others' wealth who has none himself. 
 
 = A11 non-wealthy persons are persons envious of 
 
 others' wealth. A. 
 
 10. Only doctors understand this subject. 
 
 = A11 non-doctors are-not persons able to under- 
 stand this subject. E. 
 
 11. The more, the merrier. 
 
 = A11 new-comers are mirth-increasers. A. 
 
 12. He has no home but Athens. 
 
 = All places outside of Athens are-not homes for him. E. 
 
 13. A few Greeks vanquished the vast army of Darius. 
 
 = A mere handful of Greeks is the force that van 
 
 quished the vast army of Darius. A. 
 
 14. A little knowledge is a dangerous thing. 
 
 = A smattering of knowledge is a dangerous thing. A. 
 
 15. The Romans conquered the Carthaginians. 
 
 = The Roman power is a power that conquered the 
 
 Carthaginians. A. 
 
 IG. The angles of a triangle are equivalent to two right 
 angles. 
 
 = The sum of the angles of a triangle is a quantum 
 
 equal to two right angles. A. 
 
 17. Two blacks won't make a wliite. 
 
 = All possible combinations of two blacks are-not com- 
 binations producing whiteness. E. 
 
 18. My friend plays golf. 
 
 = ]\Iy friend is a golfer. A. 
 
 19. Not all the gallant efforts of the veterans availed any- 
 
 thing. 
 = The gallant fighting of the veterans is-not fighting 
 
 that availed anytliing. E. 
 
 20. Few dogs are not fond of fetching sticks. 
 
 (Force Affirmative) 
 = All dogs not fond of fetching sticks are exceptional 
 dogs. A. 
 
 * As tho original proposition is negative in meaning, the logiceJ form must be 
 sympathetically negative.
 
 170 THE PROBLEM OF LOGIC [V. iii. 
 
 21. A few dogs are fond of cats. 
 
 = Some dogs are fond of cats, I. 
 
 22. Only a few politicians are statesmen. 
 
 (Exponible) 
 = Some politicians are statesmen and some politicians 
 
 are-not statesmen. <^. 
 
 23. Only ignorant persons hold such opinions. 
 
 (Exclusive) 
 = A11 non-ignorant persons are-not persons holding 
 
 such opinions. E. 
 
 Or— 
 = A11 persons holding such opinions are ignorant 
 
 persons. A. 
 
 24. Some men are not incapable of telling falsehoods. 
 
 (Force Affirmative) 
 = Some persons are persons quite capable of telHng 
 
 falsehoods. I. 
 
 25. Scotchmen are level-headed. 
 
 Is ' level-headed ' part of the connotation of 
 ' Scotchman ' ? Obviously not, unless by 
 ' Scotchman ' we mean ' typical Scotch- 
 man.' It appears to be a problematic pro- 
 perty. We therefore have as the strict 
 logical form : 
 = Some Scotchmen are level-headed persons. I. 
 
 26. To be or not to be, that is the question. 
 
 = Whether life is worth Mving or not is the question 
 
 I must answer. A. 
 
 27. Scarcely anyone got through. 
 
 = The number of persons who passed is a very small 
 number. A. 
 
 28. Men are not what they were. 
 
 = The manhood of to-day is-not manhood as it used 
 to be. E. 
 
 29. The side and diagonal of a square are incommensur- 
 
 able. 
 = The ratio of the side of a square to its diagonal is 
 
 an incommensurable ratio. A. 
 
 30. The only interested persons are candidates and ex- 
 
 aminers. 
 = A11 interested persons are either candidates or 
 
 examiners. A. 
 
 31. Am I my brother's keeper ? 
 
 (Rhetorical question implying negative state- 
 ment. The logical subject here is ' The 
 guardianship of my brother.') 
 = Looking after my brother is-not my business. E.
 
 Chap. XVIII.] REDUCTION OF CATEGORICALS 171 
 
 32. Fain would I climb, but that I fear to fall. 
 
 = My wish to climb is a wish checked only by my fear 
 
 of falling. A. 
 
 (N.B.— ' But ' has the force of ' only.') 
 Other alternative renderings are : 
 = A11 my impulses to climb are impulses inhibited 
 
 by the single fear of falling. A. 
 
 Or— 
 = My fear of falling is the only thing which prevents 
 
 me from climbing. A. 
 
 33. All the travellers were not provided with pass- 
 
 ports. 
 = Some travellers in the company specified are-not 
 
 travellers who were provided with passports. 0. 
 
 34. All but Noah and his family were drowned. 
 
 — All persons of Noah's day who were not of his house- 
 hold arc persons who were dro^vned. A. 
 
 35. Affictions are often salutary. 
 
 = Some visitations of afSiction are salutary experi- 
 ences. I. 
 
 36. All these claims upon my time overpower me. 
 
 = This multitude of claims upon my time is a burden 
 that overwhelms me. A. 
 
 CHAPTER XIX. 
 
 V. (iv.) THE OPPOSITION OF PROPOSITIOXS. 
 
 It is customary to say that two propositions are logically opposed 
 when, having the same subject and predicate, they differ in quan- 
 tity, in quahty, or in both quantity and quahty. From tliis point 
 of view the relation between a7iy two of the propositions A, E, I, and 
 is treated as an Opposition. 
 
 The definition, though convenient, is superficial and arbitrary, 
 and it necessitates using the term ' Opposition ' in an entirely 
 technical sense, for the relation between ' All men are mortals ' 
 and ' Some men are mortals ' is, according to the definition given 
 above, an opposition. 
 
 A much sounder method is to guide ourselves by principle, and 
 to hold that propositions are in opposition only when they violate 
 the requirement of non-contradiction. Where this requirement 
 is respected, as in the so-called ' subcontraries ' and ' subalterns,' 
 and the relation ceases to be one of opposition, we may suitably 
 speak of Subcontrariety and Subaltcrnation, but not of Sub-
 
 172 
 
 THE PROBLEM OF LOGIC 
 
 [V. iv. 
 
 contrary Opposition or of Subaltern Opposition.* The only two 
 forms of genuine opposition between propositions are known as 
 Contradictory Opposition and Contrary Opposition respectively. 
 They are relations between genuine opposites, because in each 
 case we have a pair of propositions juxt-aposed which cannot 
 logicaUy be entertained together. If one of them is accepted, the 
 otlier must be rejected. 
 
 We proceed now to the more minute consideration of the four 
 propositions, A, E, I, 0, in respect of those relations between 
 them which are customarily known as Oppositions. The relations 
 we have to deal ^ith are the following : 
 
 (a) Contradictory Opposition : 
 
 A. All S's are P's. )(t 0. Some S's are-not P's. 
 E. All S's are-not P's. )( I. Some S's are P's. 
 
 (b) Suhcontrariety : 
 
 I. Some S's are P's. )( 0. Some S's are-not P's. 
 
 (c) Co7itrory Opposition : 
 
 A. AU S's are P's. )( E. All S's are-not P's. 
 
 (d) Suhalternation : 
 
 A. All S's are P's. )( I. Some S's are P's. 
 
 E. All S's are-not P's. )( 0. Some S's are-not P's. 
 
 These last two pairs are kno\^Ti as Subalterns. A is usually 
 called the Subalternans of I, and I the subalternate of A. Similarly, 
 E is the subalternans of 0, and the subalternate of E. 
 
 The above-named relations may be diagrammatically represented 
 in what is knoMH as the Square of Opposition : 
 
 Contraries 
 
 -^e:' 
 
 \ 
 
 / 
 
 \ 
 
 / 
 
 I 
 
 I' 
 
 Co" 
 I 
 
 ^ 
 
 .c'p/ 
 
 / 
 
 
 M 
 
 A<y% 
 
 % 
 
 I 
 I 
 
 I 
 
 ) 
 
 I • 
 
 I ' 
 
 c 
 
 3 
 
 Co 
 
 Sub-contranes ^ 
 
 * Mr. Joseph (' An Introduction to Logic,' chap, ix., p. 207, footnote) draws 
 attention to Aristotle's own statement on this point : ' For some are is only verbally 
 opposed to some are not ' (Anal. Pri. B., xv. 636, 27). At the same time ilr. Joseph 
 holds that if subcontraries ' are not opposed, they are anyhow contrasted, and 
 that may justify their continued inclusion ' among ' forms of opposition.' 
 
 t We adopt the grammatical sign )( as signifying any kind of so-called 
 Opposition.
 
 CnAP. XIX.] OPPOSITION OF PROPOSITIONS 173 
 
 Contradictory Opposition. 
 
 The Rules or Laws of Contradictor tj Opposition are identical with 
 the formulae already given (p. 98) for the Princijjle of Non-Con- 
 tradiction : 
 
 Rule 1. Contradictories cannot both be accepted. 
 Rule 2. Contradictories cannot both be rejected. 
 
 These rules cannot be proved by means of principles more 
 ultimate than themselves. But they admit of the most cogent 
 proof possible, in another sense of the term ' proof.' If the proof 
 of a law lies in its indispensableness, then the Rules of Contra- 
 dictory Opposition may be proved to the hilt. For if they do not 
 hold good, there is no such thing as consistency, and it becomes 
 unreasonable to tliink at all. Moreover, the very attempt to deny 
 these laws refutes itself. The statement ' Contradictories can 
 be accepted together ' implies that the contradictory of tliis very 
 statement may itself be accepted. Finally, as we have already 
 seen, the Principle of Non-contradiction being a Law of Thought, 
 to violate it is absolutely impossible. To have succeeded in doing 
 so would be to have thought the unthinkable. 
 
 If a proposition is stated in the general form ' S is P,' its contra- 
 dictory may be stated in the form ' S is P ' or ' Not (S is P).' Some 
 care is needed in the interpretation of tliis contradiction-formula. 
 It is misinterpreted, for instance, whenever the denial is directed 
 upon some assumption which the proposition ' S is P ' takes for 
 granted, instead of being directed upon the proposition itself. 
 
 Thus, the denial of the statement that ' Japan's President is 
 revered by his people ' cannot be identified with the assertion that 
 ' The ruler of Japan is not its president.' For this simply denies 
 the impHcit assumption that Japan is represented by a president ; 
 it does not deny the given proposition. So, again, ' The Mikado 
 is the President of the Japanese Repubhc ' is not expressed by 
 asserting that the realm of Japan is not a presidency. This denies, 
 not the proposition itself, but an assumption wliich the statement 
 has taken for granted. 
 
 The denial of ' S is P ' must then itself be a statement concerning 
 the relation of S to P, and not the denial of some assumption whicli 
 the assertion ' S is P ' presupposes. But insistence on this should 
 not be carried to the point ofjisserting that the denial of ' S is P ' is 
 simply ' S is-not P,' and that the clumsier form ' Not (S is P) ' may 
 therefore give way to the form ' S is-not P,' as the typical form 
 of the denial of ' S is P.' 
 
 Those who support this equivalence appear to take the singular 
 proposition as representative of all the rest. The statement that 
 ' Socrates is-not wise ' may be taken as the denial of ' Socrates is
 
 17i THE PROBLEM OF LOGIC [V. iv. 
 
 ^se,' and therefore as the equivalent of ' Not (Socrates is wise).' 
 But when quantitative rehations are involved, or when the propo- 
 sition is indesii^nate.* nothins; but confusion can result from equating 
 ' Xot (S is P) ' to ' S is-not P.' 
 
 Let us consider tlie case where ' S is P ' is indesignate, and take, 
 by way of illustration, the proposition ' Frenchmen are vivacious.' 
 It is surely misleading to assert forthwith that the denial of ' French- 
 men are vivacious ' is ' Frenchmen are-not vivacious.' For if the 
 indesignate proposition is here taken as particular, then, in deny- 
 ing ' Some Frenchmen are vivacious ' by ' Some Frenchmen are-not 
 vivacious,' we should be mistaking what is a mere subcontrary 
 opposition for logical denial. 
 
 We conclude, then, that ' Not (S is P) ' or ' S is P,' and not the 
 declaration ' S is-not P,' is the only general form which the denial 
 of ' S is P ' can logically take ; and that when this denial is specified 
 in definite propositional form, it is a statement having the same 
 subject and predicate as the proposition wliich it denies. 
 
 As regards the more special application of the laws of Contra- 
 dictory Opposition, since A and O are contradictories, and also 
 E and I, we see that, in the case of the first of the two laws, it con- 
 sists in arguing that if a certain proposition is accepted, then another 
 proposition, having the same subject and 'predicate, but differing from it 
 in quality and quantity, cannot be accepted; and so, mutatis mutandis, 
 for the application of the second law of Contradictory Opposition. 
 
 Suhcontrariety. 
 
 The Rules of Suhcontrariety : 
 
 Rule 1. Of two Subcontraries, if one is accepted, there is no 
 logical ground either for the acceptance or for the 
 rejection of the other. 
 
 Rule 2. Of two Subcontraries, if one is rejected, the other must 
 be accepted. 
 
 Proof of Rule 1. — To accept the I proposition is to state that 
 one S at least is a P, wliile making no statement about all the S's. 
 In accepting I we do not state or imply either that the proposition 
 ' All the S's are P's ' must be rejected, or that it must be accepted. 
 The acceptance of I does not involve either the rejection or the 
 acceptance of A. Consequently the acceptance of I does not 
 involve either the acceptance or the rejection of the contradictory 
 of A. But the contradictory of A is 0. Therefore the acceptance 
 of I does not involve either the acceptance or the rejection of 0. 
 Similarly it may be shown that the acceptance of does not 
 involve either the acceptance or the rejection of I. 
 
 * Vide p. 165.
 
 Chap. XIX.] OPPOSITION OF PROPOSITIONS 175 
 
 Proof of Rule 2 : 
 
 If I is rejected, E must be accepted (Contradiction). 
 If E is accepted, must be accepted (Identity, vide infra on 
 Subalternation). 
 .-. If I is rejected, must be accepted. 
 
 So, again, 
 
 If is rejected, A must be accepted (Contradiction). 
 If A is accepted, I must be accepted (Identity). 
 .*. If is rejected, I must be accepted. 
 
 The rules of Subcontrariety, as above enumerated, cease to hold 
 when, in place of the undistributed reading for ' Some,' we substi- 
 tute either the exclusive or the indefinite reading, or the reading 
 which expresses a limitation of knowledge. 
 
 If ' Some ' = ' One at least, but not all' (exclusive), then not 
 only are the I and propositions not inconsistent — i.e., not only 
 can they be accepted together — but if one of them is accepted, then 
 the other must also be accepted. For to say that Some S's are P's 
 is to state that not all the S's are P's ; and to say that Some S's 
 are-not P's is to state that not all the S's are-not P's — i.e., that 
 Some S's are P's. 
 
 If ' Some ' = ' One at least, possibly all ' (indefinite), the I and 
 propositions cannot be accepted together. If the proposition 
 ' One S at least is a P, possibly all the S's are P's ' is accepted, then 
 the proposition ' One S at least is not a P, possibly all the S's are 
 not P's ' must be rejected, and vice versa. For the statement ' One 
 S at least is a P ' is inconsistent with the statement ' possibl}^ all 
 the S's are-not P's '; and the statement ' One S at least is-not a P ' 
 is inconsistent with the statement ' Possibly all the S's are P's.' 
 If I state that at least one member of a class is certain to pass (or 
 not to pass) a specified examination, I cannot consistently go on to 
 say that all may possibly fail (or pass). 
 
 In the case also of the reading ' One at least, but not assuredly 
 (or " knowedly ") all,' there is an inconsistency involved in accept- 
 ing the I and propositions together as true, though the incon- 
 sistency is here less easy to detect. Thus, as a despairing member 
 of a class of examinees, I might assert : ' One at least of us will 
 fail, though I do not know that we all shall,' and then, meeting a 
 feUow-candidate in whose ability I have full confidence, I might 
 say to him : ' One at least of us will pass,* though I don't know that 
 we all shall,' and my two statements would be verbally compatible, 
 for, knowing that I shall fail, I certainly do not knoic that we all 
 shall pass. But my second statement, if not actually disingenuous, 
 at least cannot be said to express a limitation of knowledge. My 
 knowledge as to the truth of the statement that we shall all pass is 
 
 * Hero ' pass * is used as an abbreviatioa for ' not fail.'
 
 176 THE PROBLEM OF LOGIC [V. iv 
 
 not limited if I know that one at least will fail. But it is precisely 
 tliis limitation of knowledge whieli the readinc; Ave are considering; 
 professes to express. There is, therefore, a contradiction involved, 
 on this reading, in accejiting the I and propositions as true 
 together. To harmonize the two I must exchange statement of 
 limitation (of knowledge) for limitation of statement. I must adopt 
 the undistributed reading, and say : ' One at least of us will pass, 
 but more than that I don't say ; about the question of our all 
 passing / make no statement.'' 
 
 Contrary Opposition. 
 The Bules of Contrary Opposition are the following : 
 
 Rvle 1. Of two contraries, if one is accepted the other must be 
 
 rejected. 
 Rule 2. Of two contraries, if one is rejected, there is no logical 
 
 ground either for the acceptance or for the rejection 
 
 of the other. 
 
 These rules may now be justified as follows : 
 
 1. Contraries cannot be accepted together ; for, supposing 
 
 tliis possible, then 
 
 As A is accepted, I must be accepted (Principle of 
 Identity : vide infra on Subalternation). 
 
 And, as E is accepted, must be accepted [ibid.). 
 .♦. A and O are accepted together, as also are E and I. 
 
 But this is impossible, by the Principle of Non-Contra- 
 diction. 
 
 2. It may be quite consistent to reject both contraries ; for 
 
 their contradictories (which are subcontraries) may 
 both be accepted. See Subcontrariety, Rule 1. 
 
 The question may be asked why, in controversy, it is preferable 
 to attempt the refutation of a statement by proving its contra- 
 dictory rather than by jjroving its contrary. The simple answer 
 is that the contrary of a proposition is harder to prove than its 
 contradictory. But this is not all. Tlie additional element of 
 assertion wliich the strengthened opposition superinduces upon the 
 pure contradiction not only plays no part in the refutation of the 
 statement, but may give the adversary an opening for counter- 
 attack which he would not have possessed had the refutation taken 
 place through contradiction simply. 
 
 Svbalter nation. 
 
 The following are the Rules of Subalternation : 
 
 Rule 1. If the universal proposition is accepted, the particular 
 projjosition must also be accepted.
 
 CiiAP. XTX.] OPPOSITION OF PROPOSITIONS 
 
 177 
 
 If A is accepted, I must be accepted (Principle of 
 
 Identity). 
 If E is accepted, must be accepted {ibid.). 
 
 Rule 2. If tlie univ^ersal proposition is rejected, there is no 
 logical ground either for accepting or for rejecting 
 the particular proposition. 
 
 Proof of Bide 2 : 
 The acceptance of affords no logical ground either for the 
 acceptance or for the rejection of I (Subcontrariety, 1). 
 But the acceptance of is logically equivalent to the 
 rejection of A (Contradiction). 
 .'. the rejection of A affords no logical ground either for the 
 acceptance or for the rejection of I. 
 
 Rule 3. If the particular proposition is accepted, there is no 
 logical ground either for accepting or for rejecting 
 the universal proposition. 
 
 For a justification of this Rule, see above, p. 174 (Proof of the 
 First Rule of Subcontrariety). 
 
 Rule 4. If the particular is rejected, then the universal must 
 also be rejected. 
 Proof : If I is rejected, E must be accepted (Contra- 
 diction). 
 .'. A must be rejected (Cofitrariety). 
 
 If is rejected, A must be accepted (Contradiction). 
 .-. E must be rejected (Contrariety). 
 
 The following Table of Opposition will serve to summarize the 
 results of the previous discussion : 
 
 1 
 
 2 
 3 
 4 
 
 Given 
 
 A 
 
 E 
 
 I 
 
 
 
 A accepted 
 
 accepted 
 
 rejected 
 
 accepted 
 neither 
 
 rejected 
 
 A rejected 
 
 rejected 
 
 neither 
 
 accepted 
 
 E accepted 
 
 rejected 
 
 accepted 
 rejected 
 
 rejected 
 
 accepted 
 
 E rejected 
 
 neither 
 
 accepted 
 accepted 
 rejected 
 
 neither 
 
 5 
 G 
 
 7 
 8 
 
 I accepted 
 
 neither 
 
 rejected 
 
 neither 
 
 I rejected 
 
 rejected 
 
 accepted 
 neither 
 
 accepted 
 accepted 
 
 accepted 
 
 rejected 
 
 neither 
 
 rejected 
 
 accepted 
 
 rejected 
 
 accepted 
 
 rejected 
 
 N.B. — ' Some ' = ' One at least, but not statedly all.' By ' neither ' we mean that 
 there is no logical ground either for the acceptance or for the rejection of the 
 proposition in question. 
 
 12
 
 178 THE PROBLEM OF LOGIC [V. iv. 
 
 Tlic Opposition of Disjunctive Propositions. 
 
 The contradictory of the disjunctive proposition ' Either P or 
 Q ' in its non-exclusive form is ' Neither P nor Q.' When the 
 ' Either ... or ' is read exclusively its contradictory is -Either 
 both P and Q, or else neither P nor Q.' In this second case ' Either 
 one or tlie other ' is contradicted by ' Either both or else neither.' 
 According to the exclusive view, ' Neither P nor Q ' could not be 
 accepted as the contradictory of ' Either P or Q,' since both pro- 
 positions could be rejected, the accepted statement being ' Both 
 P and Q.' 
 
 Again, according to the exclusive reading, ' Neither P nor Q ' 
 and ' Both P and Q ' may both be considered as contraries of the 
 disjunctive proposition ' Either P or Q.' For ' Either P or Q ' 
 and ' Neither P nor Q ' cannot both be accepted, and yet, as we 
 have already seen, it may be quite consistent to reject both. Simi- 
 larly, ' Either P or Q ' and ' Both P and Q ' cannot both be accepted, 
 but it may be quite consistent to reject them both — namely, when 
 the accepted statement is ' Neither P nor Q.' 
 
 The Opposition of Hypotheticals. 
 
 The Scheme of Opposition here takes different forms correspond- 
 ing to the different types of Hypothetical Proposition. 
 
 1. (a) The Apodeictic Scheme (Formal). 
 
 H^. If P is accepted, the acceptance of P is necessary. 
 
 Hg. If P is accepted, the rejection of P is necessary. 
 
 Hj. If P is accepted, the rejection of Q is not necessary. 
 
 Hj,. If P is accepted, the acceptance of Q is not necessary 
 
 Example. — H^. If ' All S's are P's ' is accepted, the acceptance 
 
 of ' All S's arc P's ' is necessary, 
 Hg. If ' All S's are P's ' is accepted, the rejection of 
 
 ' All S's are P's ' is necessary. 
 Hj. If ' All S's are P's ' is accepted, the rejection of 
 
 ' All S's are P's ' is not necessary. 
 {I.e., Either ' All S's are P's ' must be accepted, or 
 
 else there is no ground either for accepting or 
 
 for rejecting it.) 
 Hq. If ' All S's are P's ' is accepted, the acceptance 
 
 of ' All S's are P's ' is not necessary. 
 {I.e., Either ' All S's are P's ' must be rejected, or 
 
 else there is no ground either for accepting or 
 
 for rejecting it.)
 
 Chap. XIX.] OPPOSITION OF PROPOSITIONS 179 
 
 1. (b) The Apodcictic Scheme (Material). 
 
 H^. If P is true, then implicitly Q is true. 
 
 Hg. If P is true, then implicitly Q is not true {i.e., is false). 
 
 Hj. If P is true, then it is not implied that Q is false. 
 
 {I.e., Either it is implied that Q is true, or else there is no 
 
 implication as to the trutli or falsity of Q.) 
 Hq. If P is true, then it is not implied that Q is true. 
 {I.e., Either it is impUed that Q is false, or else there is no 
 
 implication as to the truth or falsity of Q.) 
 
 2. The Assertoric Scheme. 
 
 Affirmative universal : If P, then {i.e., in that case it is a 
 
 matter of ascertainable fact that) 
 in all eases we have Q. 
 Or, more briefly : If P, then in aU 
 cases Q. 
 Negative universal : If P, then in no case Q. 
 Affirmative particular : If P, then in some cases Q {i.e., In 
 
 some of the possible instances of 
 P's presence Q will also be 
 present). 
 Negative particular : If P, then in some cases not Q. 
 
 In discussing the Opposition of Hypotheticals, Dr. Keynef 
 contends that, where the hypothetical is assertorial* (the truth of 
 Q following upon the truth of P, but not necessarily from it), the 
 true contradictory of ' If P, then Q ' is ' P, and not Q.'f He argues 
 that, if the antecedent is not categorically posited in the contra- 
 dictory, and the contradictory of ' If P, then Q ' is stated in the 
 form ' If P, then not Q,' we are not in a position to say that one or 
 other of these must be false. For let us sujipose that, as a matter 
 of fact, P happens not to be true. Then no consequent which has 
 P as its antecedent can be definitely labelled as false. When once 
 the assumption represented by the ' if ' clause is discredited, we 
 may say anything we Hke on the imaginative supposition that the 
 antecedent which the ' if ' introduces is true. Thus, to take Dr. 
 KejTies' own example, | the two propositions ' If tliis pen is not 
 cross-nibbed, it is corroded by the ink,' and ' If this pen is not 
 cross-nibbed, it is not corroded by the ink ' cannot be contra 
 dictories, since, if, as a matter of fact, the pen happens to be cross- 
 nibbed, we cannot regard either proposition as false. 
 
 The difficulties which beset the foregoing argument disappear 
 so soon as we insist — (1) on testing contradictories by the principle 
 
 * Dr. Koyncs makes use of the word ' assertoric' 
 
 t 'Studios and Exercises in Formal Logic,' fourth edition, Part II., ch. is.., 
 pp. iCi. 263. 
 
 X Ibid., p. 207, footnote. 
 
 12—2
 
 ISO THE PROBLEM OF LOGIC [V. iv. 
 
 of Logical Consistency instead of by an appeal to the Law of Ex- 
 cluded Middle,* and (2) on respecting the true nature of the Hypo- 
 thetical Proposition as a connexion of possibilities. 
 
 Dr. Keynes maintains that, when the antecedent in the ' if ' 
 clause is not credible in the hght of actual facts, we cannot say of 
 the two statements ' If P, then Q,' ' If P, then not Q ' that one or 
 the other must be false. But the interconsistency of the two 
 statements does not depend on the testimony of actual fact. The 
 C|uestion is whether the acceptance of the one statement is consistent 
 with the acceptance of the other. When once the connexion of 
 possibihties asserted by the proposition ' If P, then Q ' is accepted, 
 the further connexion of possibilities asserted by the proposition 
 ' If P, then not Q ' must be rejected. f From the point of view of 
 consistent tliinking the relation in which the possibilities may stand 
 to realities is entirely irrelevant. 
 
 And this brings us to our second criticism, which is that Dr. 
 Ke^mes fails to respect the nature of the Hypothetical Proposition 
 as a connexion of possibihties. For a main objection that he brings 
 against the hypothetical contradictory is that we can conceive cases 
 in wliich the antecedent clause in both the contradictories cannot 
 be seriously intended as an assumption at all. Thus, when I say 
 ' If this pen is not cross-nibbed,' though all the time aware that 
 the pen is cross-nibbed, what I am really saying is ' If this pen is 
 not cross-nibbed, as I know it is, then . . .,' etc. But the hypo- 
 thetical proposition, so understood, no longer asserts a connexion 
 of possibilities. It no more deserves the title of ' hypothetical ' 
 than the proposition ' Some S's are P's ' {' some ' being taken to 
 mean ' one at least, possibly all ') deserves the title of ' particular ' 
 when the ' possibly all ' is replaced by ' actually all ' [vide p. 156). 
 
 When Mr. Grim wig in ' Oliver Twist' says ' If ever that boy 
 returns to this house, sir, I'U eat my head,' he does not — argues 
 Dr. Keynes — intend the antecedent to be taken seriously ; for what 
 he really means is that the boy is quite certain not to return. But 
 in this case — and the reasoning holds good for all the parallel 
 instances Dr. Keynes adduces — the proposition is hypothetical in 
 form only, not in function. It is a ' rhetorical ' hypothetical, of 
 which the true form is the categorical statement : ' That boy's 
 return is an event as impossible as my eating my own head.' 
 
 We conclude, then, that the objection to the hypothetical con- 
 tradictory has not been sufficiently justified. 
 
 * The disjunctive statement that * of two contradictory propositions one must 
 be false ' is, in our view, an expression of the Law of Excluded Jliddle {vide p. 99). 
 
 t I>r. Keynes' qualification of hypothetical propositions as 'true' and 'faLfe' 
 is a further difficulty in his treatment of this fjroblem. In the proposition 'If P, 
 then (,! ' is it not only P and Q that can be either true or false ?
 
 CiiAP. XIX.] OPPOSITIOx^ OF PROPOSITIONS 181 
 
 Examples. 
 
 Example 1. — Give all the logical ' opposites ' of the proposition, 
 ' All officers are citizens.' 
 
 By its logical ' opposites ' are meant the corresponding proposi- 
 tions, in the forms E, 0, and I, which have the same subject and 
 predicate, and are related to it respectively as its contrary, con- 
 tradictory, and subaltern : 
 
 Contrary, E : All officers are-not citizens. 
 Contradictory, : Some officers are-not citizens. 
 Subaltern, I : Some officers are citizens. 
 
 Example 2. — Give the logical ' opposites ' of the proposition, ' It 
 is too late to mend.' 
 
 Opposition presents a certain difficulty in the case of singular 
 propositions. Thus, it is not at first sight quite obvious whether 
 the counter-assertion ' It is-not too late to mend ' should be re- 
 garded as the contrary or as the contradictory of the given pro- 
 position. That it is an E proposition, whereas the given proposition 
 is an A proposition, constitutes a Formal ground for reckoning it 
 as the contrary. But tliis merely technical argument has no weight 
 as against the contention, based on logical principle, that the pro- 
 positions must be contradictories, since they cannot both be 
 rejected. The contradictory of ' It is too late to mend ' is, there- 
 fore, ' It is-not too late to mend.' There is no technical contrary, 
 though the proposition ' It is too soon to mend ' would be the 
 natural contrary. There is no subaltern. 
 
 If, however, there is what Professor Keynes has called ' secondary 
 quantification,' then, though the propositions will no longer be 
 opposites in the sense already defined, we may apply the Square of 
 Opposition in a shghtly modified form. 
 
 We have secondary quantification whenever the attribution of a 
 predicate to a subject is limited with reference to times or condi- 
 tions. Thus, in the following sentence from the opening of ' The 
 Pied Piper of Hamelin ' — 
 
 ' When begins my ditty — 
 Almost five hundred years ago — 
 To see the townsfolk suffer so 
 From vermin — 'twas a pity ' — 
 
 the first two lines express a secondary quantification of the predi- 
 cate, and we have secondary' quantification also in the singular 
 proposition, ' It is never too late to mend.' 
 
 Here the Contradictory is : 'It is sometimes too late to 
 mend.' 
 
 The Contrary is : 'It is always too late to mend.' 
 
 And the Subalternate is : ' It sometimes is-not too late to mend.'
 
 182 THE PROBLEM OF LOGIC [V. iv. 
 
 Example 3.* — Give the contradictory of the U proposition. 
 
 The V proposition takes the form, ' All S's are P's and all P's 
 are S's.' The contradictory of tliis is ' Either some S's are-not P's, 
 or some P's are-not S's.' If either of these alternatives is accepted, 
 the orijjjinal proposition must be rejected ; and, on the other hand, 
 if the latter is rejected, one, at least, of these alternatives must be 
 accepted. 
 
 We may observe that we cannot say tliat ' Some S's are-not P's ' 
 is a contradictory of ' All the S's are all the P's,' for it may be quite 
 consistent to reject both propositions, and to accept the proposition, 
 ' Some P's are-not S's.' 
 
 What is true of the U proposition holds good of all compound 
 propositions. A compound proposition can have only one contra- 
 dictory — not more than one — and the contradictory must affirm a 
 number of alternatives, of which one or other must be accepted if 
 the original proposition is rejected. 
 
 Example 4. — Test the following : ' Epimenides says that the 
 Cretans are Hars, and Epimenides is a Cretan, Therefore what he 
 says is not true. Therefore the Cretans are not hars. Therefore 
 Epimenides is not a Har, and what he says is true.' — Fallacy of 
 Mentiens. 
 
 This argument obviously involves a fallacy, for a proposition is 
 here deduced from its own contradictory, ' What Epimenides says 
 is true ' is deduced from ' What Epimenides says is not true.' 
 
 We may point out the fallacy most clearly as follows : 
 
 The following are the different propositions involved in the 
 argument : 
 
 (a) Epimenides says that the Cretans are hars. 
 {b) Epimenides is a Cretan. 
 
 (c) Therefore what he says is not true. 
 
 (d) Therefore the Cretans are not hars. 
 
 (e) Therefore Epimenides is not a liar.' 
 
 (/) Therefore what Epimenides says is true. 
 
 Let us concentrate attention on the logical connexion between 
 (c) and (d). If we can show that tliis connexion is logically invahd 
 when (c) and (d) are so interpreted as to free the rest of the argu- 
 ment from fallacy, we shall have sufficiently proved the logical 
 invalidity of the given argument. 
 
 The logical connexion in question depends on the assumption 
 that what Epimenides says in (a) finds its true contradictory 
 in (d). 
 
 Can we, then, fmd out without ambiguity — 
 
 (1) What Epimenides does say in (a) ; 
 
 (2) What (ri) means ? 
 
 * After Dr. Keynes.
 
 Chap. XIX.] OPPOSITION OF PROPOSITIONS 183 
 
 We can do neither of these things. But we can find out what 
 Epimenides must mean in (a) if there is to be no fallacy in the 
 inference of (c) from {h), and what {d) must mean if there is, again, 
 to be no fallacy in the inference of (/) from (e). 
 
 For (c) only follows from {h) provided that by (a) we mean 
 * Epimenides says that all statements made by Cretans are false 
 statements.' Unless (a) means this, the acceptance of [h) does not 
 necessitate the acceptance of (c). 
 
 Similarly, (/) only follows from (e) provided that by {d) we mean 
 ' Therefore, all statements made by Cretans are-not false state- 
 ments.' Unless [d) means tliis, the acceptance of (e) does not 
 necessitate the acceptance of (/). 
 
 If, then, there is to be no fallacy in the above-mentioned infer- 
 ences, the original statement of Epimenides must be ' All state- 
 ments made by Cretans are false statements '; and the statement (rZ), 
 which is put forward as its contradictory, must take the form, ' All 
 statements made by Cretans are-not false statements.' 
 
 The sole remaining question, then, is whether the proposition 
 ' All statements made by Cretans are-not false statements ' is the 
 true contradictory of ' All statements made by Cretans are false 
 statements.' It is obviously not its contradictory, but its contrary, 
 the true contradictory being ' Some statements made by Cretans 
 are-not false statements.' 
 
 What we have shown, then, is that the logical connexion between 
 (c) and {d) is invalid when the interpretations given to (c) and {d,) 
 are the only meanings that render the rest of the argument free 
 from fallacy. The argument, therefore, contains a fallacy which 
 can be driven home in the manner above described.
 
 VI. 
 i:\DIEDIATE INFERENCE.
 
 CHAPTER XX. 
 
 IMMEDIATE INFERENCE. 
 
 Formal Inference and its Logical Principle. 
 
 Formal Inference is reasoning from accepted statements or pre- 
 misses to conclusions implied in them. The Principle of Formal 
 Inference may be called the Law of Formal Validity. It may be 
 formulated as follows : 
 
 // a given 'proposition or set of propositions is accepted, then the 
 further propositions which are implied in what is thus admitted, 
 these, and these only, must also he accepted ; and the further proposi- 
 tions tvhich are in contradiction ivith any one of the admitted proposi- 
 tions, or with any one of their implications, these, and these only, mu^t 
 he rejected. 
 
 The Law of Formal Validity is no new addition to the Laws of 
 Thought. It simply interprets the Laws of Formal Identity' and 
 Non-contradiction as formulated in relation to Inference. 
 
 Identity, in its relation to Inference, is a matter of Implication. 
 If one statement is implied in another, the two must belong to one 
 and the same identical system. This systematic intimacy between 
 them constitutes their logical Identity. The Law of Formal 
 Identity in its relation to Inference may be formulated as 
 follows : 
 
 // a given proposition or set of propositions is accepted, then the 
 further jyropositions which are implied in what is thus admitted, these, 
 and these only, must also he accepted. 
 
 So the Law of Non-contradiction in its relation to Liference may 
 be formulated thus : 
 
 // a given proposition or set of propositions is accepted, then the 
 further propositions which are in contradiction with any one of the 
 admitted propositions, or ivith any propositions implied hy them, 
 these, and these only, 7nust he rejected. 
 
 The Law of Formal Inference is sometimes stated in the form 
 of the concise though negative precept 'Not to go beyond the 
 premisses.' We may connect this injunction with the Law of 
 Formal Validity by showing that in going beyond the premisses 
 we necessarily contradict certain propositions implied by these 
 
 1S7
 
 188 THE PROBLEM OF LOGIC [VI. 
 
 premisses. Thus, from the accepted statement ' All S's are P's ' 
 we might conclude that ' All P's are S's.' In drawing this 
 conclusion we should be going beyond the accepted premiss, 
 though we sliould not be contradicting it. But though we do 
 not contradict the premiss ' All S's arc P's ' Avhen we accept the 
 conclusion ' All P's are S's ' as if this were an inference from it, 
 we do thereby contradict certain implications of that premiss. 
 Thus, when we posit ' All S's are P's ' — i.e., ' All S's are some P's ' — 
 we expressly don't state anytliing about all the P's. Hence no 
 statement dealing with all the P's is implied in the original state- 
 ment. Consequent 1}% no statement dealing with all the P's can 
 be disimplicated from the original statement. But tlie incorrectly 
 dra\\m conclusion tells us that at least one statement dealing with 
 all the P's can be disimplicated from the original statement. Thus 
 we see that the proposition ' "All P's are S's" is implied in the 
 original statement "All S's are P's" ' contradicts an implication of 
 the accepted premiss, and must be rejected as inconsistent with it. 
 (It is not, of course, the proposition ' All P's are S's ' which contra- 
 dicts an implication of the accepted premiss. Were this the case, 
 we should be compelled, after accepting ' All S's are P's,' to reject 
 ' All P's are S's ' as inconsistent with it.) 
 
 We see, then, that, in going beyond the premiss, we have dis- 
 regarded the Law of Validity, and fallen, at one point at least, 
 into meanincrless self-contradiction. 
 
 o 
 
 The most simple expression of the Principle of Formal Identity 
 considered as a principle of Inference, an expression but one degree 
 removed from the blank formula of Tautology — ' If A is accepted, 
 then A is accepted ' — is provided by the First Law of Subalternation, 
 the law which states that if the universal proposition is accepted, 
 the particular proposition of tlie same quality must be accepted 
 also. 
 
 The attempt to prove this law is instructive, as it serves to bring 
 out the fact that the Principle of Formal Identity cannot be proved 
 by means of the Principle of Xon-contradiction. 
 
 Let us suppose that in accepting A we do not thereby logically 
 bind ourselves to accept I. On this supposition, when we accept A 
 we disable ourselves from rejecting the contradictory of I. But 
 this disability involves us in a logical inconsistency, since A and 
 E are contraries. 
 
 This proof, however, presupposes the truth of the law it is 
 endeavouring to prove ; for it assumes a law of Contrary Oppo- 
 sition which can be proved only by the liclp of the very law of 
 Subalternation which we are considering. We have therefore 
 committed the fallacy of reasoning in a circle. Hence the Principle 
 of Identity, in this its simplest form, cannot be proved by means 
 of the Principle of Xon-contradiction.
 
 CiiAP. XX.] IMMEDIATE IXFEREXCE 189 
 
 Inference may be mediate or immediate. With Mediate Infer- 
 ence, or Syllogism, we shall be concerned in due course. We 
 propose to make a beginning witli the study of Immediate Inference. 
 An Immediate Inference may be defined as the inference from the 
 acceptance or the rejection of a proposition to the acceptance or 
 the rejection of a further proposition on the sole basis of the Laws 
 of Identity and Xon-Contradiction. Thus we see that the relations 
 of ' Opposition ' between the four propositions A, E, I, are estab- 
 lished through processes of Immediate Inference. There is at least 
 one other form of immediate inference, as above defined, which 
 we shall presently consider. 
 
 It is customary to include, under the name of Eductions, two 
 processes of inference of which one alone, as we shall see, is an 
 immediate inference according to the definition as above stated. 
 These two processes are respectively known as Obversion and 
 Conversion. Of these Conversion alone can strictly be called an 
 immediate inference. All other so-called immediate inferences — 
 e.g.. Contraposition and Inversion — simply involve alternating 
 repetitions of Conversion and Obversion. We should not, there- 
 fore, refer to Inversion, for instance, as an eduction. The two 
 sole eductions are Obversion and Conversion. 
 
 Educts — i.e., the propositions inferred through processes of 
 Eduction — may be either ' strong ' or ' weak.' A strong educt is 
 one which, for purposes of Inference, may be taken as equivalent 
 in meaning to the proj^osition from which it is inferred. In order 
 to be ' strong ' an educt must be of the same quantity as the original 
 proposition. 
 
 A weak educt is one which presents the meaning of the original 
 proposition in a weakened form, and therefore cannot be substi- 
 tuted for the original proposition without weakening the content. 
 Whenever a proposition and its educt differ in quantity, the educt 
 is weak. Thus, if the original proposition be ' All candidates are- 
 not examiners,' then ' All examiners are-not candidates ' is a 
 strong educt, and ' Some non-candidates are examiners ' is a weak 
 educt. We shall lay especial stress on the strong educts, as, from 
 tlie point of view of Inference, they are the more serviceable and 
 important. 
 
 The processes of Immediate Inference and Eduction have, from 
 our present strictly Formal point of view, a purely Foimal interest. 
 We are solely interested in discerning what are the logical implica- 
 tions of a single proposition. The whole attention being concen- 
 trated on the validity of the reasoning, the matter of the propo- 
 sition ceases to interest us, and, as an important consequence, the 
 whole question of the existential import of the propositions in- 
 volved in the inference is appropriately ignored [vide p. 145). It is, 
 of course, always possible to concentrate interest on the premiss or 
 jremisses of an inference as material evidence. But in that case,
 
 190 THE PROBLEM OF LOGIC [VI. 
 
 as we have already appealed to knowledge, more or less systematic, 
 such k:iowledge must inform us wliethcr the subjects of our propo- 
 sitions exist or not, and, if existent, in what sense they exist. There 
 seems to be no ground for perplexing a purely Formal treatment of 
 the kind we are at present undertaking with a theory of existential 
 implication. 
 
 Our sole concern is with Formally stated propositions and the 
 inferences that can be necessarily drawn from these. We there- 
 fore assume, in our treatment of Eduction, the extensive view of 
 Import and the fourfold division of propositions according to 
 quantity and quality wliich is based upon it. 
 
 Eductions. 
 1. Obversion.* 
 
 It has been said that Obversion is the process of substituting for 
 any affirmative proposition its ' ecjuivalent ' in negative form, or 
 else the process of expressing the meaning of a negative proposition 
 as an affirmative ; thus the obverse of ' A is B ' is ' A is-not not-B,' 
 and the obverse of ' A is-not B ' is ' A is not-B ' We may take this 
 as a correct account of the process as eductive. But to accept it as 
 describing a process of Immediate Inference implies a miscon- 
 ception of the genuine meaning of Immediate Inference which it 
 is important to notice. 
 
 As we have seen, an inference is immediate if it can be drawn 
 from a single proposition on the sole ground of the ]>rinciples of 
 Formal Identity and Non-contradiction. But wlien, for instance, 
 we say that the proposition ' All S's are-not not-P's ' is the obverse 
 of ' All S's are P's,' we are maintaining what cannot be supjjorted 
 by reference to these prmciples alone. The use of the term ' not- 
 P's ' implies a universe of discourse G (let us say ' coloured object,' 
 P being ' blue-coloured object '), and the proposition on which the 
 reasoning rests is the following : ' Either an S (in respect of G) is 
 a P, or else it is a not-P.'f With this we combine the statement 
 ' All S's are P's,' and thence infer that ' All S's are-not not-P's.' 
 This is what is known as a disjunctive inference, and the process 
 of Obversion, as we find it presented in the ordinary Theory of 
 Immediate Inference, is really not an immediate inference at all, 
 but an inference based on a given disjunction. 
 
 In tliis criticism the essential points are (1) that the statement 
 
 * More or less obsolete equivalents are ' Permutation,' ' yEquipollence,' 
 ' Immediate Inference by Privative Conception.' 
 
 t This is not infrequently expressed in the form : ' All S's are either P's or not- 
 P's ' ; but, as we have seen (p. 132), the meaning of this categorically expressed 
 prop^jiition is disjunctive, and therefore can be appropriately stated only in 
 disjunctive form.
 
 Chap. XX.] 
 
 IMMEDIATE INFERENCE 
 
 191 
 
 * Either an S is a P, or else it is a not-P ' cannot be identified witli 
 the Law of Excluded Middle ; and (2) that even if it could be so 
 identified, Obversion would still rest on a Reality-Principle and 
 not on a mere Validity-Principle. Let us consider these two 
 objections in turn, beginning with the first. 
 
 As ' not-P ' means ' some specification of G which is other than 
 P ' {vide p. 49), the statement in question presupposes the more 
 fundamental statement ' All S's are G's.' Thus the statement 
 ' Either a spirit is a pink being, or it is a not- (pink being) ' pre- 
 supposes the proposition ' All spirits are coloured beings.' But the 
 integrity of our thought does not depend on our believing that 
 spirits have colour. It is not meaningless to deny that all spirits 
 are coloured. And a statement that it is not meaningless to deny 
 cannot represent a presupposition involved in the enunciation of 
 a Law of Tliought. We may, however, press this point home in a 
 more radical manner. We have seen that the positing of the Law 
 of Excluded IMiddle in the form given above necessitates the con- 
 clusion ' All S's are G's.' But evidently, from the Law of Ex- 
 cluded Middle, we cannot infer more than ' All S's are either G's 
 or not-G's ' in a sense which gives no hint as to which alternative 
 should be selected. There is, then, a latent contradiction involved 
 in the statement of the Law in the form ' Either an S is a P, or 
 else it is a not-P.' Hence the statement ' Either an S is a P, or 
 else it is a not-P ' cannot be identified with the genuine law of 
 thought to the eflfect that ' Either S is P or S is P.' 
 
 There remains the further objection that even the genuine Law 
 of Excluded Middle is not a Validity-Principle. To appeal to 
 it is therefore to appeal, beyond the Principles of Formal Infer- 
 ence, to the Intelligibility of Reality. But this is to transgress not 
 only the limits of a Formal treatment of Logic, but also the limits 
 of the Law of Formal Validity, apart from which we possess no 
 criterion for testing the soundness of any Formal inference. 
 
 The practical application of the principle of Obversion to the 
 four propositions, A, E, I, 0, gives the following results : 
 
 Obvertcnd. 
 
 Obverse. 
 
 A 
 
 All S's are P's. 
 
 All S's are-not not-P's. 
 
 E 
 
 E 
 
 All S's are-not P's. 
 
 All S's are not-P's. 
 
 A 
 
 I 
 
 Some S's are P's. 
 
 Some S's are-not not-P's. 
 
 
 
 
 
 Some S's aro-not P's. 
 
 Some S's are not-P's. 
 
 I 
 
 These inferences follow at once from the disjunctive major 
 Either an S is a P, or else it is a not-P,' where the P's and the
 
 U.2 
 
 THE PROBLEM OF LOGIC 
 
 [VI. 
 
 not-P's are mutually exclusive, and, taken together, exhaustive 
 alternatives of some given universe of discourse. Thus we 
 have : 
 
 Either an S is a P, or else it is a not-P. 
 A. But all S's are P's. 
 
 .-. All S's are-not not-P's. 
 
 Either an S is a P, or else it is a not-P. 
 But all S's are-not P's. 
 .-. All S's are not-P's. 
 
 E. 
 
 And so also for I and 0. 
 
 Obversions, therefore, are not immediate, but mediate in- 
 ferences. 
 
 The rule whereby these disjunctive obverses may at once be 
 obtained from the corresponding obvertends is usually enunciated 
 thus : 
 
 Negative the predicate, and change the quality of the proposition, 
 hut leave the quantitTj unclianged. 
 
 The fact that in Obversion the quantity is left unchanged shows 
 that all obverses are strong educts. 
 
 If we apply tlie rule of Obversion to the four obverses of A, E, I, 
 and 0, we find that the obverse of the obverse of each proposition 
 — i.e., its obverted obverse — is the proposition itself. Generally, 
 if Y is the obverse of X, then X is the obverse of Y. Obversion is 
 a reciprocal process. 
 
 Examples of Obversion. 
 
 Obvertend. 
 
 Obverse. 
 
 A 
 
 All men are mortals. 
 
 All men are-not non-mortals. 
 
 E 
 
 E 
 
 All men are-not monkeys. 
 
 All men are non-monkeys. 
 
 A 
 
 I 
 
 Some men are misers. 
 
 Some men are-not non-misers. 
 
 
 
 
 
 Some men arc-not magistrates. 
 
 Some men are non-magistrates. 
 
 I 
 
 Exercise.* — Express the following propositions in their simplest 
 forms : 
 
 1 . Not all S's are not-P's. 
 
 2. Not all S's are-not not-P's. 
 
 3. Not some S's are not-P's. 
 
 4. Not some S's are-nct not-P's. 
 
 A ' not ' placed before the sign of quantity contradicts the whole 
 proposition. Thus (1) is equivalent to the contradictory of ' All S's 
 
 * Borrowed from Minto's ' Logic,' pp. 149, 150, footnote.
 
 CiiAP. XX.] IMMEDIATE INFERENCE 193 
 
 are not-P's ' — i.e., to the contradictory of this proposition's obverse, 
 ' All S's arc-not P's ' — i.e., to ' Some S's are P's,' the I propo- 
 sition. 
 
 Similarly, 2, 3, 4 are respectively equivalent to the 0, A, and E 
 propositions. 
 
 2. Conversion. 
 
 Pure Conversion, as applied to categorical propositions, is the 
 educing from a proposition expressed in strict logical form, and on 
 the sole ground of the Law of Logical Validity, another proposition 
 in strict logical form in which the original subject and predicate 
 terms are transposed. 
 
 The Rules of Conversion : 
 
 Rule 1. The converse must be of the same quality as the 
 
 convertend. 
 Rule 2. No term must be distributed in the converse if not 
 
 distributed in the convertend. 
 
 We have now to see what is the connexion between these rules 
 and the fundamental principle of Logical Validity ; for the logical 
 justification of rules can lie only in their being rooted in rational 
 principles. 
 
 The first of these rules appears, at first sight, to be nothing more 
 than an arbitrary convention. But, if so, it has obviously no place 
 in the Theory of Immediate Inference, which is based solely on 
 the Principles of Identity and Non-contradiction. Prof. Welton 
 attempts to meet the difficulty by saying that as the converse 
 simply makes the same assertion as the convertend, looked at, as 
 it were, from the other side, ' it is clear ' that the quality of the two 
 propositions will be the same.* But is this, then, so clear ? Is it 
 quite obvious, for instance, that ' Some P's are S's ' may not be a 
 converse of ' Some S's are-not P's ' ? 
 
 The justification of the first rule of Conversion depends on the 
 recognition (based on the privative view of negation as hare nega- 
 tion, blank denial) that affirmation and negation do not imply each 
 other. The fact that the obverse of an affirmative proposition is 
 negative (and vic£, versa) does not show that an affirmative can 
 imply a negative. If the inference of ' All S's are-not not-P's ' 
 from ' All S's are P's ' rested solely on the Laws of Thought, then 
 the statement that the inference of a negative from an affirmative 
 is contrary to the Laws of Thought would obviously be untenable. 
 To accept the validity of the first rule of Conversion is therefore 
 to admit that Ob version does not rest exclusively on the Laws of 
 Thought. And tliis we have already admitted — indeed, we have 
 
 * J. Welton, 'A Manual of Logic,' Bk. III., ch. iii., vol. i., p. 2oG. 
 
 13
 
 194 THE PROBLEM OF LOGIC [VI. 
 
 insisted on it. We are therefore prepared to admit the necessary 
 character of the first rule of Conversion, and to regard it not as a 
 convention, but as an expression of the fundamental logical fact 
 that a negation cannot be immediatelv inferred from an affirmation. 
 The second rule of Conversion is clearly fundamental, for it 
 simply expresses the essential requirement of the guiding principle 
 of Inference. It is characteristic of all Formal Inference, mediate 
 or immediate, that the conclusion must not go bej'ond the pre- 
 miss or premisses. This requirement would be ignored were we to 
 allow any term undistributed in the convertend to appear dis- 
 tributed in the converse. Hence the second rule of Conversion is 
 logically justified, being, in fact, the Law of Formal Identity ex- 
 pressed in terms of distribution. 
 
 The Conversion of the A Proposition. — That the proposition 
 ' Some P's are S's ' is a converse of the proposition ' All S's are 
 P's ' follows at once from the Principle of Logical Identity in its 
 simplest form. If we accept the statement ' All S's are P's,' we 
 must accept ' Some S's are P's ' (Subalternation). But we cannot 
 accept ' Some S's are P's ' without accepting ' Some P's are S's ' 
 (Identity). 
 
 Moreover, it is the only possible converse, for ' All P's are S's ' 
 breaks the distribution-rule, ' Some P's are-not S's ' breaks the 
 quality-rule, and ' All P's are-not S's ' breaks both. 
 
 Thus the only converse of ' All S's are P's ' is ' Some P's are 
 S's.' This is a weak educt. It is technically called a ' converse 
 by limitation ' or a ' converse per accidens '*: it would be simpler 
 to call it a ' weak converse.' 
 
 The Conversion of the I Proposition. — ^That ' Some P's are S's ' 
 is a converse of ' Some S's are P's ' may be shown as above. That 
 it is the only converse may again be shown precisely as above. 
 Thus the only converse of the proposition ' Some S's are P's ' is 
 the strong equivalent ' Some P's are S's.' 
 
 The Conversion of the E Proposition. — The converse of ' AH S's are- 
 not P's ' is the E proposition ' All P's are-not S's.' For if we 
 accept the statement ' All S's are-not P's,' we must reject the state- 
 ment ' Some S's are P's,' and therefore also (by the Principle of 
 Identity) the converse statement ' Some P's are S's.' But if we 
 reject the statement ' Some P's are S's,' we must accept the con- 
 tradictory statement ' All P's are-not S's.' 
 
 Moreover, ' All P's are-not S's ' is the only converse possible if we 
 exclude the weak converse ' Some P's are-not S's.' For, since the 
 acceptance of ' All S's are-not P's ' necessitates the acceptance of 
 
 * For the ju^^tification of this expression, see H. W. B. Joseph, ' An Introduction 
 to Logic,' ch. X., pp. 211, 212. The justification, however, does not extend to 
 the use of ' accidental ' in our sense of the word {vide supra, p. 27), and we 
 therefore a^lopt the phrase ' weak converse ' instead of ' converse per accidens.'
 
 Chap. XX.] IiADIEDIATE INFERENCE 195 
 
 * All P's are-not S's,' it necessitates also the rejection of the con- 
 trary and the contradictory of ' All P's are-not S's ' — i.e., the re- 
 jection of ' All P's are S's ' and of ' Some P's are S's.' These, 
 therefore, cannot be converses of ' All S's are-not P's.' Moreover, 
 they are affirmative propositions, and therefore ruled out by virtue 
 of their quality. 
 
 The true converse of E is thus a strong educt. It is technically 
 called — as the converse of I also is called — a simple converse. It 
 would be better to call them both strong converses. 
 
 Converse of 0. — The proposition is inconvertible. Given the 
 proposition ' Some S's are-not P's,' we are unable to infer Formally 
 anything about the P's. We can, as we shall see, draw an in- 
 ference concerning the ' not-P's ' — viz., ' Some not-P's are S's '; 
 but this is not a pure converse — i.e., its subject is not the same as 
 the predicate of the convertend. 
 
 That is inconvertible may be shown by a direct application of 
 the two rules of Conversion, for of the possible converses the 
 negatives break the distribution-rule, and the affirmatives break 
 the quality-rule. It may, however, be useful to point out other 
 methods of proving the inconvertibility of 0. Thus we may pro- 
 ceed as follows : Writing down the various possible converses — 
 
 (1) Some P's are-not S's, 
 
 (2) Some P's are S's, 
 
 (3) All P's are-not S's, 
 
 (4) AU P's are S's, 
 
 we see at once that if we can prove (1) and (2) to be illegitimate or 
 ' non-illative ' converses, it will follow, a fortiori, that (3) and (4) are 
 illegitimate. 
 
 (1) Is the proposition ' Some P's are-not S's ' inferible from 
 * Some S's are-not P's ' ? The distribution-rule of Conversion 
 shows that this is impossible ; but we may vary the proof thus : 
 The two propositions are not inconsistent. For instance — 
 
 ' Some men are«not swimmers ' . . - (i.) 
 is not inconsistent with 
 
 * Some swimmers are-not men.' . , . (ii.) 
 
 But the second of these statements is not inferible from the 
 first. For if (ii.) followed necessarily on (i.), its contradictory 
 would be inconsistent with (i.) — i.e., the two statements — 
 
 ' All swimmers are men ' 
 and 
 
 * Some men are-not swimmers * 
 
 could not be entertained together. But that they obviously can 
 be accepted together is shown by the accompanying diagram, 
 
 13—2
 
 196 THE PROBLEM OF LOGIC [VI 
 
 which represents, not, indeed, the statements in question, but a 
 single fact about which they might both be truly made. 
 
 (2) ' Some P's are S's ' cannot bo a converse of ' Some S's are- 
 not P's.' This may be proved as above, or by a direct application 
 of the quality-rule cf Conversion ; or we may argue thus : 
 
 If the statement ' Some P's are S's ' can be validly inferred from 
 ' Some S's are-not P's,' then so can ' Some S's are P's ' ; for the state- 
 ment ' Some S's are P's ' must be accepted whenever the statement 
 ' Some P's are S's ' is accepted. But we know that the proposi- 
 tion does not imply its subcontrary I. Therefore, ' Some P's are S's ' 
 is not a vaUd inference from ' Some S's are-not P's.' 
 
 We now proceed to give some examples of Conversion. The 
 following extract from Professor Bowen's ' Logic ' (p. 157) will 
 here serve as a useful introduction : 
 
 ' In Conversion of Judgments, the learner must remember that 
 the whole Predicate must change places with the whole Subject — 
 that is, whatever belongs to the Predicate must be transferred to 
 the Subject's place, and whatever relates to the Subject to the 
 Predicate's place. For example, Some temple is in the city is not 
 converted into Some city is in the temple, but into Something in the 
 city is a temple. Again, tlie Predicate of Every old man has been a 
 boy is not boy, but has been a boy ; therefore it is not converted into 
 Some boy has been an old man, but into Some one who has been a boy 
 is an old man. To avoid mistakes of this sort, every proposition 
 before Conversion — or, indeed, before it is subjected to any logical 
 treatment whatever — should be reduced to its simplest logical form 
 — that is, to the formula A is B, or A is not B. Then no error can 
 arise, if we remember that all* which precedes the Copula is or is 
 not is the Subject, and that all* which follows the Copula is Predi- 
 cate.' 
 
 Examples of Conversion : 
 
 1. All organic substances contain carbon. 
 
 Some substances which contain carbon are organic sub- 
 stances. 
 
 * The ' all ' here must not be taken as including the quantity-mark.
 
 Chap. XX.] DDIEDIATE INFERENCE 197 
 
 2. Tlic poor have few friends. 
 
 Some persons who have few friends are poor people. 
 
 3. A wise man makes more opportunities than he finds. 
 
 Some persons who make more opportunities than they find 
 are wise men. 
 
 4. Warm-blooded animals are without exception air-breathers. 
 
 Some air-breathers are warm-blooded animals. 
 
 5. Some good men have not the courage to appear as good as 
 
 they are. 
 Some persons who have not the courage to appear as good 
 as they are are good men. 
 G. Some crystals are-not symmetrical. 
 No converse. 
 
 7. All men have not faith. 
 
 No converse. 
 
 8. All mathematical works are not difiiculL 
 
 No converse. 
 
 9. Mrs. BrowTi was IVIiss Smith. 
 
 One who was Miss Smith is Mrs. Brown. 
 
 10. An equilateral triangle is a plane rectilinear figure of three 
 
 equal sides. 
 This is a logical definition, which should apply only to the 
 class defined. Therefore simple tra.nsposition is in the 
 case of definitions not only admissible, but logically 
 requisite. If our purpose is definition, w^e stultify that 
 purpose if we attempt to convert b}^ limitation. There- 
 fore the converse of the given proposition is : 
 A plane rectilinear figure of three equal sides is an 
 equilateral triangle. 
 
 11. A is followed by B. 
 
 Something that is followed by B is A. 
 
 12. P struck Q. 
 
 Somebody who struck Q is P. 
 
 Contrapositive Converse* or Converse "by Negation, 
 
 The contrapositive, or Contrapositive Converse, of a given pro- 
 position may be defined as the converse of the obverse, or as thu 
 converted obverse. The definition is genetic, informing us how 
 the contrapositive may be obtained. Thus we have the following 
 table : 
 
 * The nomenclature here varies greatly in different logical treatises. Mr. 
 Joseph (' An Introduction to Logic,' ch. x., pp. 215. 210) restricts ' Converse by 
 Negation ' to the converted obverse, and reserves the term ' Contrapositive ' for 
 the ' Obverted Contrapositive ' of our own scheme (vide infra, p. 201).
 
 19S 
 
 THE PROBLEM OF LOGIC 
 
 [VI 
 
 Given Proposition. 
 
 Obverse. 
 
 Contrapositive. 
 
 A 
 E 
 I 
 
 
 All S's are P's. 
 
 All S's are-not not-P's. 
 
 All not-P's are-not S's. 
 
 E 
 
 All S's arc-not P's. 
 
 All S's are not-P's. 
 
 Some uot-P's are S's. 
 
 I 
 
 Some S's are P's. 
 
 Some S's are-not not-P's. 
 
 (No contrapositive). 
 
 
 Some S's are-not P's. 
 
 Some S's are not-P's. 
 
 Some not-P's are S's. 
 
 I 
 
 The rule of Contraposition may be succinctly stated thus : 
 ' First obvert, then convert.' 
 
 We have seen that only E and I have strong converses. But if 
 we first substitute for A and O their obverse equivalents, and then 
 convert these, we find that these indirect converses, which are called 
 contrapositive converses, do render the original propositions in 
 unweakened forms. On the other hand, the contrapositive con- 
 verses of E and I are not strong converses. One is iceak, the other 
 non-existent. 
 
 Table of Strong and Weak Converses. 
 
 Given 
 Proposition, 
 
 strong Converse. 
 
 Weak Converse. 
 
 A 
 
 Contrapositive Converse. 
 
 Simple Converse. 
 
 E 
 
 Simple Converse. 
 
 Contrapositive Converse. 
 
 I 
 
 Simple Converse. 
 
 (None.) 
 
 
 
 Contrapositive Converse. 
 
 (None.) 
 
 Exj>onihle Propositions in Relation to Eduction. [After Dr. Keynes.) 
 
 An exponihle proposition is a proposition which can be resolved 
 into two or more simpler propositions which are independent of one 
 another. For instance, the U proposition ' All S's are all P's ' is 
 an exponible, since it may be resolved into the two simpler and 
 mutually independent propositions, ' All S's are P's ' and ' All P's 
 are S's.' 
 
 Now, we ask, is the Exclusive Proposition ' Only S's are P's ' an 
 exponible proposition ? 
 
 I. Can we say that it is equivalent to the two propositions 
 ' All not-S's are-not P's ' and ' Some S's are P's ' ? Yes, argues 
 Dr. Keynes, but the two propositions are not mutually independent.
 
 Chap. XX.] I^DIEDIATE INFERENCE 199 
 
 The statement ' Some S's are P's ' is an educt from ' All not-S's 
 are-not P's ' (SeP, .-. PeS, .-. PaS, .*. SiP). ' If a proposition were 
 considered exponible simply because it could be resolved into two 
 propositions, whether or not these propositions were independent 
 of one another, then every proposition would be exponible.'* 
 
 2. Can we say that the given exclusive proposition is equivalent 
 to the two propositions ' All P's are S's ' and ' Some S's are P's ' 1 
 No doubt ; but the latter is the mere converse of the former. 
 
 We conclude, then, that the Exclusive Proposition is not 
 exponible. 
 
 Examples in Eduction. — Give, where possible, the Converse, 
 Obverse, and Contrapositive of the following propositions : 
 
 1. Only Protestant princes can sit upon the throne of England. 
 
 Given propositio7i= All persons who can sit upon the 
 
 throne of England are Protes- 
 tant Princes. A. 
 Converse : Some Protestant Princes are persons who 
 
 can . . . etc. L 
 
 Obverse : All persons who can . . . etc. are-not non- 
 
 (Protestant Princes). E. 
 
 Contrapositive : All non- (Protestant Princes) are-not 
 
 persons who can . . , etc. E. 
 
 2. Unasked advice is seldom acceptable. 
 
 Given proposition = Some instances of unasked advice 
 
 are-not instances of acceptable 
 counsel. 0. 
 
 Converse : None. 
 Obverse : Some instances of unasked advice are not- 
 
 (instances of acceptable counsel). I. 
 
 Contrapositive : Some suggestions which are not in- 
 stances of acceptable counsel are 
 instances of unasked advice. I. 
 
 3. No admittance except on business. 
 
 Given propositio7i= All persons unintent on business 
 
 are-not persons admitted. E. 
 
 Converse : All persons admitted are-not persons mi- 
 
 intent on business. E. 
 
 Obverse : All persons unintent on business are non- 
 
 (admitted persons). A. 
 
 Contrapositive : Some non- (admitted persons) are 
 
 persons imintent on business. I. 
 
 * J. N. Keynes, ' Studies and Exercises in Formal Logic,' 3rd edition, p. 75.
 
 200 THE PROBLEM OF LOGIC [VI. 
 
 4. The -^rriter of the document was A.B. 
 
 Tliis is a singular proposition with both terms distributed. 
 It is therefore not an ordinary A proposition. Our most 
 straightforward course will be to convert it into a pro- 
 position of the same type — i.e., into — 
 Converse : A.B. is the person who wrote the document. 
 
 We shall then have as the 
 Obverse : The person who wrote the document is-not 
 
 a person other than A.B, E. 
 
 Cojitrapositive : All persons other than A.B. are-not 
 
 the person who wrote the document. E. 
 
 5. ]\Iore haste, less speed. 
 
 This might be treated as follows : 
 Given proposition = M\ cases of increased haste are 
 
 cases that tend to diminish speed. A. 
 Converse : Some cases that tend to diminish speed 
 
 are cases of increased haste. L 
 
 Obverse : All cases of increased haste are-not cases 
 
 not tending to diminish speed. E. 
 
 Cojitrapositive : All cases not tending to diminish speed 
 
 are-not cases of increased haste. E. 
 
 It seems, however, more satisfactory to regard the 
 given proposition as having an abstract subject. 
 Thus : 
 Given 'proposition= GreSiteT haste is a circumstance 
 
 tending to produce correspond- 
 ingly loss speed. A. 
 Converse : A circumstance tending to produce corre- 
 spondingly less speed is greater haste. I. 
 And so on. 
 
 6. Some portraits of some celebrated men are rare. 
 
 Given proposition = Some portraits of certain cele- 
 brated men are rarities. I. 
 Converse : Some rarities are portraits of certain 
 
 celebrated men. I* 
 
 Obverse : Some portraits of certain celebrated men 
 
 are-not non-rarities. O. 
 
 Contrapositive : (None). 
 
 Systematic Eduction. 
 
 The problem of systematic eduction may be stated as follows : 
 Having taken the propositions A, E, I, and 0, plot down all the 
 legitimate educts, strong or weak, which can be obtained by succes- 
 sive processes of conversion and ob version.
 
 Chap. XX.] IMMEDIATE INFERENCE 201 
 
 For the convenient carrying out of this requirement, a suitable 
 symbolic nomenclature is essential. The following substitutions, 
 as used by Dr. Keynes, are now generally recognized : 
 
 A. AllS'sareP's=SaP. 
 E. All S's are-not P's=ScP. 
 I. Some S's are P's = SiP. 
 0. Some S's are-not P's=SoP. 
 
 The term ' not-x ' or ' non-x ' is represented by the symbol x. 
 
 
 
 Eduction 
 
 -Scheme. 
 
 
 A. 
 
 SaP 
 
 /\ 
 
 (PiS) SeP 
 
 E. 
 
 SeP 
 
 /\ 
 
 PeS SaP 
 
 1 
 
 I. 
 
 SiP 
 
 /\ 
 
 PiS SoP 
 
 0. 
 SoP 
 
 /\ 
 
 SiP 
 1 
 
 (PoS) PeS 
 
 PaS 
 
 (PiS) 
 
 PoS 
 
 PiS 
 
 PaS 
 
 (SiP) 
 
 (PoS) 
 
 
 PoS 
 
 (SiP) 
 
 (SoP) 
 
 
 
 ^ 
 
 (SoP) 
 
 N.B. — The process in everj^'case is checked by the inconvertibility 
 of tlie proposition. ( ) indicates a weak educt. 
 
 Nomenclature : 
 
 S-P : Original proposition, the convertend or obvertend 
 
 P-S : Converse. 
 
 P-S : Obverted Converse. 
 
 S-P : Obverse. 
 
 P-S : Contrapositive (Converse), or Converted Obverse. 
 
 P-S : Obverted Contrapositive. 
 
 S-P : Contrapositive of Contrapositive, or Obverted Inverse. 
 
 S-P : Inverse. 
 
 Note that from SaP, as accepted premiss, we are able to infer 
 both PoS and SoP, so that, given SaP, the proposition PoS and SoP 
 (its pseudo -con verse) must both be accepted. But this does not 
 prove that the proposition is convertible. For PoS and^oP are 
 accepted, not unconditionally, but only on the condition that SaP 
 is first accepted. The point to notice is that when two propositions.
 
 202 THE PROBLEM OF LOGIC [VI. 
 
 by means of the acceptance of a third proposition, can be shown 
 to be necessarih' accepted together, it does not therefore follow 
 that the acceptance of one of the two, without the acceptance of 
 the tliird proposition, necessitates the acceptance of the other. 
 
 Examples in Systematic Eduction. 
 
 1. Examine the inference : 
 
 All men are mortals. 
 .-. All non-men are non-mortals. 
 
 The inference is obviously faulty, but its examination will 
 serve to illustrate the general method of solving problems of 
 this kind. 
 
 The problem is equivalent to the following : Given SaP as the 
 accepted statement of the relation between S (subject) and P 
 (predicate), find the inferible statement of the relation between S 
 (subject) and P (predicate). 
 
 The educt here required is the obverted inverse. Therefore, 
 ha\T[ng posited the given proposition SaP, we start with an obver- 
 sion, and then, by means of alternate processes of conversion and 
 obversion, we push forward to the inferible statement which 
 has S for subject and P for predicate. 
 
 SaP 
 
 L 
 
 SeP 
 PeS 
 
 _L 
 
 PaS 
 
 SiP 
 
 Thus, since SiP is the strongest form of stated relation between S 
 (subject) and P (predicate) that can be inferred from SaP, it follows 
 tliat the subaltemans SaP cannot be inferred. 
 
 2. ' All that love success love work.' 
 
 Arrange tlie following propositions in the tnree groups : 
 
 (1) Those which can be inferred from the proposition 
 
 given above. 
 
 (2) Those which are not inconsistent with it, but cannot 
 
 be inferred from it.
 
 Chap. XX.] IIMMEDIATE INFERENCE 203 
 
 (3) Those which are inconsistent with it. 
 
 (a) None that loves not success loves work. 
 
 (6) All that love work love success. 
 
 (c) All that do not love work love success. 
 
 {d) None that docs not love work loves success. 
 
 (e) Some that do not love success love work. 
 
 (/) Some that do not love success do not love work. 
 
 ig) Some that do not love work love success. 
 
 (/i) Some that do not love work do not love success. 
 
 Condensed solution : 
 
 (i.) Symbolic Dictionary. 
 
 Lovers of successes. 
 Lovers of work = P. 
 
 (ii.) Symbolic equivalents of the propositio7is given above. 
 
 (a) SeP. (e) SiP. 
 
 (6) PaS. (/) SoP. 
 
 (c) PaS. (g) PiS. 
 
 (d) PeS. {h) PoS. 
 
 (iii.) Scheme of Eductions from the original proposition 
 SaP. 
 
 SaP 
 
 PiS SeP 
 
 L _l 
 
 PoS PeS— PoS 
 PaS 
 
 _L 
 
 SiP 
 
 I 
 SoP 
 
 (iv.) Classification of the given propositions, as per 
 problem, on the basis of the Rules of Opposition. 
 
 Result : {d), (/), {h) can be inferred from 
 
 SaP. 
 (^)5 (^)j (fi) a-re neither inconsistent 
 with SaP nor infer- 
 ible from it. 
 (c), {g) arc inconsistent vriih. 
 SaP.
 
 204 THE PROBLEM OF LOGIC [VI 
 
 3. Arrange the following propositions so as to show whether or 
 not the acceptance* or the rejection* of one can be inferred either 
 from the acceptance* or from the rejection* of another : 
 
 (a) No intelligent persons are prejudiced. 
 (6) AU uni^rejudiced persons are intelligent. 
 
 (c) Some unintelligent persons are imprejudiced. 
 
 (d) Not every prejudiced person is miintelligent. 
 
 (Intermediate Arts Examination, London, 1902.) 
 
 (i.) Symbolic Dictionary. 
 
 Intelligent person EeS. 
 Prejudiced person^?. 
 
 Unintelligent person r^^S. 
 Unprejudiced person = P. 
 
 This assumes that ' un ' 
 is equivalent to the 
 ' contradictory ' par- 
 ticle 'non,' and is not 
 the prefix of a 'con- 
 trary term.' 
 
 (ii.) Scheme of Strong Eductions, drawn out for the pur- 
 pose of selecting those strong educts from the given 
 propositions, SeP, PaS, SiP, PoS, which are 
 requisite for a straightforward solution of the 
 problem. 
 
 SeP PaS SiP PoS 
 
 PeS SaP 
 
 PeS 
 
 1 
 
 PiS SoP 
 
 PiS 
 
 1 
 
 PaS 
 
 
 1 
 SeP 
 
 SaP 
 
 PoS 
 
 SiP 
 SoP 
 
 The purposive selections are : (a) SaP. 
 
 (6) PaS. 
 
 (c) PoS. 
 
 (cZ) SoP. 
 (iii.) Solution. 
 
 Arranging these educts in the two groups : 
 
 J /SaP{a) jj jPaS(6) 
 
 ■ iSoP(cZ) ■ iPoS(c), 
 
 * We have substituted the words 'accej^tance ' and ' rejection ' for the original 
 ' truth ' and ' falsity.'
 
 Cfl^p XX.' 
 
 IMMEDIATE INFERENCE 
 
 205 
 
 \re see that there can be no inference from tlie 
 acceptance or rejection of any proposition in tlie 
 first group to the acceptance or rejection of any 
 proposition in the second, and vice versa. This 
 follows at once from the Rules of Conversion and 
 of Subcontrariety. 
 
 Hence, with the additional help of the Law of 
 Contradiction, we can at once tabulate our re- 
 sults thus : 
 
 Given. 
 
 (f) 
 
 (i) 
 
 (0 
 
 (.d) 
 
 (a) accepted 
 (b) 
 
 accepted 
 
 neither 
 
 neither 
 
 rejected 
 
 neither 
 
 accepted 
 
 rejected 
 
 neither 
 
 (c) 
 
 neither 
 
 rejected 
 
 accepted 
 
 neither 
 
 (d) „ 
 
 rejected 
 
 neither 
 
 neither 
 
 accepted 
 
 (a) rejected 
 
 rejected 
 
 neither 
 
 neither 
 
 accepted 
 
 {b) „ 
 <c) 
 
 neither 
 
 rejected 
 
 accepted 
 rejected 
 
 neither 
 
 neither 
 
 accepted 
 
 neither 
 
 (d) „ 
 
 accepted 
 
 neither 
 
 neither 
 
 rejected 
 
 N.B. — By ' neither ' we mean that there is no logical ground either for 
 the acceptance or for the rejection of the proposition in question. 
 
 4. What is the logical relation between the first and each of the 
 following propositions given below ? 
 
 (a) Only the sensitive are sympathetic. 
 
 (6) Some sensitive people are-not unsympathetic. 
 
 (c) AU unsympathetic people are sensitive. 
 
 (d) No sensitive people are sympathetic. 
 
 (e) No unsympathetic people are insensitive. 
 (/) All sympathetic people are insensitive. 
 
 (g) iSome unsympathetic people are-not insensitive. 
 {h) Some insensitive people are sympathetic 
 (i) No insensitive people are imsympathctic. 
 {k) Some sympathetic people are sensitive. 
 
 Logical Dictionary : (a) eeAII sympathetic people are sensitive 
 (peojDle). 
 
 Sympathetic person = S. 
 Sensitive person = P. 
 Unsympathetic =1 non-sympathetic. 
 Insensitive = non-sensitive.
 
 206 THE PROBLEM OF LOGIC [VI. 
 
 Eduction-Scheme. 
 
 SaP 
 
 / \_ 
 \7eak converse. PiS SeP Obverse. 
 
 I I 
 
 Ob verted converse. PoS PeS Contrapositive. 
 
 _L 
 
 PaS Obverted contrapositive. 
 
 I 
 ^.'p f Contrapositive of contraposi- 
 
 I \ tive, or obverted inverse. 
 
 _l 
 
 SoP Inverse. 
 
 Symbolic equivalents of given propositions, and results : 
 
 (a) = SaP. 
 
 (6) = PoS = obverted converse of (a). Inferible from (a). 
 
 (c)= SaP = contradictory of SoP = contradictory of inverse 
 of (a). Inconsistent with (a). 
 
 (<Z) = PeS = contradictory of converse of (a). Inconsistent 
 with (a). 
 
 (e)= SeP = contradictory of obverted inverse of (a). Incon- 
 sistent. 
 
 (/) = SaP= contrary of obverse of (a). Inconsistent. 
 
 ({7) = SoP=subcontrary of obverted inverse of (a). Neither 
 inferible nor inconsistent. 
 
 (A) = PiS = contradictory of contrapositive of (a). Incon- 
 sistent. 
 
 (i) = PeS = contrary of obverted contrapositive of (a). In- 
 consistent. 
 
 (^) = SiP=Subalternate of (a). Inferible. 
 
 5. What statements about candidates who pass in Logic can you 
 infer from the statement ' All candidates who do not pass in Logic 
 are-not successful candidates ' ? 
 
 Logicnl Dictionary : 
 
 Candidates who pass in Logic = S. 
 Successful candidates = P. 
 
 Problem : Given SeP, what can we infer about S ? 
 Solution : The Eduction-Scheme is the following :
 
 Chap. XX.J DLMEDIATE INFERENCE 207 
 
 SeP 
 
 PeS SaP 
 
 I _L 
 
 PaS PiS 
 
 I _l 
 
 SiP PoS 
 
 l_ 
 SoP 
 
 Thus we see that from the accepted statement SeP we can 
 infer SiP and SoP — that is : 
 
 ' Some candidates who pass in Logic are successful 
 candidates,' 
 and 
 
 ' Some candidates who pass in Logic are not non- 
 successful candidates.' 
 
 6. From the assertion ' The unknowable is-not unthinkable,' 
 what can we learn, if anything, about (1) the unthinkable, and 
 (2) the not-unknowable ? 
 
 Logical Dictionary : 
 
 Unthinkable "= S. 
 Not-unknowable = P. 
 
 Therefore, given assertion = PeS 
 
 / \ 
 
 SeP PaS 
 
 1 
 
 SaP 
 
 SiP 
 
 PiS 
 
 SoP 
 
 PoS 
 
 Hence we learn SeP and SaP concerning S, and PiS and PoS 
 concerning P. 
 
 I.e. — (1) All unthinkables are-not unknowables. 
 All unthinkables are knowables. 
 (2) Some not -unknowables are unthinkables 
 
 Some not-unknowables are-not not-unthinkables.
 
 208 THE PROBLEM OF LOGIC [VI 
 
 The startlinfr character of these conchisions is, of course, due to 
 the self-contradictory character of tlic data. This is concealed by 
 the fact that when we say that ' the unknowable is-not unthinkable ' 
 we really mean to sa}'" that the unknowable is not necessarily un- 
 thinkable — that, for instance, the pure subject which, it might be 
 argued, is unknowable, since only objects can be kno^vn, is still 
 something of which we can think. We can tliink the Self, it might 
 be urged, though we cannot know it. But in saying this, we do not 
 wish to intimate — as we logically do when we urge that the un- 
 knowable is-not unthinkable — that whatever is unknowable is still 
 thinkable. This is, in fact, self-contradictory. For the unthink- 
 able itself must be unknowable, and therefore, according to our 
 statement, still thinkable. 
 
 7. Examine the inference ' A St. Bernard is a dog. Therefore a 
 small St. Bernard is a small dog.' 
 
 This is a fallacious instance of what is known as Inference by added 
 Determinants, the added determinant here being the epithet 
 ' small.' These inferences are customarily classed as immediate ; 
 but, as we shall presently see, they are really mediate inferences. 
 
 The given argument runs thus : 
 
 ' xA.'s are B's. Therefore small A's are small B's,' 
 
 ' small A's ' being the objects common to the extension of class A 
 and the extension of the class ' small things,' or class S. 
 
 Now, if tliis class had an absolute, fixed extension, the reasoning 
 would be perfectly correct. The accompanying^diagram shows the 
 
 facts as the premiss and conclusion would represent them. The 
 validity of the argument may be shown analytically thus : 
 
 All A's are B's. 
 Certain S's are A's. 
 
 Therefore these same S's are B's — i.e., the small A's are also 
 small B's. 
 
 But if the class S is ambiguously S^ or S2, according as it concerns 
 the class A or the class B, there is no reason wliy any part of the 
 extension of AS^ should coincide with any part of the extension of 
 BS2.
 
 CuAP. XX.J 
 
 i:\DIEDIATE INFERENCE 
 
 20J 
 
 B (Dog). 
 
 S2 (Small -for a 
 St.Bernarddoq). 
 
 -S, (Small -for a 
 Uu(j generally). 
 
 (St Bernard dog). 
 
 [This clia,£rain exhibits not the argument itself, but the facti 
 which the statements of the argument might collectively represent.! 
 
 Hence, if an inference by added determinant is to be valid, the 
 determinant added must have precisely the same apphcation in 
 both cases. Its apphcation must not vary with the significance 
 of the term it qualifies. The following inference by added deter- 
 minant is perfectly valid : 
 
 A St. Bernard is a dog. 
 
 Therefore a hungry St. Bernard is a hungry dog 
 
 8. Examine the inference : 
 
 ' All judges are lawyers. 
 Therefore a majority of judges is a majority of lawyers.' 
 
 This is a fallacious instance of a kind of inference (miscalled 
 immediate) usually known as Inference by Complex Conception. 
 Here the subject and predicate are made determhiants of a third 
 expression. Here again the validity of the inference depends on 
 the unambiguous fixity in the application of tliis third expression. 
 The following inference ' by complex conception ' is valid ; 
 
 Oranges are fruit. 
 
 Therefore a barrel of oranges is a barrel of fruit. 
 
 But fallacy arises so soon as the expression (Uke the word ' majo- 
 rity ' in the given example) is used in a relative, adjustive sense, 
 varying in its import with the words that determine it 
 
 14
 
 VII. 
 THE SIMPLE CATEGORICAL SYLLOGISM. 
 
 (i.) Formal Preliminary (ch. xxi.). 
 
 (ii.) The Rules of the Syllogism — The Valid Forms (ch. xxii.). 
 (iii.) Exercises on the Structure of the Syllogism (ch. xxiii.). 
 (iv.) The Analysis of Syllogisms, and the Reduction of Arguments into 
 
 Syllogistic Form (ch. xxiv.). 
 (v.) Uses and Characteristics of the Four Figures — The Special Rules 
 
 (ch. XXV.). 
 (vi.) The Dicta (ch. xxvi.). 
 (vii.) The Problem of Reduction (ch. xxvii.). 
 (viii.) Unorthodox Syllogisms (ch. xxviii.). 
 
 14—2
 
 CHAPTER XXI. 
 
 VII. (i.) rORlL\L PRELIMINARY. 
 
 Instead of considering the valid inferences which, with the help of 
 the Princijiles of Consistency and Identity, or an assumed disjunc- 
 tion, we can make from some one given proposition, let us now 
 take two propositions, and see what can be inferred from these taken 
 together. 
 
 It is not, of course, possible to draw a conclusion from any given 
 pair of propositions. Thus, from the two propositions 
 
 All bullfinches are birds. 
 All flounders are fishes, 
 
 no conclusion can be drawn. We might suppose that the conclusion 
 ' AU flounders are-not bullfinches ' legitimately followed, but tliis 
 is not the case. To render that conclusion logically sound, a 
 further premiss would be required — namely, the statement ' All 
 fishes are-not birds.' 
 
 To put the matter more generally, notliing can bo inferred from 
 the premisses 'All S^'s are P^'s' and 'All 83 's are Pg's' unless 
 another premiss is given which states some connexion between P^'s 
 and Pg's. This is the central postulate involved in Aristotle's 
 great discovery of the Syllogism, for his discovery essentially con- 
 sisted in finding out that, if a conclusion is to be logically drawn 
 from a pair of given propositions, these must include a common 
 element, i.e., they must contain a common term. Thus, from the 
 two statements 
 
 All birds are vertebrates. 
 
 All bullfinches are birds, 
 
 where the two ideas ' vertebrates ' and ' bullfinches ' are connected 
 through the mediating link ' birds,' we at once infer the conclusion 
 ' All buDfinches are vertebrates.' 
 
 We may say, then, that a conclusion can be drawn from two 
 premisses,* only when the premisses have a common term. The 
 common term is called the Middle Term. The terms related through 
 the ^liddle Term are known as the Major and the j\Iinor Term 
 respectively. 
 
 * Tho propositions from which the conclusion is drawn are technically known 
 as premisses, propositiones prcemissce. 
 
 •213
 
 214 THE PROBLEM OF LOGIC [VII. i- 
 
 The Conclusion in a Svlloffism is the statement of a relation 
 between the major and minor terms. The minor term is always 
 the subject, and the major term the predicate, of the conclusion. 
 
 Of the two premisses, one is called the major and the other the 
 minor premiss. The major premiss is that which contains the 
 major term (P), and the minor premiss is that which contains the 
 minor term (S). The middle term, M, occurs in both. The major 
 premiss, then, states a relation between P and INI, the minor a 
 relation between S and M, and the conclusion a relation between S 
 (subject) and P (predicate). 
 
 In order to be vaUd, the conclusion in a syllogism must follow 
 necessarily and exclusively from the premisses — i.e., it must be 
 implied in the premisses themselves. Consider the argument 
 
 All Roses are Flowering-plants. 
 All Sweetbriars are Flowering-plants. 
 .•• AU Sweetbriars are Roses. 
 
 This is no valid syllogism, though the truth of the conclusion is 
 indisput«,ble ; for the conclusion is not a conclusion from the 'pre- 
 misses, but from other extraneous sources of knowledge. 
 Again, take the argument 
 
 All bats are birds 
 All bantams are bats. 
 .-. All bantams are birds. 
 
 This is a perfectly sound syllogism, though both the premisses are 
 entirely false. If we accept the statement ' All bats are birds,' and 
 also the statement ' All bantams are bats,' we are bound to accept 
 the further statement ' All bantams are birds.' 
 
 We must carefully note the relation in which this j)urely Formal 
 reasoning of the Syllogism stands to tlic question of Definition. 
 For where the nature of the reference to Reality becomes — as it 
 does in the Formal Syllogism — an irrelevant consideration, and the 
 truth or falsity of the premi>?ses a matter of indifference, all distinc- 
 tions in the definition of terms — e.g., the distinction between 
 formal and real — become correspondingly irrelevant. The purely 
 Formal treatment is also the merely verbal in this sense : that the 
 words may take on any desired meaning, provided that the meaning 
 is so given as not in any way to endanger the interests of Non- 
 ambiguity and logical Consistency. 
 
 The Form of the Syllogism. 
 
 The Syllogism, in its most schematic form, may be represented 
 thus : ]yj_p 
 
 S-M 
 
 S— P
 
 Chap. XXI.] THE SIMPLE CATEGORICAL SYLLOGISM 215 
 
 ' ~ ' indicating that the order of terms is reversible, ' — ' indi- 
 cating that the order of terms is fixed. 
 
 Tlie form of a syllogism, however, is not definitely fixed till we 
 know its ' figure ' and its ' mood.' 
 
 The four figures represent the four possible types of Syllogism, 
 each type being distinguished by a certain order of the terms in the 
 two premisses. For, as in each premiss M may be either subject or 
 predicate, we have the four possible figures : 
 
 I. XL III. IV. 
 
 M— P P— M M— P P— M 
 
 S— M S— M M— S M— S 
 
 S— P S— P S— P S— P 
 
 Figure, then, is ' the form of a syllogism as determined by the 
 position of the middle term m the two premisses.' 
 
 In Fig. 1, M is subject in the major premiss, and predicate in the 
 minor. 
 
 In Fig. 2, M is predicate in each premiss. 
 
 In Fig. 3, M is subject in each premiss. 
 
 In Fig. 4, M is predicate in the major premiss, and subject in the 
 minor. 
 
 Mood is the form of a syllogism as determined by the quality and 
 quantity of the three constituent propositions. Thus, the scheme 
 
 M— P 
 S— M 
 
 S— P 
 
 becomes indicative of mood as well as figure when it takes such a 
 
 form as tlie following : iv/r -r» 
 
 ° MaP 
 
 SaM 
 
 SaP 
 
 which is the mood AAA in Fig. 1. 
 
 Each of the four figures may be tentatively developed into sixty- 
 four different forms of Syllogism. For the major premiss may be 
 A or E or I or 0. Suppose it is A. Then with major premiss A 
 we may have four different minor premisses — A or E or I or 0. So 
 also with major premiss E or I or 0. Thus we have sixteen con- 
 ceivable combinations of premisses in Fig. 1 . And as each of these 
 combinations, again, may concoivabl}'' be associated with a con- 
 clusion that is A or E or I or 0, we have sixty-four conceivable 
 forms in the First Figure. So also in each of the other figures. 
 Thus there are 256 conceivable forms in the four figures taken to- 
 gether.
 
 210 THE PR0BLE:\I OF LOGIC [VII. ii 
 
 CHAPTER XXII. 
 
 MI. (ii.) THE RULES OF THE SYLLOGISM— THE VALID FORMS. 
 
 From tliese 256 conceivable forms we have now to eliminate those 
 whicli, in one way or another, violate the requirements of Syllogistic 
 Inference. 
 
 We do not propose at tliis stage to enter into any detailed dis- 
 cussion of the theory of Syllogistic Inference, but it is essential that 
 we should develop it sufficiently to provide ourselves with the 
 requisite ]iractical criteria for discriminating the valid from the 
 invalid forms of syllogistic reasoning. 
 
 The fundamental principle of all Inference we have seen to be 
 embodied in the injunction not to go beyond the premisses. To infer 
 syllogistically, in the widest sense of the process, is to disimplicate 
 from certain interrelated "premisses such conclusions as the said pre- 
 misses collectively necessitate. 
 
 But for our present purpose a more restricted definition is called 
 for. We are deahng with a certain restricted type of Syllogism, 
 the Simple Categorical Syllogism, and tlie definition must, accord- 
 ingly, include the further marks which stamp the Syllogism as 
 simple and categorical. Quu ' simple,' the Syllogism must contain 
 tliree propositions, and three only ; and quu ' categorical,' its proposi- 
 tions must consist of elements which arc not clauses, but terms. 
 Finally, in the Simple Categorical Distributional Syllogism these 
 elements, or terms, are read in extension, and may be referred to 
 cs distributed or as undistributed. 
 
 The Syllogism, then, with whose definition we are here concerned 
 mav be labelled as the S.C.D. Syllogism — ' S.' standing for 
 ' Simple,' ' C for ' Categorical,' and ' D.' for ' Distributional.'* 
 
 From the point of view of Inference, and accepting the definition 
 of propositions in terms of distribution, we may define the Simple 
 Categorical Syllogism as A form of reasoning according to which from 
 two accepted projjositions called premisses, which contain a common 
 term, constitutiiu/ a common link between them, we infer or disimplicate 
 a third proposition, called the conclusion. 
 
 This definition embodies not only the general Principle of Infer- 
 ence but also the Postulate of Mediation or Mediate Inference in 
 its simfilcst form — the postulate of Aristotle : the two terms of the 
 conclusion, S and P, are brought together through the mediation of 
 the third or common term to which they are severally related. 
 Reasoning of tlxis kind is accordingly known as Mediate Inference. 
 
 * The term ' Syllogism,' as uscfl in Division VII., should, Irom this point omrards, 
 be understood a.s an abbreviation for 'S.C.D. Syllogism,' and tho word 'Sj'llo- 
 giatic ' should be similarly undf-r^itood. 'J'hroiighout the treatment of Formal Infer- 
 ence, ' Syllogi.sm ' will be u.sed in the sense of ' Formal Syllogism.'
 
 Chap. XXII.] RULES OF THE SYLLOGISM 217 
 
 The definition posited, we have simply to unravel its import in 
 order to obtain, in explicit form, the so-called Rules of the Syllogism, 
 by the aid of wliich we propose to undertake the logical sifting of 
 the 256 abstractly possible forms of syllogistic reasoning. 
 
 The Rules of the S.C.D. Syllogism. 
 
 (i.) Structural Rules. 
 
 Rule I. : Every S.C.D. syllogism contains three propositions, 
 
 and three only. 
 Rule II. : Every S.C.D. syllogism contains three different 
 
 terms, and three only. 
 
 (ii.) Rules of Distribution. 
 
 Rule III. : The middle term must be distributed once at least. 
 Rule IV. : If a term is distributed in the conclusion, it must 
 
 have been previously distributed in one of the 
 
 premisses. 
 
 (iii.) Quality Rules. 
 
 Rule V. : If both premisses are affirmative, the conclusion, if 
 any, is affirmative. 
 
 Rule VI. : If one premiss, and one only, is negative, the con- 
 clusion, if any, is negative. 
 
 Rule VII. : From two negative premisses nothing can be in- 
 ferred. 
 
 Rule VIII. : If the conclusion is negative, one of the premisses 
 is negative. 
 
 Rule IX. : If the conclusion is affirmative, both premisses are 
 affirmative. 
 
 (iv.) Quantity Rules. 
 
 Rule X. : Two particular premisses prove nothing. 
 Rule XI. : If one premiss be particular, the conclusion is 
 particular. 
 
 (i.) Struoiural Rules. 
 
 The requirement that the Syllogism shall be simple provides us 
 with the first rule. 
 
 Rule I. : Every S.C.D. Syllogism contains three propositions, 
 and three only. 
 
 The second rule is similar in character to Rule I.
 
 21S THE PROBLEM OF LOGIC [VIL ii. 
 
 Bide II. : Every S.C.D. Syllogism contains three different terms, 
 and three only. 
 
 Qua categorical, the Sj'llogisin must contain terms as its structural 
 elements ; nor can it contain more than four terms, since, according 
 to the Principle of Formal Validity, the conclusion can add no new 
 term to those contained in the premisses. But (given that the 
 Syllogism is distributional*) the Postulate of ^Mediation requires that 
 two of these terms shall coalesce so as to mediate between the two 
 extremes. There must, therefore, be three terms, and three only, 
 in the S.C.D. Syllogism. 
 
 For the keeping of this rule it is obviously necessary that the 
 middle term be not ambiguous ; for if it is ambiguous, it is not one 
 term, but two, so that the syllogism contains four terms in all. 
 The fallacy caused by using four terms in a syllogism is technically 
 called quaternio terminorum, or the fallacy of four terms. 
 
 (ii.) Rules of Distribution. 
 
 A third rule — the Rule of Mediation (as expressed in terms of 
 distribution) — is ordinarily and conveniently known as the FmST 
 Rule of Distribution. 
 
 Eule III. : The middle term must he distributed once at least. 
 
 The fallacy involved in breaking this rule is known technically as 
 the fallacy of Undistributed 3Iiddle. 
 
 When we say that the middle term has not been distributed in 
 either premiss, our statement, in brief, comes to this : that some of 
 the objects extensively indicated by the middle term have been 
 identified with, or distinguished from, objects indicated by one of 
 the extremes, and some, again, identified with objects indicated by 
 the other ; but the^ 'statement does not imply that objects indicated 
 bj' the two extremes have been referred to the same objects indicated 
 by the middle term — does not, in fact, imply that there has been 
 any mediation at all. 
 
 Consider the following premisses : 
 
 ' All swallows are fond of insects.' 
 ' All hedgehogs are fond of insects.' 
 
 Here the predicate ' creatures fond of insects ' is undistributed 
 in both premisses. Swallows are stated to be some of these crea- 
 tures, and so are hedgehogs ; but there is no ground for supposing 
 that the ' some ' in the two cases refers to the same creatures. 
 Between ' All P's are ^M^'s ' and ' All S's are IVIg's ' there is no logical 
 connexion possible. To insist that there is would be to violate the 
 Postulate of Mediation and to go beyond the accepted data. 
 
 • See Chapter XXVIII., ' Unorthodox Syllogistna.'
 
 Chap. XXII.] RULES OF THE SYLLOGISM 219 
 
 The Principle of Inference, though implied in all the other rules, 
 finds its explicit embodiment in what is customarily called the 
 Second Rule of Distribution. 
 
 Rule IV. : If a term is distributed in the conclusion, it must 
 have been previously distributed in one of the 
 premisses. 
 
 To say that no term must be distributed in the conclusion that 
 was not distributed in one of the premisses is only another way of 
 saying that we must not, in the conclusion, make any statement 
 about all the objects indicated by a term when in the premisses we 
 have only referred to some of them — i.e., that we must not intro- 
 duce in the conclusion a reference to objects concerning wliich the 
 premisses make no statement. We must not trespass beyond the 
 accepted data. 
 
 The fallacy involved in breaking this rule is called an illicit 
 process of the m/xjor when it is the major term that is used first in 
 its partial, then in its total, extension, and an illicit process of the 
 minor when it is the minor term that is treated in this way. Both 
 fallacies are included under the name of Illicit Process. 
 
 (iii.) Quality Rules. 
 
 In discussing these five rules we propose, for brevity's sake, to 
 use the expressions ' a class,' ' two classes,' etc., in the sense of 
 ' the objects contained in a class,' ' the objects contained in two 
 classes respectively,' etc. ; and where such expressions as ' state- 
 ment of relation,' ' statement of identity,' ' statement of non- 
 identity ' would be inconveniently cumbersome, we propose to 
 substitute for them the words ' relation,' ' identity,' ' non- 
 identity,' etc. 
 
 We may justify the Quality Rules of the Syllogism by the following 
 considerations : 
 
 There are but two possible ways in which, from statements 
 regarding the relations of two classes to one and the same third 
 class, we can infer a relation of identity or non-identity between 
 those two classes. We must argue from identity of relation to a 
 relation of identity, or from Tion-identity of relations to a relation 
 of non-identity. 
 
 Either the relations of P to M and S to M, as stated in the pre- 
 misses, must be so far identical as to necessitate the inference that 
 the classes S and P are (partly or wholly) identical, 
 
 or those same relations must be so far non-identical as to necessi- 
 tate the inference that the classes S and P are (partly or wholly) 
 non-identical. 
 
 An inference from identity of relations to a relation of non- 
 identity is as inconceiv^able as an inference from non-identity of
 
 -■20 THE PROBLEM OF LOGIC [VII. ii 
 
 relations to a relation of identity. From statements which imply 
 the identity of the relations of P to M and S to M it is evidently 
 impossible to infer a relation of non-identity between S and P ; 
 and from statements impljnng the non-identity of those relations 
 it is evidently impossible to infer that S and P stand to one-another 
 in a relation of identity. 
 
 Thus, if premisses of the same quahty give any conclusion, it 
 must be affirmative ; and if premisses of differing quality give any 
 conclusion, it must be negative. And, vice versa, an affirmative 
 conclusion can be inferred onl}^ from premisses of like quality, and 
 a ncirativc conclusion onlv from premisses of unlike quality. This 
 justities Pules V., VI., and VIII. 
 
 With regard to Rule VII., we see that if two negative premisses 
 can give any conclusion, that conclusion must be affirmative — that 
 is, it must state an identity-relation between S and P. Now, such 
 an identity-relation can be inferred only from premisses which 
 impl}^ the identity of the relations between P and M and between 
 S and M. But two negative premisses imply no more, in this 
 respect, than that these relations so far resemble one another that 
 they are both relations of non-identity. And the relation of non- 
 identity, being purely negative, admits of infinite variety, so that 
 the implied statement of such resemblance as this involves no 
 implication that the relations between P and M and between S and M 
 have with one another any positive identity whatever. Thus, two 
 negative premisses do not imply an identity of relations from which 
 any relation of identity can be inferred. An identity-relation 
 between S and P can be inferred only from statements that the 
 classes S and P both stand in identity -xclsktionB with one and the 
 same third-class, M ; and these statements are not furnished by 
 negative premisses. Thus, from two negative premisses no affirma- 
 tive conclusion can be drawn. And since the premisses are of like 
 quality, and therefore, as we have seen, a negative conclusion is 
 impossible, it follows that no conclusion can be draA\T3 at all. This 
 justifies Rule VII. 
 
 With regard to Rule IX., we have seen that if the conclusion is 
 affirmative the premisses must be of like quality — i.e., they must 
 be either both affirmative or both negative. But in discussing 
 Rule VII. we have shown that they cannot both be negative. 
 Hence they must both be affirmative, and Rule IX. is justified. 
 
 Of the five quality-rules, the two last are derivative. Rule VIII. 
 is implied in Rule V,, for if the conclusion is not affirmative, then, 
 according to Rule V,, botii premisses cannot be affirmative ; were 
 they so, the conclusion would be affirmative. 
 
 Again, Rule IX. is implied in Rule VI., for if the conclusion is not 
 negative, then, according to Rule VI., there can be no negative 
 premiss ; were there such a premiss, the conclusion would be 
 negative.
 
 Chap. XXII.] RULES OF THE SYLLOGISM 221 
 
 The Rule that nothing can be inferred from two negative pre- 
 misses is apt to cau.se difficulty. Arguments can be framed which, 
 though they violate tliis rule, seem at first sight to be syllogistically 
 
 ^^"'^^^- E.f]., What is not M is not P. 
 
 S is not M. 
 .-. S is not P. 
 
 Here, however, tlie middle term is not ' M,' but ' Not-M,' so 
 that the minor premiss is really affirmative. Otherwise we should 
 have a quaternio terminorum ; and though the argument would, no 
 doubt, in this case be valid — the two middle terms being here 
 reducible to one — yet it would not be stated in the correct form of 
 an S.C.D. syllogism. 
 
 The Rule, in fact, applies only so long as we keep to strict syllo- 
 gistic form, and deal with three terms only. If we loosely allow 
 four terms, the rule ceases to hold. 
 
 E.g., All bats are-not birds. 
 
 All bats are-not unable to fly. 
 .♦. Some creatures that can fly are-not birds. 
 
 This argument makes use of four terms — ' birds,' ' bats,' ' crea- 
 tures unable to fly,' and ' creatures that can fly.' And the reasoning 
 is, therefore, not amenable to rules which apply only to arguments 
 expressed in strict syllogistic form. Obverting both the premisses 
 and also the conclusion, we obtain the following strict and valid 
 
 ^ ° ■ All bats are not-birds. 
 
 All bats are not- (unable to fly) 
 
 {i.e., are creatures that can fly). 
 .*. Some creatures that can fly are not-birds. 
 
 But now the premisses are no longer negative. 
 
 (iv.) Quantity Rules. 
 
 Rule X. : Two particular premisses prove nothing. 
 
 The abstractly possible combinations of two particular premisses 
 are 00, II, 10, 01. 
 
 00 breaks the rule of Negative Premisses (Rule VII.). 
 
 II breaks the first rule of Distribution (Rule III.). 
 
 10 and 01, containing, as they do, only one distributed term 
 apiece, leave no term to be distributed in the conclusion 
 if the first rule of Distribution (Rule III.) is to be observed. 
 Hence the conclusion must be I (Rule IV.) — an affirmative 
 conclusion inferred from a negative premiss — which is 
 impossible (Rule IX.).
 
 ooo 
 
 THE PROBLEM OF LOGIC [VII. ii 
 
 Though nothing can be inferred from two particular propositions 
 in the usual sense of the word ' particular,' yet from the two 
 ■pluratives ' Most M's are P's,' ' Most M's are S's,' the conclusion 
 ' Some S's are P's ' can be drawm, — ' most ' being equivalent to 
 ' more than half.' The conclusion, however, is really drawn, in 
 last resort, from two universal propositions. For let C represent 
 the ^I's that the premisses assert to be both S's and P's,* and 
 wliich have thus been implicitly stated to belong to the extensions 
 of both classes. Then the above-given argument is equivalent to — 
 
 All C's are P's. 
 
 All C's are S's. 
 
 Some S's are P's (Fig. 3, Darapti). 
 
 Example : 
 
 Most dogs are fond of worrying cats. 
 Most dogs are fond of fetching sticks. 
 ,-. Some creatures fond of fetching sticks are creatureu fond of 
 worrying cats. 
 
 Let C represent the dogs that are implicitly asserted in the pre- 
 misses to be members of both the majorities in question. Then the 
 argument is virtually equivalent to the following : 
 
 All the dogs C are fond of worrying cats. 
 All the dogs C are fond of fetching sticks. 
 
 Again, m the syllogism — 
 
 Some M's are P's. 
 Some M's are S's. 
 
 .*. Some S's are P's. 
 
 if the ' Some M's ' in the one premiss are intended by the arguer 
 to be the same M's as the ' Some M's ' in the other premiss, then 
 the argument, though incorrectly expressed, is valid. ' Some 
 M's ' represents a definite class, and all the objects which this class 
 contains are stated to be both P's and S's. Thus the fallac)'' of 
 Undistributed Middle does not occur. Let C represent the common 
 middle term which has been incorrectly labelled ' Some M's.' Then 
 the argument runs : ' All C's are P's,' ' All C's are S's '; therefore 
 ' Some S's are P's ' — a syllogism in Fig. 3 (Darapti). 
 
 * If two majorities within one and the same extension did not overlap, the whole 
 extension would be less than the sum of its parts, and this is impossible. There- 
 fore some of the .M's have been implicitly asserted to be both S's and P's, and 
 Ihcee. M's we call C's.
 
 Chap. XXII.] RULES OF THE SYLLOGISM 
 
 223 
 
 Rule XI. : If one premiss be particular, the conclusion is par- 
 ticular. 
 
 The premisses must be either both affirmative, or one affirmative 
 and one negative (Rule VII.)- 
 
 1. If both are affirmative, then A and I, as II is impossible 
 
 (Rule X.). 
 This combination contains one distributed term, necessarily 
 
 the middle tei*m (Rule III.), leaving no tcrin to be dis- 
 
 tributed in the conclusion. 
 Therefore the conclusion is I (Rule IV.). 
 
 2. If one is affirmative and one negative, then either and A 
 
 or I and E. In either case the premisses distribute only 
 two terms. One of these must be the middle term 
 (Rule III.), leaving one term, and one only, to be dis- 
 tributed in the conclusion (Rule IV.). But the con- 
 clusion must be negative (Rule VI.). 
 Therefore it must be 0. 
 N.B. — The reverse of this rule — i.e., the statement that a par- 
 ticular conclusion necessitates a particular premiss — is not true. 
 The only cases, however, in which we find a particular conclusion 
 without a particular premiss are those in which the premisses 
 assume more than is required in order to prove the conclusion. 
 
 Our development of the rules of the S.C.D. Syllogism from its 
 definition will have served to bring out the following main point : 
 that these rules are an expression of the Postulate and the Prin- 
 ciple of Inference, the Postulate insisting on the necessity of a 
 mediating link, and the Principle forbidding us to draw any con- 
 clusion not fully implied in the premisses. The two fundamental 
 rules of the S.C.D. Syllogism are the two Rules of Distribution 
 which respectively embody the requirements of the Postulate and 
 the Principle. 
 
 But our presentment of the unity of syllogistic process may be 
 carried one stage further. The Postulate and the Principle are 
 not independent of each other. The Principle, in its application 
 to the S341ogism, involves the Postulate ; and the Principle of 
 Inference (in association with the Postulate, where the reasoning is 
 Mediate) is just the Principle of Formal Validity which, in its more 
 positive aspect, as an Identity Principle, may be formulated thus : 
 // a given proposition or set of propositions is acc&pted, then whatever 
 further propositions are implied in what is thus admitted, these, and 
 these alone, mxist he accepted. And the significance of the word 
 ' implied ' is twofold. It includes a direct reference to the rule not 
 to go beyond the accepted data ; and, as applied to Mediate In- 
 ference, indirectly presupposes the conditions for there being any 
 data at all — i.e., it implies the Postulate of Mediation.
 
 224 THE PROBLEM OF LOGIC [Vn. ii. 
 
 We proceed now to the practical application of the criteria of 
 louical validity wliich we possess in the Rules of the S.G.D. Syllo- 
 gism to the logical sifting of tlie 256 possible forms of Syllogism. 
 
 If any one of these forms of reasoning is such as to break none of 
 the rules of the S.C.I). Syllogism, the form is valid ; if it breaks 
 any one of them, it is invaUd. 
 
 The Discovery of the Valid Forms. 
 
 I. Disregarding varieties of figure, we first plot down the sixty- 
 four abstractly possible moods, with a view to eHminating those 
 which are illogical in every figure.* 
 
 AAA 
 
 lAA 
 
 EAA 
 
 OAA 
 
 AAI 
 
 lAI 
 
 EAI 
 
 OAI 
 
 AAE 
 
 lAE 
 
 EAE 
 
 OAE 
 
 AAO 
 
 lAO 
 
 EAO 
 
 OAO 
 
 AIA 
 
 IIA 
 
 EIA 
 
 OIA 
 
 All 
 
 III 
 
 EH 
 
 on 
 
 AIE 
 
 HE 
 
 EIE 
 
 OIE 
 
 AIO 
 
 HO 
 
 EIO 
 
 010 
 
 AEA 
 
 lEA 
 
 EEA 
 
 OEA 
 
 AEI 
 
 lEI 
 
 EEI 
 
 OEI 
 
 AEE 
 
 lEE 
 
 EEE 
 
 OEE 
 
 AEO 
 
 lEO 
 
 EEO 
 
 OEO 
 
 AOA 
 
 lOA 
 
 EOA 
 
 OOA 
 
 AOI 
 
 lOI 
 
 EOI 
 
 001 
 
 AOE 
 
 lOE 
 
 EOE 
 
 OOE 
 
 AOO 
 
 100 
 
 EOO 
 
 000 
 
 We now proceed to cancel all moods guilty of the following 
 fallacies : 
 
 (1) Two negative premisses. 
 
 (2) Two particular premisses. 
 
 (.3) One negative premiss, affirmative conclusion. 
 
 (4) Negative conclusion, two affirmative premisses. 
 
 (5) One particular premiss, universal conclusion. 
 
 As the result of these eliminations, we are left with twelve moods 
 still to be tested in each of the four figures. 
 
 So far we have not had recourse to the two rules of distribution, 
 the reason being that they are not generally applicable unless the 
 figure is known as well as the mood. But in the single case of the 
 mood lEO we have a mood which may be rejected for breaking a 
 rule of distribution apart from any reference to a particular figure. 
 
 * For a much neater, but, from our point of view, less appropriate method, see 
 Wclton, ' A Manual of Logic,' Bk. IV., ch. iii., vol. i., pp. 319-322.
 
 Chap. XXII.] RULES OP THE SYLLOGISM 225 
 
 For the conclusion distributes its predicate, the major term ; 
 but as this term cannot in any figure be distributed in the major 
 pr?,miss, wliich is an I proposition^ there is invariably Illicit Process 
 of the Major. 
 
 Eliminating lEO, we are left with the following eleven unrejected 
 moods : 
 
 AAA, AAI, AEE, AEO, All, AOO ; EAE, EAO, EIO ; 
 lAI ; OAO. 
 
 II. We have now to find out in which of the four figures these 
 eleven unrejected or ' legitimate ' moods are valid — i.e., we have 
 to find out how many forms of Syllogism are valid in any given 
 figure. 
 
 Plotting down the figure-schemes, we have — 
 
 I. 
 
 II. 
 
 III. 
 
 IV. 
 
 M P 
 
 P M 
 
 M P 
 
 P M 
 
 S M 
 
 S M 
 
 M— S 
 
 M— S 
 
 S— P S— P S— P S— P 
 
 We now take each of the eleven moods in turn, and put it in 
 distributive form. 
 
 du dd 
 
 Thus, AAA = du ; EIO = uu, etc. 
 du ud 
 
 Comparing the distribution-scheme of AAA with each of the 
 figure-schemes in turn, we cancel this mood in any figure in which 
 it is guilty either of undistributed middle or of illicit process of the 
 major or the minor term. 
 
 du M— P 
 
 Thus, comparing du vvith S — M 
 
 du S— P 
 
 we see that none of these rules is broken. AAA is therefore 
 vahd in Fig. I. 
 
 du P— M 
 
 Again, comparing du with S — M 
 
 du S^ 
 
 we see that there is undistributed middle. AAA is therefore 
 invalid in Fig. II. 
 
 15
 
 ■226 THE PROBLEM OF LOGIC [VII. ii. 
 
 du M— P 
 
 Again, comparing <^^^i with ^I — S and with 
 
 du S— P 
 
 P— M 
 
 M — S we see that there is in each case the fallacy of 
 
 S— P 
 
 ilhcit minor. AAA is therefore invalid in Figs. III. and IV. 
 
 Proceeding in this way, we find that in each figure six of the 
 legitimate moods are vahd, so that we have in all twenty-four valid 
 forms of Syllogism. Of these, five are known as subaltern or 
 weakened forms — i.e., forms which draw particular conclusions 
 from premisses which warrant conclusions that are universal, 
 Tliis can, of course, be realized only when a universal conclusion is 
 legitimate ; and to every form of Syllogism that has a universal con- 
 clusion there will correspond a weakened form, with conclusion of 
 the same quality. 
 
 Thus, AAA gives AAIi in Fig. I. 
 EAE gives EAO in Fig. I. 
 EAE gives EAO in Fig. II. 
 AEE gives AEO in Fig. II. 
 AEE gives AEO in Fig. IV. 
 
 Since all conclusions in Fig. III. are particular, Fig. III. can have 
 no weakened form. 
 
 The nineteen forms which are not only valid but strong are 
 found conveniently tabulated in the following mnemonic verses of 
 the Traditional Logic : 
 
 Barbara, Celarcnt, Darii, Ferioque prioris ; 
 Cesarg, Camestres, Festino, Baroco sccundse ; 
 Tertia DaraptI, Dis2,mis, Datisi, Felapton, 
 Bucardo, FCrison habet ; Quarta insupcr addit 
 Bramantip, CamenGs, Dimaris, Fcsapu, Fresison. 
 
 Each of these time-honoured names contains three vowels, 
 which inform us, concerning each of the three propositions of the 
 form of Syllogism that answers to the name, whether it is A, E, I, 
 or 0. Thus, the mnemonic verses tell us that Ferison, for instance, 
 is a form in Fig. III., having major premiss E, minor premiss I, 
 and conclusion O. 
 
 Of these nineteen forms four are known as strengthened forms, 
 or strevrjthened syllogisms . A strengthened form is a strong or
 
 Chap. XXII.] RULES OF THE SYLLOGISM 227 
 
 non-weakened form whicli employs tivo universal premisses to 
 prove a particular conclusion when only one is needed. Darapti 
 and Felapton in Fig. III., Fesapo and Bramantip in Fig. IV. are 
 strengthened forms. If these are excluded from tiie list we arc left 
 with fifteen ' fundamental ' forms, valid forms that are neither 
 weakened nor strengthened. The following comparison between 
 each of the strengthened forms and the fundamental forms of the 
 same figure that have the same conclusion will serve to bring out 
 the superfluous character of the former. 
 
 MaP 
 
 Thus, Darapti, MaS, is obviously not so adequate or economical 
 SiP 
 
 MiP MaP 
 
 a form as Disamis, MaS , or Datisi, MiS . 
 
 SiP_ SiP_ .. 
 
 MeP MoP 
 
 Again, Felapton, MaS , is less efifective than Bocardo, MaS , or 
 SoP SoP 
 
 MeP ' 
 
 Ferison, MiS. 
 SoP 
 
 PeM 
 So, in Fig. IV., Fesapo, MaS , is less effective than Fresison, 
 
 SoP 
 
 PeM PaM PiM 
 
 MiS ; and Bramantip, MaS ^ less effective than Dimaris, MaS . 
 SoP SiP SiP 
 
 CHAPTER XXIII. 
 
 Vn. (iii.) EXERCISES ON THE STRUCTURE OF THE S.C.D. SYLLOGISM. 
 
 Exercise I. — Show that there cannot be more than four syllogistic 
 figures. 
 
 The number of figures is limited to the number of wavs in which 
 the position of the middle term can be varied in the two premisses. 
 M may be subject or predicate in either premiss, thus offering four 
 possible variations in position. 
 
 15—2
 
 22S THE PROBLEM OF LOGIC [VII. iii. 
 
 Exercise 2. — Prove that there must always be in the premisses 
 one distributed term more than in the conclusion. 
 
 If no term is distributed in the conclusion, yet one term — the 
 Middle — must be distributed in the premisses. And if the major 
 or the minor term is distributed in the conclusion, it must be dis- 
 tributed also in the premisses, as well as the middle term. 
 
 Exercise 3. — If tlie major premiss in Fig. I. were particular 
 affirmative, Avhat fallacy would be committed ? 
 
 P, being undistributed in the major, must be undistributed in 
 the conclusion. The conclusion must therefore be affirmative, and 
 therefore the minor must also be affirmative. The middle term is 
 therefore undistributed in both premisses, and the fallacy is that 
 of undistributed middle. 
 
 Exercise 4. — If the major premiss in Fig. II. were particular, 
 what fallacy would be committed 1 
 
 P, being undistributed in the major, must be undistributed also 
 in the conclusion, which is therefore affirmative. Both premisses 
 are therefore affirmative, and we have the fallacy of undistributed 
 middle. 
 
 Exercise 5. — Why is it that the moods EAO, EIO are valid in 
 all the four figures ? 
 
 The question is equivalent to the following : Given that the 
 conclusion is 0, why is it that the premisses EA, EI can be utilized 
 to prove it in all the four figures ? 
 
 The conclusion being 0, P is distributed in the major, but, as 
 the major is E, P may be either subject or predicate there. Again, 
 M, being already distributed in the major, need not be distributed 
 in the minor. Therefore the minor may be either A or I ; and be 
 either MS or SM, as S is undistributed in the conclusion. 
 
 Hence the major may be MP or PM, and the minor MS or SM. 
 Therefore, etc. 
 
 Exercise 6. — Detect the fallacy in the following solution of the 
 problem : Given the major premiss particular, 
 find out whether the minor is A, E, I, or 0. 
 
 Since the major is given particular, the minor must be universal. 
 Again, since the major is particular, the conclusion is particular. 
 Therefore S is undistributed in the conclusion, and therefore also 
 in the minor. Therefore the minor must be an affirmative universal, 
 since the negative universal distributes hoik terms. Therefore the 
 minor is A. 
 
 This conclusion, though correct, has been incorrectly drawn. 
 The fallacy occurs in inferring that, because S is undistributed in 
 the conclusion, it must therefore be undistributed in the minor
 
 Chap. XXIII.] EXERCISES OX THE SYLLOGISM 229 
 
 premiss.* This inference would, however, be valid if the conclusion 
 were known not to be a weakened conclusion. It is true that in 
 tlie ease of Bramantip the major term, wliich is undistributed 
 in the conclusion, is distributed in the major premiss. But Braman- 
 tip may be regarded as a weakened form in this sense : that its 
 conclusion is the weak converse of the conclusion wliich can be 
 drawn in Fig. I. from the same premisses. 
 
 In connexion with the working out of exercises on syllogistic 
 structure in relation to the S.C.D. Syllogism, the following liints 
 may be found to be of service. 
 
 (i.) In the working out of these exercises, it is permissible to 
 take for granted the Rules of the S.C.D. Syllogism, and 
 these only, 
 (ii.) What is not to be taken as given : 
 
 (o) Knowledge of tlie valid forms in each figure. The 
 mnemonic verses are to be used only as a reference, 
 for purposes of identification and nomenclature, or 
 as a check in testing the correctness of one's answers. 
 Hence a result based solely on a mere reference to 
 the list of the moods which are valid in any given 
 figure is of no value at all. Results must always 
 be reasoned out from the rules of the S.C.I). Syllo- 
 gism. 
 (h) The Rules of the Figures, sometimes called the Special 
 Rules of the Syllogism. These should not be used 
 as a substitute for direct reasoning based on the 
 general Rules of the Syllogism, 
 (iii) The solution of a problem in Formal Logic should not 
 be regarded as an experimental process. It should 
 proceed by developing the implications of the given data 
 till the required solution actually unfolds itself. The 
 discussion of a syllogistic problem should always take 
 the form of a direct series of necessary inferences, and 
 should not consist in experimentally testing a number of 
 moods. 
 
 Thus if, in an exercise on the S.C.D. Syllogism, we are given 
 certain quite general data, which do not involve any specific 
 reference to mood or figure, then the best way of discovering the 
 figure or figures in which the conditions are fulfilled is to infer 
 from the given data the respective positions of the middle term in the 
 
 * Cf. H. W. B. Joseph, 'An Introduction to Logic,' ch. xii., p. 252, footnote: 
 ' Beginiiors imagine sometimes tliat the fallacy of illicit process is committed if 
 a term which is distributed in the premiss is undistributed in the conclusion. 
 This is, of course, not the case. I must not presume on more information than ig 
 given me, but there is no reason why I should not use less.'
 
 230 THE PROBLEM OF LOGIC [VII. iii. 
 
 iico premisses. Tliis is equivalent to discovering the figure from 
 first principles. It is invariably the shortest, neatest, and surest 
 method of procedure, and should always be aimed at whenever 
 possible. 
 
 The two following exercises may serve to illustrate this important 
 point : 
 
 1 . If the minor premiss of a syllogism be 0, what is the figure 
 
 and the mood ? 
 
 The minor is 0. Therefore the conclusion is 0. Therefore 
 the major term must be distributed in the major premiss. But 
 since the minor premiss is negative and particular, the major must be 
 affirmative and universal. Thus the major is A ; and, since P is 
 distributed, the order of its terms is PJM. 
 
 Again, 31, being undistributed in the major, must be distributed 
 in the minor, and must therefore be the predicate in the minor, 
 since the minor is an proposition. Hence the order of terms in 
 the minor is SM, and the syllogism is Baroco in Fig. II. 
 
 2. Prove that a universal affirmative proposition can form 
 
 the conclusion in Fig. I. only. 
 
 Both premisses must be universal affirmative, and consequently 
 distribute only their subjects. 
 
 Now S, being distributed in the conclusion, must be distributed 
 in — i.e., in this case, be the subject of — the minor premiss. 
 
 This leaves M to be distributed in the major, of which it is 
 therefore the subject. 
 
 We therefore get MaP 
 
 SaM, i.e., Barbara in Fig. I. 
 
 SaP 
 
 CHAPTER XXIV. 
 
 m. (iv.) THE ANALYSIS OF SYLLOGISMS. AXD THE REDUCTION OF 
 ARGUMENTS INTO SYLLOGISTIC FORM. 
 
 A Categorical argument, as presented for logical handhng, may 
 either be already expressed in .syllogistic form or require reducing 
 into such form. Reduction to strict syllogistic form is an indis- 
 pensable preliminary to the application of the Rules of the S.C.D. 
 Syllogism as tests of logical validity. 
 
 An important form of unsyllogized argument goes by the name 
 of Enthymeme.
 
 Chap. XXIV.] THE ANALYSIS OF SYLLOGISMS 231 
 
 An Enthymeme is u.sually defined as a defective syllogism, as 
 a syllogistic argument with one proposition suppressed. The sup- 
 pressed proposition may be either premiss or conclusion. 
 
 Enthymemes are of three orders : 
 
 In an Enthymeme of the first order the major premiss is omitted. 
 
 Example. — The soul is a simple substance, consequently it is 
 indestructible. 
 
 Here the omitted major is ' All simple substances are indestruc- 
 tible substances.' 
 
 In an enthymeme of the second order the minor premiss is omitted. 
 
 Example.— Some plants are insectivorous, for the Sundew eats 
 insects. 
 
 Here the completed syllogism runs : 
 
 The Sundew is an insectivorous organism. 
 The Sundew is a plant. 
 .'. Some plants are insectivorous organisms (Darapti). 
 
 In an enthymeme of the third order the conclusion is omitted. 
 
 Example. — A man of fashion is a fop, and all fops are fools. 
 
 In our definition of the Enthymeme we have restricted the 
 omission or defect to a single proposition. There is, however, an 
 important variety of ' defective ' argument in which two out of the 
 three propositions essential to a syllogism are omitted. Here the 
 single proposition does duty for an inference. 
 
 By way of illustration we may take the answer of the railway 
 ticket -clerk who, when asked by on old lady whether she need 
 take a separate ticket for her pet tortoise, promptly replied : ' No, 
 ma'am ; cats is dogs, and rabbits is dogs, but tortoises is hinsecs.' 
 
 Here the two propositions, ' Cats is dogs ' and ' Rabbits is dogs,' 
 are arguments reduced each to a minor premiss. For the major 
 premiss, ' Dogs are creatures that must have tickets,' and the 
 conclusion, 
 
 ' Cats 1 
 
 4 T> ui,-f ( s-re creatures that must have tickets,' 
 
 are intended, but not expressed. As for the sentence, ' No, ma'am. 
 Tortoises is Hinsecs,' wliich is equivalent to ' Since tortoises are 
 insects, they do not need tickets,' it is an enthymeme of the first 
 order, the omitted major being ' Insects are-not creatures that need 
 tickets.' 
 
 The importance of the Enthymeme lies in its being the natural 
 mode of reasoning, the Syllogism being comparatively artificial. 
 ' The argument,' says Dr. Gilbart, ' first occurs to our mind in the 
 form of an enthymeme, but when we wish to make it clearer, we 
 extend it to a syllogism.'* From the point of view of Formal 
 analysis, however — and we are not here concerned with any other 
 
 * * Logic for the Slillion.' p. 254.
 
 232 THE PROBLEM OF LOGIC [VII. iv. 
 
 interest save that of validity — the Enthymeme is a defective 
 syll' gism, and its logical correction consists in supplying the omitted 
 l')roposition in such a way that the argument becomes syllogistically 
 valid. This is, of course, not always possible. If a particular 
 premiss is given, for instance, with a universal conclusion, the 
 pseudo-enthymeme cannot possibly be expanded into a valid 
 syllogism. If a negative premiss is given witli an affirmative 
 conclusion, there is not the same hopelessness. E.g., 'All children 
 are non-combatants, for no cliildren are soldiers.' Here the conclu- 
 sion maj'" be obverted into the form ' All children are-not com- 
 batants,' and the omitted major is ' All combatants are soldiers,' 
 the argument being in Camestres (Fig. II.). 
 
 As transitional between the Enthymeme and the Sjdlogism proper 
 we may note such arguments as the following : 
 
 ' Some plants are parasites — for example, IVIistletoe.* 
 
 Here the expression ' for example ' implies both that jMistletoe is 
 a plant and also that it is a parasite. Hence there is no premiss 
 actually omitted ; the argument is abbreviative rather than defec- 
 tive. It may be fully stated thus : 
 
 Mistletoe is a parasite. 
 Mistletoe is a plant. 
 .'. Some plants are parasites (Darapti). 
 
 A similar argument is supplied by the following sentence : 
 
 The example of Demosthenes shows that some orators are made, 
 not born. 
 
 WJiere the given argument is not logically defective, it may or 
 may not be valid. Enthymemes reduced to syllogisms are neces- 
 sarily valid, for they are made valid through the very process of 
 remedying their defectiveness. But it is, of course, otherwise 
 where the argument is from the outset comj)letely stated. In this 
 case the first step must be to reduce the argument into strict 
 syllogistic form, particular care being taken to place the three 
 propositions in their proper relative places. 
 
 Example. — Xo patience is pleasant, for no painless experience 
 is patience, and no painful experiences are pleasant. 
 
 Substituting for the proposition ' Xo painful experiences are 
 pleasant (experiences) ' its obverted converse, ' All pleasant ex- 
 periences are non-painful experiences,' we obtain the argument : 
 
 All pleasant experiences are non-painful experiences. 
 All painless experiences are-not instances of patience. 
 .*. All instances of patience are-not pleasant experiences. 
 
 If we may assume that ' painless ' is equivalent to ' non-painful,' 
 this is a valid argument in Camenes (Fig. IV.). Otherwise it ex- 
 hibits quattrnio terminorum.
 
 Chap. XXIV.] THE ANALYSIS OF SYLLOGISMS 233 
 
 When an argument is presented in what appears to be .syllogistic 
 form, we must not suppose that the first, in order, of the premisses 
 is necessarily the major premiss. The major premiss is the premiss 
 that contains the major term. 
 Example : aU fungi are plants. 
 
 Some fungi are microscopic organisms. 
 .'. Some plants are microscopic organisms. 
 
 By considering the conclusion of this argument we are able at once 
 to state which is the major and which the minor term, for the 
 major term is always the predicate of the conclusion — in this case 
 the term ' microscopic organisms.' Thus the second premiss, 
 since it contains the major term, must be the major premiss. The 
 true syllogism, then, stands thus : 
 
 Some fungi are microscopic organisms. 
 All fungi are plants. 
 .*. Some plants are micrcicopic organisms (Disamis, Fig. III.). 
 
 The order of the premisses cannot, of course, affect the validity 
 of the argument or the applicabihty of tlie rules of the S.C.D. 
 Syllogism ; nor does it even affect the labelling of the mood, pro- 
 vided that we take the preliminary precaution of deciding, as above, 
 which premiss is the major and which the minor. 
 
 In the reduction of arguments to strict syllogistic form, the 
 mood and figure should be stated if the argument is valid, and the 
 mood, figure, and Formal fallacy if it is invalid. 
 
 Examples. 
 
 Example 1. — Since we must admit that all plants are not petunias, 
 and that no poodles are petunias, it follows that some plants at 
 least are poodles. 
 
 This argument may be written thus : 
 
 All poodles are-not petunias. 
 Some plants are-not pettmias. 
 .*. Some plants are poodles. 
 
 Here we have the pseudo-syllogism EOI in Fig. II., the fallacy being 
 that of two negative premisses. 
 
 Example 2. — No one can be a great logician without being a philo- 
 sopher as well. Now we must admit that Aristotle was a pliilosopher. 
 It follows, then, that Aristotle is a great logician. 
 
 Tliis argument, when expressed in strict logical form, runs thus : 
 
 All great logicians are philosophers. 
 Aristotle was a philosopher. 
 .♦. Aristotle was a great logician. 
 
 This is AAA in Fig. II. Fallacy : Undistributed middle.
 
 234 THE PROBLEM OF LOGIC [VII. iv. 
 
 Example 3. — Only suns are stars. Therefore Sirius, being ad- 
 mitted to be a sun, is also a star. 
 Logical form : 
 
 (a) All stars are suns. 
 Sirius is a sun. 
 .'. Sirius is a star. 
 
 This, again, is AAA in Fig. II. Fallacy : Undistributed middle. 
 Or— 
 
 (6) All not-suns are-not stars. 
 
 Sirius is-not a not-sun. 
 .-. Sirius is a star. 
 
 This is EEA in Fig. I. Fallacy : Two negative premisses. 
 
 Example 4. — Since only plants are seaweeds, it follows that some 
 plants are not green, for all seaweeds are not green. 
 
 Here tlie conclusion of the argument must be the proposition 
 introduced by ' it follows that.' Therefore the major term is 
 ' green (things).' Therefore the major premiss is ' All seaweeds 
 are not green things,' which, in strict logical form, reads ' Some 
 seaweeds are-not green things ' ; and we obtain as our syllogism : 
 
 Some seaweeds are-not green things. 
 All seaweeds are plants. 
 ••. Some plants are-not green things. 
 
 This is a vahd argument in Bocardo (Fig. III.). 
 
 The Fallacy of Four Terms. 
 
 A given argument is not necessarily invalid because the form 
 in wliich we happen to express it has four terms. It may be that 
 what is incorrect is not the argument, but our pseudo-distributional 
 syllogism. If so, the syllogism must be recast into a form that 
 does not involve the fallacy. Thus an argument will frequently be 
 found to contain the four terms S, P, M, and not-M. In this 
 case the ' not ' may frequently be transferred to the copula by ob- 
 version, leaving a three-term syllogism. Again, a not infrequent 
 mistake is that of stating, with regard to a syllogism, that it not 
 only has four terms, but also breaks such and such rules. But if 
 by syllogism we mean an S.C.D. Syllogism,* then, when once we 
 have convicted a so-called ' syllogism ' of a quaternio terminorum, 
 it is sufficiently stamped as non-syllogistic, and it must then be 
 futile to convict it of breaking any other rule of the Syllogism. 
 
 * Vide supra, p. 216.
 
 Chap. XXIV.] THE ANALYSIS OF SYLLOGISMS 235 
 
 Example : 
 
 No one is contemptible but the contemptuous. 
 Some cowards are not contemptuous. 
 .*. Some cowards are not contemptible. 
 
 Formal restatement : 
 
 All non-contemptuous persons are-not contemptible persons. 
 Some cowards are-not contemptuous persons. 
 .*. Some cowards are-not contemptible persons. 
 
 Here we have four terms ; but the inference is not that the 
 ' syllogism ' is invalid, but simply that we have not yet a syllogism 
 at all. We must recast this seeming syllogism into the correct 
 form of an S.C.D. Syllogism : 
 
 All non-contemptuous persons are-not contemptible persons. 
 Some cowards are non-contemptuous persons. 
 .-. Some cowards are-not contemptible persons (Fig. I., Ferio). 
 
 Or— 
 
 All contemptible persons are contemptuous persons. 
 Some cowards are-not contemptuous persons. 
 .'. Some cowards are-not contemptible persons (Fig. II., Baroco). 
 
 CHAPTER XXV. 
 
 VII. (v.) USES AND CHARACTERISTICS OB" THE FOUR FIGURES— 
 
 THE SPECIAL RULES. 
 
 Figure I. 
 
 A GLANCE at the mnemonic verses will show that Fig. I. gives 
 conclusions in all the four forms, and therefore serves every purpose 
 of affirmation or denial, partial or total. We are therefore able, 
 in Fig. I., to make universal statements, affirmative or negative, and 
 to support them ; we are also able to contradict any statement, 
 affirmative or negative, and to support our denial in tliis same 
 figure. 
 
 Further, Fig. I. is the only figure in which the subject and predi- 
 cate of the conclusion are respectively subject and predicate in the 
 premisses in which they occur, so that, as Professor Carveth Read 
 puts it, ' the course of argument has, in its mere expression, an easy 
 and natural flow.'
 
 236 THE PROBLEM OF LOGIC [VII. v. 
 
 Figure II. 
 
 In Fig. II. only negative conclusions can be proved, so that its 
 use is greatly restricted as compared -with that of Fig. I. Still, 
 where suggested assertions about a given subject have to be rejected, 
 the use of the Second Figure is more natural than that of the First. 
 Thus the question maj'' be ' Can we accept the statement that this 
 plant is a Dicotyledon ?' and the answer ' No, we must reject it ; 
 for we have already accepted the statement that this plant does 
 not possess a character common to all Dicotyledons.' 
 
 All Dicotyledons are plants possessing two primary leaves, and 
 
 only two. 
 This plant is-not a plant possessing two primary leaves and 
 
 only two. 
 . . Tliis plant is-not a Dicotj-ledon (Camestres). 
 
 Thus Fig. II. is essentially the figure of denial. By means of it 
 we can go on denpng a succession of disjunctively accepted predi- 
 cations until, by a process of exclusion, we are enabled to accept 
 the one predication that remains. Thus Fig. II. may play an im- 
 portant part in the service of the Disjunctive Syllogism. 
 
 Figure III. 
 
 In this figure onh' particular conclusions can be proved. Its one 
 undoubted advantage is seen in deahng with singular propositions ; 
 for it is the only figure in which the singular name can be subject in 
 both premisses. 
 
 Example : 
 
 Socrates is a warrior. 
 Socrates is a philosopher. 
 .*. One philosopher at least is a warrior (Darapti). 
 
 Figure IV. 
 
 The Fourth Figure is tolerated by logicians only because its 
 inclusion is necessary for the structural completeness of syllogistic 
 theory. It formed no part in the logical system of Aristotle, but 
 was added by Galen some centuries later. It is, however, a per- 
 fectly valid form of reasoning, and, as such, resists the effort to 
 exclude it from the Syllogism. Thus, Mr. Joseph, who starts by 
 rejecting the forms of the fourth figure, is compelled to introduce 
 tliem as ' indirect moods of Fig. I.'* 
 
 It is easy to show that in no case can Fig. IV. improve on Fig. I. 
 
 • See 'An Introduction to Logic,' ch. xii,, pp. 246, 258, 261 ; ch. xiv., p. 301.
 
 Chap. XXV.] THE FOUR FIGURES 237 
 
 For, in the case of Bramantip, Camenes, and Dimaris, the pre- 
 misses, when transposed, at once give conclusions in Fig. I. in tlie 
 forms Barbara, Celarent, and Darii respectively, and the reasoning 
 in these latter forms has that easy flow which is entirely lacking in 
 Fig. IV., where Subject and Predicate in tlie conclusion are respec- 
 tively Predicate and Subject in the premisses. Moreover, in the 
 case of Bramantij}, the conclusion is weaker than the conclusion 
 drawn in Fig. I. from precisely the same premisses. Thus, given 
 the statements, ' All grasses are monocotyledons,' and ' All mono- 
 cotyledons are flowering-plants,' the conclusion in Fig. IV. is 
 * Some flowering-plants are grasses,' the weak converse of the con- 
 clusion in Fig. I., ' All grasses are flowering-plants.' 
 
 As regards Fesapo and Fresison, the former, as a ' strengthened ' 
 form, is inferior in effectiveness to Fresison ; and Fresison itself is 
 Ferio over again with the premisses simply converted ; but it is 
 Ferio with its easy flow rendered awkward, and to that extent is 
 inferior to it. Hence Fig. IV. is in all its forms a less effective 
 instrument than Fig. I. 
 
 The Special Rules of the Four Figures. 
 
 Each of the four figures has its owti special rules, deducible from 
 the more general rules of the Syllogism in conjunction with the 
 specific form of the figure. 
 
 Figure I. 
 
 Eule 1 . The minor premiss must he affirmative. 
 
 Proof : If possible, let it be negative. 
 
 Then the major must be affirmative (Rule 
 
 VII.), and the conclusion must be negative 
 j^j_p (Rule VI.). 
 
 q ,T Therefore the major term will be undistri- 
 
 buted in the premiss and distributed in the 
 
 S — P conclusion, and we have the fallacy of illicit 
 
 ~^~" major (Rule IV.). Hence the minor premiss 
 
 cannot be negative in Fig. I., and must therefore 
 
 be affirmative 
 
 Hule 2. The major premiss must he universal. 
 
 Proof : The minor premiss being affirmative, the middle 
 term is there undistributed, and must therefore 
 be distributed in the major premiss (Rule III.). 
 But it is subject in this premiss. Hence the 
 major premiss must be universal.
 
 23S THE PROBLEM OF LOGIC [VII. v. 
 
 Figure II. 
 Bide I. One premiss must be negative. 
 
 -P -^^ Otherwise there will be undistributed middle. 
 ^ ^^ Hence the conclusion in Fig. II. must always be 
 S— P negative (Rule VI.). 
 
 Fulc 2. The major premiss must be universal. 
 
 Proof : The conclusion being negative, P is distributed 
 there, as, therefore, also in the major premiss 
 (Rule IV.). Hence, since P is subject there, that 
 premiss is universal. 
 
 Figure III. 
 
 Bule 1. The minor premiss must be affirmative. 
 
 Proof : If it were negative, the conclusion would be 
 
 j^j p negative, and the major premiss affirmative. 
 
 jyj g But tliis involves illicit major (Rule 4). Hence 
 
 r — — the minor premiss must be affirmative. 
 
 Rule 2. The conclusion must be particular. 
 
 Proof : As the minor premiss is affirmative, S is undis- 
 tributed there, being predicate, and is therefore 
 undistributed in the conclusion also (Rule IV.), 
 which must therefore be particular. 
 
 Figure IV. 
 
 Bule I. If the major is affirmative, the minor is universal. 
 Bule 2. If the minor is affirmative, the conclusion is particular. 
 Bule 3. If the minor is negative, both premisses are universal. 
 Bule 4. If either premiss is negative, the major is universal. 
 
 The proof of these rules may suitably be taken as an exercise.
 
 Chap XXVL] THE DICTA 239 
 
 CHAPTER XXVL 
 
 V^I. (vi.) THE DICTA. 
 
 The view of propositional import which we have assumed as 
 fundamental for a strictly Formal treatment of Logic has been an 
 Identity-view. According to this view, the proposition ' All S's 
 are P's ' means 'Each of all S's is ( = "is identical with") a P.' 
 The Identity-view, so interpreted, requires (a) that both subject 
 and predicate terms be read in extension, and (b) that the extensive 
 reading be itself understood in a distributive, and not in a collective, 
 sense. 
 
 At this point it will be convenient to consider a view of the 
 import of a categorical proposition which is in some respects akin 
 to that of Formal Identity-import, though by no means to be con- 
 fused with it. The two views of Import are akin in this respect, 
 that in each the two terms of the proposition are read in extension. 
 The difference between these views depends on the way in which 
 this extension is referred to. In the case of the Identity-view 
 both terms are referred to distributively. In the case of the other 
 view — usually known as the Class-inclusion view — the extension of 
 the subject-term may be referred to either distributively or col- 
 lectively, but that of the predicate-term is always collectively under- 
 stood. 
 
 The definition of the Class-inclusion view of Propositional Import 
 includes the following essential marks : 
 
 1. The relation between subject and predicate must be that of 
 
 inclusion in, or exclusion from, a class. 
 
 2. The extension of this predicate-class must be referred to 
 
 collectively. 
 
 The proposition ' All S's are P's,' read on tliis view, is equivalent 
 to the statement ' All the S's (distributively) are included in the class 
 P,' or, ' All S (collectively) is included in the class P.' If we assume 
 that both terms are read collectively, the fourfold scheme of propo- 
 sitions will be thus expressed : 
 
 A. All S (collectively) is included in P. 
 E. All S (collectively) is excluded from P. 
 1. Some S (collectively) is included in P. 
 0. Some S (collectively) is excluded from P. 
 
 These propositions do not profess to express the relations which, 
 in fact, may hold between two classes, S and P ; what they formulate 
 is not ' relations,' but statements about relations. Were it the 
 intention to consider the relation directly, we should require, for
 
 240 
 
 THE PROBLEM OF LOGIC 
 
 [VII. vi. 
 
 exhaustively representing the possible relations between S and P, 
 a fivefold scheme of propositions : 
 
 1. S wholly includes P. 
 
 2. S is wholly included by P. 
 
 3, S coincides with P. 
 
 4. S partially includes and 
 partially excludes P. 
 
 5. S entirely excludes P. 
 
 The So-called ' Principles ' of the Syllogistic Figures. 
 
 1. Formal Enunciations of the Dicta, on the Class-inclusion View 
 
 of Statement-import. 
 
 The principle of Fig. I. has been called the Dictum de Omni et 
 Nullo, and may be enunciated as follows : 
 
 // we can assert of the whole of a class that it is included in, or 
 excluded from, a second class, we can assert the same of any 
 part of that whole class. 
 
 The principle of Fig. II. has been called the Dictum de Diverse, 
 and may be enunciated as follows : 
 
 // we can assert of one class that it is included in, and of another 
 tfuit it is excluded from, a third class, we may also assert the 
 mutual exclusion of the two classes.
 
 Chap. XXVI. THE DICTA 241 
 
 The principle of F\rr. III. has been called the Dictum de Exemplo 
 et de Excepto, and may be enunciated as follow.s : 
 
 // we can assert of two classes that they both inclufle one and the 
 same third class, we can also assert that they 'partially include 
 each other ; and if we can assert of two classes tluit the one 
 includes a third class which the other excludes, we can also 
 assert that part of one of these classes is excluded from the 
 other. 
 
 The principle of Fig. IV. has been called the Dictum de Reciproco. 
 Its formulation is necessarily so cumbrous that it seems idle to 
 present it. 
 
 These dicta have been very variously enunciated, the enunciations 
 varying with the view of import adopted. In the formulations 
 given above the class-inclusion view of statement-import has been 
 adopted, both subject and predicate being treated as classes, and 
 both referred to collectively. It follows from this way of interpreting 
 the Dicta that the subject-class is the class as indicated by the 
 subject-term as quantified in the premiss. Thus, in the premiss, 
 ' AD S is included in P,' ' All S ' is the one class, and ' P ' the other. 
 So, again, in the ji remiss, ' Some S is included in P,' ' Some S ' is 
 the one class and ' P ' the other. Where, as sometimes in Fig. III., 
 the subject-term M is differently quantified in the two preTiiisses, 
 the ' third class ' referred to in the Dictum must be taken to be 
 that mdicated by ' Some M.' 
 
 2. The Logical Status of the Dicta. 
 
 A logical principle, in order that it may be efficiently regulative, 
 must satisfy two conditions : 
 
 (i.) Its application must lead to unambiguous results, 
 (ii.) It must admit of being used as a criterion to test the sound- 
 ness of all relevant cases. 
 
 Of these two requirements it is the first alone which is satisfied by 
 the Dicta. 
 
 (i.) That the application of the Dicta furnishes unambiguous 
 results may, in the case of the Dictum de Omni ct Nulla, be set forth 
 as follows : 
 
 The whole class (All M) is statedly 
 All M is P included in the class P. 
 
 Barbara : All S is M = .'. Since (All S) is statedly con- 
 
 .-. A\\ S is P tained in the whole class 
 
 * ' ^-^ (All M), we must admit that 
 
 it is included in the class P. 
 
 IG
 
 242 
 
 THE PROBLEM OF LOGIC 
 
 [VII. vi. 
 
 Celarent 
 
 All M is-not P 
 All S is M 
 
 All S is-not P 
 
 The whole class (All M) is statedly 
 excluded from the class P. 
 
 Since (All S) is statedty con- 
 tained in the whole class 
 (All M), wc must admit tliat 
 it is excluded from the class 
 
 Darii 
 
 All M is P 
 Some S is M = 
 
 Some S is P 
 
 The whole class (All M) is statedly- 
 included in the class P. 
 
 Since (Some S) is statedly con- 
 tained in the whole class 
 (All M), we must admit that 
 it is included in the class P. 
 
 All M is-not P 
 Ferio : Some S is M 
 
 .-. Some S is-not P 
 
 The whole class (All M) is statedly 
 excluded from the class P. 
 
 Since (Some S) is statedly con- 
 tained in the whole class 
 (All M), we must admit that 
 it is excluded from the class 
 
 Thus the Dictum de Omni et Nullo, when applied to the four valid 
 forms of Syllogism in Fig. I., leads in each case to an unambiguous 
 result ; it enables us to draw from the premisses the correct con- 
 clusion in regard to both quality and quantity. ' The whole class ' 
 in all four moods is ' All M.' The ' part ' is either 'All S ' or 
 ' Some S,' and is stated in each minor premiss to be included in 
 ' All M.' And we see that in all four moods, in strict accordance 
 with the Dictum, what is asserted of ' All M ' may be asserted in 
 the same sense of the part ' All S ' or ' Some S.' 
 
 The Dictum de Diverse may similarly be shown to fulfil the first 
 requirement of a logical principle. 
 
 Thus, if tlie class All P is statedly contained in, and the class 
 All S statedly excluded from, a third class M, wc must admit that 
 they are excluded from each other — i.e.. All S is excluded from 
 All P, or All S is-not P. This is the reasoning of the form Camestres. 
 
 The application of tlie Dictum may similarly be shown to yield 
 unambiguous results in the case of the remaining forms in the 
 Second Figure. Thus, in the case of Baroco, the conclusion is that 
 All P and Some S exclude each other — i.e., Some S is-not P. 
 
 The first part of the principle of Fig. III., known as the Dictum 
 de Exemplo, concerns the three affirmative forms of Syllogism in the 
 Third Figure. We may illustrate its application in the case of 
 Darapti : 
 
 We assert of the two classes S and P that each includes a third
 
 < HAP. XXVL] THE DICTA 243 
 
 class (All M). Hence S and P must partially include each other — • 
 i.e.. Some S is P. 
 
 The second part of the principle, known as the Dictum de Excepto, 
 concerns the three negative forms of the Third Figure. We may 
 illustrate its application in the case of Bocardo : 
 
 We assert of the two classes, S and P, that S includes a third 
 class (Some M) which P excludes. Hence we can also assert that 
 part of S is excluded from P — i.e., that Some S is-not P. 
 
 (ii.) We have now to point out that as tests or regulative principles, 
 even within the narrow limits of their own appropriate provinces, 
 the Dicta have no value whatsoever. They cannot be used to sift 
 the pseudo-forms of Syllogism in their respective figures from the 
 logically valid forms. They are, at best, mere descriptive state- 
 ments of the type of argument characteristic of the valid forms in 
 their several figures. They are abstracts from these forms, not 
 rules for them. In a word, they are not principles but figure-heads. 
 
 Let us take the Dictum de Omni et Nulla. If it be a logical 
 principle, we must expect it to be regulative with regard to all 
 syllogisms, valid or invalid, that are of the form characteristic of 
 Fig. I. ; for is it not put forward as the ' principle of Fig. I.' ? 
 
 But suppose the pseudo-syllogism AEE, Fig. I., expressed as 
 follows : ' All M is P, All S is-not M, therefore All S is-not P,' desires 
 to be tested by this principle. The Dictum can only reply that 
 AEE, Fig. I., having, unfortunately, a negative minor, does not 
 satisfy the conditions requisite for the Dictum's applicabihty. 
 ' But,' says AEE, Fig. I., ' am I not in Fig. I., and are you not 
 the principle of Fig. I. V ' No,' answers the Dictum, ' I am not 
 the principle of Fig. I. altogether, but only a principle of the valid 
 forms in Fig. I. I was only put in office over these moods after 
 they had been sifted from the others. You therefore do not come 
 under my jurisdiction.' 
 
 Tliis is perfectly just. AEE, Fig. I., should have applied to the 
 ' Special Rules,' Fig. I., to have its inference tested. The Special 
 Rules of any figure, so far as they serve any useful function other 
 than that of the General Rules of the Syllogism, constitute the 
 test of admission to the figure in question. No mood can be ad- 
 mitted among the valid forms of Fig. I. unless (1) its major premiss 
 is universal, and (2) its minor premiss is affirmative. 
 
 Precisely similar reasoning would show that the Dicta of the 
 Second and Third Figures are but ' principles ' of the valid forms 
 in their respective figures, and therefore cannot be employed to 
 sift the valid forms from the invahd. 
 
 Our conclusion, therefore, is that the sole function of the Dicta 
 is to draw the proper conchisions from whatever premisses success- 
 fully run the gauntlet of the Rules of the Syllogism and the Special 
 Rules of the Figures. 
 
 16—2
 
 2U THE PROBLEM OF LOGIC [VIL vi. 
 
 The right order of precedence, then, is the following : 
 
 L The Principle of Logical Validity and the Postulate of 
 Mediate Inference as embodied in the definition of the 
 S.C.D. Syllogism. These are the true principles of syllo- 
 gistic reasoning. 
 
 •2. The General Rules deduced from these. 
 
 3. The Special Rules of the figures, deduced from the General 
 
 Rules. 
 
 4. The Dicta. 
 
 X.B. — The Dictum de Omni et Nullo should not under any cir- 
 cumstances be claimed as the source of the authority of the special 
 Rules of the First Figure. It is impossible to infer the latter from 
 the former (as it is customary to do) without putting the cart 
 before the horse. The Sjiecial Rules can be justified only by the 
 General Rules taken collectively — i.e., by tlie system of general 
 rules. 
 
 3. The Dictum de Omni et Nullo in its relation to Deduction. 
 
 The function of this Dictum has been identified with that of 
 Deduction or Deductive Method. Deduction is defined as ' the 
 applying of a general law or rule to particular cases,' and it is then 
 pointed out that this is precisely the function of the Dictum. It is, 
 therefore, important for us to examine whether this contention 
 can be reasonablj'' supported. Is it, we ask, the function of the 
 Dictum to apply general laws to particular cases ? 
 
 Assuming for the moment that it is logically permissible to 
 characterize the function of a principle of Inference as that of apply- 
 ing a universal to particulars, we have still to ask whether the uni- 
 versal which the Dictum applies is any universal or only a special 
 kind of universal. 
 
 This question can be unambiguously answered. In so far as the 
 dictum is enunciated on the class-inclusion view, it is clear that the 
 only universals or laws which it can possibly be made to apply 
 are the laws which can be expressed in class-inclusion form. 
 Where a law can be adequately represented, diagrammatically, by 
 the total inclusion of one circle in another, or by the exclusion, 
 mutual and total, of both, its application to particular cases may 
 then be guaranteed by the Dictum. 
 
 Now, if Science were still based on the scholastic metaphysics of 
 the Middle Ages, it might be possible to apply its laws uniformly 
 in this very simple way. But as Science stands to-day the sugges- 
 tion is ridiculous. Laws that could be effectively ajiplied on the 
 Dictum principle would have to be sought, if anywhere, in those 
 classification-tables in which species are still subsumed under
 
 Chap. XXVI.] THE DICTA 245 
 
 genera ; but as they are ' eonnotation-tables,' the Dictum lias even 
 here no natural, direct applicabihty. 
 
 But there is a further and a more deep-reaching hmitation which 
 makes it impossible to allow that the Dictum can apply, in an 
 effective scientific way, even the small group of laws to which, on 
 any given view of propositional import — be it the Class-inclusion 
 view, the Identity view, or any other — its function is restricted. 
 We express this limitation when we say that a law, applied in accord- 
 ance with the Dictum, cannot account for the differences among the 
 particulars to which it is intended to apply. The distinction 
 between the ways in which the application of Law is understood 
 by the Dictum and by Science respectively is in this respect radical. 
 
 Let us imagine that the Law of Gravitation had been ' applied ' 
 to the movements of Jupiter and Saturn in the way prescribed by 
 the Dictum. It could have been applied to these movements only 
 so far as they resembled each other. It could have given no account 
 whatsoever of the points of difference. It would have been a law 
 of the heavenly bodies in general, but of no heavenly body in par- 
 ticular. It would have enabled astronomers to conclude that 
 Jupiter, moving as it did, moved like a heavenly body, and that 
 Saturn did the same. Can we suppose that the science of Astro- 
 nomy eould have profited much from a theory of gravitation so 
 applied as to be unable to explain the differences in the movements 
 of the different planets ? It must surely be obvious that the 
 Dictum has no power of giving effect to the explanatory function 
 of the laws which it pretends to apply. 
 
 If we ask for the reason why the application of a general law to 
 particular cases is understood so differently by the Dictum and 
 by the principles of Science, and why it is that the function of the 
 Dictum cannot be styled ' deductive ' in the sense already indicated, 
 the answer is to be found in the fundamental importance which 
 Deductive Science attaches to System. Hence arises the impossi- 
 bility of identifying with the Principle of Deduction any principle 
 which represents the process as an inference proceeding from some 
 single, isolated generality. Thus, the movements of Saturn and of 
 Jupiter cannot be deduced from the sole theory of gravitation, 
 apart from the laws of motion and of equilibrium with which it is 
 inextricably involved. So, again, the theorem of Euclid I. 47 
 could not be deduced from any other single geometrical theorem 
 taken in isolation from the rest. 
 
 It is true that, in reasoning from a plurality of theories, the fact 
 that we do not reason from any one of them in isolation in no way 
 precludes our reasoning from each of them singly. As Professor 
 G. F. Stout has said : ' The logical interconnexion of the theories 
 lies precisely in the fact that each supplies specifying conditions for 
 the application of the others. Consider the theory of gravitation. 
 Here laws of motion and of equilibrium, etc., are already recognized
 
 246 THE PROBLEM OF LOGIC [YIl. vi. 
 
 in understanding what this theory means. So far as this is the 
 ease, the single theory of gravitation is treated as being in itself 
 a system of theories. But so far as this is not the case, the 
 laws of motion, equilibrium, etc., with their special applications, 
 supply determining conditions which logically specify the mode 
 in which the law of gravitation must work in this or that 
 instance. They define the conditions of its application. Doubt- 
 less it also defines the conditions of their application. But we 
 may start from the single theory of gravitation, and regard these 
 others merely as specifying conditions, just as we may start from 
 the others, and regard the theory of gravitation as a specifying 
 condition.' 
 
 So far we have been proceeding on the assumption that the func- 
 tion of a logical principle might be adequately characterized as 
 that of applying universals to particulars. The assumption, how- 
 ever, involves an important ambiguity, and tends to misleading 
 conceptions as to the nature of Logical Inference. The work of 
 ' application ' involves, in fact, imaginative and constructive activi- 
 ties of mind which cannot be reckoned as processes of logical infer- 
 ence, so long as Inference is defined as a process of disimplication, 
 pledged not to trespass beyond the data. We must distinguish 
 between Method and Inference, between Deduction-process and 
 Deductive Inference. If, dissatisfied with the Dictum, we seek for 
 a conception that can inspire the systematic application of Law to 
 Fact, we must turn, not to any mere principle of Deductive Infer- 
 ence, but to that larger process of Deduction of which the aim may 
 correctly be defined as ' the valid application of systematized 
 knowledge to unsystematized fact.' 
 
 ' Application,' then, is a function of Deduction. As such, it is 
 not limited bv the Principle of Logical Inference. Through Deduc- 
 tion we bring laws to bear at the right points, and so combine in 
 fruitful co-operation the premisses from which we then draw our 
 deductive inferences. The application of System to Fact demands 
 not only inference from given evidence, but also the purposive 
 arrangement of the evidence itself. Thus, the proof of Euclid 
 I. 47, or of any geometrical theorem, cannot be reduced, without 
 remainder, to a chain of deductive inferences. It includes con- 
 structions and combinations, apart from which the purely inferen- 
 tial process would be directionless and ineffective. 
 
 That the definition of Deduction which we have given does not 
 apply to Deductive Inference may also be shown as follows : 
 Premisses may be given which cannot be said to stand to each 
 other in the relation of statement of law to statement of fact. 
 Thus, from the two premisses ' The Nile is a blessing to Egypt ' 
 and ' The Nile is infested with crocodiles ' we may infer a neces- 
 sary conclusion ; but in so doing we cannot be ."-jaid, in any sense of 
 the words, to have applied a general law to a particular fact. So,
 
 Chap. XXVL] THE DICTA 247 
 
 a»ain, in considerins^ the two premisses ' Socrates was a warrior,' 
 ' Socrates was a philosopher,' we ask in vain which of them repre- 
 sents the universal or the general law, and which the particular to 
 which it is applied. 
 
 CHAPTER XXVII. 
 
 VTI. (vii.) THE PROBLEM OF REDUCTION. 
 
 Redfction may be defined as the process through which the reason- 
 ing in a given syllogistic form is restated in another syllogistic 
 form. 
 
 There is, liowever, a narrower sense in which the term is cus- 
 tomarily used. According to the more restricted definition, Reduc- 
 tion is the process through which a syllogism in the Second, Third, 
 or Fourth Figure comes to be restated in the form of a syllogism 
 of the First Figure. Regarded from this point of view, the First 
 Figure becomes the standard or the ' perfect ' figure, and the 
 others are called ' imperfect.' There is no reason why we should 
 quarrel with the preference here given to Fig. I., so long as the 
 reasons given for it are reasons of convenience only. Fig. I. 
 expresses the principle of mediate inference in its most straight- 
 forward form, and is for that reason to be preferred. It should not 
 be supposed, however, that the reasoning in Fig. I. is a whit more 
 valid than the reasoning in Figs. II. and III., or even than that in 
 Fig. IV. The validity of the reasoning in Fig. IV. is as surely 
 guaranteed by the Rules of the Syllogism as is the validity of the 
 reasoning in Fig. I. 
 
 We accept Fig. I., then, as the standard or ' perfect ' figure on 
 the secondary grounds already alleged. We also take it as proved 
 that the only four strong and valid forms of Syllogism in Fig. I. 
 are Barbara, Celarent, Darii, and Ferio. It is on the basis of these 
 presuppositions that the processes of Reduction are developed. 
 It has been customary to recognize two kinds of Reduction : 
 (1) Direct or Ostensive Reduction ; (2) Indirect Reduction, or 
 Reductio per Impossibile. But of these two types the latter is 
 inapplicable within the limits of a strictly Formal treatment. In 
 its essence it is equiv^alent to Indirect Proof, and seeks to establish 
 the truth, and thence the validity, of a sj^llogistically inferred con- 
 clusion in an ' imperfect figure ' by proving that the denial of the 
 same leads to contradiction with the assumed truth of one or other 
 of the premisses. This whole procedure, however, presupposes 
 the truth-interest, and cannot appropriately be considered till we 
 come to treat of the theory of Deductive or Truth-Inference in a
 
 24S THE PROBLEM OF LOGIC [VII. vii. 
 
 later chapter. We shall then be meeting it imder the name of 
 ' Indirect Proof,' or Bcductio ad Ahsurdum. We need only add 
 that the fact that the Scholastic Logicians resorted to Indirect Re- 
 duction in their attempt to confirm the validity of two ' imperfect ' 
 forms of Syllogism, Baroco and Bocardo, is sufficient evidence 
 that their treatment of Logic was not so strictly and consistently 
 Formal as has sometimes been supposed. 
 
 Direct or Ostensive Reduction. 
 
 We may illustrate the operation of direct Reduction by reducing 
 Baroco and Bocardo to the First Figure. 
 
 PaM 
 
 Baroco may thus be represented in symbolic form : SoM. 
 
 SoP 
 ]\I— P 
 We wish to reduce this to the form: S— M . Conversion * per 
 
 S— P 
 
 accidens ' of the major premiss would readily give the desired 
 reduction, did it not leave us with two particular premisses. Reduc- 
 
 M— P 
 
 tion, however, to the form S — M would, if logically possible, 
 
 S— P 
 
 answer the requirement just as well. From the eduction-schemes 
 of PaM and So]\I we select those inferences that suit the above 
 form. They are MeP, the contrapositive of PaM, and SiM, the 
 obverse of SoM ; and since from our new premisses we are able 
 to draw the old conclusion SoP, we have a syllogism in Ferio, 
 Fig. I. 
 
 MoP 
 Bocardo may be represented thus in symbolic form : MaS . 
 
 SoP 
 I\I— P "^ 
 
 We ^vish to reduce this to the form : S — M . But as in 
 
 S— P 
 
 Fig. I. the major premiss must be universal, and we cannot 
 reduce the given particular major into a universal, our first step 
 
 .,„ .„ jL — ^,-_ _^. ^ „ ,, p. In 
 
 order to place the middle term in the position of predicate in the
 
 Chap. XXVIL] THE PROBLEM OF REDUCTION 249 
 
 minor premiss, we must convert tliis premiss ; but, as it is an 
 proposition, we cannot do this without obverting it tirst. 
 
 We take, then, the contrapositive of our minor premiss, and we 
 
 obtain | ^ . From these premisses in Fig. I. we are able to draw 
 
 the conclusion PiS, of which the obverted converse is the original 
 conclusion SoP. 
 
 MaS 
 
 Thus, the valid syllogism PiM is the equivalent of Bocardo, 
 
 PiS 
 
 and it is in the First Figure (Darii). Bocardo has therefore been 
 reduced to the First Figure, as desii'ed. 
 
 The reduction of the remaining forms is simpler than that of 
 either Baroco or Bocardo, and may readily be obtained by reasoning 
 out the steps as above. It seems ungracious, however, to pass 
 over without remark the ingenious key to these reductionswhich the 
 traditional Logic has preserved for us in the Mnemonic Verses. 
 
 Each of the curious names preserved in these Latin hexameters 
 contains, in addition to the vowels which give the mood, other letters 
 significant for the purpose of reduction. Thus, to take the typical 
 case of CAmEstrEs, the initial letter C indicates the initial letter 
 of that form of Syllogism in the First Figure — namely, Celarent — 
 to which this form can be reduced. The small letters m and s 
 refer to the vowels just before them, and tell us how we must handle 
 the propositions rej)resented by these vowels in order to effect the 
 desired reduction. 
 
 Thus : s { = simpliciter) indicates simple conversion of the E 
 proposition in premiss and conclusion. 
 m { = muta) indicates that the premisses have to be 
 ' changed ' — i.e., transposed. 
 
 PaJ^I 
 
 Following out these indications, we see at once that SeM 
 
 SeP 
 
 MeS "^ 
 
 reduces to F^M — {.g. to Celarent in Fig. I. 
 PeS 
 
 Of the other letters in the mnemonic verses, we should note the 
 significance of 'p, which signifies ' conversion per accidens.'* 
 
 It will have been noticed tliat Baroco and Bocardo do not reduce 
 directly to Barbara, as the initial letter B would lead us to expect. 
 
 * For an ingenious revision of the mnemonic verses, which, while freeing the 
 scheme from all meaningless letters, shows at once the figure to which any given 
 form belongs, see Professor Carveth Read, ' IvOgic Deductive and Inductive,' 
 third edition, pp. 12G, 127.
 
 250 THE PROBLEM OF LOGIC [VII. vii. 
 
 As a matter of fact, the initial B, in the names of these two forms, 
 refers to the Barbara syllogism whicli enters into their indirect 
 reduction. The c { = per contradictionem) indicates the necessity of 
 resorting to this indirect reduction. 
 
 The refusal of the ancient logicians to admit the soundness of 
 the direct reduction of Baroco and Bocardo appears to have been 
 due to their distrust of the negative term. It is true that, in so far 
 as the reduction involves the use of obversion, it cannot be justified 
 by the sole, unconditional appeal to the principles of Identity and 
 Xon-contradiction, and we are therefore bound to recognize that the 
 validity of these reductions extends no further than the adequacy 
 of the formula : ' Either an S is a P or else it is a not-P,' 
 
 CHAPTER XXVIII. 
 VII. (viii.) UNORTHODOX SYLLOGISMS. 
 
 According to Archbishop Whately,* the Syllogism is the form to 
 which all correct reasoning may ultimately be reduced. So, also, 
 J. S. !Mill writes : ' All vahd ratiocination, all reasoning by which, 
 from general propositions previously admitted, other propositions 
 equally or less general are inferred, may be exhibited in some of the 
 above forms 'j — i.e., the valid forms of the traditional Syllogism. 
 
 Now, if Whately and Mill were right, all Formal Inference what- 
 soever would ultimately be reducible to three-term syllogisms. 
 Dr. Ke_\Ties,J however, shows quite clearly that so long as we retain 
 the orthodox copula-mark, many types of argument are irreducibly 
 four-termed. Consider, for instance, the argumentum a fortiori : 
 
 B is greater than C. 
 A is greater than B. 
 •*• A is greater than C. 
 
 The irreducibility of this argument to ' syllogistic ' form is suffi- 
 ciently shown by the fact that it cannot, through reduction or other- 
 wise, be made to conform to the Dictum de Omni et Nullo. A new 
 Dictum is required, and one which must be placed on a par with the 
 old, not in subordination to it. The new Dictum may be expressed 
 in the form, ' Whatever is greater than a second thing which is greater 
 than a third thing is itself greater than that third thing.'' ^ 
 
 * ' LoRic' Book I., § 2. 
 
 t ' A System of Logic,' Book IL, ch. ii., § 1. 
 
 X ' Studies and Exercises in Formal Logic,' fourth edition, part iii., ch. vii., 
 pp. 384-388. 
 
 § The endless number of such Dicta that would be necessary in order to do 
 justice to all the possible forms of argument is in itself sufficient proof of their 
 lutility as principles.
 
 Chap. XXVIIL] UXORTHODOX SYLLOGISMS 251 
 
 ' There are an indefinite number of other arguments,' continues 
 Dr. Ke^Ties, ' which for similar reasons cannot be reduced to syllo- 
 gistic form. For example : A equals B, B equals C, therefore A 
 equals C ; X is a contemporary of Y, and Y of Z, therefore X is a 
 contemporary of Z ; A is a brother of B, B is a brother of C, there- 
 fore A is a brotlier of C ; A is to the right of B, B is to the right 
 of C, therefore A is to the right of C ; A is in tune with B, and B 
 with C, therefore A is in tune with C. All these arguments depend 
 upon principles which may be placed on a par with the dictum de 
 omni et nullo, and which are equally axiomatic in the particular 
 systems to which they belong.'* ' The claims,' adds Dr. Keynes 
 in conclusion, ' that have been put forward on behalf of the syllo- 
 gism as the exclusive form of all deductive reasoning must accord- 
 ingly be rejected.' 
 
 If we endeavour to go a step further, and consider the relation in 
 which these unorthodox arguments stand to the traditional syllo- 
 gism, we cannot do better than follow the indication given by Pro- 
 fessor Keynes, t and, instead of opposing these arguments to the 
 S.C.D. Syllogism as irreducibly ' four-termed ' — the S.C.D. Syllo- 
 gism being three-termed — admit at once that the vital difference 
 between the two forms of argument lies in a difference of copula. 
 All simple Categorical Syllogisms are three-termed by virtue of the 
 Postulate of Mediation. | For if a ' four-termed ' sj^llogism is a 
 valid mediate inference, how is the mediation effected ? If there 
 are two middle terms, where is the ' common link ' ? In last resort, 
 the two middle terms must themselves be mediated by a third, 
 through a three-term syllogism. But if the various types of Simple 
 Categorical Syllogism do not differ from each other in number ore 
 terms, it must be allowed that in these different forms of syllogism 
 the three terms are differently related. Thus, in the a fortiori 
 syllogism the symbol for the copula is the sign ^, whereas in the 
 S.C.D. Syllogism the corresponding symbol is the sign = . 
 
 The conception of a number of syllogistic types, differentiated 
 from each other by distinctions of copula, finds its fruitful develop- 
 ment in what is now kno^vn as the ' calculus of relations ' or the 
 ' logic of relatives.' Such a calculus, however, can be satisfactorily 
 elaborated only by the methods of Symbolic Logic. 
 
 * 'Studies and Exercises in Formal Logic,' fourth edition, part iii., ch. vii., 
 pp. 386, .387. 
 
 t Ibid., pp. 387, 388. J Vide above, pp. 213, 216.
 
 VIII. 
 OTHER FORMS OF SYLLOGISM. 
 
 (i.) Complex Categorical Syllogisms : Sorites and Epicheirema (ch. xxix.). 
 (ii.) The Disjunctive Syllogism (ch. xxx.). 
 (iii.) The Hypothetical Syllogism (ch. xxxi.). 
 (iv.j The Dilemma (ch. xxxii.).
 
 CHAPTER XXIX. 
 
 Vin. (i.) COMPLEX CATEGORICAL SYLLOGISMS : SORITES AND 
 
 EPICHEIREMA. 
 
 The Sorites {in Categorical Form). 
 
 A Sorites may be generally defined as a chain of reasoning which 
 links syllogism on to syllogism, but without explicitly drawing any 
 conclusions till the argument reaches its final terminus. If we 
 understand by a polysyllogism a series of syllogisms, of which each 
 proves a premiss of the syllogism that succeeds it, we may saj% 
 with Mr. Joseph,* that a Sorites is a ' polysyllogism . . . with the 
 intermediate conclusions suppressed.' The premisses which enter 
 mto such a train of argument will be of the form either of first 
 figure premisses or of fourth figure premisses. In the latter case, 
 we have the so-called Aristotelian Sorites, better known as the 
 ' Progressive ' Sorites, seeing that Aristotle does not discuss the 
 Sorites. t The form, when categorically expressed, is as follows : 
 
 S— Xi 
 Xi — Xg 
 
 J\J} xLg 
 
 X„-P 
 
 .-. S— P| 
 
 When the premisses are of the first figure, the Sorites is known as 
 the Goclenian or Regressive Sorites, Rodolphus Goclenius, Professor 
 at Marburg at the end of the sixteenth century, having first called 
 attention to this form of the argument.§ We then get the form 
 or schema : -^ p 
 
 Xo Xg 
 
 X-X^ 
 
 S-Xi 
 
 S— P 
 
 * 'An Introduction to Logic,' oh. xvi., p. 327. f Hid., p. 327. footnote. 
 X A particularly fuie illustration of this type of Sorites is given by Mr. Joseph 
 {ibid., p. 32S). ' § Ibid., p. 327. footnote. 
 
 255
 
 256 
 
 THE PROBLEM OF LOGIC 
 
 [VIII. i. 
 
 The fact that a sorites may have all its premisses in the fourth 
 figure by no means implies that the argument can be actually 
 analysed out into arguments in Fig. IV. As a matter of fact, the 
 component syllogisms of the Progressive Sorites, no less than those 
 of the Regressive Sorites, are, as they stand, syllogisms in Fig. I. 
 
 Hence there are four, and only four, possible forms of the Aristo- 
 teUan, as of the Goclenian Sorites, corresponding respectively to 
 Barbara, Celarent, Darii, and Ferio. Assuming four premisses, 
 the forms may be specified as follows : 
 
 1. Forms of the Aristotelian Sorites : 
 
 (1) All S's are X's. 
 All X's are Y's. 
 All Y's are Z's. 
 All Z's are P's. 
 
 .'. AU S's are P's. 
 
 (3) Some S's are X's. 
 All X's are Y's. 
 All Y's are Z's. 
 AU Z's are P's. 
 
 .*. Some S's are P's. 
 
 (2) All S's are X's. 
 All X's are Y's. 
 All Y's are Z's. 
 AU Z's are-not P's. 
 
 .'. AU S's are-not P's. 
 
 (4) Some S's are X's. 
 All X's are Y's. 
 AU Y's are Z's. 
 AU Z's are-not P's. 
 
 .'. Some S's are-not P's. 
 
 2. Forms of the Goclenian Sorites 
 
 (1) All Z's are P's. 
 AU Y's are Z's. 
 All X's are Y'.«. 
 
 AU S's are X 's. 
 
 .-. AUS'sareP'sT 
 
 (3) AU Z's are P's. 
 AU Y's are Z's. 
 AU X's are Y's. 
 Some S's are X's. 
 
 .•. Some S's are P's. 
 
 (2) AU Z's are-not P's. 
 AU Y's are Z's. 
 All X's are Y's. 
 All S's are X's. 
 
 .'. All S's are-not P's. 
 
 (4) All Z's are-not P's. 
 All Y's are Z's. 
 All X's are Y's. 
 Some S's are X's. 
 
 .*. Some S's are-not P's. 
 
 Analysis of the Sorites. 
 
 The analysis of the Sorites takes place as follows : The Sorites 
 must be broken up into as many syllogisms in Fig. I. as there are 
 propositions between the first and the last. In the analysed Aristo- 
 telian Sorites, S, the subject of the conclusion, is the subject in the 
 conclusion of each of the component syllogisms. In the analysed
 
 Chap. XXIX.] SORITES AND EPICHEIREMA 257 
 
 Goclenian Sorites, P, the predicate of the conclusion is the predicate 
 in the conclusion of each of the component syllogisms. 
 
 It is important in these analyses not to confuse the major and the 
 minor premisses. In the Aristotelian Sorites the first syllogism, in 
 order to be in Fig. I., must have as major the second premiss, not 
 the first. Thus, analysing the first form of Aristotelian Sorites 
 given above, we have as the first syllogism : 
 
 All X's are Y's. 
 All S's are X 's. 
 
 .-. All S's are Y^ 
 
 Then, similarly, this conclusion must be the minor and not the 
 major premiss of the next syllogism : 
 
 All Y's are Z's. 
 All S's are Y's. 
 
 .-. All S's are Z^ 
 
 So, again : 
 
 All Z's are P's. 
 All S's are Z's. 
 
 .-. All S's are P's (conclusion). 
 
 Prosyllogisms and Episyllogisms. — In analysing a complex 
 syllogism like a Sorites into its component syllogisms, we see that 
 each component (except the last) is a prosyllogism with respect to 
 the syllogism which succeeds it, and that each component (except 
 the first) is an episyllogism with respect to the syllogism that pre- 
 cedes it. 
 
 A prosyllogism is a syllogism of which the conclusion is a premiss 
 in another syllogism with which it is connected. 
 
 An episyllogism is a syllogism of which one of the premisses is 
 the conclusion of another syllogism with which it is comiected. 
 
 Special Rules of the Aristotelian Sorites : 
 
 1. No premiss but the first can be particular. 
 
 2. No premiss but the last can be negative. 
 
 Proof of Rule 1. — The only premiss in an Aristotehan Sorites 
 which is a minor in one of the component syllogisms is the first. 
 Each of the others is, in its turn, a major premiss. Hence, if an\ 
 premiss in the Sorites, save the first, were particular, it would lead 
 to a particular major in the First Figure. A jDarticular major 
 premiss in Fig. I., however, gives illicit process of the maior or 
 undistributed middle. 
 
 17
 
 25S THE PROBLEM OF LOGIC [VIH. i. 
 
 Proof of Rule 2. — Let us take a t^-pical Aristotelian Sorites, of four 
 
 premisses, say, and anah'se it out into its component syllogisms. 
 
 This -will serve to steady the subsequent reasoning. 
 
 Thus • 
 
 All A's are B's. 
 
 All B's are C's. 
 
 AU C's are D's. 
 
 All D's are E's. 
 
 .-. All A's arc E's. 
 
 This reduces to the following series of syllogisms in Fig. I. : 
 
 All B's are C's. All C's are D's. All D's are E's. 
 
 All A's are B's. AU A's are C's. All A's are D 's. 
 
 .-. All A's are C's. .-. Ail A's are D's. .-. All A's are E's. 
 
 Now, any premiss of the given Sorites would, if negative, require 
 the conclusion of the syllogism into which it enters as a premiss to 
 be negative.* 
 
 But this means illicit major in the episyllogism in the case of all 
 but the final premiss, where the component syllogism is not followed 
 by any episyllogism. For the conclusion of each syllogism — 
 except, of course, the last — becomes the minor premiss of the epi- 
 syllogism ; and in Fig. I., if the minor premiss be negative, the con- 
 clusion being negative and the major premiss affirmative, illicit 
 major follows. 
 
 Hence it is onh^ that premiss which enters into the final syllogism 
 — i.e., only the last premiss of the Sorites — that can be negative 
 without fallacy. 
 
 The special rules of • the Goclenian Sorites may be formulated 
 as follows : 
 
 1. Xo premiss but the last can be particular. 
 
 2. No premiss but the first can be negative. 
 
 We leave the proof of these two rules to the consideration of the 
 reader. 
 
 It will be seen that, in resolving this Sorites into first-figure syllo- 
 j^isms, the two leading premisses must not be transposed, and the 
 conclusion of each prosyllogism serves as major, not as minor, 
 premiss of the epi.svllogism. 
 
 A Sorites cannot he framed in Figs. II. ami III. — A Sorites differs 
 in this respect from a simple syllogism : that it has many ' middle 
 terms.' Now, any two consecutive premisses of a Sorites in a given 
 
 * If the first premiss were negative, this negative conclusion could not be drawn 
 at all. The attempt to draw it would involve illicit major (or else two negative
 
 Ch4p. XXIX.] SORITES AND EPICHEIRE^L\ 259 
 
 figure must show the form of that figure. Thus, if we start with two 
 premisses in Fig. III. — say, e.g., 
 
 All S's are X's, 
 All S's are Y's. 
 
 the third premiss must take the form ' All S's are Z's ' ; for it is 
 only on condition that S is again subject of the premiss that we can 
 arrange that the second and third premisses together shall supply 
 the data in Fig. III. But the premisses — 
 
 All S's are X's, 
 All S's are Y's, 
 All S's are Z's, 
 
 involve, after the first stage, a quatemio terminorum. For from the 
 first two premisses we obtain the conclusion ' Some Y's are X's,' 
 and this has no point of contact with the third premiss, so that 
 the third premiss cannot be said to belong to the same train of 
 argument as the first two premisses. The reasoning, that is, breaks 
 off after the first syllogism, and we have consequently no polysyllo- 
 gism and no Sorites. A precisely similar argument would show 
 that a Sorites could not be framed in the Second Figure. 
 
 But though it is not possible to frame a Sorites in the Second or 
 the Third Figure, it is quite possible, by the aid of Conversion, to 
 resolve those forms of the Aristotelian Sorites which have particular 
 conclusions into Third Figure Syllogisms, and those forms of the 
 Goclenian Sorites which have negative conclusions into Second 
 Figure Syllogisms. The resolution of the Aristotelian Sorites into 
 Second Figure Syllogisms is not possible ; for, in order to carry it 
 out, we should have to convert the second premiss, and this would 
 give in every case component syllogisms exhibiting the fallacy of 
 undistributed middle. Similarly, it is impossible to resolve any 
 Goclenian Sorites into syllogisms of the Third Figure. 
 
 Referring back to the four forms of the Aristotelian Sorites given 
 on page 256, we see that if we convert the first premiss of (3), we 
 shall change the form of the Sorites somewhat, without affecting 
 in any way the nature of the argument, for the I. proposition is 
 simply convertible. 
 
 But when we come to analyse the Sorites whose premiss has been 
 thus converted, we shall find that the component sjdlogisms are 
 no longer in Fig. I,, but in Fig. III., as is clearly shown below : 
 
 Some X's are S's. 
 All X's are Y's. 
 
 ••. Some Y's are S's. 
 All Y's are Z's. 
 
 .*. Some Z's are S's. 
 All Z's are P's. 
 
 Some P's are S's, .•. Some S's are P's. 
 
 17^2
 
 2G0 THE PROBLEM OF LOGIC [VIIL i. 
 
 Form (4) of the Aristotelian Sorites may be treated in the same 
 manner. 
 
 So, afrain, the resohition of the Goclenian Sorites : 
 
 All Z's are-not P's. 
 All Y's are Z's. 
 All X's are Y's. 
 All S's are X's. 
 
 .-. All S's are-not P's, 
 
 into its component syllogisms gives a series of syllogisms in Fig. II., 
 if we start by converting the leading premiss, and convert all the 
 negative conclusions as they occur. 
 
 All P's are-not Z's. 
 All Y's are Z's. 
 
 .-. All Y's are-not P's. 
 
 Converting this conclusion, we have : 
 
 All P's are-not Y's. 
 All X's are Y's. 
 
 .-. All X's are-not P's. 
 
 Converting this conclusion, we have : 
 
 All P's are-not X's. 
 All S's are X's. 
 
 .-. All S's are-not P's. 
 
 The Goclenian Sorites with a particular negative conclusion may 
 be similarly resolved. 
 
 We see, then, that wc can resolve some forms of Sorites into 
 syllogisms of Fig. II., and others into syllogisms of Fig. III. We may 
 also {e.g., by converting the final premiss of the Celarent type of 
 Aristotelian Sorites) resolve a Sorites into syllogisms which are 
 partly of the first and partly of another Figure {e.g., partly Fig. I. 
 and partly Fig. II.). 
 
 The Epicheirema. 
 
 The Epicheirema is a chain of reasoning of an abridged and con- 
 centrated kind. Chains of reasoning may be either progressive or 
 regressive. When, as in the case of the Sorites, the conclusion of 
 each component syllogism becomes a premiss in an episyllogism, the 
 chain is progressive ; when the movement of thought takes place 
 in the contrary direction — namely, from syllogism to prosyllogism —
 
 CuAP.XXIX.] SORITES AND EPICHEIREMA 261 
 
 the chain is regressive. The Epicheirema is essentially regressive 
 in character. The main argument consists of an ordinary syllogism, 
 but the premisses in the argument are, one or both of them, brought 
 forward as conclusions from other premisses. Thus, the Epi- 
 cheirema is a syllogism of which each premiss, or one of the premisses, 
 is stated as the conclusion of a prosyllogism (as when the premiss 
 ' All S's are P's ' is given thus : ' All S's are P's, for All M's are P's, 
 and All S's are M's '), or, as is more usual, is stated as the con- 
 clusion of an enthymeme. When there are two prosyllogisms or 
 enthymemes, the Epicheirema is called ' double ' ; when only one, 
 it is ' single.' 
 
 The following illustrates a single Epicheirema : 
 
 ' All rational beings are to be treated with respect, inasmuch 
 as they are made in the image of God. 
 Slaves are rational beings. 
 Therefore slaves should be treated with respect.' 
 
 (Father Clarke.) 
 
 The following illustrates a double Epicheirema : 
 
 ' All Malays are cruel, because all savages are. 
 All the aboriginal inhabitants of Singapore are Malays, because 
 
 all the natives of that part of Asia are. 
 Therefore all the natives of Singapore are cruel.' 
 
 (Welton.) 
 
 The foUowmg scheme, given by Professor Welton, illustrates 
 what he refers to as a Complex Epicheirema : 
 
 Every M is P, because it is X, and every X is Y. 
 Every S is M. 
 .-. Every S is P. 
 
 We may anatyse this Complex Epicheirema into its three com- 
 ponent syllogisms. These are : 
 
 (1) Every X is Y. (2) (Every Y is P.) (3) Every M is P. 
 Every M is X. Every M is Y. Every S is M. 
 
 (.-. Every M is Y.) .-. Every M is P. .-. Every S is P. 
 
 It should be noted, in connexion with the form of the major pre- 
 miss, that the addition ' and every X is Y ' is not another reason 
 added to the first reason ' because it is X,' but rather a proposition 
 which, in combination with it, jaelds a reason as conclusion. Every 
 Complex Epicheirema, in fact, must be either a single or a double 
 Epicheirema. It cannot be construed into a fresh form distinct 
 from both of these two forms.
 
 2G2 THE PROBLEM OF LOGIC [VIII. ii 
 
 CHAPTER XXX. 
 
 Vni. (ii.) THE DISJUNCTIVE SYLLOGISM. 
 
 The Disjunctive Syllogism, where there are two alternatives only, 
 may be developed from the Disjunctive Proposition, as basis of 
 inference, in two ways, giving two distinct moods. 
 
 1. The Modus Tollendo Ponens — i.e., the mood which posits 
 
 the one alternative by rejecting the other. 
 
 Either P or Q. Either P or Q. 
 
 Not P. Not Q. 
 
 2. The Modus Ponexdo Tollens — i.e., the mood which sublates 
 
 or rejects the one alternative by accepting or positing the 
 other. 
 
 Either P or else Q. Either P or else Q. 
 
 P. Q. 
 
 .-. XotQ. .-. Not P. 
 
 [Note. — The minor premiss may posit or reject an alternative in a form which is 
 more definite than that in which it is presented in the major premiss. This is 
 by no means an arbitrary provision, for a syllogism, like any other form o 
 thinking, is essentially a development of meaning. What is relatively indefinite 
 in the major premiss may receive clearer definition in the minor premiss and con- 
 clusion. 
 
 It should be noticed that the Disjunctive Proposition as an integral element in 
 the Disjunctive Syllogism requires to be read from the point of view of mere state- 
 ment-import. Thus, ' Either P or Q ' would read : ' Either P or Q, but not 
 statedly both.' So ' Either P or else Q ' would read : ' Either P or Q, but statedly 
 not both.'] 
 
 In a disjunctive syllogism in which there are two, and only two, 
 alternatives, the 'major' is always disjunctive, the 'minor' cate- 
 gorical, and the conclusion categorical. Where there are more than 
 two alternatives in the major, either the minor premiss or the con- 
 clusion may be disjunctive. Thus : 
 
 Either P or Q or R. 
 
 Not P. 
 
 .'. Either Q or R. 
 
 Either P or else Q or else R. 
 Not P. 
 .-. Either Q or else R. 
 
 Either P or Q or R. 
 
 Neither P nor Q f compound categorical). 
 .-. R.
 
 Chap. XXX.] THE DISJUNCTIVE SYLLOGISM 2G3 
 
 Either P or else Q or else R. 
 
 P. 
 
 .'. Neither Q nor R. 
 
 Either P or else Q or else R. 
 Either P or else R. 
 .-. Not Q. 
 
 Of the two disjunctive moods, the Modus ToUendo Ponens is 
 necessarily valid, whether ' either ... or ' be taken to imply 
 ' statedly not both ' or ' not statedly both.' It is the mood which 
 brings out the exhaustive character of the disjunctive major. The 
 Modus Ponendo Tollens, on the other hand, is vahd only when 
 ' either ... or ' imphes ' not both ' — i.e., when the exclusive reading 
 is understood. If we have accepted no statement to the effect that 
 So-and-So, while he is either a knave or a fool, is not both together, 
 it is invalid to argue that because he is a knave he cannot be a fool. 
 
 CHAPTER XXXI. 
 
 VIII. (iii.) THE HYPOTHETICAL SYTXOGISM. 
 
 Hypothetical Syllogisms may be either Mixed or Pure. 
 
 A Mixed Hypothetical Syllogism has the first premiss hypo- 
 thetical, the second categorical, and the conclusion categorical ; 
 whereas a Pure Hypothetical Syllogism has both premisses and 
 the conclusion hypothetical. 
 
 The two fundamental forms of the Mixed Hypothetical Syllogism 
 are known respectively as Constructive and Destructive. Con- 
 sidered as Moods of the Syllogism, they are respectively known as 
 the Modus Ponens and the Modus Tollens. 
 
 1. Constructive : 
 
 If A, then C 
 A 
 
 This is the Modus Ponens, the mood which in the minor premiss 
 posits the antecedent of the major 
 
 2. Destructive : 
 
 If A, then C 
 not C 
 
 .'. not A
 
 261 THE PROBLEM OF LOGIC [Vin. iii. 
 
 Tliis is the Modus Tollens, the mood which in the minor premiss 
 removes or sublates the consequent of the major. 
 
 The Categorical premiss in the modus ponens need not be affirma- 
 tive, nor. in the modus tollens, need it be negative. 
 Thus; 
 
 If not A, then C 
 not A 
 
 .-. C {modus 'ponens). 
 
 So again : 
 
 If not A, then not C 
 C 
 
 {modus tollens). 
 
 The Rules of the Mixed Hypothetical Syllogism. 
 
 When once the h\-pothetical premiss is accepted, then — 
 L // the Antecedent is accepted, the Consequent must he accepted. 
 This follows at once from the consideration that to disallow the 
 rule is to refuse to accept the hypothetical premiss from which 
 the reasoning starts. 
 
 2. // the Consequent is rejected, the Antecedent must he rejected. 
 Granted that the Consequent is rejected, then our data are seen to 
 necessitate the rejection of the Antecedent. For the acceptance 
 of the Antecedent would lead to the contradiction of accepting the 
 Consequent which we have just been rejecting. And what it is 
 contradictory to accept, we are logically compelled to reject. 
 
 Fallacies of the Mixed Hypothetical Syllogism. 
 
 1. Bejection of the Antecedent. — Rejection of the Antecedent does 
 not entitle us to the inference that the Consequent must be rejected. 
 
 For, the rejection of a proposition being identical with the 
 acceptance of its contradictor}', we see that to assert that from 
 the rejection of the Antecedent we can infer the rejection of the 
 Consequent is to say that if a proposition C can be inferred from 
 another propo.sition B, then from the contradictory of B we can 
 infer the contradictor^' of C. But this is manifestly not the case. 
 For instance, from an A proposition we can infer the corresponding 
 I ; but from 0, the contradictory of A, we cannot infer E, tlie 
 contradictory of I. Therefore, from the rejection of the Antecedent 
 we cannot infer the rejection of the Consequent. 
 
 2. Acceptance of the Consequent. — Acceptance of the Consequent 
 does not entitle us to the inference that the Antecedent must be 
 accepted.
 
 Chap. XXXI.] THE HYPOTHETICAL SYLLOGISM 2G5 
 
 For to assert that from the acceptance of the Consequent we 
 can infer the acceptance of the Antecedent is to say that if a pro- 
 position C can be inferred from another pro])osition B, then from 
 C we can infer B, But this is manifestly not the case. For 
 instance, the Subalternate can be inferred from the Subalternans, 
 but not tlie Subalternans from the Subalternate. 
 
 Summing U]), we see that, while acceptance of the Antecedent 
 involves acceptance of the Consequent, and rejection of the Conse- 
 quent involves rejection of the Antecedent, nothing can be logically 
 inferred either from the rejection of the Antecedent or from the 
 acceptance of the Consequent. 
 
 The So-called Immediate Inferences from Hypotheticals. 
 
 We ma}', perhaps, venture the following suggestions with regard 
 to the so-called ' Immediate Inferences ' from Hj'potlieticals. 
 
 L There can be no logical ' converse ' of the proposition ' If A 
 then C,' for there can be no inference from the acceptance 
 of the consequent. 
 
 2. Nor can there be any logical ' inverse ' of the proposition 
 
 ' If A, then C,' for there can be no inference from the 
 rejection of the antecedent. 
 
 3. There can be no logical ' obverse ' of the proposition ' If A, 
 
 then C,' not even the proposition ' If A, then not C ' — 
 i.e., ' If we accept the antecedent, we must reject the 
 contradictory of the consequent.' For, as the rejection of 
 the contradictory of any proposition is precisely identical 
 with the acceptance of that proposition, this obverse has 
 no logical value. 
 
 4. There can be no logical ' contrapositive ' of the proposition 
 
 ' If A, then C,' for where there is neither converse nor 
 obverse, there can manifestly be no converted obverse. 
 
 The drawing of the inference ' If A, then C, .-. if C, 
 then A,' which usually goes by the name of contra- 
 positive inference, is just another form of the hy])othetical 
 syllogism 
 
 If A, then C 
 Note 
 
 .'. not A, 
 
 and is therefore not immediate. For the proposition 
 ' If C, then A,' as an inference from ' If A, then C,' simply 
 states that granted ' If A. then C,' then, if we also grant C, 
 we must necessarily grant A.
 
 26U THE PROBLEM OF LOGIC [VIII. iii. 
 
 It remains, of course, perfectly legitimate to posit the 
 proposition ' If C, then A ' as the contrapositive of ' If A, 
 then C provided that the contrapositive is not here taken 
 as an immediate inference, found by converting the 
 obverse of ' If A, then C This reservation should be 
 borne in mind in connexion with the problem which 
 immediately follows. 
 
 Give the contrapositive of the following proposition : 
 
 If either no P is R or no Q is R, then nothing that is both 
 P and Q is R (KejTies). 
 
 This takes the form ' If A is accepted, then C is accepted,' of which 
 the contrapositive is ' If C is rejected, A is rejected.' 
 
 The rejection of C means the rejection of ' Nothing that is both 
 P and Q is R,' and is therefore equivalent to ' Some things both 
 P and Q are R.' 
 
 The rejection of A means the rejection of ' Either no P is R or 
 no Q is R.' 
 
 This disjunctive proposition is of the form ' Either X or Y,' 
 of which the contradictory, on the exclusive view, is : ' Either both 
 X and Y, or neither X nor Y.' 
 
 Hence the rejection of A is given by ' Either no (P's and Q's) 
 are R, or some P's and some Q's are R.' 
 
 The required Contrapositive is then the following : 
 
 • If some things that are both P and Q are R, then either no 
 P is R and no Q is R, or some P's are R and some Q's 
 are R.' 
 
 On the non-exclusive view it takes the simpler form : 
 
 ' If some PQ's are R, then some P's are R, and some Q's 
 are R.' 
 
 Examples. 
 
 1. WTiich of the following arguments are logically correct ? 
 
 (a) A is B if it is C ; it is not C, therefore it is not B. 
 
 (6) A is not B, unless it is C ; as it is not C, it is not B 
 
 (cj If A is not B, G is not D ; but as A is B, it follows that 
 
 Cisl>. 
 {d) A is not B if C is D ; C, then, is not D, for A is B. 
 
 (Jevons.)
 
 Chap. XXXL] THE HYPOTHETICAL SYLLOGISM 267 
 
 Put into strict logical form, these run : 
 
 (a) If A is C, A is B 
 A is not C 
 
 .-. A is not B 
 
 '^ Fallacy of Rejection of Antecedent. 
 
 (Assuming that ' A is not C '= 
 AisC, or not (A is C), and simi- 
 larly as regards ' A is not B.') 
 (6) If A is not C, A is not B 
 A is not 
 
 .-. A is not B 
 
 Valid. Modus Ponens. 
 
 (c) If A is not B, C is not D 
 A is B 
 
 C is D 
 
 {d) If C is D, A is not B 
 AisB 
 
 .*. C is not D 
 
 Fallacy of Pvcjection of Antecedent. 
 (Assuming that 'A is B ' = 
 A is not B, and similarly as 
 regards ' C is D.') 
 
 Valid. Modus Tollens. (Assum- 
 
 ing that ' A is B ' = A is not B 
 . and 'CisnotD' = Cls~I).) 
 
 2. Arrange the following in proper logical form, and test their 
 validity : 
 
 (a) It was agreed that unless the weather turned fine, we were 
 
 to postpone the match ; so, as the weather has not turned 
 
 fine, the match must be postponed. 
 (6) Men are not pleased if their meals are not served punctually ; 
 
 thus, as Mr. X. is never kept waiting for his meals, he must 
 
 necessarily be pleased, 
 (c) If the cat's away, the mice are everywhere ; the cat must 
 
 then be about, for the mice are nowhere. 
 
 Reduced to proper form, the given arguments read as follows : 
 
 (rt) If the weather does not turn fine, the match is to be post- 
 poned. 
 
 The weather has not turned fine. 
 .-. The match is to be postponed. 
 
 {Modus Ponens.)
 
 26S THE PROBLEM OF LOGIC [VIII. iii. 
 
 {b) If men are not served punctually, they are displeased. 
 Mr. X. is served punctuall}'. 
 .-. He must be pleased. 
 
 (Rejection of Antecedent.) 
 
 (c) If the cat is away, the mice are everywhere. 
 But the mice are not anywhere. 
 .-. The cat must be about. 
 
 Here the argument is sound {Modus Tollens), though the 
 minor, in rejecting the consequent through its contrary 
 instead of through its contradictory, states more than is 
 necessary for drawing the conclusion. The mood is there- 
 fore a ' strengthened mood.' 
 
 o 
 
 Annex the proper conclusion (if any) to the following premisses : 
 
 (a) If the earth did not rotate on its axis, there would be no 
 alternation of day and night, whereas this alternation does 
 actually occur (Welton). 
 {Modus Tollens.) Ergo s The Earth does rotate on its 
 axis. 
 
 (6) If no men were mad, asylums would be useless ; but they 
 are not useless (Welton). 
 {Modiis Tollens.) Ergo : ' Some men are mad,' the con- 
 tradictory of the antecedent. 
 
 (c) If all men were reasonable, all would be contented ; but some 
 are unreasonable. 
 
 This involves rejection of the antecedent ; for even if 
 ' unreasonable ' be taken as the term contrarij to reason- 
 able, the ' unreasonable ' would still be included under 
 ' those who are not reasonable.' Thus no conclusion can 
 be dra%vn. 
 
 The Transversion of the Hypothetical Syllogism into Categorical Form. 
 
 The 3Iodu8 Ponens can be transverted as follows :* 
 
 The case of A (being accepted) is the case of C (being accepted). 
 This is the case of A (being accepted). 
 .-. This is the case of C (being accepted). 
 
 (Barbara, Fig. I.) 
 
 • It should not be forgotten that these transversions do not give us, in the 
 place of the given hypotneticals, genuine categoricals — i.e., propositions with a 
 caU-gorical meaning. What they give us are simply hypothetical propositions in 
 categorical form. If taken to be a genuine bridge from one type of proposition to 
 the cither, the clumsy artifice through which the transversion is effected is no more 
 than ' a transparent piece of self-deception. "The case of " must be taken to mean 
 *' the po=<sible case of." Thus, it is simplv a .synonym for "in case" — i.e., for 
 " if " ' (Professor Stout).
 
 Chap. XXXI.] THE HYPOTHETICAL SYLLOGISM 2G9 
 
 Again, in so far as the rejection of a statement is taken as 
 equivalent to its non-acceptance, the transverse of the Modus Tollens 
 takes the following form : 
 
 The case of A (being accepted) is the case of C (being accepted). 
 This is-not the case of C (being accepted). 
 .'. This is-not the case of A (being accepted). 
 
 (Camestres, Fig. II.) 
 
 These transversions are useful, if only because they serve to em- 
 phasize the mediate character of hypothetical inference (which has, 
 curiously enough, been disputed) ; and they afford justification for 
 the convenient practice of referring to the hypothetical premiss as 
 the major, and the categorical as the minor, since, in the transversion, 
 the hj^othetical premiss becomes the proposition containing the 
 major term, and the categorical premiss, the premiss containing 
 the minor term. 
 
 Again, by these transversions we are able to show that the fallacy 
 of rejecting the antecedent is analogous to illicit major, and the 
 fallacy of accepting the consequent corresponds to undistributed 
 middle. 
 
 Thus (a), the rejection of the antecedent, in the transverse, gives : 
 
 The (case of A being accepted) is a (case of C being accepted). 
 This is-not a (case of A being accepted). 
 .-. This is-not a (case of C being accepted). 
 I.e., a categorical pseudo-syllogism, involving illicit major. 
 
 So again (6), the acceptance of the consequent in the transverse 
 gives : 
 
 The (ease of A being accepted) is a (case of C being accepted). 
 This is a (case of C being accepted). 
 ••. This is a (case of A being accepted). 
 I.e., a categorical pseudo-syllogism, involving undistributed 
 middle. 
 
 The Pure Hypothetical Syllogism, 
 
 This takes the form : 
 
 If A, then C^ 
 If C i, then C 3 
 
 .-. iTx. then C, 
 
 The Law of Inference here exemplified may be expressed as 
 follows : 
 
 ' The consequent of the consequent is the consequent of the 
 ground.' 
 
 It is a direct embodiment of the principle of Identity.
 
 270 THE PPvOBLE:M OF LOGIC [VIII. iii. 
 
 This same principle of reasoning applies equally to sj^llogisms of 
 this form, however we may increase the number of the premisses. 
 Thus : 
 
 If A, then C, 
 
 If Cj, then a 
 
 If C.„ then C3 
 
 If Cn-i, then Cn 
 .*. If A, then C^ 
 
 We may conveniently refer to this form as the Standard Sorites 
 (Hypothetical). 
 
 If we alter the order of the premisses, starting with the last and 
 proceeding gradually to the first, we get the following form of 
 reasoning, which we may call the Inverted Sorites. 
 
 If Cn_i, then On 
 If Cn_2, then Cu_i 
 
 If Ci, then C2 
 If A, then 0^ 
 
 .'. If A, then Cq 
 
 Example. — If humble-bees are abundant, the red clover which 
 humble-bees alone pollinate is also abundant. 
 If field-mice are few, humble-bees are abundant. 
 If oats are abundant, field-mice are few. 
 .*. If cats abound, red clover is also abundant. 
 
 (Darwin, ' Origin of Species,' p. 53.) 
 
 The analysis of a Sorites of this inverted form presents no diffi- 
 culty. When analysed out, the argument runs as follows : 
 
 (1) Transposing the first two premises, we have 
 
 If Cn_2, then C„_i 
 If Cn_i, then C., 
 
 .'. If Cn_2, then C„ 
 
 (2) Combining this conclusion with the third premiss, we get 
 
 If C^_3, then C,,_2 
 If Cn_2, then Cn 
 
 ••. If C„_3, then C„ 
 
 and so on.
 
 Crup.XXXI.] THE HYPOTHETICAL SYLLOGISM 271 
 
 The analysis of tlic Standard Sorites proceeds on similar and still 
 simpler lines, for there is here no need for any transposition of 
 premisses. Thus : 
 
 (1) If A, thenC, 
 If Ci, then Ca 
 
 and so on. 
 
 .-. If A, 
 
 then C2 
 
 (2) If A, 
 IfCa, 
 
 then C, 
 , then C3 
 
 .-. If A, 
 
 then C3 
 
 CHAPTER XXXII. 
 
 Vin. (iv.) THE DILEMilA. 
 
 ' The Dilemma is an argument in which one or other of two alterna- 
 tives is offered, the admission of either of which involves something 
 further which an opponent will commonly be unwilhng to admit. 
 Thus, the adversary is said to be placed in a " dilemma " — that is, 
 he is in a position in which he cannot help admitting that one or 
 other of the two alternatives offered to liim holds good, while yet 
 the admission of either involves liim in the acknowledgment of some- 
 tliing further as dependent on it, which he is unwilling to allow, 
 and so he has only a choice of evils left ' (Karslake, ' Aids to the 
 Study of Logic,' ii.,p. 94). 
 
 Formal Definition. 
 
 The Dilemma is a compound hypothetical syllogism, and is there- 
 fore governed by the laws of the hypothetical syllogism. It must, 
 however, be partly disjunctive in form, seeing that the essence of 
 a Dilemma is to present a choice of evils. The disjunction may be 
 offered in the consequent of the hypothetical major premiss, but is 
 normally presented in the minor premiss and in the form of a dis- 
 junctive proposition. 
 
 A Dilemma may be either Constructive or Destructive. It is 
 constructive if the argument proceeds from the acceptance of 
 antecedent to the acceptance of consequent, and destructive if 
 the argument is from the rejection of consequent to the rejection 
 of antecedent. These two standard forms of Dilemma may be 
 formulated as follows :
 
 o-.-> 
 
 THE PROBLEM OF LOGIC [VIII. iv. 
 
 1. The Constructive Dilemma, 
 
 If Pj, then Qj ; and if Pg, then Q2. 
 But either P^ or Po. 
 .'. Either Q^ or Q,. 
 
 Example. — ' If I cross tlie field, I shall meet the bull ; and if I 
 go up the lane I shall meet the farmer. 
 ' Either I shall cross the field or I shall go up the 
 lane, 
 .'. ' Either I shall meet the bull or I shall meet the 
 farmer.' (Stock.) 
 
 2. The Destructive Dilemma. 
 
 If P^, then Qi ; and if P,, then Qo. 
 But either not Q^ or not Qg. 
 .*. Either not P^ or not Po. 
 
 Example. — ' If he were clever, he would see liis mistalte ; and if 
 he were candid, he would acknowledge it. 
 ' Either he does not see his mistake or he will not 
 acknowledge it. 
 .♦. ' Either he is not clever or he is not candid.' 
 
 (Stock.) 
 
 The two fundamental forms of Dilemma — the Constructive and 
 the Destructive — may take certain limiting forms wliich are suffi- 
 ciently important to call for special consideration. 
 
 Thus, in the case of the Constructive Dilemma, the two conse- 
 quents Qi and Qg may coincide, and the Dilemma then takes the 
 following simplified form : 
 
 If either Pj or P2, then Q. 
 But either Pj or P,. 
 .-. Q. 
 
 Example. — ' Whether a man acts in accordance with his own 
 judgment or is guided by the opinions of others, 
 his action will be criticized. 
 ' But either he acts in accordance with his own judg- 
 ment or he is guided by the opinions of others. 
 .*. ' In any case, his action will be criticized.' 
 
 It should be noticed that the ' either ... or ' of the major 
 premiss does not stand for a genuine disjunction. That the dis- 
 junction is merely verbal is shown by the fact that we have but 
 slightly to modify the statement of the major premiss to get rid of 
 the disjunction altogether. Thus, instead of saying ' If either Pj 
 or P2, then Q,' we may say ' If P^, then Q ; and if Pg, tlicn Q.'
 
 CiTAP. XXXIL] THE DILEMMA 273 
 
 In the case of the Destructiv^e Dilemma the two antecedents may 
 coincide. The major premiss then reads, ' If P, then Q^ ; and if P, 
 then Q2,' and tlie Dilemma takes the following simplified form : 
 
 If P, then both Q^ and Qg. 
 But either not Qj or not Qg. 
 .-. Not P. 
 
 Examfle. — ' If table-rappers are to be trusted, the departed are 
 spirits, and they also exert mechanical energy. 
 ' But either the departed are not spirits or they do not 
 exert mechanical energy. 
 ' .-.Table-rappers are not to be trusted.' (Carveth Read.) 
 
 These simplified forms of the two standard types of Dilemma are 
 sometimes referred to as the Simple Constructive Dilemma and the 
 Simple Destructive Dilemma respectively ; and by way of contrast 
 the standard forms are respectively referred to as the Complex 
 Constructive Dilemma and the Complex Destructive Dilemma. It 
 is, however, essential to bear in mind, when these titles are used, 
 that the fundamental forms of Dilemma are the complex forms, 
 the simpler forms being mere derivatives from these, owing their 
 simplicity to the coincidence either of the two consequents or of the 
 two antecedents. As a diagnostic mark for distinguishing between 
 the complex and the simplified varieties, we may mention the differ- 
 ence in the form of the conclusion. The conclusion, in the case 
 of the Complex Dilemma, is disjunctive ; in the case of the Simple 
 Dilemma, categorical. 
 
 We have spoken of the Simple Destructive Dilemma as derived 
 fpom the corresponding complex form by making the two ante- 
 cedents coincide. If, in addition to this coincidence of antecedents, 
 we substitute for Qj ' Either R^ or Rg,' and cancel Q2, the Destructive 
 Dilemma takes the following form : 
 
 If P, then either R^ or R^. 
 But neither H^ nor R.,. 
 .-. Not P. 
 
 The peculiarity of this form of dilemma is that the disjunction, 
 instead of being presented in the minor j) remiss, is presented in 
 the major. 
 
 Example. — ' If a body moves, it must either move where it is, or 
 where it is not. 
 ' But a body caimot move where it is ; neither can it 
 move where it is not. 
 ' .-. It cannot move at all.' 
 
 {I.e., Motion, being unintelligible, is impossible.) 
 
 [Zcno's Dilemma].* 
 
 * Vide Chapter XXXIII., p. 290 ; ami cf. Joseph, ibid., pp. 332-.'^34. 
 
 18
 
 274 THE PROBLEM OF LOGIC [Vlll. iv. 
 
 In testing the Formal validity of dilemmas, we have to bear in 
 mind three essential criteria : 
 
 1. The argument must, either in major or in minor premiss, 
 present genuine alternatives. Otherwise, however correct the 
 reasoning may be in itself, we have only a pseudo-dilemma. Thus 
 the following, though valid as an argument, is not a dilemma : 
 
 ' Whether Geometr}'- be regarded as a mental discipline or as 
 a practical science, it deserves to be studied. 
 
 ' But Geometry may be regarded as both a mental discipline 
 and a practical science. 
 .-. ' It deserves to be studied.' (Dr. Fowler, quoted by Mellone.) 
 
 Of this reasoning we cannot say that it is ' an argument in which 
 one or other of two alternatives is offered.' For the major premiss, 
 in wliich alone there is any semblance of a disjunction, is not 
 genuinely disjunctive, but only a compound hypothetical proposi- 
 tion of the form ' If Sj or Sg, then P.' The disjunctive form of the 
 antecedent is quite illusory, the whole proposition being equivalent 
 to ' If S^, then P ; and if So, then P.' Here there is no choice of alter- 
 natives offered as in the case of a disjunctive consequent (' If S, 
 then either Pj or Pj). 
 
 2. The principles of hypothetical reasoning must be strictly 
 applied. Where these are infringed we have a Formally invalid 
 dilemma. 
 
 3. The logical structure of the disjunctive proposition, and its 
 import, whether exclusive or non-exclusive, must be steadily kept 
 in view. 
 
 The ' Rebuttal ' of a Dilemma. 
 
 The metaphors used to describe dilemmatic argument seem to have 
 been taken from the old pastime of bull-fighting. Thus, the adver- 
 sary who is unable to escape the force of a dilemma is said to be 
 ' fixed on the horns of a dilemma.' He is said to rebut the dilemma 
 if he can oppose to the argument with wliich he is assailed a relevant 
 and equally convincing counterpart. 
 
 We must attempt to show in what sense and to what extent such 
 rebuttal is possible. Let us start by considering what we can logi- 
 cally understand by the rebuttal of a typical Constructive Dilemma, 
 — a dilemma, that is, of the following form : 
 
 If Pj, then Qi ; and if Pg, then Qj. 
 But either P^ or Pg. 
 .♦. Either Q^ or Qg. 
 
 Now, if we were to suppose that the objector who endeavours to 
 rebut this dilemma is restricted, as regards data, to the statements 
 brought forward in the premisses of the dilemma, the only premisses
 
 Chap. XXXII.] THE DILEIVDIA 275 
 
 which he could relevantly utilize for the purposes of rebuttal — 
 utilize, that is, in such a way as to win his opponent's assent to them 
 — would be inferences from the original premisses. But even if it 
 were possible to draw such inferences, they would be useless for the 
 objector's purpose ; for starting, as he would then be doing, from 
 the very same premisses as his opponent, though perhaps differently 
 expressed and arranged, he could not possibly extract from these 
 any conclusion that would contrast with the conclusion of the 
 original dilemma. 
 
 The possibility of a cogent rebuttal depends, then, on the data 
 common to the two disputants not being exhaustively stated in 
 the premisses of the original dilemma. The premisses of both the 
 dilemma and its counterpart must be drawn from the same set of 
 facts, each disputant selecting from them the data that suit liis 
 purpose, 
 
 A merely Formal treatment of rebuttal is therefore necessarily 
 inadequate. We must presuppose a certain common fund of 
 material evidence — a certain given situation. But more than this 
 is necessary to account for the peculiar form which dilemmatic 
 argument takes. 
 
 If we consider a classical example of Rebuttal such as the Litigi- 
 osus {vide pp. 293-295), we see that the Counter-dilemma derives its 
 cogency from an adroit jugghng with two standards the determina- 
 tions of which are in every instance precisely opposed. If we sym- 
 bolize these standards as S^ and S2, and explicitly refer to them 
 wherever reference to them is implied, we may formulate the 
 Dilemma (Constructive) and its Counter-Dilemma as follows : 
 
 Dilemma : If Pj, then S^Q^ ; and if Pg, then 8202- 
 But either P^ or P,. 
 .-. Either S^Q^ or S2Q2' 
 
 Counter-Dilemma : If Pj, then S2Q2 ; and if Pg, then S^Q^. 
 
 But either Pj or Pg. 
 .•. Either S^Q^ or SoQa- 
 
 Example — Dilemma : 
 
 ' If I live for others, neglecting myself, then (to judge by 
 egoistic standards)* I shall make myself unhappy ; and if I 
 live for myself, neglecting others, then (to judge by altruistic 
 standards) others will make me unhappy. 
 
 ' But either I shall live for others or I shall hve for myself. 
 
 ' .-. Either (to judge by egoistic standards) I shall make 
 myself unhappy, or (to judge by altruistic standards) others 
 will make me unhappy.' 
 
 * The additions in parentheses would not, of course, appear in the actual 
 presentation of the Dilemma, nor even in its Retort. 
 
 18—2
 
 276 THE PROBLEM OF LOGIC [VIIL iv. 
 
 Counter-Dilemma : 
 
 ' If I live for others, neglecting myself, then (to judge by 
 altruistic standards) others will not make me unhappy ; and 
 if I live for myself, neglecting others, then (to judge by 
 egoistic standards) I shall not make myself unliappy. 
 
 ' But either I shall live for others or I shall live for myself. 
 ' .-. Either (to judge by altruistic standards) others will not 
 make me unhappy, or (to judge by egoistic standards) I 
 shall not make mj'self unhajipy.' 
 
 We seem, then, to be justified in concluding that, so far as the 
 Constructive Dilemma is concerned, it is possible, where two stan- 
 dards are alternately appealed to according to the convenience of 
 the reasoner, to devise a relevant Counter-Dilemma that shall meet 
 the original argument upon its own ground by playing its own game. 
 The Retort, then, serves the useful purpose of equalizing matters, 
 and the onus of taking any further steps is appropriately left to the 
 propounder of the original dilemma. Where there is no such shifting 
 of the point of \'iew, no retort is required, and no relevant counter- 
 dilemma is possible. A Rebuttal, so understood, is not a refutation. 
 It is the logical means for exposing the dilemmatic fallacy of Sliifting 
 the Standard of Reference, by making a counter-move of a com- 
 plementary' kind. 
 
 If we simplify these results by making Qj and Q2 coincide, we 
 obtain the formula for the Simple Constructive Dilemma and its 
 Counterpart or Retort : 
 
 Dilemma : If Pj, then S^Q ; and if Pg, then SgQ. 
 But either P^ or F^. 
 .-. Either SjQ or S2Q ; .-. Q. 
 
 Counter-Dilemma : If Pj, then SgQ ; and if Pgj then S^Q. 
 
 But either Pj or P2. 
 .-. Either SjQ or SgQ ; .-. Q. 
 
 Example — Dih:7nma : 
 
 ' If I am altruistic, then (to judge by egoistic standards) I 
 am unhappy (because whatever happiness I bring about is 
 other 7>oople's, and not mine). 
 
 ' If I am egoistic, then (to judge by altruistic standards) 
 I am unhappy (because to seek after one's own happiness, 
 says the Altruist, is the surest way of missing it). 
 
 ' But either I shall be altruistic or I shall be egoistic. 
 
 ' .-. I am bound to be unhappy.'
 
 Chap. XXXIL] THE DILEMMA 277 
 
 Counter -Dilemma : 
 
 ' No ! I am bound to be happy. 
 
 ' For, if I am altruistic, then (to judge by altruistic 
 standards) I am sure to find my own happiness (since I don't 
 seek after it). 
 
 ' And if I am egoistic, then (to judge by egoistic standards) 
 I am sure to be happy (because whatever happiness I bring 
 about is my own). 
 
 ' But either I shall be altruistic or I shall be egoistic. 
 
 ' .*. I am bound to be happy.' 
 
 The Rebuttal of the Destructive Dilemma. 
 
 We may formulate the Complex Destructive Dilemma and its 
 Counter-Dilemma as follows : 
 
 Dilemma, : If P^, then S^Q^ ; and if P2, then S2Q2-* 
 But either S^Qj or S2Q2- 
 .'. Either P^ or Pg. 
 
 Counter-Dilemma : If Pj, then S^Qg ; and if P2, then SiQ^. 
 
 But either S2Q2 ^^ S^Q^. 
 .'. Either Pj or Pg. 
 
 The corresponding formulations in the case of the Simple Destruc- 
 tive Dilemma and its Counter-Dilemma will then be : 
 
 Dilemma : If Pj, then SjQi and SgQa-* 
 But either S^Q^ or SgQg. 
 • P 
 
 Counter-Dilemma : If P^, then S2Q2 and S^Qi 
 
 But either S2Q2 or S^Qi- 
 • P 
 
 * The reference to the Standards Sj and S2 would not, of course, be expUciily 
 made either in the presentation of the Dilemma or in that of the Counter-Dilemma.
 
 IX. 
 FALLACIES.
 
 CHAPTER XXXIII. 
 
 FALLACIES. 
 
 Logical Fallacies are infringements of logical principle, and any 
 classification of fallacies based upon this definition would be appro- 
 priate. But what are we to accept as the logical principles, infi- 
 delity to which spells fallacy ? Here differences in arrangement are 
 inevitable. But we would suggest the following classification of 
 logical principles as most in harmony with the distinctions upon 
 which, in dealing with the fundamentals of Logic, we would lay the 
 greatest stress : 
 
 1. The Principle of Definition. 
 
 2. The Principle of Inference. 
 
 3. The Principle of Proof. 
 
 4. The Principle of Inductive Method. 
 
 Corresponding to these we should have four main sources of 
 fallacy : 
 
 1. Ambiguity. 
 
 2. Invalidity. 
 
 3. Inconclusiveness. 
 
 4. Breach of Method. 
 
 Fallacies are, then, of four main kinds. They may be Verbal, 
 Formal or Inferential, Demonst rational. Methodological. Let us 
 consider these types in turn, begining with the Verbal. 
 
 L Verbal Fallacies. 
 
 Ambiguity, as we have already seen, is not to be confused with 
 an appropriate indcfxnitcness in our use of words. All indefiniteness 
 does not call for definition, but only such indefbiiteness as is not 
 sufficiently definite for our purpose. In practical intercourse it 
 would be pedantic to insist on a purposeless refinement of our 
 meaning. In literature, the infinite — i.e., the indefinite — sugges- 
 tiveness of a word or phrase is often its main title to excellence. 
 But in scientific terminolo2;v and nomenclature, in the enunciation 
 
 181
 
 2S2 THE PROBLEM OF LOGIC [IX. 
 
 of all fundamental laws, and in all reasoning that aims at proof, 
 precision in the meanings we give to the terms we use is absolutely 
 indispensable. Here to be indefinite is to be ambiguous and to 
 supply to the reasoned superstructure, that labours to complete 
 itself on the security of undefined meanings, the sandy foundation 
 of tlie Gospel parable. As an illustration we may take the following 
 argument : 
 
 If A is true, I is true. 
 
 If I is true, may be true. 
 
 .'. If A is true, may be true. 
 
 Here the conclusion is, to all appearance, self-contradictory, though 
 the premisses are sound and the reasoning valid. The appearance 
 of contradiction vanishes, however, when once the premisses and the 
 conclusion are rid of their ambiguities. The argument then takes 
 the following form : 
 
 If A is true, I is true. 
 
 If I is true, then (on the ground of I being true) we are unable 
 
 to state anything certain concerning the 
 
 truth or falsity of 0. 
 
 .'. If A is true, then {on the ground of I being true) we are unable 
 
 to state anj^thing certain concerning the 
 truth or falsity of 0. 
 
 T\Tien one and the same word or phrase is used repeatedly (twice 
 or more than twice) in the same argument, it is particularly essential 
 that such indefiniteness as it may legitimately possess should not 
 result in any such variation of its meaning as would amount to its 
 being used in different senses in the various contexts. 
 
 \Vlien in dififerent parts of an argument the same word or phrase 
 is used in different senses, while the argument proceeds as if these 
 senses were the same, we have the important type of ambiguity 
 known as Equivocation. 
 
 In illustration of this fallacy we may cite the following example : 
 
 ' All able men are consistent with themselves. 
 ' He who changes his opinions is not consistent with himself. 
 '.-. He who changes his opinions is not an able man.' 
 
 (Father Clarke.) 
 
 Here ' consistent,' in the major, refers to opinions held together 
 and at the same time, whilst in the minor premiss it refers to opinions 
 held at different times. Cf. Emerson's dictum that ' a foolish 
 con.sistency is the hobgoblin of little minds.' 
 
 Where the Equivocation arises through a confusion of grammatical 
 form, as between masculine and feminine gender, active and passive
 
 Chap. XXXIII.] FALLACIES 283 
 
 voice, etc., the fallacy is called a fallacy of Flexion. Thus the 
 participle of a verb may have acquired a meaning? quite distinct 
 from that of otlicr parts of the verb — e.g., ' All presuming men are 
 contemptible ; this man, therefore, is contemptible, for he presumes 
 to believe his opinions are correct ' (Jevons). 
 
 As a further instance, we may take the following argument of 
 J. S. Mill: 
 
 ' The only proof capable of being given that an object is visible 
 is that people actually see it. The only proof that a sound 
 is audible is that people hear it. And so of the other 
 sources of our experience. In like manner, I apprehend, 
 the sole evidence it is possible to produce that anything 
 is desirable is that people do actually desire it ' (' Utili- 
 tarianism,' chap, iv., § 2). 
 
 In this passage Mill is endeavouring to prove that happiness is 
 the Summum Bonum. He must therefore understand ' desirable ' 
 in the sense of ' worthy to be desired.' But if the word, as Mill's 
 argument requires, is to have a meaning analogous to that of the 
 words ' visible ' and ' audible,' it must mean ' able to be desired.' 
 Now, if this meaning is given to it, the argument can in no way 
 help Mill to show that happiness is a Moral End or Good. 
 
 Where the Equivocation arises through a confusion of the dis- 
 tributive and collective uses of a term, the fallacy is kno\^Ti as a 
 fallacy of Composition or of Division. There is fallacy of Com- 
 position when, having j)redicated something of a term used dis- 
 tributively, we proceed as if the term had been used collectively. 
 
 The following will serve as illustrations of this fallacy : 
 
 1. All drops of water are small objects. 
 
 The Pacific Ocean is (nothing but) drops of water. 
 .-. The Pacific Ocean is a small object. 
 
 2. All atoms are invisible. 
 
 All material objects are (composed of) atoms. 
 .-. All material objects are invisible. 
 
 3. No human beings (singly) are 200 j'ears old. 
 
 The British race is (an aggregate of) human beings. 
 .'. The British race is not 200 years old. 
 
 There was, doubtless, a latent Fallacy of Composition in Sidney 
 Smith's remark during the discussion as to whether it would be 
 possible to pave St. Paul's Churchyard with blocks of wood. ' The 
 Dean and Chapter,' he said, ' need only put their heads together, 
 and the thing would be done.' But though to infer from those 
 dignitaries' being i)idividually blockheads that therefore collectively 
 their heads would serve the useful purposes of a wood-pavement
 
 -5-t THE PROBLEM OF LOGIC [IX. 
 
 was doubtless an instance of this fallacy, yet another kind of 
 Equivocation also contributed to the argument. 
 
 There is fallacy of Division when, having predicated sometliing 
 of a term used collectively, we proceed as if the term had been 
 used distributively. 
 
 The following may serve as examples of this fallacy : 
 
 1. All men are an aggregate including many millions of persons. 
 
 Hence at least one such aggregate is a man. 
 
 2. All coals are atoms of carbon. Therefore some atoms of 
 
 carbon are coals. 
 
 3. English people are not cowardly. Therefore no cowards 
 
 are EngUsh people. 
 
 4. The angles of a triangle (collectively) are equal to two right 
 
 angles. A, B, C (distributively) are the angles of a 
 triangle (distributively). Therefore A, B, C (distributively) 
 are equal to two right angles. 
 
 Perhaps the commonest form of the fallacy is that which it takes 
 in such arguments as ' It must be wrong for you to act in this 
 manner, because if every one did so, the coi^equences would be 
 disastrous.' We start by urging that if A and B and C . . . (con- 
 jimctive) acted in some specified manner — e.g., all studied Logic 
 to the neglect of business — the welfare of the world would be 
 fatally affected, and we go on to argue that no less fatal conse- 
 quences must follow when A or B or C . . . (disjunctive) act in the 
 manner specified — e.g., become so enamoured of Logic as to allow 
 the study of it to lead to the neglect of the interests of business. 
 
 A good illustration of this fallacy occurs in the ' Imitatio Christi ' 
 attributed to Thomas a Kempis (Lib. I., cap. xx., § 8). Here the 
 writer, who is exhorting the ' good monk ' to seek no earthly 
 delight, but to remain alone in his cell, reasons with liim thus : 
 ' What canst thou see elsewhere which here thou seest not ? 
 Behold the sky and the earth, and all the elements ; for of these 
 all tilings are made.' Expressing this argument sj'-llogistically, 
 we see that the fallacy lies in the ' division ' of the iVIiddle Term : 
 
 The elementary substances (in organized combination) are the 
 
 whole material world. 
 The objects included in the prospect from your cell are the 
 
 elementary substances (not so combined). 
 .*. The objects included in the prospect from your cell are the 
 
 whole material world. 
 
 A more important illustration of the fallacy of Equivocation 
 is supplied by the fallacy of Accident and the Converse fallacy of 
 Accident, 
 
 The fallacy of Accident consists in first employing a term or 
 proposition in a relatively indeterminate or unconditioned sense,
 
 CiiAr. XXXIIL] FALLACIES 285 
 
 and then proceeding as though it had been used in a sense rela- 
 tively determinate or conditioned. The nature of this fallacy is 
 sufficiently described by its Latin name, ' Argumentum a dicto 
 simpliciter ad dictum secundum quid ' — that is, an equivocal 
 transition from a statement in its general or indeterminate form 
 to the same statement ' Avith a modification.' Thus, in discussing 
 ethical questions, some one may insist that all men by nature seek 
 after the good, meaning that each man acts sub specie boni, and is 
 therefore seeking after some kind of good, even when he is pursuing 
 his own pleasure. But, in continuing the argument, he may use 
 the term ' good ' in a differentiated sense — e.g., in the sense of tlie 
 ' common good,' or the good of humanity — and come thereby, 
 through a fallacy of accident, to the conclusion that all men by nature 
 seek the general good. 
 
 The fallacy can be easily exposed, once the ambiguity in the 
 use of the term ' the good ' is remedied, and the equivocation thus 
 eliminated. In the light of a Formal criticism, the fallacy of 
 Accident, when used in Syllogism, is seen, like all the other fallacies 
 of equivocation, to reduce to a quaternio terminoriim. As a further 
 instance of this fallacy, we may cite the following : 
 
 The killing of living creatures is sometimes necessary. 
 Murder is the killing of living creatures. 
 IMurdcr is sometimes necessarv. 
 
 In the Converse Fallacy of Accident we first employ a term or 
 proposition in a relatively determinate or conditioned sense, and 
 then proceed as though it had been used in a sense which is relatively 
 indeterminate or unconditioned. We argue ' a dicto secundum 
 quid ad dictum simpliciter.' 
 
 Example. — To drink wine in excess is injurious to health. Hence, 
 since it is wrong to injure our health, we see that 
 to drink wine is wrong. 
 
 Ambiguity may lie not only in the use to which a word or phrase 
 is put, but in the very structure of a sentence. The corresponding 
 fallacy is then known as the fallacy of Amphibole, or Ambiguous 
 Structure, a sentence being ' ampliiboHc ' when it admits of a doublo 
 interpretation. 
 
 Example. — ' The wolf the shepherd slew.' 
 
 Here we are left uncertain whether it was the wolf 
 or tke shepherd that was killed. 
 
 As another example of this fallacy we may take the response of 
 the oracle to Pyrrhus : ' Aio te, ^Eacida, Romanos vincere posse.' 
 ' Pyrrhus the Romans can, I say, subdue.' It was, in fact, the 
 business of the oracle to devise plausible Amphiboles.
 
 2S6 THE PROBLEM OF LOGIC [IX. 
 
 The fallacy of False Parenthesis is an important form of Amplii- 
 bole — e.g., ' I will go and return to-morrow.' This may either 
 mean ' I will (go and return) to-morrow,' or it may mean ' I will 
 go (and return to-morrow).' Cf. also the following : ' I ruined the 
 Cause and injured my ox^ti prospects, which I deep!}'' regret.' 
 
 The following advertisements accurately illustrate the fallacy of 
 Amphibole : 
 
 (a) Wanted : a groom to look after two horses of a pious turn 
 
 of mind. 
 (6) For sale : a Newfoundland dog ; will eat anything — very 
 
 fond of children, 
 (c) Lost : a valuable silk umbrella belonging to a gentleman 
 
 with a curiously carved head (Welt on). 
 
 2. FoEMAL OR Inferential Fallacies. 
 
 The main breaches of the Principle of Formal Inference, not to 
 go beyond the accepted premisses, we have already considered. To 
 break any rule of the Syllogism is to commit a Formal fallacy. As 
 these breaches of rule have already been discussed and illustrated 
 in dealing with the rules themselves and their appHcation, there 
 c-an be no call here for their further consideration. 
 
 We need only allude to one important point, and that is the relation 
 of Formal to verbal fallacy. To apprehend that relation logically, 
 we must see it in the light of the following rule : ' Never consider 
 whether an argument presents a Formal fallacy or not, until it 
 has been adequately cleared of all its verbal ambiguities.' All 
 verbal fallacies must, in fact, be rectified before the argument 
 can be logically convicted of any Formal fallacy. In other words, 
 the Principle of Logical Consistency depends for its correct appli- 
 cation upon a due preliminary observance of the Principle of Non- 
 Ambiguity. Thus, if we wish to discuss the validity of the following 
 argument, 'Some men are selfish; therefore it may be true that 
 some men are not selfish,' we must first of all know precisely what 
 we mean by the words ' some ' and ' may.' 
 
 A further reason why the verbal fallacies must be disposed of 
 first is that this is part of the process of reducing to strict logical 
 form. L'ntil tliis is done, it is irrelevant to speak of Formal Fal- 
 lacies — or, at least, it is quite impossible to detect them. 
 
 We may add that all Verbal Fallacies in argument would, if 
 the argument were written syllogistically, reduce to Quaternio 
 Terminorura, except Amphibole, which would simply vam'sh. 
 
 3. Demonstrational Fallacies. 
 Demonstrational fallacies may be divided under two main 
 heads : ^^ Fallacies of Irrational Evidence. 
 
 (2j Fallacies of Illicit Proof.
 
 Chap. XXXIII.] FALLACIES 287 
 
 (1) Fallacies of Irrational Evidence. 
 
 The characteristic appeal which Science, in its processes of 
 reasoning, makes to the mind is the argumentum ad judicium, or 
 appeal to the reason. When the appeal is not to the impartial 
 reason but to the feelings, passions, prejudices of men, it is, from 
 the logical point of view, radically irrelevant, and involves the 
 fallacy of Irrational Evidence. 
 
 In the Essay concerning Human Understanding (Book IV., 
 ch. xvii., § 22) Locke contrasts what he terms the argumentum 
 ad judicium with certain so-called arguments which, instead of 
 making an appeal to impartial reason, address themselves to the 
 modesty, ignorance, passions, and prejudices of men. ' Of all the 
 arguments,' he writes, ' that men ordinarily make use of, the 
 argument ad judicium alone brings true instruction with it, and 
 advances us in our way to knowledge.' 'For,' he adds, ' (1) it 
 argues not another man's opinion to be right, because I, out of 
 respect, or any other consideration but that of conviction, will not 
 contradict him. (2) It proves not another man to be in the right 
 way, nor that I ought to take the same vnih. him, because I know 
 not a better. ' (3) Nor does it follow that another man is in the 
 right way because he has sho\vTi me that I am in the wrong. I may 
 be modest, and therefore not oppose another man's persuasion ; I 
 may be ignorant, and not be able to produce a better ; I may be in 
 an error, and another may show me that I am so. This may dispose 
 me, perhaps, for the reception of truth, but helps me not to it ; 
 that must come from proofs and arguments, and light arising from 
 the nature of things themselves, and not from my shame-facedness, 
 ignorance, or error.' 
 
 Varieties of Irraiiotial Evidence — {\)The Argumentum ad Hominem. 
 — ^This fallacy has received two different interpretations that should 
 not be confused with each other. 
 
 (i.) The first implies in the so-called argument a very glaring 
 irrelevancy. Instead of defining our position rationally, we merely 
 show that our opponent is not the man to attack it. The cliief 
 characteristic of this form of argument consists in its evading the 
 real issue by means of some personal calumny or rejoinder. Pro- 
 fessor Minto gives the following illustration : 'A story is told of 
 O'Connell that on one occasion, when he had to defend a man 
 who was clearly in the wrong, the counsel for the prosecution was 
 a certain Mr. Kiefe, who had come in for some money in rather a 
 questionable way, and had taken the name of O'Kiefe. O'Comiell 
 commenced his defence by addressing his opponent : 
 
 "Mr. Kiefe O'Kiefe, 
 I see by your brief o' brief 
 That you are a thief o' thief," 
 
 which so disconcerted ]\Ir. O'Kiefe and so tickled the jury that a 
 verdict was returned for the defendant.'
 
 2SS THE PROBLEM OF LOGIC [IX. 
 
 Occasionally tliis appeal to the individual is justifiable. Thus, 
 if a man is zealous in some cause which brings him in a large income, 
 we are quite justified in urging his interestedness against his right 
 to speak on the subject. But this simjily comes to refusing to discuss 
 the matter with him. It should, moreover, be borne in mind that 
 the rejoinder is a mere retort and not an argument, and should 
 therefore be advanced as a retort. If advanced as an argument, it 
 exhibits the fallacy of irrational evidence. 
 
 The argximcntum ad }mssiones, or tlie argumcntum ad 'populum, is 
 an argument similarly irrelevant with the type of argumentum ad 
 hominem we have just been considering. Here it is not the judg- 
 ment that is convinced, but the inclinations and passions. 
 
 (ii.) As a statement of the second interpretation of the argu- 
 mcntum ad hominem, I borrow the following from Dr. Gilbart,* 
 who liimself refers to Dr. Watts' ' Improvement of the Mind ' 
 (c/. Part I., ch. x., § xii.). It will be seen that it differs essen- 
 tially from the interpretation already considered in that it does 
 imply an appeal to the reason. It cannot, therefore, be classed as a 
 fallacj'' of irrational evidence. If we call it a fallacy of irrelevant 
 evidence, it is because its reasoning is not ad rem, but ad hominem. 
 But tliis defect, though it prevents the evidence from having any 
 relevance for scientific purposes, does not jjreclude its possessing a 
 certain formal relevance. As a strictly formal argument, the 
 argumentum ad hominem, thus interpreted, may be perfectly valid. 
 ' Sometimes we may make use of the very prejudices under which 
 a person labours in order to convince him of some particular truth, 
 and argue with him upon his own professed principles as though 
 they were true. This is called argumentum ad hominem, and is 
 another way of dealing with the prejudices of men. SupjDose a 
 Jew lies sick of a fever, and is forbidden flesh by his physician, 
 but hearing tbat rabbits were provided for the dinner of the family, 
 desired earnestly to eat of them ; and suppose he became impatient 
 because his physician did not permit him, and he insisted upon it 
 that it could do him no hurt : surely, rather than let him persist in 
 that fancy and that desire, to the danger of his life, I would tell 
 him that these animals were strangled, which sort of food was 
 forbidden by the Jewish law, though I myself may believe that law 
 is now abolished. 
 
 ' Encrates used the same means of conviction when he saw a 
 Mahometan drink wine to excess, and heard him maintain the law- 
 fulness and pleasure of drunkenness : Encrates reminded him that 
 his own prophet Mahomet had utterly forbidden all wine to his 
 followers, and the good man restrained his vicious appetite by his 
 superstition wlicn lie could no otherwise convince him that drunken- 
 ness was unlawful nor withhold him from excess !' 
 
 * ' Logic for the Million,' p. 125.
 
 Chap. XXXIII.] FALLACIES 289 
 
 (2) Argumentum ad Verecundiam. — This is an appeal to a man's 
 modesty. It consists in alleging, as of more weight than a reasoned 
 proof, the opinion of the majority, the wisdom of the aged, or the 
 tradition of the elders. ' When men are established in any kind of 
 dignity,' writes Locke,* ' it is thought a breach of modesty for 
 others to derogate any way from it, and question the authority of 
 men who are in possession of it. This is apt to be censured, as carry- 
 ing with it too much of pride, when a man does not readily yield 
 to the determination of approved authors, which is wont to be 
 received with respect and submission by others ; and it is looked 
 upon as insolence for a man to set up and adhere to his own opinion 
 against the current stream of antiquity, or to put it in the balance 
 against that of some learned doctor, or otherwise approved writer. 
 Whoever backs his tenets with such authorities, thinks he ought 
 thereby to carry the cause, and is ready to style it "impudence" 
 in anyone who shall stand out against them. This, I think, may 
 be called argumentum ad verecundiam.^ 
 
 (3) Argumentum ad Ignorantiam, or Address to our Ignorance. — 
 ' Another way that men ordinarily use to drive others, and force 
 them to submit their judgments, and receive the opinion in debate, 
 is to require the adversary to admit what they allege as a proof, 
 or to assign a better. And this I call Argumentum ad Ignorantiam.^ \ 
 
 (2) Fallacies of Illicit Proof. 
 
 The two main fallacies to which we are liable when, in tlie light 
 of a truth-interest, we seek to prove conclusions from material 
 evidence are the Fallacies of Petitio Princiyii and Ignoratio Elenchi. 
 The Dilemma, however, is liable to proof-fallacies of a peculiar 
 kind, and we shall therefore treat of these separately. 
 
 (i.) Petitio Principii. — ^Mr. Alfred SidgwickJ has aptly defined 
 Petitio Principii as ' the surreptitious assumption of a truth you 
 are pretending to prove.' The procedure is popularly known as 
 Begging the Question. The terras ' Circular Reasoning ' and 
 ' Arguing in a Circle ' are sometimes reserved to indicate an argument 
 in which a premiss is used that cannot be proved except by means 
 of the very conclusion that it is to assist in proving. Such reason- 
 ing, however, involves precisely the same fallacy as a Petitio 
 Principii, though in a form that is less direct and less easy to detect, 
 and we therefore propose to use Petitio Principii in the larger 
 sense of the term, which includes both the direct and the indirect 
 forms of begging the question at issue. 
 
 That the fallacy is Material and not Formal will be apparent 
 when we consider what we mean by Formal Proof, what we mean 
 
 * 'Essay concerning Human Understanding,' ch. xvii., § 19. 
 
 t Locke, ihid., ch. xvii.. § 20. 
 
 X ' The Use of Words in Reasoning,' p. 131.
 
 290 THE PROBLEM OF LOGIC [IX. 
 
 when we say that the conclusion is proved to be valid on the as- 
 sumption that certain premisses are accepted. There is here no 
 pretence of proving the conclusion as a tnatter of fact ; we profess 
 onlj" to prove it as a conclusion validly inferred. We cannot, how- 
 ever, in the premisses assume the conclusion qua conclusion, for 
 until the premisses are given no conclusion can be drawn. And 
 yet, in a Formal argument, the conclusion cannot be conceived out 
 of relation to the premisses from which it is inferred — cannot be 
 conceived, that is, in any relation other than that of a conclusion 
 from the premisses in question. 
 
 What, in fact, is meant by a petitio principii is that, in our 
 endeavour to establish the material truth of a proposition, we 
 actuall}' utilize for this proof, at one stage of it or another, the 
 very proposition we wish to prove. The attempt to prove the 
 Principle of Identity by means of the Principle of Non-Contra- 
 diction* is a case in point. Another instance is supphed by the 
 famous dilemma of Zeno the Eleatic, in which that paradoxical 
 thinker undertook to prove the impossiblity of motion. 
 
 Zeno's Dilemma : 
 
 ' If a body moves, it must move either where it is or where 
 
 it is not. 
 ' But a body cannot move where it is ; neither can it move 
 
 where it is not. 
 ' .-. It cannot move at all. 
 
 ' I.e., Motion is impossible.' 
 
 The conclusion cannot be gainsaid if once we grant the major, 
 seeing that the major covertly assumes it. The truth, however, 
 is that bodies move neither where they are nor where they are not, 
 but ' from wliere they are to where they are not. Motion consists in 
 change of place ; the major assumes that the place is unchanged — 
 that is, that there is no motion ' (c/. !Minto, ' Logic,' pp. 224, 225). 
 
 The argument is really equivalent to the following : 
 
 ' If a body moves, it must move under conditions which render 
 
 motion impossible. 
 ' But a body cannot move under conditions which render 
 
 motion impossible. 
 ' .-. It cannot move at all.' 
 
 Here the petitio principii is manifest. 
 
 Let us take as a further illustration the following argument, by 
 means of which Father Clarke illustrates this fallacy : 
 
 ' The Catholic Church is infallible. Therefore its sayings are 
 
 infallibly true. 
 ' The Catholic Church maintains the inspiration of the Bible. 
 ' .-. Therefore the Bible is inspired. 
 
 * Vide supra, p. 188.
 
 Chap. XXXIIL] FALLACIES 291 
 
 ' Now, the Bible being inspired, all its statements are infallible, 
 ' But the Bible states that the Catholic Church is infallible. 
 ' .-. The Catholic Church is infallible.' 
 
 If this be taken as a merely Formal inference, there is no Petitio 
 Principii. There is, of course, that futile form of tautology which 
 consists in the respondent's saying that, as the infallibility of the 
 Catholic Church is accepted by the questioner, the conclusion that 
 it is so, on that assumption, follows irresistibly. The conclusion is 
 as irresistible as it is tautological. 
 
 But if the questioner is anxious to discover a cogent proof of the 
 infallibility of the Catholic Church, based on real evidence, the 
 attempt in question involves Petitio Principii in its most unveiled 
 form. The real ground for supposing that the Catholic Church is 
 infallible, it says in effect, is that it is infaUible. This is the funda- 
 mental fallacy of Dogmatism — the fallacy of mistaking assertion 
 for proof — and this is the fallacy of Petitio Principii in its blankest 
 and absurdest form. 
 
 As an instance of a Petitio Principii only one degree less glaring, 
 we have the case in wliich the conclusion, in an argument intended 
 as a real proof, is exactly synonymous with one of the premisses. 
 Thus, a Sophist may bring forward a proposition expressed in 
 words of Saxon origin, and give as a reason for it the very same 
 proposition stated in w^ords of classical origin. 
 
 E.g. : ' To allow every man an unbounded freedom of speech 
 must always be, on the whole, advantageous to the 
 State, for it is liighly conducive to the interests of the 
 community that each individual should enjoy a liberty 
 perfectly unhmited of expressing his sentiments ' 
 (Whately). 
 
 (ii.) Ignoratio Elenchi. — ^The Ignoratio Elenchi may be described as 
 an evasion of the point at issue. In a long argument, for instance, 
 the ground may be shifted, and the actual conclusion reached may 
 thus be different from the conclusion wliich the issue requires. 
 Where the issue is deliberately evaded, the Ignoratio Elenchi 
 becomes a fallacy of ' surreptitious conclusion.' 
 
 This view of the Ignoratio Elenchi does not exactly reproduce 
 the original interpretation of the fallacy, of which it is a modernized 
 modification. The ' elenchus ' was the technical name given to the 
 final syllogism, in which the contrary or contradictory of the oppo- 
 nent's thesis was shown to be true, and the thesis thereby disproved. 
 Hence Ignoratio Elenchi is literally the ignorance of the syllogism 
 required for ' clinching ' a point in tliis special way. 
 
 The famous paradox of Acliilles and the Tortoise, invented by 
 Zeno the Eleatic, may serve as an illustration of Ignoratio Elenchi. 
 
 Achilles and a tortoise run a race together. The details of the 
 race are not stated, but we may suppose, for clearness' sake, that 
 
 10—2
 
 292 THE PROBLEM OF LOGIC [IX. 
 
 Acliilles runs ten times quicker than the tortoise, and accepts, in 
 consequence, a handicap of 100 yards. Under these conditions, 
 says the argument, Achilles will never overtake the tortoise ; for 
 wlien the tortoise has gone 10 yards, Achilles will still be 10 yards 
 behind liim. When these 10 yards are caught up, the tortoise will 
 still be ahead by 1 j^ard. When this yard is caught up, ^^ of a yard 
 will still separate the two, the advantage resting with the tor- 
 toise. When this ~ yard is covered, the lead dwindles to y^ yard ; 
 and 5'et, though it thus decreases continuously from y^y yard to 
 ToVzT ya-rd; and then to ttj-otttj- y^-rd, it still finds the tortoise in front 
 and Achilles behind. Achilles, then, though he will be continually 
 drawing nearer to the tortoise, will never actually overtake him. 
 
 The main gist of Aristotle's criticism of this paradox is that it 
 involves a confusion of infinite length with infinite divisibility of 
 length. The argument aims at proving that the space which Achilles 
 must cover before overtaking the tortoise is an infinite magnitude ; 
 but what it does prove is not this, but simply that the space in 
 question is divisible ad infinitum. It is in this failure to lead the 
 reasoning to its right terminus tliat the Ignoratio Elenchi consists. 
 
 (iii.) Proof -Fallacies in the Dilemma. — When A presents B with 
 a choice between two alternatives, and points out that, whether 
 he choose the one alternative or the other, disastrous conse- 
 quences wiU follow, B is threatened with being fixed on the 
 horns of a dilemma. If he can show, however, that the conse- 
 quences which attach to the acceptance of the alternatives are 
 not really disastrous, or that there are legitimate alternatives other 
 than the two to which he is restricted, he will have freed himself 
 from an embarrassing situation. In technical phraseology, he will 
 have succeeded in his attempt either ' to take the dilemma by the 
 horns 'or 'to escape between the horns of the dilemma.' He takes it 
 by the horns when he successfully disputes the truth of its hypo- 
 thetical propositions ; he escapes between the horns when he shows 
 that the disjunctive premiss is false. 
 
 For instance, if A informs B that if he takes wine he will gradu- 
 ally become a drunkard, and that if he refrains from wine he will 
 necessarily become dull and anaemic, B may justly deny the truth of 
 both the propositions. He may insist that he may still take wine 
 and not lose his self-control, and that he may also cease taking wine 
 and yet remain spirited and full of life. Or, again, A may urge that 
 men are either fools or knaves, and point out that whether they 
 belong to the one class or the other, the outlook for humanity is a 
 dark one. In this case B may attack A's division of men into 
 cheaters and cheated, and protest that the non-cheated need not be 
 cheaters, but rather men who refuse to allow themselves to be 
 cheated because they have a right sense of the dignity of human 
 nature, and believe in a common good.
 
 Chap. XXXIII.] FALLACIES 293 
 
 In further illustration of a dilemma which can be defeated only 
 by escaping between its horns, we may cite the Fatalistic Dilemma, 
 commonly known as the Ignava Ratio. 
 
 The Fatalistic Dilemma {Ignava Ratio) : 
 
 ' If it is fated that you die, you will die whether you call in 
 a doctor or not ; and if it is fated that you recover, you 
 will recover whether you call in a doctor or not. 
 
 ' But it must be fated either that you die or that you recover. 
 
 ' .-. You will either die or recover, independently of the ques- 
 tion whether you will call in a doctor or not.' 
 
 The disjunctive premiss in this dilemma is faultily constructed. 
 The action of an inexorable fate is assumed as the basis upon which 
 the disjunction is built up. When the alternatives are stated in 
 the correct form, ' Either it is fated that you die, or it is fated 
 that you recover,' it is clear that a third alternative is pos- 
 sible — namely, ' or men's destinies are not predetermined by an 
 inexorable fate.' But the admission of tliis third alternative 
 entirely breaks up the logical force of the dilemma. 
 
 The mere rebuttal of a dilemma, as we have seen, is not a refuta- 
 tion of it. It serves, however, as an effective check upon the 
 pretensions of the argument rebutted, and if the counter-dilemma 
 set up to meet the original dilemma is as cogent as its rival, the 
 result will be a logical dead-lock. 
 
 To transcend the dead-lock, we must weigh the respective claims 
 or values of the two standards wliich are at once used and abused 
 by the two dilemmas. If we can decide which of the two standards 
 is the more obligatory, our logical duty will be to apply it rigorously 
 to the case at issue, and so, by the substitution of a Single for a 
 Double Standard of Reference, effectively clear up the situation. 
 If the two standards should turn out to be equally obligatory in 
 relation to the issue with which the dilemmas are concerned, we are 
 confronted with a perplexity which, in its abstractly logical way, is 
 analogous to the perplexities from which tragedies arise, both in 
 drama and in real life. We can but deepen the issue, and revise 
 in the light of our clearer insight. 
 
 Many dilemmas, however, lead to no such tragical dead-lock. 
 There are comedy-dilemmas as well as tragedy-dilemmas. As an 
 instructive example of the Comedy-dilemma, we may cite the famous 
 and ancient dilemma known as Litigiosus. 
 
 Liiigiosus. — Protagoras the Sophist is said to have engaged with 
 his pupil Euathlus that half the fee for instruction should be paid 
 down at once, and the other half remain due till Euathlus should 
 win his first cause. Euathlus deferred his appearance as an advo- 
 cate till Protagoras became impatient and brought him into court. 
 The sophist then addressed his pupil as follcws : ' Most foolish
 
 294 THE PROBLEM OF LOGIC [IX. 
 
 young man, whatever be the decision, you must pay your money ; 
 if the judges decide in my favour, I gain my fee by the decision 
 of the Court ; if in yours, bj'- our bargain.' This dilemma Euathlus 
 rebutted by the following : ' Most sapient master, whatever be the 
 decision, you must lose j^our fee ; if the judges decide in my favour, 
 you lose it by the decision of the Court ; if in yours, by our bargam, 
 for I shall not have gained my cause.' 
 
 Putting the dilemma presented by Protagoras into strict logical 
 form, we get the following : 
 
 ' K the judges decide in favour of Protagoras, then Protagoras 
 
 gains his fee by the decision of the Court ; 
 ' And if the judges decide in favour of Euathlus, Protagoras 
 
 gains his fee bj' the terms of the Agreement. 
 ' But either the judges will decide in favour of the one disputant, 
 
 or they will decide in favour of the other. 
 ' Hence, whether through the decision of the Court or the 
 
 terms of the Agreement, Protagoras will gain his fee.' 
 
 The Retort presented by Euathlus will then run as follows : 
 
 * If the judges decide in favour of Protagoras, Protagoras will 
 
 not gain his fee by the terms of the Agreement ; 
 ' And if the judges decide in favour of Euathlus, Protagoras will 
 not gain his fee by decision of the Court. 
 
 * But either the judges will decide in favour of the one disputant, 
 
 or they will decide in favour of the other. 
 ' Hence, whether through the terms of the Agreement or the 
 decision of the Court, Protagoras will not gain his fee.' 
 
 With a view to meeting the perplexities of the situation, as ex- 
 pressed through these two dilemmas, let us consider the respective 
 values of the two standards relative to the point at issue. The 
 question at issue is whether Protagoras is or is not to gain his fee. 
 The standards are (1) the deed of agreement, (2) the verdict of the 
 Court. With regard to the claims of these two standards, it seems 
 obvious that the deed of agreement has the prior claim. In fact, 
 it is only on the supposition that the terms of the deed have been 
 ignored or violated by Euathlus that the recourse to law can be 
 justified. But it is quite plain that, whilst plaintiff and defendant 
 are pleading their claims before the law, the deed of agreement 
 remains unviolated. Euathlus has not yet won a case, and, until 
 he has done so, the verdict of the Court, whether favourable or 
 unfavourable, can have no legal value. 
 
 The root-error, which consists in the appeal to a second standard 
 when the first is still fully competent to meet all the requirements 
 of the situation, may be detected in another form in the very 
 structure of the disjunctive minor premiss from which both dilemmas 
 are developed. For not only were the judges not obliged to decide
 
 Chap. XXXIII.] FALLACIES 295 
 
 in favour of one or other of the disputants, but it was logically 
 impossible that they should give any verdict at all. There was 
 simply no case to be judged ; for the condition that alone could have 
 justified an appeal to law over the matter — the winning of a case 
 by Euathlus — had never been fulfilled. But where there is no case 
 to be heard, there can be no case to be lost or gained. In particular, 
 a refusal to decide on the part of the judges cannot be interpreted 
 as a verdict in favour of Euathlus. For the judges do not refuse 
 to decide on a case they have considered and found baffling ; they 
 refuse to accept the plaintiff's appeal as constituting a legal case 
 at all. Their verdict is not on the case, but on the nature of tlie 
 conditions requisite to constitute a case. ' There is absolutely 
 no way out of the difficulty except one. The Court can only post- 
 pone the case until the conditions on which a decision depends 
 are fulfilled. Until then, a verdict is impossible. Protagoras had 
 made payment depend on the occurrence of a case which could be 
 decided. No such case had occurred. Hence he must still wait 
 till it did occur. And if it never occurred, he would never get his 
 money. The assumption, of course, is that the verdict is based on 
 the facts of the case. If the Court give an arbitrary verdict, they 
 are simply robbing one or other of the disputants ' (Professor 
 Stout). 
 
 4. Methodological Fallacies. 
 
 The discussion of fallacies in Method falls naturally within the 
 scope of the Theory of Induction. Here we would simply draw 
 attention to the fact that under fallacies in Method we do not 
 include mere inaccuracies in the application of a Method. Such 
 inaccuracies have importance only for the special science which they 
 concern : their significance is not methodological. More generally, 
 errors of fact are not fallacies. 'If the falsity of the premiss,' 
 says Mr. Joseph {ibid., ch. xxvii., p. 532), ' can only be ascertained 
 empirically, there is error, but not fallacy.' Thus it would be an 
 error, but not a fallacy, to assert that the whale is a fish, or that a 
 dromedary has two humps. If, on the other hand, instead of ob- 
 serving all instances relevant to the testing of an idea, we were to 
 observe only such as were favourable to it ; or if, again, we were to 
 ignore the limitations of Inductive Method, and speculate as to the 
 purposes of things instead of inquiring after the laws which they 
 obey, we should commit breaches of Method, and the fallacy would 
 be methodological.
 
 X. 
 
 THE PROBLEM OF INFERENCE. 
 
 (i.) Mill's Estimate of the Syllogism (eh. xxxiv.). 
 (ii.) The Function and Value of a Formal Discipline (oh. xxxv.). 
 (iii.) Truth-Inference, formal and real (ch. xxxvi.).
 
 CHAPTER XXXIV. 
 
 X. (i.} iHLL'S ESTDIATE OF THE SYLLOGISM. 
 
 An Inference, according to Mill, if it is to be an inference at all, 
 must lead us to new truth, must bring us from the known to the 
 hitherto unknown. Now since what we do know, according to Mill, 
 are the particulars of sense-observation and the observed resem- 
 blances between them, these constitute the natural starting-point 
 of Inference ; and the procedure whereby we improve on such initial 
 knowledge is formulated by Mill as follows : 
 
 ' Certain individuals have a given attribute. 
 
 ' An individual or individuals resemble the former in certain 
 
 other attributes. 
 ' .•. They resemble them also in the given attribute.' 
 
 This, says Mill, may be taken as ' an universal type of the reasoning 
 process.'* 
 
 All inference, then, according to Mill, is from individuals to indi- 
 viduals, from particulars to particulars. The child whose fingers 
 have once been burnt makes this particular experience his reason 
 for not touching the grate with his fingers again, and has no thought 
 of any general maxim such as ' Fire burns.' So the village matron, 
 consulted as to how to treat her neighbour's sick child, ' pronounces 
 on the evil and its remedy simply on the recollection and authority 
 of what she accounts the similar case of her Lucy.'f 
 
 After stating that all Inference is from particulars to particulars. 
 Mill goes on to identify such inference with inductive inference. 
 All inference, according to IMill, is inductive, and always consists 
 in reasoning from particulars to particulars. Xow the usual view 
 of inductive inference is that it consists in arguing from particulars 
 to universals.J Mill quite accepts this usual view, harmonizing it 
 with his o\^'Ti through his o^Ti peculiar view of the meaning of 
 generalization. 
 
 * J. S. Mill, 'A System of Logic Ratiocinativo and Inductive,' Book II., 
 ch. iii.. § 7, init. f Id., ib.. § 3. 
 
 + This, of course, is not the same as arguing from the particular proposition 
 I or O to the universal proposition A or E. A particular instance would naturally 
 be expressed in the form of a singular proposition. 
 
 299
 
 300 THE PROBLEM OP LOGIC [X. i. 
 
 If we prefer, argues ^Mill, as a practical convenience, to argue 
 first from a number of known particulars, such as ' Socrates died,' 
 ' Plato died,' etc., to a general proposition wliich includes these 
 particular statements — ' All men are mortal ' — and then to argue 
 syllogistically from this general proposition to the particular pro- 
 position, 'The Duke of Wellington [then living] is mortal,' we may- 
 do so, even with advantage, but we must not fancy that the validity 
 of the inference is in any way increased by taking this circuitous 
 route via the universal proposition. For we do not reason from 
 this universal proposition, even when we have got it, but only 
 according to it. What we do reason from are the particular facts 
 from which the general proposition was first drawn. The universal 
 premiss in a syllogism, according to Mill, contains two elements : 
 a proved part and an unproved part. The proved part is that 
 which registers our previous observations of particular cases ; the 
 unproved part adds the inferences from these, and ' instructions for 
 making innumerable inferences in unforeseen cases.'* ' The un- 
 proved part is bound up in one formula with the proved part in 
 mere anticipation, and as a memorandum of the nature of the 
 conclusions which we are prepared to prove.' f 
 
 With the assertion of the general proposition the inference, on 
 Mill's view, is complete. Hence, when we conclude, according to 
 the major ' All men are mortal,' that Socrates is mortal, or that the 
 Duke of Wellington must eventually die, we make no inference, 
 but simply interpret a memorandum. ' The inference is finished 
 when we have asserted that all men are mortal. What remains to 
 be performed afterwards is merely deciphering our own notes.' J 
 To syllogize, in a word, is not to infer, but to decipher and interpret. 
 
 Mill chnches this statement of his, that Syllogism is not Inference, 
 by showing that, if it be considered as an argument to prove the 
 conclusion, it involves the fallacy of Petitio Principii. He takes 
 the example : 
 
 'All men are mortal, 
 Socrates is a man, 
 .'. Socrates is mortal;' 
 
 and points out that ' we cannot be assured of the mortality of all 
 men, unless we are already certain of the mortality of every indi- 
 vidual man.'§ 
 
 In thus accusing the Syllogism (qiui proof process) of a Petitio 
 Principii, Mill does not go so far as to assert that the conclusion 
 to be proved must have formed one of the particular cases through 
 the observation of which the major premiss was first inferred. 
 He expressly guards himself, in fact, against this misinterpretation 
 
 ♦ J. S. Mill, 'A System of Logic Ratiocinative and Inductive.' Book II., 
 ch. iii., § .3. t Id., ih., § 8, last footnote. 
 
 Xld.,ib.,%^. § Id., ib., § 2. init.
 
 Chap. XXXIV.] MILL ON THE SYLLOGISM 301 
 
 of his meaning. ' Whoev^er pronounces the words " All men are 
 mortal," ' he writes, ' has affirmed that Socrates is mortal, though 
 he may never have heard of Socrates ; for since Socrates, whether 
 known to be so or not, really is a man, he is included in the words 
 "All men." '* Hence, in asserting all men to be mortal, we are 
 implicitly asserting Socrates to be mortal, though Socrates may 
 have had notliing whatever to do with the establishment of the 
 general proposition. Its establishment is an inductive inference 
 from the observations, ' My father and my father's father . . . 
 and . . . other persons were mortal '; f and may have been quite 
 independent of Socrates. 
 
 Criticism. 
 
 If we could agree with Mill's use of the term ' Inference,' there 
 would be no reason to quarrel with his views on the function of the 
 Syllogism. For IVIiU admits that, as a process of the disimplication 
 of premisses (the deciphering or interpreting of a memorandum, as 
 he puts it), the Syllogism is a perfectly valid form of reasoning. 
 There is notliing fallacious about the Syllogism, on his view, unless 
 it be regarded as a process of Inference; for Inference, in Mill's 
 sense, is a process of reasoning from the known to the unknown. 
 In his clear insistence on the point that Syllogism is one thing and 
 Induction another. Mill deserves the gratitude of all who profit 
 from his labours. 
 
 It would be possible, no doubt, to maintain that jMill goes too 
 far in pressing his charge of Petitio Principii against the Syllogism 
 qua proof -process ; for, granted that the conclusion is implied in 
 the premisses adduced in its support, there is surely some novelty 
 in the transition from the implicit to the explicit. Yet, as Mill 
 justly insists, the major premiss ' All men are mortal ' covers the 
 case of Socrates, even though we never heard of that particular 
 case, and though the minor premiss has not yet been stated and may 
 not yet be known to be true. In asserting the major premiss, we 
 do in fact (whether consciously or not) assert Socrates to be a 
 mortal ; for whether we have ever heard of liira or not, or whether, 
 having heard of him, we know whether he is a man or not, still, as 
 a matter of fact, he is a man. Hence, though we may justly claim 
 that there is a distinction between a ' stated ' and a ' covered ' 
 case, and that Mill fails to do justice to the distinction, this should 
 not blind us to the fact that the attempt to treat a syllogism as 
 a material proof or inference (in Mill's sense) is fundamentally 
 fallacious, and that if the fallacy is not a Petitio Principii strictly 
 SD-called, it is a fallacy so closely related to it that it is barely worth 
 while to differentiate the two. 
 
 * J. S. Mill, 'A System of Logic Ratiocinative and Inductive,' Book II., 
 cli. iii., § 8, last footnote. f Id., ib-, § 6.
 
 302 THE PROBLEM OF LOGIC [X. i- 
 
 MilVs View of the Nature of Inference. 
 
 Inference, saj's Mill, is from particulars to particulars. But at 
 the same time he recognizes the possibility of making inferences 
 from particular facts of observation to general propositions or laws ; 
 indeed, his whole theory of Induction is a setting forth of the 
 methods of such inference. Induction is for Mill an inference 
 from particulars to particulars, because he regards the universal 
 propositions arrived at as mere summaries of particular facts — not, 
 however, of the observed facts only, but also of all others that may 
 be inferred from these. 
 
 ]\Iill clearly recognizes two kinds of inference : 
 
 1. Inference from particular facts observed {e.g., ' A, B, C, 
 
 and others, have died ') to a particular conclusion {e.g., 
 ' Socrates is mortal ') ; and 
 
 2. Inference from particular facts observed to a general or 
 
 universal proposition {e.g., ' All men are mortal '), which, 
 though general, is a summary of particulars. 
 
 The second kind of inference is for Mill quite as truly inference 
 as the first. All that he contends for is that the second is not 
 indispensably necessary (as an intermediate step) to the first, 
 though he admits that the circuitous route — via the universal — 
 to the particular conclusion is of the utmost importance and ad- 
 vantage in many cases, and that it is absolutely indispensable to 
 the testing of the validity of the inference. 
 
 We may take it, then, that the substance of Mill's view of Inference 
 is given in the contention that it proceeds from particulars to par- 
 ticulars, and may always move from particular data to a particular 
 conclusion without the intervention of a general 'pr opposition. 
 
 It is against the idea expressed in these italicized words that our 
 criticism must be directed. It is necessary to maintain that 
 without such intervention there can be no logical reasoning from 
 particulars to particulars, that an indispensable part of the logical 
 jjrocess is the disengaging of some universal from the particulars, 
 through the help of a number of particular instances. 
 
 The instances which j\Iill chooses for justifying the process of 
 inference from particulars to particulars, without the intervention 
 of any universal proposition, are almost exclusively selected from 
 the reasonings of children, dogs, savages, and ignorant people. 
 This is no oversight on Mill's part. He is quite aware that the 
 inference which proceeds without generalization is necessarily of a 
 primitive type, or, to use his own expression, ' the rudest and 
 most spontaneous form ' of mental operation.* Generalization, he 
 
 ♦ J. S. Mill, ' A Syritem of Logic Ratiocinative and Inductive,' Book II., 
 ch. iii., § 3.
 
 Chap. XXXIV.] MILL ON THE SYLLOGISM 303 
 
 insists, is ' the most important of all helps ' in reasoning, and 
 not to generalize ' is a defect and often a source of errors.' 
 Mill is, indeed, so conscious that to reason without the help of 
 universal propositions represents a low intellectual level that he 
 makes admissions which clearly show that the reasoning has no 
 logical character at all. Thus, when speaking of the man who 
 reasons from particulars to particulars without using intermediate 
 general propositions, he says of him : ' Though he may conclude 
 rightly, he never, properly speaking, knows whether he has done 
 so or not.'* But the truth is that, in all the cases adduced by 
 Mill in support of his thesis, the reasoning, if even it rises at all 
 above the level of mere association, is certainly not logical — i.e., 
 it is not actuated by any logical ideal consciously held, it is not 
 reasoning conscious of the grounds that justify it. The mental 
 process which keeps the once burnt child from allowing itself to 
 be burnt again is probably the result of mere association. Inference 
 is reasoning in its logical aspect, reasoning conscious of the grounds 
 upon which it ventures its conclusions, so that it is not illuminating 
 to have instances of ' inductive instinct,' the tendency to learn 
 somehow by experience, offered to us in place of instances of ' in- 
 ductive inference.' This inference of one particular fact from 
 another, says Mill, is a case of induction ; and he adds that it is of 
 this sort of induction that brutes are capable — ' Not only the 
 burnt child, but the burnt dog, dreads the fire.'f 
 
 If Mill had considered what really happens when we argue, on 
 the basis of real evidence, from one particular concrete fact to another, 
 he would have seen, as he practically did see in writing his later 
 chapters, that we are logically justified in so doing only when we 
 have first discovered some general law, or laws — preferably some 
 causal law — that shall bind the two instances together. 
 
 Consider the argument, ' All men who have so far lived have also 
 died. Therefore So-and-so, now living, will also die in his turn.' 
 Tliis conclusion can be logically reached only by a process which, 
 through observation and analysis, passes from the observed instances 
 to some general law indissolubly connecting mortality with the 
 conditions of human existence. The process of explanation (as 
 we should prefer to name it) moves, therefore, not from one particular 
 to another, but from particular instances to a systematic network 
 of general laws, and back again from this network of laws to other 
 particular instances. 
 
 It may not be amiss to close this discussion concerning ^Mill's 
 view of Inference with a brief statement, retrospective and pro- 
 spective, of our o^vn view. 
 
 Inference, according to our definition, is the valid disimpli- 
 cation of the meaning of the premisses according to the Laws of 
 
 * J. S. Mill, 'A System of Logic Ratiocinative and Inductive,' Book II., 
 ch. iii., § 8. -j- Id., ib., § 3.
 
 304 THE PROBLEM OP LOGIC [X. i. 
 
 Thought — of Identity and of Xon-Contradiction. Mill, as we have 
 seen, persuaded that the conclusion so reached is not radically 
 novel, denies that this is Inference at all. But the essential re- 
 quirement of Inference (in our sense of the word) is not novelty, 
 but logical irresistibility. For the really novel and fruitful, we 
 must turn to Explanation. Nor should we forget that the inability 
 of an inferred conclusion to trespass beyond the accepted data does 
 not render it superfluous or tautological. Mill himself admits that 
 the conclusion brings out explicitly what is only implied in the 
 premisses. 
 
 We hold, then, that the disengaging of the universal from its 
 expression in particular instances, the disengaging which issues in a 
 generalization or the formulation of a hypothesis, is not Inference, 
 but only the first stage in scientific Explanation of facts. It is 
 certainly not Inference in the strictly logical sense in wliich we have 
 defined the term. Its principle, in fact, is to go beyond the evi- 
 dence, from the known to the unknown. It is, therefore, not 
 Inference. Nor can it be Explanation, for it includes no verification. 
 It is simply generalization, tentative generalization, as yet unde- 
 veloped and untested. 
 
 CHAPTER XXXV. 
 
 X. (ii.) THE FUNCTION AND VALUE OF A FORMAL DISCIPLINE. 
 
 Formal Logic is the Logic of Validity as distinguished from the 
 Logic of Truth, and its central concern is the proper interpretation 
 and correct application, in Opposition, Eduction, and Syllogism, 
 of the fundamental principles of thought. 
 
 As a study in mere Validity, Formal Logic has definite positive 
 value. But it is more than a study in correct thinking. It is the 
 indispensable propaedeutic for the study of scientific Method. 
 For the deductive reasoning which enters so vitally into the at- 
 tempt to interpret Fact through Hypothesis, though it is there 
 exercised in the service of material truth, can render this service 
 effective only through respecting the requirements of Validity. 
 A hypothesis is of little use to Science unless its consequences can 
 be inferred with logical precision. And such precision can be 
 adequately learnt only in that school of Validity, the domain of 
 Formal Logic. 
 
 Further, though framed in the interests of Validity, the Logic of 
 Form sets us at least on our way toward the Logic of Science. 
 Thus Syllogism, properly understood, introduces us, in the simplest 
 and most instructive way, to the fundamental ideas of scientific 
 theory and system.
 
 Ciur. XXXV.] VALUE OF FORMAL LOGIC 305 
 
 The rules of the Syllogism, for instance, form a ' science ' in 
 miniature. Dependent as they are on the single principle of 
 logical Validity, which they express in a variety of forms, they 
 foreshadow to the student the principles of Science and the coherent 
 systems of laws into which these principles can be developed. They 
 are, moreover, collectively applied in the solution of difficulties, 
 and in this way bring home the important truth that in solving any 
 difficulty in any special science all the systematic resources of the 
 science are available, and may be requisite, for the solution. 
 
 At the same time it cannot be too strongly urged that tlie dis- 
 tinction between the procedure of Formal Logic (of Syllogism in 
 particular) and that of the Logic of Science or Methodology is 
 radical. 
 
 This will be clear if we consider for a moment the Enthymeme 
 as the point of departure of the two systems of logical treatment. 
 A man says 'This plant has milky juice; therefore it is poisonous.' 
 Formal logicians, with their interest centred solely in Vahdity, 
 ' complete ' the reasoning as follows : 
 
 ' All plants with milky juice are poisonous plants. 
 ' This plant is a plant with milky juice. 
 ' .•. This plant is a poisonous plant.' 
 
 The Logic of Science, or Methodology, completes the reasoning by 
 analysing the data of the situation. Has the observation been 
 correctly made ? And is the milky appearance of the juice of a 
 plant a trustworthy index to its jDoisonous character ? In a word, 
 whilst a Formal treatment seeks to express the general conditions 
 upon which the validity of the reasoning depends, whateve? the 
 particular circumstances may be, scientific treatment analyses the 
 conditions upon Avhich depends the truth of the conclusion. 
 
 It has sometimes been stated that Formal Logic is useless, since 
 people can reason without its help. Let us briefly consider the 
 justice of the statement. 
 
 It is, of course, true that Logic does not teach us to reason. As 
 Prof. Carvetli Read puts it, ' Wc learn to reason, as we learn to walk 
 and talk, by the natural growth of our powers, with some assistance 
 from friends and neighbours. But, to be frank,' he adds, ' few of 
 us walk, talk, or reason remarkably we41.'* It is the business of 
 Logic to train us into reasoning well. 
 
 But it may be said that Mathematics, and indeed any study 
 that depends on close reasoning for its evidence, is a discipline in 
 good reasoning. This is true enough ; but in none of these sciences 
 are we explicith' taught the principles upon which sound reasoning 
 depends. Apart from logical theory we can give no final or con- 
 clusive justification of our reasoning. We may fall back, as in 
 
 * ' Logic Deductive and Inductive,' third edition, cli. i., p. 6. 
 
 20
 
 306 . THE PROBLEM OF LOGIC [X. iu. 
 
 Mathematics, on Axioms and Postulates and Definitions ; but it is 
 only Logic that concerns itself with studying the nature and value 
 of these first jirinciples. 
 
 W. S. Jevons, in the Preface (p. x) to his ' Studies in Deductive 
 Logic,' lays stress on the fact that 'a mathematical education requires 
 to be corrected and completed ... by a logical education.' The 
 conditions of quantitative reasoning arc, after all, sj^ecial conditions, 
 and the pure mathematician is apt to confuse these with the con- 
 ditions of argumentation in general. Thus, having proved that, if ' 
 a triangle have two sides equal to one another, the opposite angles 
 will also be equal, a mathematician might be tempted to give a 
 separate proof of the proposition ' If a triangle has two unequal 
 angles, the sides subtended by these angles will also be unequal,' V 
 not recognizing tliat the second statement is an immediate inference 
 from the first (c/. ibid., p. ix). 
 
 CHAPTER XXXVI. 
 
 X. (iii.) TRUTH-IXFEREXCE, FORJIAL AND REAL.* 
 
 Where the process of Inference is associated, as it habitually is 
 both in Science and in ordinary life, with an interest in knowing 
 whether the evidence from which a conclusion is validly drawn 
 is itself true or false, the new interest, which is essentially an interest 
 in verification and proof, tends at once to introduce a radical change 
 into the character of the inference. It is not, however, the form 
 of the argument which is thus essentially changed. So long as the 
 logical process consists simply in disimplicating the meaning of 
 given premisses, the process of inference remains structurally the 
 same, whether we are or are not interested in our premisses as 
 items of knowledge. In either case the start is made with given 
 premisses, and the goal is the conclusion drawn with logical necessity 
 from these same premisses. What is changed is the function. 
 With the expansion from the validity-interest to the truth-interest, 
 Inference ceases to be an end in itself, and becomes an episode 
 in the inquiry through which we seek the truth of material 
 fact. 
 
 There are two well-marked stages in the evolution of the 
 ti-uth-interest, and, corresponding to these, two main varieties of 
 
 * The reader will find an interesting and suggestive estimate of the historical 
 connexion between the so-called Traditional Logic and the Logic of Induction in 
 Mr. Joseph's ' Introduction to Logic,' ch. xvii., particularly pp. 34-4-349. Cf. 
 also Prof. Minto's important Introduction to the second book of his Logic.
 
 Chap. XXXVI.] TRUTH-INFERENCE 307 
 
 truth-inference. The interest in Fact and in the truth of it may 
 be either formal or real. 
 
 The potential significance of a given range of fact may be con- 
 ventionally limited in the service of some definite human interest, 
 or it may be regarded, as in the various departments of Natural 
 Science, with a disinterestedness that has no limit except its respect 
 for a reign of law. In the former case, the Truth-inference will be 
 formal ; in the latter, real. 
 
 Of the formal tj^es of truth-inference, we may specify three as 
 of special importance. The world of fact within which the truth- 
 interest is restricted may be a world of make-believe, the make- 
 believe world of a child, or the world which is real for legend or 
 romance. Again, it may be a world of closed beliefs, in which 
 authority supplies the facts and the premisses of inference. The 
 truth-interest of the scholastic Logic moved witliin a world of this 
 kind. Truth, it was held, had already been found. It lay bound 
 up in the creed of the Church and the wisdom of Aristotle. All that 
 was needed was consistently to unravel the implications of these 
 fundamental dogmas and systematically apply them to the needs 
 of life. 
 
 Finally, the truth-interest may be strictly limited in the interest 
 of discussion. The hold on Reality in this case may be very feeble 
 indeed. In the game of Question and Answer with which Professor 
 Minto associates the origin of the Aristotelian Logic,* the dominating 
 interest seems still to have been the n'.terest in Validity. Here the 
 questioner, in his endeavour to entangle the respondent in self- 
 contradiction, was limited by this one consideration, that he 
 could not go beyond the admissions of the respondent. His data 
 consisted simply in the respondent's explicit admissions. These 
 admissions it was the questioner's function not to criticize, but to 
 disimplicate. 
 
 The interests of discussion may, however, be far more closely 
 bound up with those of truth than was the case in this plaj'ful 
 dialectic, or in the formalities of the scholastic discussion-class. 
 They may even be integrally allied with that impartial pursuit of 
 truth which marks the procedure of the Natural Sciences. Scientists 
 may thrash a matter out in the laboratory, with Nature herself as 
 referee, and such discussions would be essentially ' real." But it is 
 customarily and naturally held that the requirements of discussion, 
 as compared with those of investigation, are relatively subjective 
 and conventional. Disputants must be ready to accept each tlie 
 other's point of view, and argue ad kominem {vide p. 28S) rather 
 than ad rem. 
 
 The distinction between a formal and a real reference to Reality, 
 between formal and real types of evidence, is, within certain limits, 
 a relative distinction. In plaj-ing with Reality — and whenever 
 
 * W. :JIinto, ' Logic,' pp. 3 et seq. 
 
 20—2
 
 308 THE PROBLEM OF LOGIC [X. iii. 
 
 we content ourselves with formal evidence our contact with Reality 
 may not unjustly be termed playful — one form of ' play ' may be 
 relatively real as compared with another. ' Within the domain 
 of Chess, Chess Problems are (relatively) formal, while Gaines are 
 relatively real. 
 
 ' In the problems, we start with isolated positions arbitrarily 
 constructed, and considered without any reference to their occur- 
 rence as stages in the development of a game. Indeed, many of 
 them could not occur in a game. 
 
 ' In a Game we proceed according to the same rules of operation 
 as in Problems, but we start with that initial arrangement of pieces 
 wliich involves all possible chess combinations — the whole system 
 of chess-reality. The subsequent development is within this 
 system, and is throughout controlled bv it ' (Professor Stout). 
 
 There is. however, a point at which the relativity of the dis- 
 tinction gives way, and that is where we pass from Reality as 
 regulated by conventional rules to Reality as governed by natural 
 law. The appeal to Fact as an embodiment of natural law sharply 
 discriminates between a real and a formal attitude towards Reality, 
 Where the reference to Reality stops short of a reference to Reality- 
 as-under-natural-law — i.e., to a general system of Reality — the 
 reference is definitely formal in the logical sense of the word. The 
 first logical conversion from a formal to a real grasp of Reality 
 takes place when material evidence is handled in the interest of 
 the scientific belief in natural law. ' To return to Cliess, both games 
 and problems are formal processes from the point of view of Logic. 
 A Game at Chess is logically a formal process, because its total 
 datum — the arrangement of the pieces and the nature of their 
 movements — is not regarded as a part or phase of a general system, 
 but as an ultimate assumption, neither requiring nor admitting 
 any further explanation. And the series of inferences involved in 
 playing a game are quite isolated ; they lead to nothing further ; they 
 do not even supply a point of departure for the next game ' (Pro- 
 fessor Stout). 
 
 We see, then, that the essential characteristic of the truth- 
 interest, in the form in which it is active in Real Inference, is that 
 the evidence upon which it ultimately relies is the potentially 
 unlimited evidence of Fact unfettered by any convention that would 
 prevent the fact from fully displaying its own peculiar nature as 
 a fact under natural law. The appeal in scientific reasoning is 
 always back to Nature and natural law. The facts of Nature, as 
 facts under law, are here the ultimate repositories of evidence ; and 
 the conviction that this evidence remains perennially fresh and full 
 of surprises, and that the fund of reality to be drawn upon is practi- 
 cally inexhaustible, is the characteristic belief of all the Natural 
 Sciences. 
 
 Real Evidence, then, differs essentially from formal evidence by
 
 Chap. XXXVL] TRUTH-INFERENCE 309 
 
 its perpetual back-reference to still undiscovered or unexliausted 
 Fact as its ultimate source of supply. It is essential to the effective- 
 ness of disputational argument based on formal evidence that the 
 evidence sliould be accepted by both disputants, otherwise the 
 deductive process is abruptly checked ; but it is quite unessential 
 tliat the facts, as they exist outside the statements of these dis- 
 putants, should confirm the evidence the disputants are prepared 
 to accept. On the other hand, in talking with Nature the condi- 
 tions are changed. Nature is a silent respondent, and even when 
 Science, the questioner, wrings from her a provisional answer, it 
 is only on the condition that the enforced disclosure be revisable 
 in the light of what may be subsequently elicited. 
 
 This back-reference to an inexhaustible source of further evidence 
 is the distinguishing and controlling feature of Inference based on 
 Real Evidence. The conclusions from Real Evidence can never 
 be considered closed, except in so far as one can claim complete 
 control over the source of supply, in the sense of understanding all 
 the possible disturbing elements and the conditions of their effective 
 appearance. In mathematical Science alone do we possess over 
 real evidence a control of this kind, and consequently ' closed ' 
 conclusions. In formal evidence, on the other hand, the reality 
 referred to may be so limited as to make closed conclusions in- 
 evitable. Thus every conclusion that necessarily follows from 
 premisses accepted by all the interested disputants as true bases of 
 argument is considered closed, so long as the consent originally given 
 to these same premisses is not withdrawn. 
 
 Evidence, then, may be either formal or real. But in either case 
 the meaning of the terms in which the evidence is presented must 
 be unambiguously understood. There is, however, a difference ; 
 for, in order to fix this meaning, we fall back in formal evidence 
 upon formal definitions, whereas in real evidence we fall back upon 
 real or scientific definitions. Again, formal definition itself implies 
 a formal reference to Fact, and real definition a real reference to 
 Fact ; so that ultimately it is always the difference in the respective 
 references to Reality that is the essential difference between the two 
 types of evidence. In the one case the reference is fragmentary 
 and subjectively conventional ; in the other, it is systematized, 
 objective, and methodical. 
 
 W'e cannot leave the question of evidence in its relation to 
 Inference without recalling the reader's attention to the important 
 fact to which reference has already been made in the Introduction 
 (p. 7) and elsewhere — to the fact, namely, that in drawing an 
 inference the interest may be entirely transferred from the nature 
 of the evidence to the logical nature of the reasoning. The interest 
 in tlie distinction between truth and falsity may entirely disappear, 
 and the strictly Formal or abstract interest in Validity take its place, 
 ill sucli event, the reference to Reality is no longer even playful ;
 
 310 THE PROBLEM OF LOGIC [X. :u. 
 
 for the interest in the material truth or falsitj^ of our statements* 
 has completely vanished. There is no further make-believe, or 
 conventional acceptance of stated premisses as true. In the strictly 
 Formal treatment of Logic, the treatment whose sole ideal is 
 Validity, the premisses maj' be true or they may be false, but their 
 truth or falsity is logically irrelevant. 
 
 * Vide Note, pp. 9. 10.
 
 XI. 
 
 INDUCTION AND TPIE INDUCTIVE PRINCIPLE. 
 
 (i.) General Theory of Induction (ch. xxxvii.). 
 
 A. The Pure Inductive Method. 
 
 B. The Essentials of Induction. 
 
 C. Induction and ' Inductive Inference.' 
 (ii.) Hj'pothosis (ch. xxxviii.). 
 
 (iii.) Generalization (ch. xxxix.).
 
 CHAPTER XXXVII. 
 
 XI. (i.) GENERAL THEORY OF INDUCTION. 
 
 A. The Pure Inductive Method. 
 
 The history of Logic, from the time of Bacon and Galileo down- 
 wards, has been mainly determined by the desire to divert Logic 
 from the word-chopping tendencies of formal discussion to the 
 requirements of natural fact. The movement in which this desire 
 found a systematized expression is commonly known as the In- 
 ductive movement. It aimed at substituting for the consistent 
 elaboration of dogma the true explanation of Nature. Its funda- 
 mental working principle was embodied in the general requirement 
 of Fidelity to Fact. 
 
 The first tendency of the so-called Inductive logicians was to 
 keep to the letter this new principle of investigation, and to uphold 
 tlie necessity of allowing the facts to si3eak for themselves, the 
 mind maintaining with regard to them a purely receptive attitude. 
 
 Hence arose the pure Inductive Method, in the stricter, clearer 
 sense of the term — the method which aims at avoiding Hypothesis, 
 for the simple reason that to make a hypothesis is, temporarily, to 
 go bej'ond tlie strict evidence of the facts. According to this 
 method ' No hypotheses or guesses are to be made ; but we must 
 wait till our tabulations of the particular phenomena reveal the 
 general "form" or principle which belongs to them' (Prof. Creighton, 
 ' Logic,' p. 29). 
 
 The natural goal of Science based on this radically cmj^rical 
 method is to become a Science of Statistics so compiled and arranged 
 as to force upon the methodical collector of observations the laws 
 which the facts require to explain them. In tliis way, laborious 
 method takes the place of the scientific imagination and the happy 
 idea. 
 
 ]Much may be said in favour of this primitive conception of the 
 Inductive Principle. The tendency to avoid the use of Hypothesis 
 in the beginnings of Inductive /Science is, as a matter of fact, justified 
 by the very nature of the conditions under whicli alone Hypothesis 
 can be usefull^^ emploj'-ed. To be fruitful and not barren, a hypothesis 
 must be rooted in a scientific system ; where there is as yet no scien- 
 tific system, we cannot expect any fruitful application of Hypotlicsis. 
 
 313
 
 314 THE PROBLEM OF LOGIC [XL i. 
 
 Moreover, we are bound to recognize the great value of that impartial 
 and unprejudiced collection of facts upon which this early em- 
 piricism laid so much stress. And yet it is imperative to add that 
 the mind cannot collect facts methodically unless it also selects 
 them. Not only is there a natural tendency to observe in the 
 liglit and under the selective guidance of a thought, but, further, it 
 is essential to the interests of scientific progress that the mind 
 should go to meet the facts armed Avith ideas. It is not enough to 
 be faithful to fact. Science must be faithful to its own facts, to 
 the facts relevant to its own ideas. What Induction needs as its 
 guiding principle is not a vague Fidelity to Fact, but Fidelity to 
 Relevant Fact.* 
 
 Speaking from the standpoint of a developed Science, it follows 
 
 (1) that, prior to any collecting, the mind should know towards 
 what end and for wliat purpose the collection is to be made ; and 
 
 (2) that during the collection of material it must be ready to seize 
 on any indication of a law embodied in the material, to make a 
 hypothesis of it, and to test it either by experiment, or, if experi- 
 ment is impossible, by further collection of material. 
 
 In illustration of the form which the Baconian method would thus 
 assume, we quote the following from Darwin's ' Autobiography ' : 
 ' After my return to England, it appeared to me that, by following 
 the example of Lj'ell in Geology, and by collecting all facts which 
 bore in any way on the variation of animals and plants under 
 domestication and nature, some light might perhaps be thrown 
 on the whole subject. ... I worked on true Baconian principles, 
 and without any theory collected facts on a wholesale scale, more 
 especially with respect to domesticated productions, by printed 
 enquiries, by conversation with skilful breeders and gardeners, and 
 by extensive reading. ... I soon perceived that selection was the 
 keystone of man's success in making useful races of animals and 
 plants. But how selection could be applied to organisms living 
 in a state of nature remained for some time a mystery to me. In 
 October, 1838 — that is, fifteen months after I had begun my sys- 
 tematic enquiry — I happened to read for amusement IMalthus on 
 Population, and being well prepared to appreciate the struggle for 
 existence which everywhere goes on from long-continued observa- 
 tion of the habits of animals and plants, it at once struck me that, 
 under these circumstances, favourable variations would tend to be 
 preserved, and unfavourable ones to be destroyed. The result of 
 this would bo tlic formation of new species. Here, then, I had at 
 last got a theory by which to work.'f 
 
 Tliis adoption of the Baconian method seems to have come 
 
 * Vide Introduction, pp. 1-8 ; also the concluding chapter on the Inductive 
 Postulate. 
 
 ■f ' Charles Darftin : his Life told in an Autobiographical Chapter, and in a 
 Selected Series of hh Published Letters. Edited by his son, Francis Darwin,' 
 ch. ii.. pp. 39, 40.
 
 Chap. XXXVIL] THEORY OF INDUCTION 315 
 
 naturally to Darwin through his being a born ' collector,' as witness 
 the following extracts : 
 
 At eight years of age ' my taste for natural history, and more 
 especially for collecting, was well-developed. I tried to make out 
 the names of plants, and collected all sorts of things — shells, seals, 
 franks, coins, and minerals. The passion for collecting which 
 leads a man to be a systematic naturalist, a virtuoso, or a miser 
 was very strong in me, and was clearly innate, as none of my sisters 
 or brother ever had this taste.'* 
 
 ' No pursuit at Cambridge was followed with nearly so much 
 eagerness, or gave me so much pleasure, as collecting beetles. 'f 
 
 During the voyage of the Beagle ' another of my occupations was 
 collecting animals of all classses, briefly describing and roughly 
 dissecting many of the marine ones.'f 
 
 It is therefore not surprising to read the following : ' My mind 
 seems to have become [at seventy- two years of age] a kind of 
 machine for grinding general laws out of large collections of facts.' 
 
 On the other hand, this Baconian method was not adopted to 
 the exclusion of the ' Newtonian.' ' I have steadily endeavoured,' 
 he says, ' to keep my mind free, so as to give up any hypothesis, 
 however much beloved (and I cannot resist forming one on every 
 subject) as soon as facts are shown to be opposed to it. Indeed, I 
 have had no choice but to act in this manner, for, with the excep- 
 tion of the Coral Reefs, I cannot remember a single first-formed 
 hypothesis which had not after a time to be given up or greatly 
 modified. '{ With this we may compare a passage from his son's 
 ' Reminiscences.' Speaking of his father, the writer remarks : 
 
 ' He often said that no one could be a good observer unless he 
 M'as an active theorizer. This brings me back to what I said about 
 his instinct for arresting exceptions ; it was as though he were 
 charged with theorizing power ready to flow into any channel on 
 the slightest disturbance, so that no fact, however small, could avoid 
 releasing a stream of theory, and thus the fact became magnified 
 into importance.' § 
 
 We conclude, then, that, whatever stress we lay on the value of 
 collecting instances with open mind and letting the facts speak for 
 themselves, we cannot dispense with Hypothesis. Hypothesis is 
 needed to give meaning to Fact. 
 
 The attitude taken by the radical empiricists of the earl}' Inductive 
 period may be explained, though not, indeed, justified, as a reaction 
 against the apotheosis of dogma in the days of Scholasticism. 
 The principle of investigation here was not fidelity to fact, but 
 
 * ' Charles Darwin : his Life told in an Autobiographical Chapter, and in a 
 Selected Series of his Published Letters. Edited by his son, Francis Darwin,' 
 ch. ii.. p. 6. 
 
 t Ihid^, p. 20. J Ibid., p. 52. 
 
 § ' Life and Letters of Charles Darwin, edited by his son, Francis Darwin,' 
 vol. i., ch. iii., p. 14D.
 
 316 THE PROBLEM OF LOGIC [XL i. 
 
 lovaltv to do^ma. The dictca of Aristotle and of Cburcli Theolof^v 
 ■ncre treated as beyond the reach of criticism, and the aim of 
 Science was construed as that of reconciUng Nature with Dogma, 
 and of proving herself the dutiful handmaid of Philosophy and 
 Theology. 
 
 Thus the discovery of Copernicus was opposed by reasoning such 
 as the followinsr : 
 
 ' Theology teaches that the sun was made to give light to the 
 
 earth, 
 ' Xow, when we wish to light our houses, we do not move the 
 
 house about the torch. On the contrary, we move the 
 
 torch about the house. 
 ' Therefore it is the sun that moves about the earth, not the 
 
 earth about the sun.' 
 
 An old scholastic professor, when his pupil one day brought to 
 his notice Galileo's discovery of sun-spots, made the following 
 characteristic remark : 
 
 ' My friend, I have read Aristotle twice from beginning to end, 
 and I know that there cannot be spots on the sun. Just wipe your 
 glasses a little more carefully. If the spots are not in the telescope 
 itself, they must be in your own eyes.'* 
 
 B. The Essentials of Induction. f 
 
 (1) Principle and Metliod. 
 
 The principle upon which the early empiricism proceeded — 
 namely, that of Fidelity to Fact — was sound at heart, and, as 
 applied by the empiricists, contained implicitly the tendencies 
 requisite to its own correction. Busied as they were with fact, 
 they were equally concerned about Method ; and fidelity to fact 
 implied, as its precondition, fidelity to the method, instrument, 
 or organon, through which the facts were to be approached and 
 studied. For them, fidelity to fact really meant fidelity to fact 
 along the lines of pure inductive method. Relevancy to scientific 
 purpose, though not explicitly recognized, was still unquestionably 
 
 * Vide Ernest Xavillc, ' La Logique de I'Hypothese,' p. 17. 
 
 f More correctly ' Scientific Induction,' for inductive procedure may be either 
 formal or real. It is formal in the service of such restricted interests as that 
 which dominated the Socratic method of finding definitions, or that which prompts 
 the guessing of a riddle. It is real in the service of Science. In the chapters that 
 follow, however, we shall treat ' Induction ' and ' Real or Scientific Induction ' as 
 synonymous terms. We should add that we regard the processes of Definition and 
 Division, whether formal or real, as operations subsidiary to Induction, rather than 
 as Inductive processes proper. The Principle of Fidelity to Relevant Fact is 
 naturally regulative, not only of Induction itself, but of all operations subsidiary 
 to it.
 
 CiiAP. XXXVII.] THEORY OF IXDUCTIOX 317 
 
 implied in tlicir Avliole attitude towards Nature. Thus tlie task 
 of the later Induction has not been that of substituting a fresh 
 principle and a fresh method, but of reconstructing both. Fidelity 
 to Relevant Fact may be accepted as an adequate principle for 
 modern inductive research, but it is simply a specification of the 
 vaguer principle of Fidelity to Fact. Still, the specification is of 
 the first importance. In explicitly introducing, through the use 
 of the term ' relevant,' a reference to scientific purpose, it draws 
 attention to the fundamental truth that fact out of relation to idea 
 is meaningless, and that it is only in so far as fact is relevant to idea 
 that idea can be true to fact. 
 
 Closely connected with this fundamental improvement in the 
 conception of the principle, we have an equally radical improvement 
 in the method of its application. The method was improved by 
 giving Hypothesis a central place in inductive procedure, whilst 
 the empirical ideal was adequately guaranteed by insistence on 
 Verification. Through the Verification-test subjective prejudices 
 are warded off quite as effectively as though they had never been 
 allowed a chance of expressing themselves, and Science is the gainer 
 by enlisting the powerful assistance of the scientific Imagination. 
 The due recognition that the limitations characteristic of the earlier 
 empirical method were not only uncalled for in the interests of 
 fict, but prejudicial to such interests, coincides with the first 
 clear perception of what we really mean by that ' Fidelity to 
 R ^levant Fact ' which is the fundamental guiding idea of Scientific 
 Explanation. 
 
 Nor must it be supposed that this jealous care for Fact implies 
 any opposition to the interests of System. On the contrary, it is 
 precisely the loyalty to Fact that explains the organization of Science 
 into System. A perpetual willingness to be lessoned by fact is, 
 indeed, the characteristic of all Science that has ever succeeded in 
 systematizing itself. And this is very significant ; for it shows that 
 the fact which Science calls Nature is itself systematically struc- 
 tured, and that the systematic character of successful Science is 
 forced upon it by the very nature of the facts it endeavours to 
 interpret. And that the facts can only be interpreted systematically 
 means just this — that Nature, being itself systematic, can only be 
 understood in the light of a system. To interpret facts is to sys- 
 tematize them. 
 
 The Principle of Induction, then, cannot be understood in any 
 sensfe that implies disparagement of System. To be controlled by 
 the facts — the relevant facts — just means to be shaped into 
 systematic coherency through the essential coherency of fact 
 itself. In refusing to go beyond the evidence, the scientific spirit 
 is assimilating the principles of an objective order, making them 
 its own, and approximating more and more to the ideal of sj-ste- 
 matic unity in interpretation.
 
 31 S THE PROBLEM OF LOGIC [XL i. 
 
 And yet it would be misleading to say that Science aims at its 
 own systematization. For the true lesson of Science is that 
 systematic coherency can be gained only indirectly. To aim directly 
 at being systematicallj'- coherent is to forfeit the true objective 
 system to which Fact is the one and only key, and to win a partial 
 systematization that is hopelessly subjective, and doomed on that 
 account to eventual barrenness. 
 
 Hence the supremacy, for Science, of Fact over System.* The 
 true interest of S^^stem is secured by subordinating the desire for 
 systematic coherency to the determination to be at all costs faithful 
 to the facts. Fact dominates System because it is, in its true nature, 
 itself the ideal and standard system. 
 
 (2) Hypothesis. 
 
 We have seen that, in the application of the new Inductive 
 Method, Hypothesis is a central and indispensable factor. It is 
 the essential medium of contact between the scientific system, 
 which is being applied to the explanation of facts, and the facts to 
 which it is being applied. As consistent with the system of which it 
 has been provisionally enrolled a corporate member, a hypothesis 
 belongs to the system. As the direct agency through which the 
 facts are to be explained, it belongs to the facts. Consistency is 
 required of it from the one point of view ; truth is required of it 
 from the other. 
 
 The hypothesis is not artificially grafted on to the scientific 
 system in question, but is its product, and presupposes it. The 
 law of gravitation, as conceived by physical Science, presupposes a 
 whole theory of dynamics, and in the formulation of it the require- 
 ments of this theory of dynamics were all along kept rigidly in view 
 (c/. Newton's treatment of Descartes' ' Theory of Vortices '). It 
 is the scientific system which gives to the h3^pothcs;s its explanatory 
 resources, its deductive vitality. Consider how forlorn and resource- 
 less a theological or pliilosophical hypothesis is when grafted on to 
 a scientific system, and vice versa. 
 
 (3) Deductive Inference. 
 We may distinguish three main functions of deductive inference : 
 
 1. Tlie development of a hypothesis. 
 
 2. The application of a hypothesis. 
 
 3. The proof of a proposition from axiomatic premisses. 
 
 * We do not say ' of Fact over Idea,' for Fact is nothing out of relation to Idea. 
 
 Our point is simply that Science attaches primary importance to being true to 
 Nature in so far as Nature is relevant to its inductive idea. Its own organization 
 as Science is a subsidiary matter.
 
 CiiAP. XXXVII.] THEORY OF INDUCTION 319 
 
 I. Deductive Inference as the Process through which a Hypothesis is 
 developed. — We define an important aspect of deductive inference 
 when we refer to it as the process by which some hj'pothesis, or 
 theory, is, through its connexion with the resources of a system, 
 developed into such necessary consequences as admit of being 
 directlv tested by the facts. 
 
 The inference here (in the simplest form of the process) consists 
 in the drawing of a conclusion from two premisses, of which one 
 is the statement of the hjrpothesis, and the other some statement 
 representing the system. Thus, the hypothesis may be that 
 ' No micro-organisms arise by " spontaneous generation." ' Look- 
 ing at this proposition in the light of our systematized knowledge 
 concerning micro-organisms and the conditions under which their 
 life can be maintained, we are able to connect it with the second 
 proposition, ' Germs produced in sealed sterilized infusions arise 
 by " spontaneous generation," ' and thence we can infer that 
 ' No micro-organisms are produced in infusions so treated.' 
 
 All germs produced in sealed sterilized solutions arise by 
 abiogenesis (premiss supplied by the system). 
 
 No micro-organisms arise by abiogenesis (undeveloped hypo- 
 thesis). 
 
 .'. No micro-organisms are produced in sealed sterilized 
 solutions (developed hypothesis). 
 
 The development of a hypothesis, or verificandum, will, as a 
 rule, be less simple than in the case of the illustration just given. 
 The development may take the form of a Sorites : 
 
 Sa]\Ii (undeveloped hypothesis) 
 IM^a^Ij \ 
 
 ]\I,aM. 
 
 3 
 
 - (system) 
 
 Mn_iaP/ 
 .•• SaP (developed In^othesis). 
 
 Or, in the equivalent pure-h}^othetical form : 
 
 If S, then ^Ij (undeveloped hypothesis) 
 But if Ml, then :Mo \ 
 
 If :M., then M3 
 
 If M„_i, then P 
 
 (system) 
 
 .*. If S. then P (developed hypothesis).
 
 320 THE PROBLEM OF LOGIC [XL i. 
 
 Thus wc may wish to test the suggestion ' If cats abound, red 
 clover is also abundant.' The development will then take the 
 form of the following Sorites : 
 
 If cats abound, red clover is also abundant (undeveloped 
 
 hypothesis). 
 If red clover abounds, humble-bees (which alone"! 
 
 pollinate this clover) must also abound. ( , , . 
 
 If humble-bees abound, then their inveterate foes, j ^ "^ 
 
 the field-mice, must be few in number. J 
 
 .*. If cats abound, field-mice are few (developed hj^othesis). 
 
 Xo one can deny that, through the development of the hypothesis, 
 the problem of verification has been considerably simplified. So 
 far we have dealt only with that form of development which 
 proceeds by analysing the predicate-concept of a categorical pro- 
 position, or the consequent of a hypothetical proposition. The 
 development — though it here assumes a different form — may equally 
 well take place through the analysis of the subject of the categorical, 
 or the antecedent of the hj'pothetical, proposition. Thus the 
 proposition to be tested may be ' S is P,' and analysis of S may 
 show us that S is M. We then substitute for the original Iwpothesis, 
 not the developed hypothesis S is M, but a new hypothesis, M. is P 
 For if we can show that M is P, then, since S is INI, we shall have 
 shoA^Ti that S is P. To recur to the illustration just cited, we may 
 wish to justify the statement that, ' If cats abound, red clover 
 also abounds.' We may then argue as follows : 
 
 If cats abound, field-mice are few. 
 If field-mice are few, humble-bees abound. 
 .*. If cats abound, humble-bees abound. 
 
 Now, if we can justify the proposition, ' If humble-bees abound, 
 red clover also abounds,' we shall also have justified the hypothesis 
 we started with, for we can then reason as follows : 
 
 If cats abound, humble-bees abound. 
 If humble-bees abound, red clover also abounds. 
 .'. If cats abound, red clover also abounds. 
 
 2. Deductive Inference as the Process through which an Hypothesis 
 is applied. — When once the hypothesis has been suitably developed, 
 the application to particular cases follows at once, so soon as the 
 case can be brought under the hypothesis through the discovery of 
 a suitable middle term. Thus we apply the hypothesis that ' No 
 micro-organisms arise by " spontaneous generation " ' to the case 
 of Bacillus suhtilis by bringing this species under the more general 
 concept of ' micro-organism ' : 
 
 No micro-organisms are produced by abiogenesis. 
 Bacillus suhtilis is a micro-organism. 
 .'. Bacillus suhtilis is not produced by abiogenesis.
 
 Chap. XXXVII.] THEORY OP INDUCTION 321 
 
 So, again, we might have occasion to apply the oecological hypo 
 thesis that the Whortleberry belongs to the Heather Association 
 of plants. Formulating our hypothesis in the proposition, ' All 
 districts in which Heather is plentiful are districts in which the 
 Whortleberry is found,' we might apply it by the help of an inference 
 such as the following : 
 
 All heather districts are whortleberry districts. 
 Dartmoor is a heather district. 
 .'. Dartmoor is a whortleberry district. 
 
 We see, then, that in the transition from a Formal Syllogistic 
 to Scientific Explanation, the central function of logical Inference, 
 the drawing of conclusions with logical necessity from stated pre- 
 misses, loses none of its importance. It is, in fact, only when 
 transported into this new inductive setting that logical Inference 
 is seen in its true light as the process through which the implications 
 of Knowledge are unfolded. In its Formal setting, Inference did 
 not develop what had been or was to be adequately verified, but only 
 what had been accepted as data in the interests of the Implication- 
 problem. Formal Inference aims only at validly developing 
 accepted premisses ; scientific Inference aims primarily at developing 
 the truth. Validity is here a secondary, though still an essential 
 requirement. 
 
 Formal Inference gives way, then, in Scientific Method to an 
 ideal of Deductive Inference, to valid inference from real grounds — 
 inference as logically necessary as was Formal Inference, but differ- 
 ing essentially from the latter in its function, which, in last resort, 
 is that of furthering the work of scientific explanation. It is through 
 Deductive Inference that the principle of logical necessity is brought 
 into relation with the investigation of fact : whilst retaining a 
 strictly logical character. Inference now becomes an integral factor 
 in the progress of Science {vide Chapter XXXVI. on ' Truth- 
 Inference, formal and real '). 
 
 Deductive Inference should not be confused with Deduction. 
 Deduction is the wider term. It includes not only the strict infer- 
 ence, but also, as in the proof of Euclid's theorems, for instance, 
 the spatial and other imaginative constructions, and those ingenious 
 combinations of the parts of a whole treatment which provide the 
 strictly novel and progressive element in the whole procedure. The 
 systematic construction of complex trains of reasoning involves, 
 to use MiU's phrases, a great deal of ' scientific dexterity ' and 
 ' artful combination.'* The use of Deduction, in the large sense 
 of the word, is always likely to involve this vital, creative clement, 
 this native tact or ingenuity, this inventive imagination, which is 
 guided by glimpses, suspicions, analogies, rather than b}^ method. 
 
 * 'A System of Logic,' Bk. II.. cli. iv., § 4. 
 
 21
 
 r22 THE PROBLEM OF LOGIC [XI. i. 
 
 In brief, Deduction includes extra-logical elements. Deductive 
 Inference, on the other hand, is a strictly logical process. It can 
 only make the implied exphcit. The inexhaustible fruitfulness of 
 mathematical reasoning, though bound up with its deductive 
 character, is in no sense a product of mere deductive inference. 
 
 Let us consider, in this connexion, the part played by construction 
 in the deductions whereby Euclid proves his theorems. Ostensibly 
 the proofs rest upon the security of definitions, postulates, and 
 axioms. But these in their turn refer us back to Euclidean Space — 
 i.e., to an intuitive basis which is as rich a storehouse of geometrical 
 fact as the living organism is fruitful of data for the biologist. Thus 
 the properties of a circle cannot be deductively inferred from its 
 definition, excepting so far as the definition is that of ' a circle in 
 Euclidean Space,' and is supported by actual constructions. To 
 prove that the angle in a semicircle is a right angle, we must construct 
 a circle in Space (as idealized to meet the purpose of our traditional 
 geometry), and draw special lines within the circle. We thus avail 
 ourselves of the inexhaustible fertiUty of intuited space when 
 penetrated by the invisible network of geometrical ideas. To realize 
 intuitively the continuity and tridimensionality of space, and its 
 susceptibihty of being determined into forms through inclosure of 
 its parts, is to be master of properties that admit of endless com- 
 bination and development through the medium of construction. 
 
 This ultimate reference to a space-intuition is concealed in 
 Euclid's Geometry. Thus, as Professor Latta has pointed out,* 
 (1) the definitions and postulates presuppose this system of space 
 ' without sho^ving how the figures described in the definitions, 
 or the right to demand these postulates, follow from the very 
 nature of the space itself ' ; and (2) the proofs are stated as though 
 they were deduced solely from these definitions, postulates and 
 axioms, whereas ' the proof of each proposition requires a " con- 
 struction " of some kind to be made, such as the producing of lines 
 or the superposition of figures. ... If you produce two sides of a 
 triangle in order to prove something about its angles, you implicitly 
 recognize that the triangle is not a self-complete system, the proper- 
 ties of which may be directly deduced from its definition, but that 
 it is an element in a surface, and that its internal properties are 
 logically dependent on its external relations, or, at least, are in the 
 most intimate connexion with them.' 
 
 3. Dedvclive Inference as the Process through which a Proposition 
 is PROVED. — In the endeavour to jjrove or demonstrate the truth of 
 a proposition, the aim must be to discover a basis of proof wliich 
 does not itself stand in need of proof, and from which the pro- 
 position may be inferred with logical necessity. 
 
 * ' On the PvC'lation between the Philosophy of Spinoza and that of Leibniz,' 
 Mind, New Seriea, vol. viii., 1S99, p. 335.
 
 Chap. XXXVII.] THEORY OF INDUCTION 323 
 
 There are three main ways in which we may thus transform our 
 demonstrandum into the conclusion of a deductive inference of 
 which the premisses are either axiomatic or can themselves bedemon- 
 strated as necessary conclusions from axiomatic premisses.* The 
 deductive inference through which the proof is in last resort effected 
 may be categorical or disjunctive or hypothetical. 
 
 Thus the demonstrandum ' All S's are P's ' may be accepted 
 as proved, provided we can accept ' All M's are P's ' and ' All S's 
 are ^I's ' as axiomatic propositions or as propositions already proved. 
 The proof will then take the categorical form : 
 
 ' All M's are P's ' is true. 
 
 ' All S's are M's ' is true. 
 
 .-. ' All S's are P's ' is also true. 
 
 Where the inference is disjunctive, the proof proceeds by a 
 Method of Exclusion, and may be called Proof by Exclusionf or 
 Proof by Exhaustion. J Suppose we wish to prove the proposition 
 ' Si is P^,' and are in a position to set down the proposition ' Either 
 Sj is Pi or So is P, or S3 is Pg ' as axiomatic or as demon strable from 
 axiomatic grounds. We have then only to show that ' Sj is Po' and 
 ' S3 is P3 ' are true in order to demonstrate the truth of ' Sj is Pj.' 
 Thus, to borrow the example cited by Mr. Milnes, suppose our 
 thesis to be that of Euclid I, 19 : 'If, in any triangle, one of two 
 angles be greater than the other, the side subtending the greater 
 
 angle is greater than that subtending the less. This we may 
 prove by arguing as follows : 
 
 Either the side AC is greater than the side AB, or else it is 
 less than the side AB, or else it is equal to it (axiomatic). 
 But AC cannot be less than AB, nor can it be equal to it. 
 Therefore AC is greater than AB — i.e., the side subtending 
 the greater angle is greater than that which subtends the 
 less. 
 
 * In the discussion that follows, v>g assume that such axiomatic or self-evident 
 propositions are obtainable. For a brief consideration of this point, see 
 Chapter XLV. There is, however, no attempt made in tiie present volume 
 to deal adequatelj' with the theory of Mathematical Knowledge, or with the 
 problem of self -evidence. 
 
 t See Dr. Christoph Sigwart, ' Logik,' vol. ii., ch. iii., § SI, 10. English Transla- 
 tion by Helen Dendy, p. 201. 
 
 I Cf. Alfred Jlilnes, ' Elementary Notions of Logic,' second edition, p. 85. 
 
 21—2
 
 324 THE PROBLEM OF LOGIC [XL i. 
 
 The exclusions through which, in the application of this method, 
 the proof of the non-excluded alternative is effected are frequently 
 justified by a form of proof usually kno\\Ti as Indirect Proof. The 
 proposition ' So is Pg ' is assumed to be true, and it is then shown 
 that on this assumption tlie proposition ' Sn is P„ ' must also be 
 accepted as true. Hence, if the proposition ' S^ is P^ ' contradicts 
 a proposition already proved to be true, if follows by the Modus 
 Tollens that ' S2 is Pg ' cannot be true. Thus the main argument in 
 Indirect Proof runs as follows : 
 
 If S, is P2, then Su is Pq 
 But S„Js Pn. 
 .♦. S2 is Pg. 
 
 The argument through which an indirect proof is effected is 
 known as a reductio ad ahsurdum, the absurdity to which the argu- 
 ment reduces us being that of self-contradiction. 
 
 The Reductio ad Ahsurdum has been utilized, under the name of 
 Indirect Reduction or Reductio per impossibile, to test the correct- 
 ness of inferences drawn in the ' imperfect ' figures of tlie Syllogism 
 (Figs. II., III., and IV.), on the assumption that inferences drawn 
 in the first or ' perfect ' figure, according to the Dictum de Omni 
 et Nullo, could be accepted as correct. The process, however, is 
 applicable to syllogistic reasoning only so far as tliis is treated 
 as a truth-inference, so that the premisses and conclusion can 
 legitimately be characterized as true or false. Indirect Reduction, 
 then, may be defined as a proof, effected by means of a syllogism 
 in the first figure, that the truth of certain conclusions drawn 
 in the ' imperfect ' figures follows with logical necessity upon the 
 truth of their premisses, because, if those premisses are true, the 
 contradictories of those conclusions are necessarily false. 
 
 The traditional Logic, following Aristotle,* who in Direct Reduc- 
 tion made use of Conversion only and did not recognize Ob version,! 
 singled out Baroco and Bocardo as the most suitable forms of 
 Syllogism for illustrating this process of Indirect Reduction. We 
 may therefore take one of these — Baroco — and use it for the illus- 
 tration of the method. 
 
 If it is suggested that the inference in Baroco is not correct — 
 so the method argues — then in that case, though we assume the 
 premisses to be true, the truth of the conclusion ' Some S's are-not 
 P's ' will not necessarily follow. That is, granted the truth of the 
 premisses ' All P's are M's ' and ' Some S's are-not M's,' still ' Some 
 S's are-not P's ' may be false. But since the denial of a proposition 
 is logically equivalent to the affirmation of its contradictory, it 
 follows that the proposition ' All S's arc P's ' may be true. Let 
 us, for the sake of the argument, assume that it is true. Taking 
 
 * Vide An., Pr., A., c. 4.j, p. 516, 1, 2. f Cf. above, pp. 189-192.
 
 CiiAP. XXXVII.] THEORY OF INDUCTION 325 
 
 this proposition, ' All S's are P's,' as the minor premiss of a syllo- 
 gism, and combining it with our original major premiss, ' All P's 
 are M's,' we can at once draw the conclusion ' All S's are M's ' 
 in the standard or ' perfect ' figure. But this conclusion contra- 
 dicts our original minor premiss ' Some S's are-not M's.' Now 
 the premisses of the original syllogism are both, ex hypoth»si, true. 
 The statement ' All S's are M's ' is therefore false. Hence, since 
 the form of reasoning in Fig. I. is admittedly valid, one at least 
 of the two premisses which necessitated this conclusion must be 
 false ; for, if both were true, then, by the Principle of Identity, the 
 conclusion ' All S's are M's ' would also be true. But of these two 
 premisses, ' All P's are M's,' being our original major premiss, is, 
 ex hypothesi, true. Hence the assumed statement ' All S's are P's ' 
 must be false, and its contradictory, ' Some S's are-not P's,' must 
 be true. But this is the conclusion of our original syllogism in 
 Baroco. Thus we see that in Baroco, granted the truth of the 
 premisses, the truth of the conclusion necessarily follows. We 
 have thus shown that Baroco is a valid form of Syllogism. 
 
 Any other of the valid forms may be justified in a precisely 
 similar way. 
 
 The INIethod of Proof assumes a peculiar form whenever the 
 premisses from which the demonstrandum is deducible are reversible 
 or simply convertible. In this case these requisite premisses may 
 be discovered by means of a process of regressive analysis based 
 upon the assumption that the demonstrandum is true. We may 
 show that, on this assumption, certain consequences necessarily 
 follow which are already known to be true. We may then reason 
 back, in Sorites form, from these known truths to the demon- 
 strandum. Thus, let D be the demonstrandum. The regressive 
 analysis will then take some such form as this : 
 
 If D is true, C is true. 
 If C is true, B is true. 
 If B is true, A is true. 
 But A is true 
 
 Now, if the premisses are of such a character as permits us to argue 
 from the truth of the consequent to that of the antecedent — i.e., 
 if the premisses of the sorites are reversible — we can present the 
 proof as follows, in the form of a series of hypothetical syllogisms 
 in the Modus Pouens : 
 
 A is true. 
 But if A is true, B is true. 
 
 .-. B is true. 
 But if B is true, C is true. 
 
 .-. C is true. 
 But if C is true, D is true. 
 .*. D is true.
 
 329 THE PROBLEM OF LOGIC [XL i. 
 
 C. Induction and ' Inductive Inference.' 
 
 (1) The Meaning of Induction.* 
 
 It is essential to note the wide sense in which we are proposing 
 to use the term ' Induction.' It is not unusual to identify In- 
 duction with the first stage in the whole process of Scientific Ex- 
 planation, the stage which starts with the observation of facts and 
 terminates in the formulation of some hypothesis. Were this 
 nomenclature adopted, Induction, Deduction, Verification would 
 be the three successive stages in a complete scientific explanation. 
 But if we do adopt this nomenclature, we must cease to talk of 
 Inductive Logic, and must speak, instead, of the Logic of Scientific 
 Explanation. For it is Scientific Explanation, qua completed 
 process, which alone is governed by the fundamental principle of 
 Fidelity to Relevant Fact. So far as the goal of a reasoning-process 
 is the mere formulation of an undeveloped, unverified hypothesis, 
 it is surely unreasonable to contend that it aims at not transgressing 
 the evidence of fact. Does it not embody a tendency to go bej^ond 
 the facts rather than not to go beyond them ? 
 
 Assuming, tlien, that we accept ' Fidelity to Relevant Fact ' as 
 the Inductive Principle, the use of the word Induction in its narrower 
 sense of a tentative passage from particulars to universals is, strictly 
 speaking, illegitimate. We cannot be faithful to fact by reposing 
 on untested generalizations from experience and dispensing with the 
 test of Verification. In the wider and legitimate sense of the 
 term, ' Induction ' covers the whole process of Scientific Explana- 
 tion, the formulation and verification of Hj-pothesis, a process 
 which ends only when, through its various methods, it has made 
 sure that its tentative explanation does not go beyond the evidence 
 of the facts. 
 
 (2) The So-called ' Inductive Infeicnce.^ 
 
 So account of Induction, however introductory in character, 
 can dispense with an allusion to the much-abused term ' Inductive 
 Inference.' We hold, for our part, to the simple conviction that 
 there is only one fundamental type of logical Inference — that, 
 namely, which consists in rendering explicit what is implied in a 
 system of given premisses, through the sole help of the principle 
 of logical Validity. We consequently view the term ' Inductive 
 Inference ' as a misnomer. Inference may be Formal or Deductive ; 
 may be drawn, that is, from Formal premisses in the fight of a strictly 
 abstract validity-interest, or else from material grounds in the 
 light of a genuine truth-interest. But in either case it is a strictly 
 
 * See also footnote, p. 3IG,
 
 Chap. XXXVIL] THEORY OF INDUCTION 327 
 
 logical process governed exclusively by tlie Law of Identity and the 
 Law of Non-Contradiction. Now, in inductive procedure, the only 
 inferential stage of this kind is that of tlie deductive development 
 and application of Hypothesis. Hence ' Inductive Inference ' is 
 either a misnomer for Deductive Inference, or it is the name for a 
 type of thinking wliich is not exclusively governed by the Law of 
 Logical Validity — that is, by tke above-named Laws of Thought. 
 
 It is in this latter sense, of a heuristic, tentative suggestion or 
 supposition, that the term is customarily used. Thus, having ob- 
 served that a large number of instances of a certain class have the 
 mark x, we are said to infer (inductively) that all the instances of 
 that class may be found to possess the mark x. So again, in the ca»se 
 of what is known as ' Analogical Inference,' there is a precisely 
 similar use of the word ' Inference ' in the sense of a tentative, 
 though it may be a well-grounded, suggestion. We are accustomed 
 to say that, since A resembles B m many important respects, we 
 infer (by Analogy) that it will resemble it in some further respect 
 also. 
 
 The meaning we have given to the term ' Inference ' prevents U3, 
 however, from making use of it to designate any tentative form of 
 argument, whether enumerative, analogical, or of any other kind. 
 We therefore, somewhat reluctantly, renounce the use of the 
 familiar and time-honoured term ' Inductive Inference,' and with 
 it the use of such cognate expressions as ' probable ' and ' analogical ' 
 inference. The Theory of Induction, as we conceive it, is Induction 
 without Inductive Inference. As a general substitute for ' in- 
 ference ' in this sense, we propose to use the term ' conjecture.' 
 Thus, on the ground that certain S's are P's, we conjecture that all 
 S's will be found to be P's ; and, on the basis of the many important 
 resemblances between the Earth and Mars, we conjecture tliat Mars 
 will resemble the Earth in being inhabited also. In this way we 
 hope to avoid confusion, though we cannot hope either to satisfy 
 the ear or to uproot the inbred prejudice in favour of drawing 
 ' inferences ' from grounds which do not imply but only suggest 
 them.* 
 
 * It may be worth while, at this point, to draw attention to the ambiguity 
 attaching to the present use of the more natural term ' Inference.' As currently 
 used, it denotes now a Formal inference, now a deductive inference, now a deduc- 
 tion, now a complete induction, now some form of tentative guess-work culminating 
 in a hypothesis. It also denotes now a process, now a product. Thus, the conclu- 
 sion of a syllogism is frequently spoken of as an inference from the premisses, whilst 
 the process of disimplicating the conclusion from the premisses is also referred to 
 as an inference. Our own use of the term ' Inference ' is intended to refer exclu- 
 sively to processes of strictly valid reasoning — that is, reasoning in which the con- 
 clusion follows from the premiss or premisses with logical necessity.
 
 32S THE PROBLEM OF LOGIC [XI. ii. 
 
 CHAPTER XXXVIII. 
 
 XI. (ii.) HYPOTHESIS. 
 
 J. S. i\IiLL defines a Hj^pothesis as follows : ' An hypothesis is any 
 supposition which we make (either without actual evidence, or on 
 evidence avowedly insufficient) in order to endeavour to deduce 
 from it conclusions in accordance with facts which are known to be 
 real.'* Xot ever}' supposition, therefore, can rank as a HyjDothesis. 
 A Hypothesis is a supposition made in view of a truth-interest. 
 It is a supposition wliich (1) admits of being devcloj)ed into its 
 consequences, and (2) requires and admits of verification. f 
 
 What ^lill thus defines is, in fact, the legitimate Scientific Hypo- 
 thesis. To be legitimate a Hypothesis has one essential condition 
 to satisfy : it must be verifiable. { But to be verifiable it must be 
 adequately developable. 
 
 A legitimate hypothesis, again, is identical with a working hypo- 
 thesis in the widest sense of that term. For a working hypothesis 
 {e.g., 'Electricity behaves as though it were a fluid'; 'Vegetable 
 mould is due to the action of earthworms ') is a hj^pothesis that 
 works — works, that is, by attempting to explain the facts. 
 
 A successful working hypothesis is a hypothesis that works well. 
 To work well, a hypothesis must be both resourceful and fruitful. 
 To be resourceful, it must be rooted directly (or indirectly through 
 the medium of a general working idea) in a reasoned system or science. 
 To be fruitful, it must be capable of continually extending its sphere 
 of verification, and of bringing more and more facts under scientific 
 control. 
 
 A working hypothesis is, as a rule, closely allied with what we 
 maj^ suitablj^ call a working idea. This is the germinal conception 
 out of which the true working hypothesis is shaped. The working 
 hypothesis is developed out of the working idea not by being deduced 
 from it with logical necessity (the ' idea ' would in that case be 
 only a more fundamental working hypothesis), but by processes 
 of imaginative construction of a purely tentative kind. The work- 
 ing idea stands to the working hypothesis thus developed from it 
 in a relation somewhat analogous to that in which the subject of 
 discourse stands to the particular proposition through which it is 
 at any moment being developed. § It stands for the relatively 
 
 * ' A System of Logic,' Book III., ch. xiv., § 4. 
 
 t The term ' Verificatioa ' is, in its current use, ambij^uous. Ordinarily, as here, 
 it means a ' test ' — a test that may result in disproving the hypothesis. In its 
 stricter u.se, ' Verification ' is the process which pro tanlo confirms the truth of 
 a hypothesis. Only sorm hypotheses would, in this sense of the term, admit of 
 being verified. Similar remarks apply to the use of the term ' verifiable.' 
 
 X For a more radical criterion of the legitimacy of a hypothesis, see the chapter 
 on the Inductive Postulate. 
 
 § Cy. pp. 118. 119.
 
 Chap. XXXVIII.] HYPOTHESIS 329 
 
 indeterminate guiding conception wliich may subsist unchallenged 
 through tlie successive failures of a great many working hypo- 
 theses which have been constructed along the general lines marked 
 out by it. 
 
 Illustration of the Importance and Significance of the 
 Working Idea from the History of Astronomical Science. 
 
 Problem : To formulate a theory or system that shall exhaustively 
 account for all the varied movements of the heaven!}'' 
 bodies. 
 
 1. Ptolemy^ s System. 
 
 In the second century of our era, the Alexandrian astronomer 
 Claudius Ptolemseus brought all the observed movements of the 
 heavenly bodies into one system. The main feature of this system 
 is that the Earth at rest is taken as the centre of the universe, while 
 sun, moon, planets, and stars revolve in various circles around it. 
 
 The main working ideas* of this system were : 
 
 1. The Earth, as the one fixed centre of the universe, is cer- 
 
 tainly at rest, and the celestial movements we see are the 
 real movements. The opposite suggestion was charac- 
 terized by Ptolemy as ' tlie height of absurdity.' 
 
 2. The heavenly bodies are divine and incorruptible, and must 
 
 therefore move in circles (the circle being considered by 
 the ancients as the most perfect geometrical figure). 
 
 On the basis of these working ideas the system was built up. 
 Now it had long been noticed that the apparent paths of the planets 
 were anything but simple. A planet watched night after night 
 would be seen first to move from west to east with a varying velocity, 
 then to stop, then to adopt a retrograde movement, then to stop 
 again, then to go forward again, and so on. 
 
 The problem was to analyse these movements into a system of 
 circular movements, the Earth at rest being the ultimate centre of 
 the whole system. 
 
 From the nature of these observed movements it was plain that 
 the simple device of making each planet go in a circle round the Earth 
 as centre would not do. It was therefore necessary to invent a 
 system of epicycles, an epicycle being a smaller circle whose centre 
 moves along the circumference of a greater. Each planet was then 
 supposed to move uniformly along the circumference of the epic3'cle, 
 whilst the centre of the epicycle itself moved uniformly in a circular 
 orbit round the Earth. Thus the system of epicycles was an attempt 
 to explain the movements of the planets ; but the attempt was not 
 
 * That Ptolemy himself should have treated these as axiomatic does not affect 
 thoir true logical character as ' working ideas '
 
 330 THE PROBLEM OF LOGIC [XL ii. 
 
 altogether successful. It was found necessary to supplement the 
 system of epicycles with a system of excentrics. 
 
 This ' excentric ' device consisted essentially in allowing the 
 planet's cycle (not its epicycle) to move about a point outside 
 the Earth, this point itself being supposed to revolve about the 
 Earth as centre, so that in last resort the sanctity of the first idea 
 should not be violated. The sanctity of the second was preserved 
 inviolate through the care taken to make all the orbits circular. 
 
 Such a sj'stem was Ptolemy's. Now, as observations became 
 more accurate, it was found continually necessary to modify this 
 system of wheels witliin wheels. New circles had continually to 
 be added (for instance, an epi-epicycle revolving on an epicycle), 
 until in Copernicus's day seventy-nine of these circles were found 
 necessary in order to represent, even with the roughest approxima- 
 tion to accuracy, the movements of the heavenly bodies. 
 
 2. The System of Copernicus. 
 
 The main advance made by the system of Copernicus on the 
 system of Ptolemj'- lay in its completely shifting the point of view 
 and shattering the first of Ptolemy's dogmatic Ideas. It showed 
 by a clear mathematical treatment (which took thirty years to 
 develop) that the observed movements of the planets were ap- 
 parent, not real; that the Sun was the centre of the system of 
 planets ; and that the Earth was only one planet like the others. 
 Copernicus showed that no amount of mere modification could 
 justify the Ptolemaic conception of things ; tliat its fundamental 
 assumption, being entirely incorrect, must be given up in favour 
 of the Heliocentric Theory. 
 
 The great work of Copernicus, then, was the shattering of this 
 first Idea. But he was still himself a slave to the second — the idea 
 that the circle, as the only perfect figure, could alone be worthy 
 to represent the path of a planet. The fundamental change which 
 transferred the centre of motion of the solar system from the 
 Earth to the Sun had made perfectly easy the explanation of the 
 stationary points and retrograde (apparent) movements of the 
 planets ; but there were still certain minor irregularities which 
 Copernicus tried to solve by means of a new system of epicycles 
 and excentrics. 
 
 Moreover, he had not completely emancipated himself from the 
 first Idea. The Sun took the place of the Earth, not only as the 
 centre of the solar system, but as the centre of the universe, so that 
 the Ideas of Copernicus — propositions which were to him axiomatic 
 — were — 
 
 L There is a centre of all things, the Sun. 
 
 2. The circle is the only perfect figure ; therefore all heavenly 
 bodies move directly or indirectly in circles.
 
 Chap. XXXVIII.] HYPOTHESIS 331 
 
 The stars were conceived by Copernicus as absolutely fixed in 
 a great all-embracing sphere. They did not shine by their own 
 light — i.e., they were not suns themselves — but, like the moon, 
 reflected the light of the Sun, the source of all the light as well as 
 the centre of all the movement of the universe. 
 
 3. Kepler. 
 
 It was left for Kepler to remove once and for all the second Idea 
 of the Copernican System ; and he also made a great step towards 
 the removal of the first. This latter step he took when he showed 
 that the Sun itself is only a star, and that the stars are suns 
 made star-like bj^ distance. He showed that the stars must be at 
 least 2,000 times as far away as Saturn, the planet wliich at that 
 time was regarded as the outside warder of the solar system. But 
 he still imagined that the stars were all of a piece, all on one sphere 
 at the same distance from the Sun. Further, it was a long time 
 before Kepler removed the hoary prejudice involved in Idea Xo. 2. 
 It was only because the fundamental Idea on wliich he himself 
 worked was seen eventually to require its suppression that he was 
 able at length to lay aside the prejudice of 2,000 3"ears. 
 
 Kepler's one leading Idea was that the Creator must have been 
 a geometer. The idea is Pythagorean. It was adopted by Plato, 
 and borrowed from Plato by Kepler. It was no longer ' The 
 Creator must have arranged the orbits of the heavenly bodies on a 
 circular pattern,' but ' He must have arranged them on a geometrical 
 pattern.' At the same time Kepler was at first quite unconscious 
 of this distinction, and all his first efforts, directed mainly towards 
 explaining the movements of Mars on the basis of Tycho-Brahe's 
 observations, were spent on making hypotheses of the old circular 
 kind. But try as he would, there still remained a large unexplained 
 error of about eight minutes of arc, about one-eighth of a degree, as 
 compared with Tycho's observations. 
 
 He then said boldly that it was impossible that so good an 
 observer as Tycho could be wrong by eight minutes, and added : 
 ' Out of these eight minutes we will construct a new theory that 
 will explain the motions of all the planets.' He then proceeded 
 to work out the theory of motion in ellipses. For he had at length 
 found to his great satisfaction that when the sun was placed not 
 at the centre of the ellipse, but at its focus, and the planet was 
 supposed to move in such a way as to describe equal areas in equal 
 times, all the irregularities were adequately explained. The exten- 
 sion of this discovery to the movements of all the other planets 
 followed very easily ; and, instead of the old cumbrous system of 
 epicycles and excentrics, Kepler produced a system of the greatest 
 simplicity, which had the paramount merit of explaining to a close 
 degree of approximation the various movements of all the planets.
 
 332 THE PROBLEM OF LOGIC [XL ii 
 
 We have, then, two main laws of planetary motion already 
 established. They are known as Kepler's first and second Laws of 
 Planetary Motion. 
 
 L Every planet revolves around tlie sun in an elliptic path, 
 the sun being at one of the foci. 
 
 2. Every planet moves round the sun with such a velocity at 
 
 every point that a straight line draAVTi from it to the sun 
 passes over equal areas in equal times. 
 
 To these Kepler added a third : 
 
 3. The squares of the periodic times are proportional to the 
 
 cubes of the mean distances from the sun (the periodic 
 time being the time which the planet requires for the 
 completion of its orbit) — 
 
 I.e., — - is constant for all the planets. 
 
 In these three laws we have the ripe expression of the great move 
 ment of scientific thought which had its root in the Working Ideas 
 of the Ptolemaic system. But the culminating point is reached 
 when Newton passes beyond these three laws to the single principle 
 of Gravitation, which at once explains and transcends them. 
 
 4. Newton. 
 
 In ' The S3'stem of the World ' Newton explains his own position 
 very clearly in the light of a historical retrospect. The ancients 
 satisfied the instinctive desire they felt for a causal explanation of 
 the planetary movements by their theory of crystal orbs or spheres. 
 These orbs served to keep the planets in their places, and held them 
 as it were by a support which, though invisible, was still material. 
 It was the comets that first broke up this old theory.* ' Above 
 all things, the phenomena of comets can by no means consist with 
 the notion of sohd orbs,' for, he adds.f ' as it was tlie unavoidable 
 con-sequence of the hypothesis of solid orbs, while it prevailed, that 
 the comets should be thrust down below the moon, so no sooner had 
 the later observations of astronomers restored the comets to their 
 ancient places in the higher heavens, but these celestial places were 
 at once cleared of the incumbrance of solid orbs, which by these 
 observations were broke into pieces and discarded for ever.' 
 
 But with these orbs disappeared also the supporting forces that 
 explained the apparently unsustained movements of the planets 
 through the heavens. Other forces had to take their place. Des- 
 cartes, in his vortex theory, made a distinction between at least 
 
 • Newton's 'Works,' American edition, p. 511. t I^id., p. 512.
 
 Chap. XXXVIIL] HYPOTHESIS 333 
 
 two kinds of matter, one very tenuous, the other heavier, the matter 
 of the phmcts. This heavy planetary matter was wliirled round 
 the sun in a vortex of the tenuous matter. This tenuous matter, a 
 mid-way form between etlier and ponderable matter, filled the 
 whole of space, and was endowed with great velocity, forming by 
 its rotatory movements the eddies in which the planets were 
 borne round. 
 
 This Cartesian vortex theory was very popular. It was adopted 
 by Fermat, Huygens, Bernoulli, Leibniz. It was a very sane and 
 ingenious idea. All storms move in vortices. It is very probable 
 that the great disturbances on the face of the sun in connexion with 
 sun-spots and faculse are cyclonic storms of fiery gas (c/. Lord Kelvin's 
 ' Vortex Theory of Matter '). But it will not work mathematically. 
 
 Now Newton, while justifying the instinct that compelled 
 astronomers to adopt the working idea of Mechanical Force, and to 
 see force at the root of movement, stated very clearly that the 
 only essential property of this force that was of any use or interest 
 to astronomers was its mathematical law. 
 
 Thus, on the one hand we read* ' From the laws of motion it is 
 most certain that these effects must proceed from the action of 
 some force or other,' and then, ' but our purpose is only to trace 
 out the quantity and properties of tliis force in a mathematical way, 
 so as to avoid all questions about its nature or quality.' Compare 
 also the following passage in the ' Principia,' p. 506 (conclusion) : 
 ' Hitherto we have explained the phenomena of the heavens and of 
 our sea by the power of gravity, but have not yet assigned the causes 
 of this power. . . . But hitlierto I have not been able to discover the 
 cause of these properties of gravity from phenomena, and I frame 
 no hypotheses {hypotheses non fingo). . . . And to us it is enough that 
 gravity does really exist, and act according to the laws which we 
 have explained, and abundantly serves to account for all the 
 motions of the celestial bodies, and of our sea.' 
 
 The force by wliich Newton sought to explain the movements 
 of the heavenly bodies he called ' attraction,' and its mathematical 
 law, with which he was solely concerned, was the following : ' That 
 all bodies tend to attract each other mutually Avith a force that 
 varies directly as the product of their masses and inversely as the 
 square of the distance between them.'f 
 
 In liis attempt, then, to explain causally the movements of the 
 heavenly bodies, Newton adopts as his Working Idea not the notion 
 of mere force, but rather that of the law according to which gravita- 
 tive force is operative. In modern Science the idea of Force has 
 given way to that of Energy or Capacity for Work, and its funda- 
 mental Working Idea is the Law of the Conservation of Energy. 
 It is essential not to forget that this great law is after all only a 
 ' Working Idea.' 
 
 * Newton's ' Works,' American Edition, p. 51-2. t Ibid., p. 385.
 
 334 THE PROBLEM OF LOGIC [XL U.\ 
 
 The Problem of Verification. 
 
 1. Negative Aspect.— U the Working Idea has only a relative 
 permanency, the working hypothesis is emphatically and essentially 
 a provisional explanation, to be laid aside at once if it fails to account 
 for the facts it professes to account for. If any fact absolutely 
 refutes it, the hypothesis must give way before the fact. But we 
 must be qitite sure that the opposing fact is genuine. Thus it is a 
 natural primitive objection to the Law of Gravitation that some 
 bodies— balloons, for instance— tend to move away from the earth. 
 But this is not a genuine objection, since the balloon really tends 
 to fall, but is thwarted by the upward pressure due to the gravitation 
 of the air. If we were to remove the air, and leave the balloon in 
 vacuo, it would fall like a stone. 
 
 But even if compelled to admit the objection, we need only correct 
 or modify the hypothesis to the extent required for the removal 
 of the discrepancy. In discussing the character of the successful 
 experimentalist, W. S. Jevons shrewdly points out (' Principles of 
 Science,' vol. ii., p. 232) that ' Readiness to reject a false theory 
 may be combined with a pecuhar pertinacity and courage in main- 
 taining an hypothesis as long as its falsity is not actually apparent ;' 
 and he quotes Leslie's remark concerning his own experimental 
 investigations into the nature of heat : ' In the course of investiga- 
 tion I have found myself compelled to relinquish some preconceived 
 notions ; but I have not abandoned them hastily, nor till, after a 
 warm and obstinate defence, I was driven from every post {ibid., 
 p. 234). There are circumstances which will justify a scientist in 
 clinging to a theory through everything. 
 
 An investigator may say : Given A, B will follow ; and if B is not 
 observed to follow under the conditions symbolized by A, the result 
 is said to be negative. But this negative result need not be accepted 
 as conclusive, and this for three reasons : — 
 
 (1) The result, though genuine, may disprove not the working 
 
 idea, but only a certain determinate form of it, the specific 
 working hypothesis under investigation. 
 
 (2) The result may not disprove even the working hypothesis. 
 
 It may only show that the conditions of the actual ex- 
 periment were not satisfactory. The effect may have 
 been produced, but in too slight a form to be detected, 
 or the arrangements of an experiment may not have been 
 suitable; just as CErsted could not detect electro-magnetism 
 so long as liis wire was perpendicular to the plane of motion 
 of his needle {ibid., p. 239). 
 (3; It may be that the fact itself has not been properly ob- 
 served. The fault in this case lies, not with the hypothesis, 
 nor with the apparatus, but with the observation.
 
 Chap. XXXVIII.] HYPOTHESIS 335 
 
 Jevons reminds us that Faraday, convinced on general grounds 
 that some mutual relation must exist between Light and Magnetism, 
 struggled against negative evidence for forty years in his attempt 
 to prove their kinship. His conviction was that the various forms 
 of energy have a common origin, and are so directly related as to be 
 mutually convertible. In this case his courage was justified when 
 a happily devised experiment showed that magnetic force had the 
 power of twisting the plane of polarization of a ray of light. The 
 whole theory was subsequently developed mathematically by Clerk- 
 Maxwell, and is now known as the electro-magnetic theory of Light. 
 Here struggle against negative evidence meant partly a revision of 
 working hypotheses, partly a further elaboration and perfection of 
 apparatus. Meanwliile the working idea remained as the germinal, 
 controlling conception which directed the whole procedure. 
 
 Again, experiments devised by Faraday to prove the connexion 
 between gravity and electricity gave only negative results. ' The 
 experiments,' he wrote, ' were well made, but the results are nega- 
 tive ;' and yet he adds : ' I cannot accept them as conclusive ' 
 (Jevons, ibid., pp. 238, 239). Here Faraday's meaning, logically 
 expreesed, amounted to this : that he did not consider that he had 
 tested all the working hypotheses by which his working idea could 
 be verified. 
 
 We see, then, that the refusal to accept a negative result as 
 conclusive does not mean refusing to take the result as a trust- 
 worthy indication that something is wrong either in the working 
 hypothesis, or in the apparatus, or in the use made of one's o^vn 
 senses. It does mean this. But a negative result only says that 
 something is wrong ; it can give no indication as to what precisely 
 is wrong. If a chain breaks, any link may have given way ; and 
 the negative result is just the signal for revision. 
 
 2. Positive Aspect. — ^The verification of hypothesis naturally 
 consists, in the first instance, in showing that the h}^)othesis in 
 question accounts for knowTi facts. Thus, in the case of the Undu- 
 latory Tlieory of Light, it had, of course, to be shown that, when 
 developed into its consequences, it could adequately account for 
 the known facts of rellection and refraction. But a more striking 
 form of verification occurs when the consequence is a novelty, and 
 experiment is required to attest or disprove its presence. This 
 derivation of a previously unobserved fact as a necessarj'' conse- 
 quence from a given hypothesis is what is knoA\'n as Prediction. 
 
 ' Prediction ' means stating what the facts are before they have 
 been observed. The time-element in prediction in no way affects 
 its logical nature. To predict 'in advance,' or to predict 'back- 
 wards,' is, so far as the thought-process is concerned, essentially 
 one with prediction that has no time-reference at all. Thus to 
 predict eclipses in time, whether backwards or forwards, is just the 
 same process as predicting from the known laws of motion that
 
 336 THE TROBLEM OF LOGIC [XL ii. 
 
 eclipses, in general, must take place under certain conditions ; or 
 a it differs, it differs only as regards precision of statement. 
 
 It is sometimes supposed that power to predict is an infallible 
 proof of the truth of a lu'pothesis. This is a perfectly ungrounded 
 statement. A successful prediction is only an interesting, striking 
 form of verification. 
 
 A hypotliesis that may afterwards prove to be erroneous may 
 furnisli verified predictions. The one general test of the truth of a 
 h^'pothesis is continued conformity with fact. There may be 
 developed two or more theories of which each sufficiently accounts 
 for all the fundamental, perhaps all the kno^vn, facts. It is the 
 agreement -snth jurther facts still untested or still to be discovered 
 which must then decide between the two hypotheses. Sooner or 
 later the insUintia cruris — the crucial instance — presents itself, and 
 the experimentum criicis — the experin ent of the finger-post — decides 
 between the rival claims. A crucial instance is a circumstance that 
 is decisive between rival hypotheses, admitting of one explanation 
 only ; a crucial experiment is a test so arranged that the result 
 is bound to approve one hypothesis, and rf{.sprove the other. 
 Professor Minto (' Logic,' pp. 347, 348) gives the following crucial 
 instance as proving decisive against the Cartesian Theory of 
 Vortices : ' The fact that comets pass into and out of spaces where 
 the vortices must be assumed to be in action without exhibiting 
 any perturbation is an instantia crucis against the hypothesis.' 
 As an instance of the experimentum crucis we may cite Foucault's 
 experiment on the velocity of light in two different media, the 
 denser water and the lighter air.* Foucault showed that light 
 took longer to travel a certain distance through glass or water 
 than it did to pass through the same distance in air or through 
 a vacuum. This confirmed the Undulatory Theory of Light, and 
 effectively demolished the Corpuscular Theory, which required that 
 light should move more rapidly through the denser medium. 
 
 It should be noticed that the crucial instance, though it negatives 
 the defeated theory, does not absolutely prove the one that is success- 
 ful. It only serves greatly to increase the possibility of its being 
 the true one. No inductive hypothesis can ever be regarded as the 
 perfected, unimprovable expression of the truth. Verification in a 
 word is not Completed Proof. We must distinguish the hypo- 
 thesis as verified from the hypothesis as proved. f 
 
 Let us suppose that the hypothesis ' A is B ' is to be tested, and 
 that the consequence ' P is Q ' is, with logical necessity, derived from 
 it and from the scientific system in which it is rooted. Let us 
 further suppose that the verification of ' P is Q ' has been conclusive. 
 This verification of ' P is Q ' does not, of course, prove the trutli of 
 ' A is B.' To suppose that it did so would be to commit the fallacy 
 
 * Cf. Prof. Welton. 'A Manual of Logic,' vol. ii., bk. v., ch. iv., p. 103. 
 t For a discussion of the process of Proof proper, see above, pp. 322-325
 
 Chap. XXXVIII.] HYPOTHESIS 337 
 
 of ' affirming the consequent.' We have shown that tlie hypotheses, 
 properly enunciated, enables us to explain the facts without trans- 
 gressing the evidence. To prove the hypothesis, we have still to 
 show that no other hypothesis will adequately account for the 
 facts. The evidence afforded by the verification must be adequate 
 to prove this, if we are to be justified in saying that the hypothesis 
 is proved. 
 
 Let us consider from this point of view the status of a typically 
 fundamental and fruitful hypothesis, that of gravitation. 
 
 Newton proved clearly by exact mathematical reasoning that, 
 given the working idea of ' attraction ' or ' gravitation,' there was 
 only one admissible form of that idea — the law that the attraction 
 varied inversely as the square of the distance, and directly as the 
 product of the masses of the bodies concerned. He proved the 
 working hypothesis on the assumption that the working idea teas 
 sound. He discovered once and for all the only form into which 
 the working idea of gravitation could be developed so as to explain 
 the facts. But the working idea itself still remains subject to 
 revision. Thus Faraday's conception of the affinity between 
 gravitation and electricity, if justified, would involve a revision of 
 the working idea of gravitation. 
 
 Barren Hypothesis. — A barren hypothesis is a supposition from 
 which, or from the functional substitute for which {vide p. 320), 
 no verifiable conclusion can be drawn. It is the opposite of what 
 is usually known as a ' permissible ' or ' legitimate ' hypothesis — 
 i.e., of a hypothesis which is workable, and workable because it 
 suggests a mode of operation which is, at least to some extent, 
 analogous to operations with which we are already famiHar. A 
 supposition such as ' Tliis havoc has been wrought by a ghost,' or, 
 again, ' Neptune's irregularities are the frolic of a demon,' would be 
 a barren hypothesis. How can we aspire to verify either of these 
 suggestions ? We know nothing sufficiently definite about ghosts 
 or demons to enable us to deduce from the hypothesis consequences 
 which could be compared with facts. Referring to the ' catas- 
 trophic ' or ' convulsion ' theory in Geology, Professor Creighton 
 points out that, in assuming its truth as a basis of investigation, 
 we are assuming the operation of incalculable forces, the positing 
 of which leads and can lead to nothing. ' Instead of these mys- 
 terious agencies, Lyell assumed that causes similar to those with 
 which we are now acquainted had been acting uniformly for long 
 ages. The nature of the causes at work being knowai, it became 
 possible to calculate the nature of the effects, and thus to reduce the 
 facts of Geology to order and system.'* 
 
 A hypothesis, however, may be barren at one stage of scientific 
 culture and yet prove fruitful later on. When we know more about 
 ghosts and their ways, the ghost-hypothesis may cease to be barren. 
 
 * ' Logic,' p. 243. 
 
 22
 
 33S THE PROBLEM OF LOGIC [XL ii. 
 
 Three thousand years ago, as Professor Carveth Read points out,* 
 the hj'pothesis that the Sun is the centre of our planetary system 
 would have been a perfectlj* barren In^pothesis. It needed the 
 observations of the Greek and Alexandrian astronomers, and the 
 discoveries of Galileo, Kepler, and Xewton to make it really 
 verifiable and fruitful. 
 
 False Hypothesis. — A barren hj'pothesis must be carefully dis- 
 tinguished from a false or erroneous hj^othesis. A barren hypo- 
 thesis is one that, under the given conditions of scientific knowledge, j 
 cannot be verified, and in this sense is illegitimate (vide above 
 p. 337), a hypothesis improperly so-called ; an erroneous hypothesis 
 — e.g., the Corpuscular Theory of Light — is a legitimate but unsuc- 
 cessful hypothesis, a supposition which can be developed and tested, 
 but does not happen to fit all the facts. It is an attempt at ex- 
 planation shown to be inadequate by means of some crucial instance 
 or experiment. At the same time an erroneous hypothesis is not 
 necessarily worthless. Its inadequacy means no more than that 
 the hj-pothesis breaks dov\Ti at a certain point, and before breaking 
 down it may have done considerable service to Science. 
 
 And yet it is essential to note that it is certainly not qua erroneous 
 that the h^-pothesis is serviceable. The Ptolemaic hj-potliesis, vsdth 
 its geocentric principle and its assumption that all the movements of 
 the planets were uniform in speed and circular in direction, gave a 
 definite steadpng-point for astronomical science. But the service 
 here rendered was due to the systematic way in which the hypo- 
 thesis was developed, and not in any way to what was false in the 
 theory. Moreover, in so far as it merely assumed that the Earth 
 was the fixed centre from which all astronomical observations 
 must be made, it assumed what all other astronomical theories 
 have been obliged to acquiesce in. Hence the quality of the 
 observations made at that period was not influenced in any way 
 by the specific character of the Ptolemaic theory. Whatever the 
 astronomical theory may be, the Earth remains, for our perception, 
 the apparent centre of the universe. 
 
 But in so far as the hypothesis that the Earth was the real centre 
 of the universe was brought into play, and the apparent movements 
 of the planets treated as real movements, the only utility of this 
 error lay in its eventually refuting itself through the bewildering 
 complexities to which it gave rise. Except as an object-lesson in 
 the truth that the path of error is a path of growing complexity, 
 it had no intrinsic utility. A theory cannot be said to have been 
 
 * ' Logic, Deductive and Inductive,' third edition, ch. xviii., pp. 250, 251. 
 
 t It is this limitation which justifies the concession of the title ' hypothesis ' to 
 these barren suppositions. Were they intrinsically unverifiable, they would not be 
 hypotheses at all. They would be intrinsically, and not merely provisionally, 
 illegitimate. The intrinsically illegitimate supposition, from the strictly scientific 
 point of view, is a supposition which defies the requirements of the Inductive 
 Postulate (c/. Chapter XLVIL).
 
 Chap. XXXVIII.] HYPOTHESIS 339 
 
 useful in so fear as it lias stood in the way of the apphcation of far 
 truer theories whose work, since it began, has shown itself to be one 
 of progressive simplification and progressive fruitfulness. 
 
 So, again, the Corpuscular Theory of Light was useful in so far 
 as it made it possible to explain the phenomena of the reflection 
 and refraction of light, and so provided a rallying-centre for the 
 advance of optical theory. But, unfortunately, it had Newton's 
 brain behind it, so that it received a development out of all propor- 
 tion to its intrinsic merits, and a prestige which gave it an artificial 
 lease of life and prevented the wave-theory, as developed by Young 
 and others, from exercising its due effect on the mind of the time. 
 The non-utility of a false hypothesis is thus evidenced by the fact 
 that the more perfectly it is elaborated, the more surely does it 
 prove a hindrance to the development of Science. 
 
 It is true that Bradley's explanation of the phenomenon of Aberra- 
 tion, according to which each fixed star appears, in the course of a 
 year, to describe a small orbit about its true position, was developed 
 on the basis of the Corpuscular Theory, and experts aver that the 
 discovery could not so readily have been made had the wave theory 
 been adopted, so that in this case an erroneous theory led to an 
 important discovery.* At the same time it is hard to convince 
 oneself that what was positively erroneous in the theory could have 
 led to the discovery in question. 
 
 CHAPTER XXXIX. 
 
 XI. (iii.) GENERALIZATION. 
 
 Generalization is a process so distinctive of the very activity of 
 thought that it is not easy to define it. To be observing a fact or 
 ' particular ' in the light of an idea or ' universal ' is already to be 
 breaking through its sense-isolation and winning it for thought. 
 A fact is indeed meaningless except in so far as it is relevant to some 
 interest, and rests upon some background, however indeterminate, 
 of questioning mental activity. We start the generalization of a 
 fact when we first realize it as a fact, when we first question its 
 meaning for us and our interests, and so transform the sense-datum 
 into a thought problem. We complete the generalization when the 
 fact is, in all its relevant relations, adequately systematized. If the 
 
 ^ ' The curious inference may be dra%m that, if the more correct modern notions 
 of the nature of light had prevailed in Bradley's time, it must have been much mora 
 difficult, if not impracticable, for him to have thought of his explanation of the 
 stellar motions which he was studying ; and thus an erroneous theory led to a most 
 important discovery ' (Arthur Berry, ' A Short History of Astronomy,' p. 265). 
 
 o.-> o
 
 3-iO THE PROBLEM OF LOGIC [XL iii. 
 
 process of generalization may be said to consist in the progressive 
 idealization of fact, in the continuous revelation of fact as an 
 ordered system, the guiding-idea of that process may be defined as 
 the systematizing of fact in the simplest and most economical way, 
 so as to bring the simplest thoughts to bear upon the widest range 
 of facts. 
 
 The fundamental form which generalization takes is that of 
 fixing the flux of facts, as immediately exj)erienced, within the 
 steadying and permanent form of a concept. This process is com- 
 monly referred to as the generalization of facts under concepts or 
 notions. It consists in starting from the observation of concrete 
 objects, and proceeding thence to a knowledge of classes. This is 
 done by progressively omitting the attributes peculiar to this object 
 or to that, and by retaining those which are attributes of all the 
 objects considered. So viewed. Generalization involves more or 
 less complex processes of Comparison and Abstraction, and cul- 
 minates in Definition. 
 
 In those discussions on the nature of the virtues with which 
 Socrates began the first European ' Philosophy of Morals,' his method 
 was to proceed by critical comparison of a number of instances, to 
 abstract the common or essential features, and then to formulate 
 these in a definition. 
 
 ' If you were in need of a dinner,' asked Socrates, ' would you 
 apply to a shoemaker ? No, but to a cook. Again, if you were 
 bestormed, and wished to make for a harbour, would you resort to 
 the soldiers on board ? Not so, but to the pilot.' After a batch 
 of such questions the learner would be in a position to see that 
 cook, pilot, and the rest shared a common quality in virtue of 
 which, on the respective occasions, application was made to them 
 ratlier than to others — namely, the quality of being technically 
 qualified. 
 
 The process of generalizing facts under concepts is beset by two 
 main difficulties, to which Dr. Venn calls attention in his ' Empirical 
 Logic ' : 
 
 1. There is, first, the difficulty of clearly detecting the common 
 quality in a number of given instances of a certain class. Thus, to 
 take Dr. Venn's instance, ' let A, B, C be Sheffield grinders, a famiUar 
 and well-marked class. It had long been known that they were 
 sickly and short-lived people, but the person who first clearly 
 recognized the character of their symptoms, so as to bring the 
 disease ' — a sort of lung disease — ' under one concept, had no easy 
 task to perform.'* 
 
 2. The quality in question may be obvious enough, but the 
 individuals A, B, C, in which the quality is recognized, may never 
 have been classed together. We have then to bring them under 
 one class-concept, the right one for our purpose. This will often 
 
 * D.-. Vean, ' The Principles of Empirical or Inductive Logic,' eh. xiv., p. 347.
 
 Chap. XXXIX.] GENERALIZATION 341 
 
 be a matter of great difficulty. Briefly, the detection of the property 
 to be generahzcd and of the class over which this property is to be 
 generalized is often the most difficult and important of the many 
 operations involved in the establishment of a complete induction. 
 
 If the problem of generalizing facts under concepts connects 
 itself with the problem of Definition, that of generalizing concepts 
 under concepts gives rise to the problem of Classification, a problem 
 with which the name of Aristotle is as closely associated as the 
 name of Socrates with that of Definition. In Classification {vide 
 Chapter V.) we have the natural development of the same general- 
 izing tendencies which find their first resting-place in Definition. 
 But the problem of Scientific Classification is far more com- 
 plicated now than it was in Aristotle's day. Aristotle worked on 
 the basis of certain simplifying assumptions, such as that of the 
 fixity of species, which modern theories of causation and develop- 
 ment have rendered more than problematic. As a consequence, 
 the nature of the generalizing processes involved in Scientific 
 Classification and Definition has been greatly modified in modern 
 times. The simpler processes of comparing attributes and abstract 
 points of agreement have given way more and more to the more 
 complex processes of analysing relations and gathering variations 
 around diagnostically defined types. And pari passu with thia 
 complication in the nature of the process itself we find that the 
 whole problem of bringing concepts under concepts becomes ever 
 more intimately connected with a further aspect of the generaliza- 
 tion-problem. This concerns the bringing of facts under laws — 
 laws of causal interaction and laws of develojm^nt. The very 
 variability of objects ' forces us beyond the statement of a fixed 
 complex of perceptible characteristics, and obhges us to include 
 causal relations or laws of development in our enumeration of the 
 attributes by which one class of things is distinguished from all 
 others. . . . Quicksilver seems to admit of a simple statement of 
 characteristics by means of which its attributes are expressed in a 
 combination which belongs to no other object ; but it is only at an 
 ordinary temperature that it is such an easily recognizable object : 
 it evaporates in heat and becomes solid in cold, it combines with 
 other metals to form amalgams, and with sulphur to form cinnabar, 
 and not until we have included these transformations in our concept 
 can we claim to have stated what quicksilver is ' (Sigwart, ' Logik,' 
 vol. ii., part iii., ch. ii., § 77, 6 ; EngUsh Translation, II., p. 163). 
 
 In the generalization of facts under laws, the first step is the formu- 
 lation of what are commonly called ' empirical ' laws. There are 
 three essential marks which must be included in the definition of 
 an Empirical Law. In the first place, it must have been gained 
 through direct observation of facts. In the second place, it must 
 not already have been explained as a particular case or specifica- 
 tion of some law more fundamental than itself ; it is a law, in fact,
 
 342 THE PROBLEM OF LOGIC [XL iii. 
 
 which has not itself been systematized. Thus Kepler's Laws of 
 Planetary Motion were empirical in this sense U7itil Newton showed 
 that they were necessary deductions from his own principle of 
 universal gravitation. They then became specifications or expres- 
 sions of the Law of Gravitation. Li the third place, the Empirical 
 Law is not an explanatory law. It is a law descriptive of the be- 
 haviour of fads, without at the same time being explanatory, or 
 descriptive of the mode of behaviour of a cause. For an ex- 
 planatory law, as Science understands it, can mean no more (and 
 nc less) than a law which necessarily implies a reference to a cause, 
 force, energy, or tendency, thougli it makes no attempt to explain 
 the nature of that cause beyond defining the law according to which 
 it works. 
 
 We may illustrate the meaning of an Empirical Law bj' means of 
 certain instances which we borrow from Dr. Sigwart {ibid., § 96 ; E. T., 
 pp. 362-3G6). We may take the law of falling bodies, which states 
 that, whenever a body falls freely from rest, it describes spaces 
 which are proportional to the squares of the times. This is a law 
 in the sense that it describes the motion of a falhng body by means 
 of a formula (2 s = gt2), but it is not a law in the causal or explan- 
 atory sense : it states the ' how,' but not the ' why.' So, again, 
 Kepler's first law tells us that planets move in ellipses about the 
 Sun as focus, tells us how and not why they move as they do ; 
 whilst his second law — the law of equal areas — states in a formula 
 the relation between the velocity of a planet and its distance from 
 the sun, and is similarly descriptive and empirical. The law 
 which states the connexion between clianges in the height of the 
 tide and the changing positions of sun and moon with respect to 
 each other and the Earth is, again, ' only a descriptive law con- 
 cerning the regular accompaniment of one change by the other, 
 and it is essentially different from the causal explanation which 
 deduces this connexion from the attraction of moon and sun upon 
 the waters of the earth ' (ibid., E. T., II., p. 365). These descrip- 
 tive statements are particularly frequent and important in Natural 
 History. Thus the following are empirical uniformities, mere laws 
 of sequence and coexistence : 
 
 A Dicotyledonous seed, when sown in suitable soil under favour- 
 able conditions of moisture, temperature, etc., will commonly 
 germinate. The embryo which it contains will protrude its radicle. 
 This will grow do^vnward, and from it will be developed the primary 
 root which fixes the developing seedling in the soil. This root 
 may branch repeatedly, and ultimately form a complex root- 
 system. The two cotyledons will (in many cases) emerge from the 
 se-id-coat and develop into green leaves. The stem will grow upward. 
 If an embryonic plumule is present, this will develop its rudimentary 
 leaves into the first foliage-leaves of the growing plant. The stem, 
 like the root, may branch repeatedly, and will give rise to more
 
 Chap. XXXIX.] GENERALIZATION 343 
 
 foliage-leaves, and ultimately to the reproductive shoots known as 
 flowers. The latter will (in many genera) be bisexual, producing 
 both ovules and pollen. If the pistil is pollinated, tliis will normally 
 give rise to further processes which will lead to the maturing of the 
 seeds. Each seed will contain an embryo capable of developing 
 into a form similar to the parent-plant. 
 
 The essential limitation of the Empirical Law is that it cannot be 
 relied on beyond the range of those facts of which it serves to register 
 the behaviour. Thus Kepler's law of equal areas, prior to Newton's 
 generalization of it, was a law of the movement of the planets round 
 the sun. In Newton's hands it became a law of the movements 
 of any rigid body revolving under the influence of any central force 
 acting, as gravitation does, according to the inverse square of the 
 distance. But such an extension of the empirical law was not 
 possible in Kepler's da,y. It was only the generalization of Kepler's 
 law under Newton's that made this extension intelligible. This 
 further problem of the generalization of laws through their pro- 
 gressive development and simplification, of the yeneralization of 
 laws under laws, is, however, substantially identical with that of 
 the generalization of facts under laws through Analysis and Syn- 
 thesis. 
 
 The essential means for passing from fact to law is Experimental 
 Analysis ; and the explanation which is based upon such analj^sis 
 takes the form of Sj^nthesis. Facts are, as a rule, complex, and 
 the function of Inductive Explanation is, through analyses and 
 synthetic reconstructions based upon these analyses, to transform 
 the complexity we start from into a strictly relevant coherency. 
 The resolution of the complex fact into its simple factors is the work 
 of tentative analysis — which must try and test itself at every step — 
 the culmination of the generalizing process being reached in the 
 formulation of the laws according to which these more general 
 factors operate. The systematic reconstruction of the fact, in idea, 
 through deductive processes which, proceeding from the ascertained 
 laws of operation of the factors, show how these factors co-operate 
 in producing the fact, is the work of S3TLthesis, a sj^nthesis wliich 
 finds its climax in successful Verification. 
 
 The distinction between Anah'sis and Synthesis may be stated in 
 a somewhat different form by reference to the purposive idea witliin 
 which the whole process of inductive inquiry takes place. The aim 
 of Analysis, we may say, is to disengage from a given complex 
 situation such elements and combinations of elements as are 
 relevant to its purposive reconstruction. From this point of view 
 Analysis is essentially a process of purposive Elimination, Synthesis 
 a process of purposive Elaboration. Through Analysis we eliminate 
 the irrelevant elements in a total complex datum ; through Syn- 
 thesis we elaborate the relevant residue of Analysis in the light of 
 the idea which has dominated and controlled the whole process.
 
 344 THE PROBLEM OF LOGIC [XL iii 
 
 The immediately preceding paragraphs contain in themselves a 
 complete programme of Inductive INIethod, a programme which it 
 ■will be the main business of the following chapters to develop. It 
 may not be out of place, however, to add, in conclusion, an illus- 
 tration or two of the general process through which facts are 
 generalized, and, as a further consequence, systematized and 
 explained. 
 
 Let us take as our complex fact a door with its framework some- 
 what out of order. We proceed to analyse the obstruction by 
 resolving the vague complex idea of a door out of order into the 
 more simple ideas of a latch, a hinge, or fitting out of order. We 
 find eventually that the upper hinge is to blame, and that its 
 looseness has caused the door to lean and graze the floor : it -presses 
 against the floor, and so scrapes against it when moved. The 
 problem is now reduced to a question of pressure and friction, and 
 we may now proceed to explain the difficulty by showing how, 
 according to the laws of pressure and friction, the obstruction 
 necessarily came about. By this simplification the fact is general- 
 ized. It is generalized as a particular case of the operation of the 
 general laws of pressure and friction. It is also systematized, or at 
 least potentially correlated with a host of other facts ; for, by 
 connecting the phenomenon with all other phenomena of pressure 
 and of friction, we have taken it out of its mere particularity and 
 isolation. 
 
 Again — to take an illustration given by Dr. Venn* — if we are 
 attempting to explain the slipporiness of ice, we at once simplify 
 and generalize the problem before us by displaying the fact of 
 slipperiness as a specific variety of the forward and backward 
 reactions that always take place between our feet and the surface 
 of tlie ground against which they press. ' We slip,' we say, ' be- 
 cause the horizontal reaction to the imj)ulse of the feet has fallen 
 below a certain minimal amount.' 
 
 Once more, if we desire to explain the succulent habit of some 
 desert-plant, we generalize the fact of succulence by showing it to 
 be a specific form of ' xeropliilous ' adaptation, and by regarding 
 this again as a special kind of that adaptive modification, in response 
 to the influence of environment, which, gradually perfected through 
 the process of natural selection, ultimately fits each species of 
 plant to the conditions of its own particular habitat. 
 
 * 'The Principltt} of Empirical or Inductive Logic,' ch. xxi., p. 498.
 
 XII. 
 
 APPLICATION OF THE INDUCTIVE PRINCIPLE TO 
 
 ' INDUCTIONS IMPROPERLY SO-CALLED ' AND 
 
 TO 'IMPERFECT INDUCTIONS.' 
 
 (i.) Inductive Inferences, improperly so-called (ch. xl.]. 
 (ii.) The ' Imperfect Inductions ' : 
 
 (A) Enumerative Induction (ch. xli.). 
 
 (B) Argument from Analogy (ch. xlii. ).
 
 CHAPTER XL. 
 
 XII. (i.) INDUCTIVE INFERENCES. IMPROPERLY SO-CALLED. 
 
 The essential distinction between popular and scientific explana- 
 tion has been expressed in the familiar saying that Science is just 
 organized common sense. This organization implies two things — 
 principle and method, the method being determined by the principle. 
 And the principle whereby Science organizes common sense in tlie 
 matter of the explanation of facts is just that of steadfast loyalty 
 to the facts in so far as they are relevant to its scientific purpose. 
 This principle gives to scientific investigation an ultimate standard 
 or criterion — ultimate since the scientific aspiration and purpose 
 does not extend beyond that of an adequate explanation of the 
 facts of the sense-world. Hence, in inquiring whether any pro- 
 posed method of dealing with the facts is scientifically adequate or 
 not, we shall have to ask : In what sense does it provide an adequate 
 Verification-test ? This is the touchstone of Scientific Method, to 
 which all else is subordinate. 
 
 For the purpose of carrying out this principle and applying this 
 inductive standard in detail, we propose to adopt ^Mill's* Inductive 
 Scheme, which may conveniently be laid out as follows : 
 
 ' Inductions.' 
 
 Improperly so-called. Properly so-callod. 
 
 ' Perfect Induction.' Parity of Reasoning. Colligation of Facts. 
 
 I I 
 
 Imperfect Inductions. Scientific Induction. 
 
 Imperfect Enumeration. Analogy. 
 
 Under the head of Inductions improperly so-culled ^lill reckons 
 three tj^pes : 
 
 1. ' Perfect Induction.' 
 
 2. ' Induction by Parity of Reasoning.' 
 
 3. Colligation. 
 
 * Vide J. S. Mill. ' A System of Logic,' Book III. 
 317
 
 348 THE PROBLEM OF LOGIC [XII. i. 
 
 1. ^Perfect Induction.'' 
 
 Suppose that we have certain knowledge that all the instances 
 belonging to a given class have been considered by us ; then, if we 
 find that a certain attribute is possessed by each of these instances, 
 it is an act of ' perfect induction ' to universalize — i.e., summarize — 
 the discovery by stating that all the members of that class possess 
 the attribute in question. 
 
 Thus : 
 
 ' January, February . . . December have each twenty-eight 
 
 days or more. 
 ' January, February . . . December are all the months in 
 the year. 
 ' .-, All the months in the year have twenty-eight days or 
 more.' 
 
 Xow, is there anything in this reasoning which could allow us to 
 consider it as a form of Scientific Explanation or Induction ? 
 
 There is Scientific Explanation, as we have seen, only where a 
 hypothetical conjecture is adequately verified. Two elements 
 are here indispensable : the conjecture (going tentatively beyond 
 the evidence), and the verification or the justifj^ing of the con- 
 jecture. 
 
 Here, we might say, we have at least verification : the general 
 proposition seems to admit of complete, final verification. But 
 this confidence is illusory ; for there has really been no conjecturing 
 at aU, and there is therefore nothing to verify. There has been no 
 tentative supposition made, no uncertainty at any point of the 
 reasoning, no temporary passage from the known to the unknoAvn, 
 which, when understood to mean a tentative passage from known 
 facts to a hj-pothesis, is essential to all inductive procedure. In 
 order that a scientific method may satisfy the criterion of Induction, 
 not only must its verification-tests be adequate, but it must involve 
 a verifiable cory'ecture. The legitimate hypothesis is the precon- 
 dition of verification. ' Perfect Induction ' lacks the legitimate 
 hj'pothesis — lacks, indeed, the hypothetical element altogether. 
 If it is something more than ' a mere short-hand registration of 
 facts knov^Ti,'* as Mill puts it, it is still essentially a self-contained 
 deductive inference, and of the verj'' simplest type. As such, it is 
 no episode in a total process of Induction, and therefore not in- 
 ductive in any sense of the word.'-) 
 
 • J. S. MiU, 'A System of Logic' Book III., ch. ii.. § 1. 
 
 t For Jevona' defence of Perfect Induction, see ' Elementary Lessons in Logic,' 
 p. 214. He appears to himself to h>e criticizing 3Iill, whereas he is only forcibly 
 repeating JLll's own words. {Cf. ' X System of Logic,' ibid. : ' The operation may 
 be very useful, as most forms of abridged notation are ; but it is no part of the in- 
 vestigation of truth, though often bearing an important part in the preparation of 
 the materials for that investigation. 'j
 
 CuAP. XL.] ' IXDUCTIVE IXFERE^X'ES ' 349 
 
 2. Parity of Reasoning. 
 
 When, after having sho^vn that a certain fact is true of A, we 
 argue that it must be true of B, not because it is true of A, but for 
 the same reason as that whicli convinced us of its truth in A's case, 
 we are said to proceed by Parity of Reasoning. Thus, ' liaving 
 sho'WTi that the three angles of the triangle ABC are together 
 equal to two right angles, we conclude that this is true of every other 
 triangle, not because it is true of ABC, but for the same reason 
 which proved it to be true of ABC* 
 
 But if this is the case, surely Parity of Reasoning is no more than 
 Identity of Reasoning. And to reason identically is to take no 
 further step in the reasoning at all, so that we have not only no 
 induction, but no logical process whatever. 
 
 But the essence of the criticism which Mill brings to bear upon 
 Parity of Reasoning, considered as an Induction, is that the con- 
 clusion drawTi by it ' is not believed on the evidence of particular 
 instances,' and consequently that the process lacks the charac- 
 teristic qualit}^ of Induction. 
 
 Instead of saying with Mill that an inductive inference is 
 grounded in the instances whose sole function has been to suggest 
 the generalization, we should rather say that an induction can be 
 grounded only through a process of progressive verification exercised 
 on the as yet unobserved or uncritically observed instances. Xor 
 is the difference a mere matter of words. Mill repeatedly, though 
 not invariably, argues as though the generahzation were believed 
 not on the evidence that verifies it, but on the evidence of the par- 
 ticular instances wliich have suggested it. 
 
 "■oo^ 
 
 3. Colligation of Facts. "^ 
 
 The term is Whewell's, and means ' the act of bringing a number 
 of facts actually observed under a general description,' as when 
 Kepler, having made a large number of observations on the suc- 
 cessive positions of Mars at different points of its orbit,brought them 
 all together under the one collective conception of an ellipse. Mill 
 and Whewell agree as to this definition of the term 'Colligation.' 
 They disagree in this : that, whilst Whewell calls it ' Induction,' Mill 
 denies that it is an induction at all. 
 
 Whewell, says Mill, is confounding a mere description of a set of 
 observed phenomena with an induction from them. The descrip- 
 tion is an operation subsidiary to Induction, not itself an induction. 
 
 * J. S. Mill, ' A System of Logic,' Book IIL. ch. ii.. §-2. 
 
 t Ibid., Book III., ch. ii., § 3 //. ; Book IIL, ch. xvi. ; Book IT., ch. i., ii. 
 Whewell, 'Of Induction.' pp. 1-4.5. Cf. also G. F. Stout, 'Analytic Psychology,' 
 ii., pp. 49-o2 ; and J. Venn, 'The Principles of Empirical or Inductive Logic,' 
 ch. xiv., pp. 3J3 ff.
 
 350 THE PROBLEM OF LOGIC [XII. i. 
 
 It is essentially a mere act of comparison. Thus, what Kepler did 
 in the case considered was to discover by careful comparisons the 
 circumstance in which all the observed positions of the planet 
 agreed. But to discover what is common to a series of observations 
 is only to give a general character to these observations : it is not 
 in any way an induction from them. 
 
 Whewell, in defence, contends that, besides giving a general 
 character to the facts, Colligation introduces, as a principle of con- 
 nexion, a conception of the mind not existing in the facts. The 
 inductive act consists in a superinduction of a conception — the 
 conception of the ellipse in Kepler's case — upon the facts. The 
 statement about the elliptical orbit, he argues, ' was not the sum 
 of the observations merely ; it was the sum of the observations 
 seen under a new point of view, which point of view Kepler's mind 
 supplied.' That the statement about the elliptical motion was not 
 merely the sum of the different observations is plain from this 
 argues Whewell : that other persons, and Kepler himself before his 
 discovery, did not find it by adding together the observations.* 
 
 We must further remember — a point not pressed by Whewell 
 himself — that the orbit of Mars is not an actual ellipse, so that if 
 Kepler had simplj^ been summarizing the actual facts, he could not 
 have arrived at the idea of the elhpse. ' Wliat Kepler did was, 
 from a finite number of observed positions to frame a rule for 
 inferring all the intermediate unobserved positions, as well as those 
 at any past or future time.'f Notwithstanding [Mill's attempt to 
 show that no generalization is involved (' Logic,' Book III., 
 eh. ii., § 5), we must agree with Dr. Venn in this statement of 
 the case, Kepler's procedure, we take it, was a true tentative 
 generalization. To this extent we hold that Whewell is justified 
 as against Mill. Colhgation does, indeed, take us a first step beyond 
 the observed facts by suggesting a tentative, descriptive generahza- 
 tion of them such as no mere observation could possibly have sup- 
 plied, seeing that it refers to the unobserved as well as to observed 
 instances. It certainly is not, as Mill maintained that it is, a 
 mere summary of the facts thrust upon one by the facts themselves. 
 But, from the point of view of the inductive criterion which we have 
 adopted, Colligation is not Induction. It is an unverified Generaliza- 
 tion. It has not been justified as a generalization true to the 
 evidence of the facts. 
 
 * Cf. ' Of Induction," p. .33. 
 
 t J. Venn, 'The Principles of Enapirical or Inductive Logic,' ch. xiv., p. 354, 
 footnote, ad fin.
 
 Chap. XLL] ' DIPERFECT IXDUCTIOXS 351 
 
 CHAPTER XLI. 
 
 XII. (ii.) THE ' IMPERFECT IXDUCTIONS.' 
 
 A. Enumeeative Induction. 
 
 The term ' Enumerative Induction,' or the equivalent expression 
 ' Induction by Simple Enumeration,' has been customarily used to 
 represent the uncontrolled tendency of the mind ' to generalize its 
 experience, provided this points all in one direction ' (J. S. Mill, ibid., 
 Book III., ch. iii., § 2), consisting, as i\Iill puts it, ' in ascribing the 
 character of general truths to all propositions which are true in 
 every instance that we happen to know of.' It is a generalization 
 based on the mere affirmation that we have never known an in- 
 stance to the contrary. 
 
 This view of Enumerative Induction presupposes, however, a 
 definition of Induction which is different from that adopted in the 
 present volume. When Bacon and J. S. Mill speak of Induction by 
 Simple Enumeration, they are identifying Induction with Generaliza- 
 tion in a sense which excludes Verification. The tendency of the 
 mind which moves from the known to the unknown by means of a 
 generalization based on the counting of instances remains uncon- 
 trolled by the appeal to facts. It is liable, of course, to find its 
 path blocked by an exception which pitilessly opposes this easy 
 invasion of the universal, but it makes no attempt to search out 
 exceptions, and never sees them unless they are thrust upon its 
 notice. 
 
 But Induction, as we have defined it, is no mere conjecture from 
 the known to the unknown, from facts to hypotheses, however 
 probable these latter may appear to be. It is a process which, 
 even in its most rudimentary form, must culminate in some kind of 
 methodical attempt at v^erification. Its aim is the explanation of 
 fact, and its means to this end is the Verification of Hypothesis. 
 
 We would therefore suggest the following scheme for the explana- 
 tion of fact through a method of Enumerative Induction : 
 
 "o* 
 
 I. Preliminary Observation. 
 
 a, b, c, d, e [generahzation instances] possess the pro- 
 perty X 
 
 II. Generalization. 
 
 (a) First stage. Generalization proper (Colligation). 
 
 a, b, c, d, e are instances that all belong to a class 
 defined by the marks p, q. 
 {jS) Second stage. JJ niversalization. 
 .'. perhaps all instances that possess the marks p, q 
 possess also the mark x.
 
 .?o2 THE PROBLEM OF LOGIC rxn. „. 
 
 iil. Deductive Development. 
 
 If all instances of tliis kind possess the mark x, then 
 a', b', c', d', e' . . . (which all possess the marks p, q) 
 must also possess the mark x. 
 
 IV. Progressive Verification. 
 
 a', b', c', d', e' . . . [test-instances], wliich all possess the 
 marks p, q, do (or do not) possess the property x. 
 
 Illustration : 
 
 I. These various individuals are suffering from this specific 
 
 luns; disease. 
 II. (a) Tliey all belong to the class defined by the mark 
 
 ' Sheffield grinders.' 
 II. (/i) .-. perhaps all the individuals belonging to the class 
 
 of Shefiield grinders, to wliicli the observed individuals 
 
 belong, also suffer from this disease. 
 
 III. But if all are thus afflicted, then Smith, Jones, Robinson, 
 
 etc., who are also Sheffield grinders, should prove to be 
 afflicted in tliis way. 
 
 IV. Attempt at progressive verification by further observa- 
 
 tion of Smith, Jones, Robinson, etc. 
 
 [Note. — In the generalization characteristic of Induction by Simple Enumeration, 
 the two essential aspects of Generalization, the singling out of the class-marks and 
 the universalizing element, take place separately and successively ; and, since the 
 class-generalization is taken for granted, it comes to pass that the second of these 
 two aspects is identified as a rule with the whole process of generalization, as 
 though (Generalization were identical M-ith Univcrsalization.'] 
 
 We must admit that this conception of Inductive Method as based 
 upon enumeration of instances does answer, in outline at least, to the 
 Inductive Idea as we conceive it. The essentials of inductive procedure 
 are all present : Preliminary Observation of instances. Generalization 
 of experience, deductive Development of hypothesis, attempt at 
 progressive Verification through further observation of instances. 
 It is therefore at least possible to attempt to solve the problem of 
 Induction by the use of the Method of Simple Enumeration. 
 
 Our discussion of this simplest form of the Inductive Process will 
 be concerned with two main questions : 
 
 1. What is the value of Enumeration in Inductive inquiry ? 
 
 2. What are the limitations of Enumerative Induction as an 
 
 Inductive Method ? 
 
 1. The Value of Enumeration in Inductive Inquiry. 
 
 It is particularly valuable in those more complex Sciences, such 
 as Meteorology and Sociology, in which the data from which 
 generalizations are drawn can often be adequately given only in
 
 Chap. XLI.] ENUMERATIVE IXDUCTION 35S 
 
 the form of aggregates and averages. The Method of Statistics. 
 though it includes an element of Ana'ysis, is founded on Enumenv- 
 tion, or tlie counting of instances. 
 
 In every statistical investigation there are certain f'efinitely 
 stated phenomena to be counted and tabulated — as, e.g., in a 
 census, the number of married men or of bachelors, the number of 
 widows and widowers, the proportion of those families that in- 
 habit tenements with more than five rooms to those that inhabit 
 tenements with five or less than five rooms, and so on. The 
 method of counting such agreements and comparing the results 
 constitutes the Method of Statistics. 
 
 ' The first rule of a statistical investigation is : the phe- 
 nomenon to be counted must be a countable fact that can serve as 
 a unit.'* Thus, supposing it is required to count the number of 
 houses in a town. What is a house ? Any place inhabited by 
 human beings ? And of any size ? Professor Scripture points out 
 that a house census taken in India gave the greatest trouble in 
 respect to this indefiniteness of the unit. We must therefore 
 qualify the object to be counted so as to render it a suitable unit. 
 Instead of counting up the ' hot ' days in the year — to take a 
 somewhat artificial instance — and contrasting these with the 
 number of cold days, we should coimt up, say, all days in which 
 the highest shade temperature exceeded 70'^ F., and contrast with 
 the days in which it was less than 40° F, 
 
 The second main rule of a statistical investigation follows naturally 
 from the first. It is that ' all things which are to be counted shall 
 correspond completely and exactly to the stated definition of the 
 counted object, and that nothing that does so correspond shall be 
 omitted. This requires that all the properties of the thing counted 
 shall be accurately determined before the count begins, and that 
 they shall not be changed during the counting. 'f 
 
 It is interesting to notice how the ancient Logic and Science dis- 
 regarded the serviceability of these statistical, merely numerical, 
 relations. The knowledge of the concept, so Aristotle argued, can 
 gain nothing by our knowing how often it is realized. But in 
 modem Science, where the desire to apprehend the Given fully and 
 accurately, as an indispensable preface to understanding its laws, 
 has been characteristically predominant, number has attained 
 definite scientific value. ' Who can explain,' says Darwin, ' why 
 one species ranges widely and is very numerous, and why another 
 allied species has a narrow range and is rare ? Yet these relations 
 are of the liighest importance, for they determine the present 
 welfare, and, as I believe, the future success and modification of 
 every inhabitant of tliis world. . . .'J 'As variations manifestly 
 useful or pleasing to man appear only occasionally, the chance of 
 
 • E. W. Scripture. ' The New Psychology.' p. 16. f Ibid., p. 13. 
 
 + ' The Origia of Specie^.' Introduction, p. 4. 
 
 23
 
 
 THE PROBLEM OF LOGIC 
 
 [XII. 
 
 11. 
 
 their appearance will be much increased bj^ a large number of indi- 
 viduals being kept. Hence number is of the highest importance for 
 success. On this principle Marshall formerly remarked, with 
 respect to the sheep of parts of Yorkshire, " As they generally 
 belong to poor people, and are mostly in small lots, they never can 
 be improved." On the other hand, nurserymen, from keeping 
 large stocks of the same plant, are generally far more successful 
 than amateurs in raising new and valuable varieties.'* 
 
 Illustration of the Value of Statistics. 
 (From Romanes' ' Darwin, and after Darwin.') 
 
 Statistics of the species of fauna found on islands (1) that have 
 long been separated by great distances of sea from the mainland ; 
 (2) that have only recently been separated from the mainland. 
 
 As tj^jical instances of (1) we take the Sandwich Islands, the 
 Galapagos Islands, and St. Helena. We get the following results : 
 
 
 
 o 
 
 CO 
 
 a 
 
 CO 
 
 K 
 
 
 CO 
 
 1l 
 
 CD 
 
 .a 
 
 CO 
 
 Vj 
 
 C 
 
 o 
 
 CO 
 
 a 
 
 CO 
 
 o 
 
 p< 
 
 P3 
 
 t3 00 
 
 CO 
 
 Sandwich 
 Islands 
 
 400 
 
 (?) 
 
 2 
 
 16 
 30 
 
 
 
 
 
 (?) 
 (?) 
 (?) 
 
 (?) 
 
 
 
 
 
 
 
 Galapagos 
 
 15 
 
 35 
 
 10 
 
 
 
 (?) 
 
 
 
 1 
 
 
 
 8t. Helena 
 
 20 
 
 128 
 
 
 
 1 
 
 
 
 i 
 
 () 
 
 
 
 
 
 Totals 
 
 435 
 
 163 
 
 12 
 
 47 
 
 
 
 
 
 
 
 1 
 
 
 
 Peculiar Species. 
 
 Non-peculiar Species. 
 
 Thus, out of a total of 658 species of terrestrial fauna, all are peculiar, 
 with the exception of a single land bird found in the Galapagos 
 Islands and on the American Continent as well. 
 
 As a typical instance of (2) we take the Island of Great Britain. 
 
 i 
 
 s 
 
 "3 
 
 i 
 
 Is 
 
 u 
 
 ^ 
 
 i 
 
 B 
 
 4 
 
 a a 
 
 
 CO 
 
 O 
 
 00 
 
 s ^ 
 
 CO -; 
 
 00 
 
 t 
 
 pq 
 a 
 
 •i 
 
 »3i 
 
 46 
 
 
 
 ^ 
 
 s 
 
 
 
 
 u 
 
 
 S 
 
 4 
 
 liO 
 
 
 
 1 
 
 
 
 1.462 
 
 83 
 
 12,551 
 
 13 
 
 130 
 
 40 
 
 V. 
 
 
 ^ 
 
 •»^ 
 
 
 
 
 
 ^' 
 
 
 Peculiar Species. 
 
 
 
 No 
 
 n-peculiar 
 
 Species 
 
 
 
 * ' The Origin of Species,' ch. i., p. 28.
 
 Chap. XLL] ENUMERATIVE INDUCTION 335 
 
 The statistics furnished by this tabular analysis are eloquent, and 
 point irresistibly towards the conclusion ' that wherever there is 
 evidence of land-areas having been for a long time separated from 
 other land-areas, there we meet with a more or less extraordinary 
 profusion of unique or peculiar species, often running up into 
 unique genera ; and. in fact, so far as naturahsts have hitherto 
 been able to ascertain, there is no exception to this general law in 
 any region of the globe. Moreover, there is everywhere a constant 
 correlation between the degree of the peculiarity on the part of 
 the fauna and flora and the time during which they have been 
 isolated.'* 
 
 The complete subordination of statistical numeration to scientific 
 explanation is made clear by the fact that, as soon as laws are 
 actually established, statistical numeration ceases to be of interest. 
 As Dr. Sigwart points out, the interest in counting how many 
 eclipses of sun and moon occurred annually has completely vanished 
 ' since the rule has been found according to which they occur, and 
 can be calculated for centuries past and to come ' {'Logic,' English 
 translation, vol. ii., § 101, p. 483). 
 
 What, then, we ask in conclusion, is the value of number of 
 instances in inductive inquiry ? We may safely say that the 
 strength of an induction is not necessarily proportional to the 
 number of instances cited. The generalizations that lead to the 
 most trustworthy hypotheses are based, not on the counting of the 
 number of instances, but on the weighing of their quality and 
 character. One crucial instance may be worth a hundred others. 
 Again, strong analogical arguments can be based on the com- 
 parison of two instances only, whilst finally the most stringent in- 
 ductions of all are those carried out by the Method of Difference, in 
 wliich only one instance in two forms or, at most, two instances are 
 needed. 
 
 We must add, however, that the work of true scientific analysis 
 is often greatly assisted by the fact that the number of instances of 
 a phenomenon or of the repetition of an event is large ; for the larger 
 the number of varied instances, the more easily can the unessential 
 elements be detected, together with the form of the general law. 
 
 2. Enumerative Induction in the Light of the Inductive Criterion. 
 
 We have seen that the counting of instances may have definite 
 scientific value during the preliminary stages of scientific obser- 
 vation. 
 
 We have now to consider to what extent a uniformity suggested 
 by the counting of instances can be satisfactorily verified by the 
 further counting of instances. Tliis inquiry, which amounts to the 
 
 ♦ ' The Origia of Species,' p, 235. 
 
 23—2
 
 356 THE PROBLEM OF LOGIC [XII. ii. 
 
 testing of Enumeration by the Inductive Criterion of Fidelity to 
 Relevant Fact, will serve to define the extent to which it is adequate 
 as an Inductive Method. 
 
 We shall see that insistence on the principle that facts (so far 
 as relevant) must have their nature respected serves to deepen in 
 ever\' direction the significance of the processes which, in a crude 
 germinal form, are present in Enumerative Induction itself, and to 
 open out the way to true Scientific Induction. 
 
 How. then, does this Method stand in relation to the requirement 
 of Verification ? 
 
 It is characteristic of Enumerative Induction that the process 
 of Progressive Verification through which we attempt to justify the 
 suggestion that all S's are P by verifj'ing that S is P in each indi- 
 vidual instance may culminate in a verification that is final and 
 complete. The conditions may be such as to admit of a complete 
 enumeration of instances, and on testing these instances indi- 
 vidually it may be sho\^Ti that thej' all possess the required mark. 
 
 This Complete Enumeration, as a genuine inductive process, 
 should not be confused with the so-called ' Perfect Induction * 
 already considered. The argument in an Induction by Complete 
 Enumeration would run as follows : 
 
 I. Preliminary Observation. — a, b, c, d, e are all instances pos- 
 sessing the mark x. 
 II. Generalization. — (1) p, q are the marks common to all these 
 
 instances. 
 (2) Perhaps all instances of this class (pq) 
 possess the mark x. 
 
 III. Deductive Development. — If all these instances possess the 
 mark x, then any individual instance must do so. 
 
 IV. Verification. — All the possible instances having been ex- 
 amined, we are able to assert categorically either that all 
 instances of the class pq are accompanied by x, or that some 
 instances lack the mark in question. 
 
 Thus in an Induction by Complete Enumeration the Verification 
 may be complete. The question then naturally suggests itself : 
 Have we not in Induction by Complete Enumeration an ideal 
 form of Induction ? To this our answer must be that it does 
 indeed realize an ideal, the ideal of Enumerative Induction ; but 
 the further question remains : What are the intrinsic limitations 
 of this kind of Induction ? 
 
 (i.) It can do no more than verify a ' that.' It cannot verify a 
 ' how.' Given the observed fact that p, q are in a certain number of 
 cases accompanied by x, the induction at best can only verify the 
 suggested fact that p, q are in all possible instances accompanied 
 by X. But this is not in any way a verified explanation of the con- 
 nexion between (p, q) and x. How [p, q) and x are connected can
 
 Chap. XLL] ENX^IERATIVE INDUCTION 357 
 
 never be ascertained by this process, thougli it may, of course, be 
 known before the process begins. Complete Enumeration then 
 (when the results are wholly favourable) is just a verified conjecture 
 from one fact (Some S's are P) to another fact (All S's are P). 
 Enumerative Induction cannot, therefore, be used for establishing 
 laws of connexion, whether of sequence or of coexistence, and so, 
 in particular, cannot establish laws of causal connexion. For all 
 scientific laws are uniformities which state how one fact is con- 
 nected with another. To state that one fact is always connected 
 with another is to state a fact of uniform connexion, but not the 
 explanatory law of that connexion. 
 
 This limitation is fundamental. 
 
 (ii.) A second limitation is almost, if not quite, as fundamental. It 
 is impossible through an Induction by Simple Enumeration, unsup- 
 ported by analysis, to deal with what we may call ' apparent 
 exceptions.' If an object possessing the marks p, q is found not 
 to possess the mark x, it must, so far as this method is concerned, 
 be put down as an exception, and with one exception the enumer- 
 ative universal ' All pg is a; ' can no longer be completely verified. 
 
 The most obvious suggestion which this method suppUes towards 
 its own reconstruction is its indication of Analysis as the method 
 required for meeting the difficulties with which it is itself unable to 
 cope. Enumeration itself contains a germ of Analysis. Closer 
 inspection, indeed, shows that, apart from that analysis wliieh 
 supplies the class-distinction and makes possible the generalization, 
 enumeration could not come into play at all. Unless I know that 
 I have to count instances characterized as pq, counting is im- 
 possible. Analysis, again, is the natural remedy for tlie deficiencies 
 of mere Enumeration. To Enumeration Nature necessarily pre- 
 sents itself as a mere aggregate of instances. The systematic 
 character of the given is by this method completely and inevitably 
 ignored. But to do justice to that systematic character is the main 
 function of the analytic method to which Enumeration by its own 
 limitations points. Thus we see that Enumeration needs to be 
 completed and transcended by the deeper method of Analysis. 
 
 A study of the Method's defects has made it clear that for the 
 true interpretation of fact we need an experimental analysis of 
 the systematic connexions of Nature. It would therefore seem 
 api^ropriate to proceed at once to the consideration of that com- 
 plete ilethod, the Method of Scientific Induction, which actually 
 works on the lines thus indicated. But we must first consider the 
 import and value of a Method which, though non-experimental, 
 and but dimly foreshadowing the systematic character of real fact, 
 is still a Method of Analysis, and, as such, takes us a full stage 
 beyond the unanalytic ilethod of Simple Enumeration.
 
 358 THE PROBLEM OF LOGIC [XII. ii. 
 
 CHAPTER XLIL 
 
 XII. (ii.) THE 'BIPERFECT INDUCTIONS.' 
 B. Argument from Analogy. 
 
 The Argument from Analogy is based upon resemblance. Tho 
 form of the argument may be roughly stated as follows : 
 
 Two things (or classes), A and B, resemble each other in one or 
 
 more properties, Rj, Rg. . . . 
 B possesses a certain other property, P. 
 We therefore conclude that A will resemble B in possessing tliis 
 
 property also. 
 
 In arguing from Analogy we no longer count instances, as we do 
 in Enumerative Induction, but we weigh properties.* In Enumera- 
 tive Induction we lay stress solely on the number of corroborative 
 instances — on the fact that, though the number of instances con- 
 sidered is great, no contrary instance has yet been met with. This 
 unanalj'tic character of Simple Enumeration is its essential weak- 
 ness. The argument by Enumeration gives way to an argument 
 from Analogy so soon as attention is turned from an enumeration 
 of observed instances to an analysis of their character. 
 
 Mill's view of the Argument from Analogy has sometimes been 
 severely criticized, as though he had treated it as founded on a mere 
 enumeration of resemblances. This, however, is hardly a fair 
 criticism ; for Mill's process of Analogy undoubtedly begins with an 
 analytic inquiry through which all those resemblances are elimin- 
 ated which can be showTi to be unconnected with the property in 
 question (' A System of Logic,' Book III., ch. xx., § 2). More- 
 over, in his analogical argument about the habitability of the 
 Moon, he shows how one single difference (the fact that the 
 Moon aptparently has no atmosphere) outweighs a large number 
 of resemblances. Indeed, he shows that, viewed in the light 
 of this one difference, ' all the resemblances which exist become 
 presumptions against, not in favour of, the moon's being in- 
 habited ' {ibid.). And all through his discussion of False Analogies 
 (Book v., ch. v., § 6) he implicitly insists on the importance of 
 weighing rather than counting resemblances, showing that the 
 important resemblances are those on which the suggested property 
 dep'in/h. 
 
 The Argument from Analogy, then, as opposed to Enumerative 
 Induction, proceeds by analysis of content. But if the analysis 
 could be sufficiently thoroughgoing to disclose an invariable causal 
 connexion between the property P and the properties R^, Rg . . . 
 
 • Of. Profesfjor Bosanquet, ' Logic,' vol. ii., ch. iii., p. 8"?.
 
 Chap. XLII.] ANALOGY 359 
 
 which are common to both the objects A and B, the argument 
 would lose all analogical significance. The gist of the argument 
 would then be the scientific induction whereby the causal con- 
 nexion had been discovered and verified. The remainder would bo 
 mere Parity of Reasoning. Thus : 
 
 There is an invariable causal connexion (as witness the instance 
 A) between the property P and the properties R^, Rj, 
 R3. ... 
 
 Now in B we have the properties R^, Rj, R3. . . . 
 
 Therefore in B we must also have the property P. 
 
 A conjecture, then, can be said to be based on Analogy only when 
 the suggested property is not known to be causally or otherwise 
 invariably connected with the properties of which the common 
 possession by the two objects forms the basis of the Analogy. 
 
 On the Value of an Arguraent from Analogy. 
 
 The worth of an Argument by Analogy depends on the im- 
 portance of the resemblances on which it is based, and the cor- 
 responding non-importance of the differences between the two 
 objects concerned. The resemblances must be essential, the 
 differences unessential. By this we mean that, if the argument 
 is to be cogent, the properties which the two objects (or classes) 
 A and B have in common must be closely related to the problematic 
 property P, while their points of difference must be but loosely 
 connected with it. We must not treat Analogy as though it were 
 a question of Enumeration, and argue as if the strength of the 
 analogical argument depended on the ratio of the number of points 
 of resemblance to the number of points of difference. Mere 
 counting of the points of resemblance and of difference is of 
 little use. 
 
 For example, we might enumerate many points of external re- 
 semblance between the Whale and the Shark, and found upon them 
 an analogical argument to the effect that the respiration-processes 
 in the two animals must be similar. The whale, we might say, 
 resembles the shark not only in all the common characters of 
 Vertebrates, but also in its submarine habitat and in being (as 
 regards many species) one of the very largest of marine animals. 
 Like the shark, it is fish-like in external form, its fusiform body 
 being well fitted for cleaving the water. Anteriorly its body passes 
 into the head without any distinct neck, and posteriorly it is fur- 
 nished with a swimming-tail into whicli the body gradually tapers. 
 It has no hairy covering. Like the shark, again, it has a wide 
 mouth, and it is of predaceous habit, feeding only on living animal 
 nutriment. Therefore we may with great probpJ)ility conclude ih\t
 
 360 THE PROBLEM OF LOGIC [XII. u. 
 
 its method of respiration is like that of the sliark — i.e., that it 
 breathes the oxygen dissolved in the water, and has no need to be 
 supplied with atmospheric air. 
 
 But this argument is unsound. The points that we ought to 
 have observed are the characters connected with the function of 
 resj)iration. The presence of gill-slits in the shark and their absence 
 in the whale is a difference so essential to the inquiry that its 
 observation would at once have been sufficient to make our analogy 
 fall to the ground. And among the still more obvious external 
 ditferences there is a single character wliich also should alone out- 
 weigh all the above-mentioned resemblances. The extremity of 
 the shark's tail is expanded vertically ; in the whale the flukes of 
 tlie tail are placed horizontally. From many points of view this 
 difference might be regarded as unimportant ; but from the point 
 of view of our analogical argument it is very important indeed, for 
 it is intimately connected with the problematic point of resemblance 
 that we are endeavouring to cstabhsh. For sea-creatures which, 
 like the whale and the shark, spend their life in swimming freely 
 through the water, the direction of movement is chiefly determined 
 by the presence or absence of the air-breathing habit. Those crea- 
 tures that have no need of atmospheric air move usually in straight- 
 forward and lateral directions, and for effecting such movements a 
 vertically expanded caudal fin is admirably fitted. But creatures 
 that need to rise frequently to the surface of the water for the pur- 
 pose of respiration are constantly moving upward and downward. 
 To movements of this kind a tail with horizontally expanded flukes 
 is precisely adapted. Thus the whale's horizontally expanded tail 
 affords a strong presumption in favour of the presence of the air- 
 breathing habit ; and this apparently trifling difference between the 
 two creatures mast be regarded as fatal to the cogency of the 
 analogical argument. 
 
 We see, then, that it is the important difference which invalidates 
 an argument from Analogy, this importance being purely relative 
 to the problematic property (P) which the argument seeks to 
 e.stabhsh. So also it is only by resemblances that are in this same 
 sense important tliat arguments from Analogy can be justified. 
 
 Thus, if we arc arguing from the habitabiUty of the Earth to that 
 of Mars, the atmospheric and temperature conditions on the surface 
 of Mars are the all-important points. Since these constitute marks 
 of resemblance to the Earth rather than marks of difference, the 
 analogical argument is proportionately strong. 
 
 It is frequently stated that, in estimating resemblances, a group 
 of causally connected resemblances should only count as one. If 
 the strength of an analogy depended on the number of independent 
 points of resemblance, this would be a reasonable precaution to 
 insist on. But in proportion as we cease to lay stress on number, 
 this precaution loses its meaning. When points are weighed
 
 Chap. XLII.] ANALOGY 301 
 
 instead of being counted, then the more complex and causally co- 
 herent any such relevant group of resemblances S (Ri) may be, the 
 more likely is it that the property P will ultimately be brought 
 within the same causal nexus. 
 
 The Logical Character of Analogical Argument. 
 
 The Analogical Argument finds its natural place as a stage in a 
 complete inductive inquiry. It has genuine inductive value as a 
 means of suggesting hypotheses and of sifting out, from an un- 
 tested heap of mere guesses at truth, such as are ' soundest ' and 
 the most likely to repay the arduous work of development and 
 verification : the true function of analogical reasoning cannot be 
 other than that of recoramendmg a hj-pothesis as worth the trouble 
 of verification. To have any inductive value, the conclusion of an 
 analogical argument must be verifiable. It is not essential that the 
 verification should be immediately practicable. The argument 
 may concern the conditions of life on Mars or on the Moon, and, in 
 the present state of astronomical knowledge, it raay not be pos- 
 sible for astronomers to put the conclusion to the test. In this 
 case the conclusion should be treated as a suspended hypothesis — 
 as a hj-pothesis of which the verification is deferred sijie die. The 
 analogical argument stands as a preliminary stage in an inductive 
 inquiry that is not as yet completed. 
 
 The true place of Analogy is in the service of Scientific Induction. 
 In relation to a complete scientific inquiry, its logical function is 
 heuristic. It plays an important part in the Logic of Discovery, 
 but has no place in the Logic of Verification. 
 
 The Value of Analogy in suggesting Scientific Hypotheses. 
 
 As Mill has said, it is when Analogy is used ' as a mere guide-post, 
 pointing out the direction in which more rigorous investigations 
 should be prosecuted,' that it has ' the liighest scientific value.'* 
 This point of view is, however, lightly passed over by Mill liimself, 
 concerning, as it does, the Logic of Discovery rather than the 
 Logic of Evidence. Jevons, on the other hand, in liis chapter on 
 Analogy! in his ' Principles of Science,' emphasizes it almost ex- 
 clusively. Quoting from Jeremy Benthara's ' Essay on Logic,' 
 ' Discovery,' he asserts, ' is most frequently accomplished by 
 following up hints received from analogy.' Thus, if a chemist is 
 testing ' what he believes to be a new element,' and sees that ' in 
 any one of its quafities the substance displays a resemblance to 
 an alkaline metal ... he will naturally proceed to tr\' whether it 
 possesses other properties common to the edkaline metals.' 
 
 * J. S. iim. ' A System of Logic.' Book in., ch. ix.. § 3. 
 t Vol. ii., pp. 283-305.
 
 3G2 THE PROBLEM OF LOGIC [XII. ii. 
 
 Jevons draws especial attention to the perfect example of Analogy 
 piTsented by the analogy between Geometry and Algebra. So 
 long as this analogy was not suspected, and the two sciences de- 
 veloped iiHlepcndently, as they practically did up to Descartes' 
 day, they made but slow progress. When Descartes, in his system 
 of Algebraic Geometry, had shown that the straight lines or 
 co-ordinates ' required for fixing the position of a point ' and the 
 algebraical symbols x, y were fundamentally analogous, in such 
 wise that a geometrical figure or curve could always be repre- 
 sented by an algebraical equation, the two sciences rapidly de- 
 veloped. Properties of curves were discovered by solving equations, 
 and equations established by geometrical investigations of curves. 
 The analogy between thing and symbol, when properly understood, 
 is the most perfect and fruitful of analogies. 
 
 Another interesting instance of historically fruitful Analogy, 
 cited by Jevons {ibid., ii., p. 298), is the analogy between Jupiter 
 and its moons on the one hand, and the Sun and its planets on 
 the other. ' While the scientific world was divided in opinion 
 between the Copernican and Ptolemaic systems, . . . Galileo dis- 
 covered, by the use of his new telescope,' four of the ' small 
 satellites which circulate round Jupiter, and make a miniature 
 planetary world. These four Medicean Stars, as they were called, 
 were plainly seen to revolve round Jupiter in various periods, 
 but approximately in one plane, and astronomers irresistibly in- 
 ferred that what might happen on the smaller scale might also be 
 found true of the greater planetary system. The relation between 
 Jupiter and its satellites is in many important respects analogous 
 to the relation between the Sun and its planets. Therefore, since 
 the satellites move round Jupiter, the planets should also move 
 round the Sun.' 
 
 The so-called ' False Analogy.'' 
 
 From tlie point of view of the Inductive Standard which we have 
 adopted the tentative conclusion drawn through an analogical 
 argument may be called true only when adequately verified by a 
 subsequent process of scientific Experiment ; jalse when disproved 
 by the process. 
 
 Let the two instances A, B be symbolically analysed as follows : 
 A=Ri, Kg, R3, . . . di, dg, dg, . . . P. 
 B = Ri, R2, R3, ...g,, §2, §3. •••(P^)- 
 Here R^, Rj, R3 . . . arc t)ic ' points ' of resemblance ; 
 
 dj, dj, dg . . . ' 7»oints ' possessed by A, but not by B ; 
 ^1' ^2' ^3 • • • ' P'^i'^ts ' possessed by B, but not by A ; 
 P a mark possessed by A ; — the analogical argument con- 
 sisting in concluding that B also possesses P.
 
 Chap. XLII.] ANALOGY 3G3 
 
 A true, conclusion from Analogy, then, would consist in a con- 
 nexion between P and R^, Rg- K3 • • • founded upon resemblance, 
 sustained by an analogical argument, and established by Scientific 
 Induction. 
 
 A false conclusion from Analogy would be one that should break 
 down under the criticism of a scientific verification, the suggested 
 connexion between P and R^, Rg, R3 . . . proving to be unwarranted. 
 
 But so long as it is not practicable to proceed to this verification, 
 tlie conclusion cannot be regarded as more than probable. ' True ' 
 and ' false,' as applied to it, are inappropriate epithets, and the 
 place of this distinction is naturally taken by the distinction be- 
 tween ' sound ' and ' unsound,^ between well-grounded and un- 
 grounded analogical conclusions. A sound or well-grounded 
 analogy — using ' analogy ' in the sense of a conclusion drawn 
 through an analogical argument — would then be one in which 
 the resemblances R^, Rj, R3 . . . were essential in relation to P, and 
 the differences between A and B (again in relation to P) were un- 
 essential. 
 
 On the other hand, an argument from Analogy should be stamped 
 as unsound or illegitimate when it can be shown that the dififerences 
 in relation to P are essential and the resemblances unessential ; or, 
 in other words, when we can show that the suggested property P, 
 far from being rooted in the known resemblances, is rooted in 
 certain differences between A and B, in certain conditions which 
 obtain in A, but not in B, Such an argument from Analogy could 
 appropriately be called ' ungrounded.' 
 
 Illustrations of Illegitimate Analogy : 
 
 1. States must decay, as individuals do.* 
 
 In the case of the individual body, decay results from ' the 
 natural progress of those very changes of structure which, in their 
 earher stages, constitute its growth to maturity,' — i.e., from the 
 conditions of old age, dj, dg, dg — whereas in the case of the State 
 these conditions are not present, for the properties h^, So, S3, upon 
 which its healthy existence at any stage depends, are such as 
 point to an indefinite continuance of healthy growth. The decay 
 of States is due, not to inevitable old age, but to disease wliich 
 may be warded off. ' Bodies politic die,' says Mill, ' but it is of 
 disease or violent death ; tliey have no old age.' 
 
 2. A and B resemble each other in possessing a certain kind 
 
 of skill. They can both make locks. A is furtlier given 
 to picking them with a view to robbery. Therefore B is 
 also addicted to lock-picking with a view to robbery. f 
 
 * J. S. Mill. ' A System of Logic.' Book V., ch. v., § G. 
 ■f- Of. Plato, ' Republic,' p. 334, A : ' What a man is cle^ 
 
 3ver at keeping ho is clever 
 at stealin;^ too.'
 
 304 THE PROBLEM OF LOGIC [XII. ii. 
 
 Here A = skill to make locks implying slcill to pick them + the 
 will to pick them (dj). 
 B= skill to make locks implying skill to pick them + a 
 will about whose tendencies, one way or the other, 
 nothing is known (Si). 
 
 The argument is that, as A actually ptc^5 locks, therefore i> does 
 too ; and the question is whether the argument is sound. 
 
 Now P, tlie addiction to robbery, is rooted in d^, A's will to pick 
 locks. Hence, as d^ is not kno^vn to be a characteristic of B, there 
 is no growid for connecting P with B. 
 
 Therefore tjie analogy is unsound. 
 
 So long as an analogical conclusion is not verified it cannot be 
 called true. But, it may be urged, an Unsound Analogy may surely 
 in the strictest sense be false — in the case, namely, in which P can 
 be proved on the given data to be incompatible with certain differ- 
 ences, Sj, So, 8^. possessed by B but not by A. In this case is not the 
 argument from Analogy strictly disproved ? Certainly ; but the 
 disproof would entail a Scientific Induction. It is as difficult to 
 prove such incompatibility as it is to prove or disprove a causal 
 connexion. 
 
 Thus, ' Is human life incompatible with absence of water V To 
 answer this question satisfactorily we must analyse out the pro- 
 perties of Water (considered as a solvent, etc.) ; we must physio- 
 loeically analyse the meaning of thirst, and the necessity of a supply 
 of liquid ; we must consider in detail the whole question of possible 
 substitutes for water. Not till these processes of analysis are 
 completed can we venture on a downright use of the term ' incom- 
 patible.' 
 
 Incompatibility would thus seem not to be a matter that concerns 
 Analog}'. Analogy deals with points of difference, certainly, as 
 with points of resemblance. But so soon as we can deepen the 
 fact of difference into the fact of incompatibility, we seem almost 
 as certainly to liave gone beyond Analogy as we have when we 
 have deepened the fact of resemblance into a fact of causal con- 
 nexion.
 
 XIII. 
 THE GOAL OF INDUCTION : CAUSAL EXPLANATION. 
 
 (i.) Cause and Causal Law (ch. xliii.). 
 (ii.) The Process of Scientific Observation (ch. xliv.), 
 (iii.) The Method of Causal Explanation (ch. xlv.). 
 
 Dlustrations of the Application of Inductive Method (ch. ilvi.}.
 
 CHAPTER XLIII. 
 
 XIII. (i.) CAUSE AND CAUSAL LAW. 
 
 We have now to consider the form which Induction takes when it 
 most adequately embodies the true interest and intention of Science. 
 Fact, for Science, as we have seen, means Fact in the hght of Law ; 
 but for Scientific Induction in particular Fact means ' Effect ' — 
 Effect that evidences the presence of Cause acting according to 
 uniform Law. Conceived in an explanatory relation to effects, a 
 law is known as a causal law or a law of causation. 
 
 The Meaning of the Term ' Law.^ 
 
 Scientific Laws take shape and develop under the inspiration of 
 the Inductive Principle of Fidelity to Relevant Fact — i.e., to Fact 
 in so far as it answers to the demand of the Inductive Postulate 
 of Uniformity or Determinism {vide Chapter XLVII.). Within the 
 limits of this postulate Fact is supreme. It is Fact that controls 
 the tentative gropings of Hypothesis, and, in proportion as the 
 latter conforms to its requirements, allows it to assume the status 
 of Law. Scientific Laws are laws on sufferance, and they hold their 
 office as interpreters of the world of real fact only so long as there 
 are no other promising hypotheses to perform that function better. 
 
 Tliis subordinate position of Law in Science is no exception to 
 the general rule. Law, when healthily operative, is every^vhere 
 subordinate, be it to the constitutional authority of a nation's 
 will, to the Principles of Reason and Conscience, or to the require- 
 ments of Relevant Fact. It is only the dead law of the Modes and 
 Persians whicli altereth not. 
 
 Wlien, in Science, we speak of a causal law, we in no way con- 
 ceive it as causing or bringins about effects. The cause which 
 is responsible for the effects is not itself the law through which those 
 effects are interpreted. The law is a tentative, though approved, 
 statement of uniformity ; it is not a force. In no sense is it a power 
 that can control the farts. 
 
 And yet, whilst submitting its laws to the sifting and refining 
 coatrol of relevant fact, Science still remains self-governed. Its 
 
 367
 
 3G8 THE PROBLEM OF LOGIC [XIII. i. 
 
 submission to Fact presupposes a fidelity to its own fundamental 
 principles, for it is these alone which determine how Science shall 
 study Fact. On the other hand, the fundamental assumption of 
 Science, the Liductive Postulate, is not arbitrary. It has its root 
 in the protest of the scientific spirit against the anthropomorphisms 
 and animisms of pre-scientific ages (vide Chapter XL VII.), and 
 testifies to the deep conviction of modern Science that Nature is the 
 expression of Natural Law. It is out of this belief in the immanent 
 laws of Nature that the ideal laws of Science have tentatively and 
 gradually taken shape, bringing Nature at last, through man, into 
 self-conscious possession of its own intrinsic orderliness. 
 
 Is all Inductive Explanation Causal ? 
 
 We may divide our answer to this question under two main 
 heads : 
 
 1. All complete Scientific Explanation is directly or indirectlf/ 
 ' causal ' in the sense which this word properly bears in Science. 
 
 It may be argued that, so long as we are simply inquiring how an 
 object or fact is constituted, asking what are the simple elements or 
 factors out of wliich it is constructed, the explanation is substantial, 
 not causal. But this mere analysis of a fact into its factors is no 
 explanation. The fact is explained only when we can show Row 
 it is the product of these factors. Indeed, Science, in last resort, 
 reduces all questions concerning the coexistence of properties in a 
 thing or substance to questions of causal connexion. So long as 
 we are studying the connexions of what we take to be systematic 
 in nature, we are studying causal connexions, whether these con- 
 nexions be coexistences or sequences. 
 
 Uniformities of coexistence are the most obvious uniformities in 
 Nature. Every natural classification, so far as it is natural, is a 
 record of Nature's correlated facts — i.e., of uniformities of co- 
 fxistence. The name of a class is the sign by wliich we recognize 
 the coexistence of a multitude of properties. For instance, by 
 ' Gold ' we understand a metal of high specific gravity, high melting- 
 j)oint, low chemical affinities, great ductility, yellow colour, etc. 
 I:i the case of the higher divisions of a natural classification we are 
 furnished with a similar clue to coexistent properties. For in- 
 stance, ' Monocotyledon ' is a sign whereby we are apprised of the 
 whole list of the coexistences expressed in its definition. Again, the 
 inferences that can be drawn from a natural classification are all 
 ex[)ressive of the coexistence of correlated properties. 
 
 Correlatioas or coexistences — e.g., that of two such properties as 
 tlie ruminating habit and divided feet — in most cases can only be 
 empirically formulated.
 
 Chap. XLIIL] CAUSATION 369 
 
 In the particular case of Geometry, however, we have a science 
 of rationah'zed coexistences, the correlated properties boini^ here 
 all deduciljle from the essential properties of Space and Figure. 
 All the properties of a circle are coexistent ; and not only so, but 
 exact reasons can be given to explain their coexistence. We have 
 here a type of coexistence in which causal explanation passes into 
 pure mathematical deduction. Explanation in Mathematics is 
 non-causal. In abstracting from any reference to substance, it 
 eo ipso abstracts from all causal considerations. 
 
 But when, as in all systems of Natural Classification, the properties 
 are not correlated in reason only, the scientific belief that tlie con- 
 nexions are not arbitrary, but must eventually admit of being 
 reasoned out, leaves the scientist with the problem of discovering 
 some common cause for all these coexistent effects. Precisely the 
 same reasoning takes place where uniformities are sequence-unifor- 
 mities, as in the sequences of day and niglit, summer and winter, etc. 
 Here again we are driven to seek for some causal explanation, 
 to find some cause of wliich the sequences are co-effects — though, 
 of course, not simultaneous effects — or else to deepen the relation 
 of sequence itself into one of causation. 
 
 All complete Scientific Explanation is, therefore, causal. If in- 
 complete, as in Enumerative Induction and Analogy, where the 
 ' that ' is investigated, but not the ' how,' the explanation can at 
 best be preliminary, or else subsidiary, to true Causal Explanation. 
 
 In what is called the explanation of laws we have a process of 
 systematization which is not direclly causal ; but, though it does not 
 deal with the relation of Cause and Effect, it does deal with the 
 interrelations of uniformities which themselves must be regarded 
 from the point of view of Causality, so that the procedure is still 
 indirectly referable to Causal Explanation. 
 
 2. If the word ' causal ' he used in an ultimate sense, no Scientific 
 Explanation is causal. 
 
 In the scientifically causal explanation of a fact we look upon 
 that fact as an effect consequent on certain ascertainable conditions. 
 Here the first step in causal explanation consists in the ascertain. 
 ment of the conditions on wliich the effect depends, and the main 
 step is the discovery and verification of the laws according to which 
 these conditions operate in bringing about the effect. In Scientific 
 Explanation we are in the habit of contenting ourselves with the 
 tracing of effects to the operation of conditions according to ascer- 
 tained laws. 
 
 Against Scientific Explanation as thus conceived it has been 
 urged that it is really no explanation at all, but mere description. 
 Only a free agent, it is said, can be a cause ; and these antecedent 
 conditions are certainly not free agents, but must themselves be 
 
 24
 
 370 THE PROBLEM OF LOGIC [XIII. i. 
 
 explicable by previous antecedent conditions. A cause that is 
 not a first cause is a cause only in the name. It is entirely deter- 
 mined by its antecedents, and is, therefore, a mere eflect. So 
 Berkeley asserts that ' The mechanical philosopher . . . inquires 
 properly concerning the rules and modes of operation alone, and 
 not concerning the cause ; forasmuch as nothing mechanical is, 
 or really can be, a cause.'* 
 
 The objection is fundamental, implying, as it does, that, far from 
 every complete scientific explanation being causal, none is causal. 
 
 The objection is valid and useful, provided its bearing be properly 
 understood ; for it simply emphasizes the restricted sense in which 
 the term ' cause ' is used for scientific purposes. Science elucidates 
 laws of connexion, but does not explain why the connexion should 
 be as it is, and not otherwise. As Causal Explanation, in Science, 
 is not causal in the true teleological sense, as it does not deal with 
 final causes (which are the concern of Philosophy), so it does not 
 deal with causes qua efiicient forces, though it is obliged to presuppose 
 them. It deals A\ith causes solely as conditions, as conditions apart 
 from which the event or effect in question would not take place, f 
 
 It remains to follow up these remarks by an attempt to analyse 
 the meaning of ' Cause ' as interpreted in Inductive Science. 
 
 The Meaning of ' Cause ' for Inductive Science. 
 
 A cause is defined by Mill as the sum total of the conditions, 
 which being realized, the consequent invariably and unconditionally 
 follows, j 
 
 Let us consider separately the two aspects of Cause suggested by 
 this definition. 
 
 1. Cause as the sum total of the conditions. 
 
 By ' the sum total of the conditions ' Mill can only mean the sum 
 of all the conditions that are both relevant and sufficiently im- 
 portant. Apart from this limitation, as Venn points out, we should 
 
 * Dr. George Berkeley, ' Siris,' § 249, Fraser'-s edition, vol. ii., p. 457 ; ' Selec- 
 tions,' fifth edition, p. 293. 
 
 t The Cau.se, or CaiLsal Antecedent, should be carefully distinguished from the 
 merely logical antecedent as it occurs in a hypothetical judgment. The latter is 
 the logical ground of a stated consequent, whereas tho former is the existential 
 ground of an effect, the ground apart from which the effect would not exist at all. 
 Of course, the consequent mat/ stand for an effect, and the antecedent which serves 
 as logical ground for the con.sequcnt tnay stand for the cause, as when we say, ' If 
 it rains, the grass will be wet.' Here the reason for our being able to infer that the 
 grass will be wet is stated to be the presence of the cause. But there are other 
 reasons besides those furnished by causes. Any siyn may furnish a reason. Thus 
 I may say : ' If the barometer falls, the grass will soon be wet.' Here the reason ia 
 nol tho cause. 
 
 : See 'A System of Logic.' Book III., ch. v.. §§ .3, G.
 
 TuAP. XLIII.] CAUSATION 371 
 
 be quite unable * to secure that repetition of occurrence which we 
 require in order to apply the sequence we have noti(;ed in the past 
 to some new instance in the future.'* If a causal connexion i.s to 
 be tested, the antecedents must be under control ; must, therefore, 
 be limited in number, and must be dealt with in abstraction from 
 such individual characteristics as are not repeatable. 
 
 Among these conditions Mill distinguishes the positive from 
 the negative. ' The negative conditions,' he says, ' may be all 
 summed up under one head — ^namely, the absence of preventin'^ 
 or counteracting causes . . . one negative condition invariably 
 understood, and the same in all instances (namely, the absence of 
 counteracting causes) being sufficient, along with the sum of the 
 positive conditions, to make up the whole set of circumstances on 
 which the phenomenon is dependent.'! 
 
 Among positive conditions we may distinguislx two classes, which 
 may be called respectively Predisposing and Excitmg or Initiating 
 Causes. Predisposing Causes are the relatively permanent con- 
 ditions whose presence may precede the effect for any length of 
 time, but which, for lack of the Exciting Cause, remain inoperative. 
 The Exciting Cause is an instantaneous change, a something which, 
 by coming into play, brings all the conditions into effective action 
 and precipitates the effect. When we say that small causes may 
 produce great effects, we have initiating causes in mind — the spark 
 that burns a city, and the speecli that brings about a war. Here 
 is a fictitious instance given by Boscovich, and quoted by Tyndall 
 (' Keat a Mode of Motion,' ninth edition. Lecture III., pp. 66-68) : 
 
 ' Boscovich . . . pictures a high mountain rising out of the sea, 
 with sides so steep that blocks of stone are just able to rest upon 
 them without rolling down. He supposes such blocks, diminishing 
 gradually in size, to be strewn over the mountain — large below, 
 moderate at the middle height, and dwindling to sand-grains at the 
 top. A small bird touches with its foot a grain on the summit ; it 
 moves, sets the next larger grains in motion ; these again let loose 
 the pebbles, these the larger stones, these the blocks ; until finally 
 the whole mountain-side rolls violently into the sea, there pro- 
 ducing mighty waves. Here the foot of the little bird unlocked 
 the energy, the rest of the work being done by gravitation.' 
 
 Similarly a ' spark acts hke the foot of the bird ; it starts a pro- 
 cess which is continued and vastly augmented by the molecular 
 forces of the fuel. . . .' Tyndall further points out that ' The action 
 of the nerves in unlocking the power of the muscles also falls in 
 admirably with the conception of Boscovich here described.' 
 Again, when a number of conditions combine to keep a body in a 
 position of unstable equilibrium — e.g., to keep an egg balanced on 
 one end — an infinitesimal determining cause is all that is needed to 
 
 * Dr. John Venn, ' The Principles of Empirical or Inductive Logic,' ch. ii.. p. 57. 
 t ' A System of Logic.' Book IIL, ch. v. § 3. 
 
 24—2
 
 372 THE PROBLEM OF LOGIC [XIII. L 
 
 decide tlie fate of the system. Hence the importance of the de- 
 termining or exciting cause even wlien, considered in isolation, it 
 might seem to be of a very trifling nature. 
 
 The predisposing causes constitute what Professor Bain calls a 
 Collocation. In considering the causes of a summer thunderstorm 
 we may regard the presence of aqueous vapour, the liigh tempera- 
 ture of the earth's surface, and the ascending currents of air ai 
 constituting, together with difference of electric potential, the 
 Collocation. The do'miward rush of cold dry air, which (in con- 
 junction with the ascensional movements of warm moist air rising 
 from the heated ground) causes a sudden condensation of great 
 quantities of aqueous vapour, would be here the determining cause. 
 
 In Pathology' the distinction between predisposing and deter- 
 mining causes is often clear!)' marked. If we are investigating the 
 causes of a case of ' nervous break-down,' we may have to reckon 
 »mong predisposing conditions an inherited neuropathic tendency, 
 insufficient or improper nourishment, bad air, insufficient exercise, 
 lite hours, over-excitement, over-work, long-continued anxiety ; 
 vhile the exciting cause may be a sudden shock or a single sleepless 
 night. 
 
 Bat these distinctions ar-3 not absolute. We might have re- 
 garded a sudden rise of temperature as the exciting cause of the 
 'heat thunderstorm,' and all the other conditions as constituting 
 the Collocation ; nor had the descending masses of cold dry air any 
 better right than the ascending columns of warm moist air to be 
 railed the determining cause. So again a long-continued suc- 
 cession of nervous shocks and an established habit of insomnia 
 might be regarded as predisposing causes of a neurasthenic ' break- 
 down,' wliile the exciting cause miglit be a day's fasting or an 
 hour's worn,'. The important point is that a change through which 
 ft total cause produces its effect can take place only when all the 
 fssential conditions are present. If n be the number of essential 
 renditions, then, if any (n — 1) of these are present, they are present 
 as predisposing causes until matters are precipitated by the arriv 1 
 «f the n**^. It is always the laH straw that breaks the camel's 
 back, no matter in what order they are successively placed in 
 position. 
 
 We have said enough, perhaps, sufficiently to emphasize the im- 
 p«>rtance of not taking the antecedent conditions in an undis- 
 enrainated heap. Clearly we must at least insist on the distinction 
 between the collateral or predisposing and the initiating or exciting 
 conditions. 
 
 Tliere are other distinctions which may at any time assume 
 importance, and require to be emphasized. Thus, wherever the 
 agency of human beings is presupposed, there is tie practically 
 important distinction between the controllable conditions and 
 those beyond human control. Take the case of a carelessly hung
 
 Chap. XLIII.] CAUSATION 373 
 
 picture which, on the clapping of a door, suddenly falls. Here the 
 conditions are : 
 
 (1) Weight of picture. 
 
 (2) Rotten cord or loose nail. 
 
 (3) Slamming of door (the precipitating cause). 
 
 Here the important distinction lies between (2), on the one hand, 
 and (I) and (3) on the other, (2) being the eminently controllable 
 condition. In hanging pictures we may see to it that the cords are 
 sound and the nails well driven in. We cannot so easily be re- 
 sponsible for the non-slamming of doors or for the weight of the 
 pictures. 
 
 When we are considering causal influences of a persistent char- 
 acter, a commonly recognized distinction is that between periodic 
 and non-periodic causes. With this is associated, in somewliat 
 complex relations, the distinction between cumulative and non- 
 cumulative effects. 
 
 A periodic cause might be defined as the rhythmic recurrence of 
 identical or similar conditions in a regular order of succession. 
 Thus the successive systole and diastole of the heart are a periodic 
 cause with regard to the circulation, and the rhythmic succession 
 of muscular contractions and relaxations which alternately expand 
 and diminish the tlioracic cavity is a periodic cause of respiration. 
 
 A non-periodic cause is the continuous persistence (or irregular 
 recurrence) of identical or similar conditions tending in one and the 
 same causal direction. Whatever action there is is non-rhythmic. 
 Thus the gravitation of bodies at the surface of the earth is a non- 
 periodic cause, and so is any steadily maintained pressure like that 
 exerted by the mainspring of a watch or by the steady pull of the 
 hand in drawing out a cork with a corkscrew. 
 
 But modern Science tends to multiply periodic causes, and if 
 we include among them all vibratory and undulator^' motions it 
 may be that the Science of the future will regard the action of every 
 physical force as in some sense periodic. 
 
 An effect is said to be cumulative when it persistently increa-es 
 in one and the same sense. Such effects may be produced either by 
 non-periodic causes or by causes of which one at least is periodic. 
 
 (a) Cumulative effects produced by Periodic Causes. 
 
 Rhythmic circulation of the blood may result in growth, and 
 daily exercise in continually increasing streUL'th. The beat of a 
 bird's wings results in continuous flight, and the alternate move- 
 ments of the legs may result in locomotion to an ever-increasing 
 distance from the starting-point. 
 
 Tyndall gives the following instance of a cumulative effect pro- 
 duced by a total cause of which one main factor was periodic. A 
 sheet of lead, covering the sloping choir-roof of Bristol Cathedral
 
 374 THE PROBLEM OF LOGIC [XIII. i. 
 
 in two years moved bodily down throuo;h a space of eighteen inches. 
 The lead was exposed to a periodically varying temperature, higlier 
 in the day-time, lower at night. Under the action of gravity it 
 expanded more freely doM-nwards than upwards, and, in con- 
 tracting, ' its upper edge was drawn more easily downwards than 
 its lower edge upwards. Its motion was therefore that of a common 
 earth-worm ; it pushed its lower edge forward during the day, and 
 drew its upper edge after it during the night, and thus by degrees 
 it crawled through a space of eighteen inches in two years.'* 
 
 We take another typical example from Sir Robert Ball's ' Time 
 and Tide,' p. lOL ' You see,' he says, ' a hea\^ weight hanging by 
 a string, and in my hand I hold a little slip of wood no heavier than 
 a common pencil ; ordinarily speaking, I might strike that heavy 
 weight with this slip of wood, and no [perceptible] effect is pro- 
 duced ; but if I take care to time the little blows that I give so that 
 they shall harmonize with the vibrations which the weight is 
 naturally disposed to make, then the effect of many small blows will 
 be cumulative, so much so that after a short time the weight begins 
 to respond to my efforts, and now you see it has acquired a swing of 
 very considerable amplitude.' 
 
 Mr. Edmund Catchpool gives similar instances : ' The regular 
 tramp of soldiers crossing a bridge will break the bridge down if 
 its period of oscillation agrees with the interval between the steps ; 
 and the vibrations of the air caused by an organ (it is said even by 
 the voice) will break a pane of glass in a window if the pane is of 
 such a size that it vibrates ' with a frequency corresponding to the 
 successive impacts of the sound-v/aves. •]• 
 
 Again, the periodic movements of the tides have a cumulative 
 effect in the retardation of the Earth's rotation on its axis. Con- 
 sequently the day is always lengthening and the moon retreating, 
 and there seems to be no counteracting agency anywhere dis- 
 cernible. 
 
 (b) Cumvlative ejects produced by Non-periodic Causes. 
 
 As an example of an effect of this kind we have the results of the 
 conduction of heat from the Earth's interior to its surface, and its 
 loss thence by radiation into space. This process is extremely 
 tardy ; but, going on incessantly and always in the same direction, 
 it produces gigantic effects. Other examples are the changes pro- 
 duced in rocks through the age-long pressure of the superincumbent 
 strata, and — in the biological sphere — the gradual adaptation, to a 
 new habitat, of a race of animals or plants. 
 
 The effects produced by both types of cause may be at least par- 
 tially non-cumulative. Thus, as a non-cumulative effect due to 
 
 * John Tyndall, ' Heat a .Moiic of .Motion,' ninth edition. Lecture IV., p. 95. 
 t ' A Text-book of Sound,' second edition, oh. v., p. 76.
 
 Chap. XLIII.] CAUSATION 375 
 
 th'? action of a total cause in which a large factor is periodic, we haro 
 vviiat Sir Robert Bill has called ' that stupendous Annus Mac,'au9 
 of hundreds of tliousands of years during wliich the Earth's orbit 
 itself breathes in and out in response to the attraction of the 
 planets.'* 
 
 As a non-cumulative effect due to the action of non-periodic 
 causes, we may instance the periodic undulations maintained on 
 the surface of the sea by the continuous pressure of a storm-wind ; 
 and all cases of the maintenance of equilibrium by a system of 
 mechanical forces might be taken as examples of this kind of effect 
 if we were to regard as negligible the progressive character of the 
 results of internal strain and pressure. 
 
 Composition of Causes and Intermixture of Effects. 
 
 Mill follows up his discussion of Cause as the totality of the con- 
 ditions by a more specific consideration of the case (in his view 
 ' almost universal ') in which ' two [or more] different agents, 
 operating jointly, are followed, under a certain set of collateral 
 conditions, by a given effect ' (Book III., cli. vi., § 1). The main 
 interest attaching to this more complex conception of Causation 
 lies in the nature of the effect which results from the co-operation 
 of the different agents. 
 
 We must first notice that, the effects of these several agents may 
 intermix or not. Where there is no Intermixture of Effects, the 
 effect of each of the separate causes is clearly distinguishable from 
 those of all tlie others, and can readily be disengaged from the total 
 effect. Two writers, pos^.essing completely different styles, may 
 co-operate in producing a book, and the handiwork of eacli may be 
 of so characteristic a quality as to be easily distinguishable within 
 the total effect due to the combined work of both. 
 
 The operating causes may, however, interfere with each other's 
 work, so that the result is Intermixture of Effects. But here Mill 
 makes an important distinction. The result of Intermixture may 
 be, with regard to the causal agencies which produce it, either 
 homogeneous or heterogeneous. It is homogeneous when the con- 
 current causes, whilst more or less modifying, or even counteracting, 
 each other's effects, still exert their full efficacy, each according 
 to its own law — its law as a separate agent. It is heterogeneous or 
 heteropathic when ' the agencies which are brouglit together cease 
 entirely, and a totally different set of phenomena arise : as in the 
 experiment of two liquids which, when mixed in certain proportions, 
 instantly become, not a larger amount of liquid, but a solid mass ' 
 (Book III., ch. vi., § 1). 
 
 Where the Intermixture of Effects produces a homogeneous 
 result, the co-operation of the causes which thus combine their 
 
 * 'Time and Tide.' p. 72.
 
 376 THE TROBLEM OF LOGIC [XIII. i. 
 
 individual effects is called by Mill a Composition. ' I shall give the 
 name of the v imposition of Causes,' he says, ' to the principle which 
 is exem]-)lified n. all cases in which the joint effect of several causes 
 is identical with the sum of their separate effects ' (ibid.). 
 
 In illustration of the Intermixture of Effects due to Composition 
 of Causes we may cite the movement of the Earth in space relatively 
 to the Sun. The distinguishable causes wliich contribute in pro- 
 ducing this single total effect are many in number. There is the 
 gravitative action of the Sun and of the Moon and of the various 
 planets, each of these bodies contributing definite items to the total 
 effect ; and there is also the Earth's own momentum in the direction 
 of the tangent to its orbit. 
 
 A more general illustration is that of the resultant effect pro- 
 duced by the composition of any forces acting according to 
 d>Tiamical laws. For instance, two forces, P, Q, acting on a particle, 
 and represented, in magnitude and direction, by the adjacent 
 sides AB, AC of a parallelogram ABDC, will cause that particle 
 to move in the direction of the diagonal AD ; and the position to 
 which the particle is eventually transported through the simul- 
 taneous action of these two forces, operating during a given time T, 
 will be precisely the same as that to which it would have been 
 brought had it been acted upon in succession first by the one force 
 and then by the other, each of them acting during a time T. 
 
 The mutual interference of homogeneous effects may, as a 
 limiting case, amount to complete Counteraction. Thus, if the two 
 forces F, Q are equal in magnitude, but act upon a jiarticlc in opposite 
 directions, the particle will remain at rest. Mill gives the following 
 illustration : ' A stream running into a reservoir at one end tended 
 to fill it higher and higher, while a drain at the other extremity 
 tends to empty it '; and he adds that ' even if the two causes which 
 are in joint action exactly annul one another, still the laws of both 
 are fulfilled : the effect is the same as if the drain had been open for 
 half an hour first, and the stream had flowed in for as long after- 
 wards ' (Book III., ch. vi., § 1). 
 
 In reference to this phenomenon of Counteraction, Mill makes 
 the important remark that laws are not broken even when the 
 causes whose mode of operation they attempt to explain, through 
 counteraction of their natural effects, apparently fail to act in accord- 
 ance with these laws. He refers to ' the popular prejudice that 
 all general truths have exceptions,' and to the fallacy underlying 
 this prejudice when it is made to bear on laws of Causation. ' There 
 are not a law and an exception to that law, the law acting in ninety- 
 nine cases, and the exception in one. There are two laws, each 
 po.ssibly acting in the whole hundred cases.' And again : ' What 
 is tliought to be an exception to a principle ... is always some 
 other and distinct principle cutting into the former ' (Book III., 
 <;h, X., § 5).
 
 Chap. XLIIL] CAUSATION 377 
 
 Mill proposes to remedy the confusion caused by this misinterpre- 
 tation of the meaning of scientific law by an appropriate use of the 
 expression ' tendency.' ' All laws of causation, in consequence of 
 their liability to be counteracted, require to be stated in words 
 affirmative of tendencies only, and not of actual results. . . . Thus, 
 if it were stated to be a law of Nature that all heavy bodies fall 
 to the ground, it would probably be said that the resistance of tiie 
 atmosphere, which prevents a balloon from falling, constitutes the 
 balloon an exception to that pretended law of Nature. But the 
 real law is that all heavy bodies tend to fall ; and to this there is no 
 exception, not even the sun and moon ; for even they, as every 
 astronomer knows, tend towards the earth, with a force exactly 
 equal to that with which the earth tends towards them ' (Book III., 
 eh. X., § 5). 
 
 2. Cause as the ' concurrence of antecedents ' on which an effect is 
 ' invariably and unconditionally consequent.'' 
 
 Mill is very firm in insisting on the point that the theory of 
 Induction requires no other notion of Cause than such as can be 
 gained from experience ; and by ' experience ' Mill means ' sense- 
 experience.'* He considers that our conviction of the invariability 
 of causal consequences is obtained by Induction from experience 
 so understood. It is important to notice that by ' invariable ' 
 Mill does not mean merely ' invariable so far as our experience has 
 gone,' or even ' unvaried within the limits of human experience.' 
 He saw quite clearly that invariability in tliis sense is not sufficient 
 to constitute any causal connexion. 
 
 ' The succession of day and night,' says Mill, is as much an in- 
 variable sequence as the alternate exposure of opposite sides of the 
 earth to the sun. Yet day and niglit are not the causes of one 
 another. Why ? Because their sequence, though invariable in 
 our experience, is not miconditionally so ; those facts only succeed 
 each other provided that the presence and absence of tlie sun 
 succeed each other, and if this alternation were to cease, we might 
 have either day or night unfoUowed by one another. There are 
 thus two kinds of uniformities of succession, the one unconditional, 
 the other conditional on the first : laws of causation, and other 
 successions dependent on those laws.'f To confuse these two is to 
 commit the fallacy of ' Post hoc, ergo propter hoc' 
 
 Thus the invariability of Mill's causal consequence is uncon- 
 ditional invariability. 
 
 In another passage Mill connects the idea of unconditionahiesa 
 
 * ' It may, therefore, safcl)' be laid down as a truth both obvious in itself, and 
 admitted by all whom it is at present necessary to take into consideration, that of 
 the outward world we know, and can know, absolutely nothing, except the sensa- 
 tions which we experience fiom it ' (Book I., ch. iii., § 7). 
 
 f ' Auguste Comte and Positivism,' second edition. Part I., pp. 57, 53.
 
 37S THE PROBLEM OF LOGIC [XIII. i. 
 
 with that of necessity. * If there be any meaning,' we read, ' which 
 confessedly belongs to the term "necessity," it is uiiconditioiialness. 
 That wliich is necessary, that which must be, means that which will 
 be, whatever supposition we may make in regard to all other things. 
 The succession of day and night evidently is not necessary in this 
 sense. It is conditional on the occurrence of other antecedents. 
 That which will be followed by a given consequent when, and only 
 when, some third circumstance also exists, is not the cause, even 
 though no case should ever have occurred in which the phenomenon 
 took place without it.'* 
 
 It follows at once from the passages quoted above that the 
 antecedent upon which an effect follows ' unconditionally,' in Mill's 
 sense of the word, must be the ' total ' antecedent. The partial 
 antecedent will always be dependent for its effectiveness on the 
 conditions that are lacking — i.e., on ' a third circumstance ' — and 
 therefore cannot be a ' cause ' according to Mill's definition of the 
 term. Hence the two reqai3ites of totality and unconditionalness 
 are essentially akin, and the question forcibly suggests itself whether 
 the two aspects of ' Cause ' that we have been considering — 
 (1) Cause as the sum total of the conditions, and (2) Cause as the 
 concurrence of antecedents on which an effect is invariably and 
 unconditionally consequent — may not event;ially prove to be but 
 two renderings of one and the same conception. 
 
 With a view to determining the relation between these two 
 aspects, let us once more consider what it is that ]\Iill understands 
 by the totaUty of the conditions. In the first place, having respect 
 to Mill's explicit statement that the cause he is considering is only 
 the ' physical ' and not the ' efficient ' cause, we must not press the 
 meaning of the expression beyond its relevance for strictly scientific 
 purposes. Again, the ' total ' antecedent might be understood to 
 include an endless chain of causes, for the immediate total ante- 
 cedent is itself dependent on a preceding totality of conditions, and 
 so backward ad infinitum ; for a first and unconditional cause would 
 never be reached. A completely sufficient reason, indeed, cannot 
 be given by Inductive Science for any of the effects that arise within 
 the endless chain of phenomena. But it is essential to add that Mill, 
 in his causal theor}^ makes no pretence of exhausting the signifi- 
 cance of the term ' Cause.' His sole aim is to interpret it in the way 
 most purposive for the true interests of inductive inquiry. ' I 
 make no research,' he says, ' into the ultimate or ontological cause 
 of anything ' (Book III., ch. v., § 2). 
 
 We can trace Mill's real meaning most clearly when we turn to 
 the sections in his Logic (Book III., ch. v., §§ 2 and 3) in which he 
 first states and develops his definitions of Cause. The idea which 
 Mill initially connects with the law of Causation is that of invariable 
 sequence. This conception, on ADll's view, is inductively reached as 
 
 * ' A System of Logic/ Book III., ch. v., § G.
 
 Chap. XLIIL] CAUSATION 379 
 
 <?, jijeneralization from experience — from observed facts of Xafurc. 
 * The only notion of a cause which the theory of induction requires,' 
 lie says, ' is such a notion as can be gained from experience ;' and 
 he adds : ' The Law of Causation, the recognition of wliich is tlio 
 main pillar of inductive science, is but the familiar truth that in- 
 variability of succession is foimd by observation to obtain betweei 
 every fact in nature and some other fact which has preceded it ' 
 (Book III., ch. v., § 2). We shall be concerned in a later chapti^-r 
 with the critical discussion of this view of the Law of Causation. 
 At present we are only concerned to notice that the notion of 
 invariable sequence is the idea wliich ^lill holds to be most essential 
 to the idea of the causal relation. 
 
 From this point Mill proceeds by an easy transition to connect 
 the idea of invariable sequence with the conception of Cause as the 
 total antecedent. ' It is seldom, if ever,' he writes, ' between a 
 consequent and a single antecedent that this invariable sequence 
 subsists. It is usually between a consequent and the sum of 
 several antecedents ' {ibid., § 3) ; and towards the end of this same 
 section we find liim specifically defining the Cause as ' the sum 
 total of the conditions positive and negative taken together ; the 
 whole of the contingencies of every description, which being realized, 
 the consequent invariably follows.' Thus, ' if a person eats of 
 a particular dish, and dies in consequence' — to take Mill's 
 own illustration — ' there cert<iinly is, among the circumstances 
 wliich took place, some combination or other on wliich death is 
 invariably consequent : as, for instance, the act of eating of the 
 dish, combined with a particular bodily constitution, a particular 
 state of present health, and perhaps even a certain state of the 
 atmosphere.' We see, then, that by the sum total of the con- 
 ditions which make up the Cause Mill means the sum of just those 
 antecedents with wliich the consequent can be connected by an 
 invariable law. But beyond tliis there is an ideal meaning of 
 causal connexion which forms no part of Mill's explicit doctrine. 
 The causal connexions which inductive inquiry must seek to establish 
 are reversible connexions. Not only must the antecedent be in- 
 variably followed by the consequent, but the consequent must be 
 invariably preceded by this same antecedent. Mill's belief in what 
 he calls 'plurality of causes' {vide p. 3S3), according to which 
 doctrine ' one fact may be the consequent in several invariable 
 sequences ' (Book III., ch. x., § 1), for ' many causes may produce 
 mechanical motion,' and ' many causes may produce death,' j)rc- 
 vents him from introducing the idea of a ' reversible connexion ' 
 into his definition of the causal relation. But Mill's own expres- 
 sions sometimes, though, as it seems, unintentionally, ]>oint us to 
 this deeper conception of the causal relation. Thus, in the ptvssage 
 (Book III., ch. v., § 2) in which he is insisting on the fact that a 
 cause must be invariably followed by the same consequent, he
 
 3S0 THE PROBLEM OF LOCxIC [XIII. i. 
 
 MTite« : ' The invariable antecedent is termed the cause ; the 
 invariable consequent the effect '; and adds : ' Let the fact be 
 what it may, if it has begun to exist, it was preceded by some fact 
 or facts with which it is invariably connected.' 
 
 ^lill's language here seems not to express his meaning quite 
 accurately. Were it not misleading to press the literal meaning 
 of the words, we might say that if a consequent is invariably 
 connected with an antecedent which is itself invariable — i.e., pre- 
 suniably, the invariable antecedent of the consequent — the con- 
 nexion must be reversible, and the causal antecedent must precede 
 the consequent, just as invariably as the consequent follows the 
 antecedent. 
 
 But when once the causal connexion is held to be ideally re- 
 versible, the meaning of the words ' total ' and ' unconditional * 
 is correspondingly modified. The ' totality ' of the relevant con- 
 ditions must mean that totality which is relevant to the ideal of a 
 reversible connexion. All other elements in the antecedent must be 
 eliminated as accidental. And it is, in fact, in this very direction 
 that Mill is feeling when, after pointing out that the sun, friction, 
 percussion, electricity, chemical action may all be causes of heat, 
 he adds that ' If, on further analysis, we can detect in these any 
 common element, we may be able to ascend from them to some one 
 cause which is the really operative circumstance in them all. Thu:i 
 it is now thought that in the production of heat by friction, per- 
 cussion, chemical action, etc. the ultimate source is one and the 
 same ' (Book III., ch. x., § 3). Again, in the light of the ideal of 
 reversibility we may give a nev/ meaning to the term ' uncon- 
 ditional.' For since antecedent and consequent in a reversible 
 connexion belong to each other as closely as they belong to 
 themselves, we may appropriately equate the two conceptions of 
 reversibility and unconditionality, and account a connexion uncon- 
 ditionally valid when it is reversible. It should, however, be borne 
 in mind that this unconditionality is not absolute, but relative to 
 the inductive point of view. The discussion in Chapter XLVI. 
 (below) should make this clear. 
 
 Again, in the light of the reversibility-ideal, we see more clearly 
 the importance of that requirement of immediacy which Science 
 is accustomed to attach to the connexion between Cause and 
 Effect. 
 
 One main distinction between the popular and the scientific 
 conception.s of ' Cause ' Ues in the indifference of popular usage to 
 this requisite of immediacy. The ordinary man, looking forward 
 rather than backward, is not interested in reversibility of causal 
 connexion. For practical purposes he often requires some view 
 ahead, and he is then opposed to any attempt to bring Cause ai.d 
 Effect so near together that this view ahead is lost. To screw 
 up Cause and Effect into close juxtaposition (to use Dr. Venn'a
 
 Chap. XLliL] CAUSATION 381 
 
 expressive phrase*) would be to secure certainty and reversibility 
 of connexion at the price of usefulness. Thus death is conceived 
 as the eifect of taking poison, ruined health as tlie effect of intemper- 
 ance, despite the long uncertain interval between cause and effect 
 in each case. We may grant tliat Dr. Venn is quite right in sup- 
 posing that the popular consciousness is perfectly'' ready to alio.-/ 
 any desired interval to elapse between cause and effect, provided 
 tliis implies practical gain. The ordinary man adapts his phrase 
 to his needs. But his interests are just as frequently furthered by 
 noticing the more nearty immediate effect as by emphasizing that 
 wliich is more remote. Tiuis the lighted match is causally con- 
 nected in the popular mind with the smoking tobacco and the 
 pleasant feeling that speedily ensues, the touching of hot iron with 
 a feeling of pain, and so forth. The practical man seems indiffer- 
 ently to emphasize either the most conspicuous after-effect or such 
 after-effect as happens at the moment specially to interest him. 
 It still remains perfectly true, as Dr. Venn points out, that the ' plain 
 man ' will not screw cause and effect tightly together in order to 
 ensure a degree of scientific certainty which is useless to liim. 
 
 But the main interest of Science in the matter is to secure at all 
 costs regularity and certainty, and, where possible, reversibility 
 of causal connexion. This entails a bringing of the two compo- 
 nents in the causal sequence, the causal antecedent and its con- 
 sequent, into such closeness of contact with each other as will 
 suffice for this purpose Now, ideal regularity, certainty, and 
 reversibility cannot be attained so long as tlie mind cannot grasp 
 and control every link in the chain that leads from antecedent to 
 consequent. If we ignore these intermediate links, we labour under 
 a disability very similar to that wliich we experience when we over- 
 look essential elements in the sum or totahty of the conditions. 
 As Mr. A. Sidgwick well observes ('The Process of Argument,' 
 ch. xi., p. 153), ' Intermediate links in a chain of causation are so 
 many opportunities for counteraction, in the same way as the 
 length of a piece of railway provides opportunities for an accident. 
 They are intermediate conditions. The i^ull on the trigger will fire 
 the shot if, and only if, the catch, the spring, the hammer, the cap, 
 and so on, all act u\ the expected manner. Therefore our forget ful- 
 ness of intermediate links takes effect just in the same way as our 
 forgetfulness of conditions generally ; it may give us a false security.' 
 Li the interests of vigorous induction it is essential to aim at brinEr- 
 ing cause and effect sufficiently close to exclude all intermediate 
 Unks of which we cannot render an intelligible accomit. Itisintiiis 
 sense that the inductive interest calls for sometliing approaching 
 to continuity between cause and effect. Tluis it would be unsatis- 
 factory to assign Clilorme as the cause of the bleaching of vegetable 
 dyes. It is true that, popularly speaking, Clilorine, added to 
 
 * ' The Principles of Empirical or Inductive Logic,' ch. ii., p. 50.
 
 382 THE PROBLEM OF LOGIC [XIII. L 
 
 vegetable dyes, bleaches them ; but it is not the Chlorine that is 
 the bleaching agent. The Clilorine, by uniting with the Hydrogen 
 of the water, liberates Oxygen ; and it is this nascent Oxygen which, 
 rciicting upon the colouring matter of the dye, forms a colourless 
 compound, and so eifects the bleaching. 
 
 The requirement of immediacy can be pressed only in so far as it 
 is called for in the interests of ensuring the unconditionahty and 
 reversibility of the causal sequence. In a chain of necessary con- 
 nexions any link may be held to be causally connected with any 
 other link, though many intermediate links may intervene between 
 the two. If, instead of being considered as a mere means for 
 ensuring unconditionalness and reversibihty, the requisite of 
 immediacy were treated as an end in itself, the only logical terminus 
 of such a view would be an insistence on the absolute simultaneity 
 or identity of cause and effect ; and indeed, as Mill points out, 
 ' Cessante causd cessat et effectus has been a dogma of the schools ' 
 (Book III., ch. v., § 7). 
 
 . With regard to tliis maxim, we would point out that, if it were 
 strictly true, there would never be any chain of causation at all ; 
 and that, though it is undoubtedly true that every cause expresses 
 itself in producing an effect, still, the direct effect, which is partly 
 contemporaneous with the action of the cause, need not be the 
 effect relevant to the scientific interest. All that Science requires 
 is that the connexion between the action of a certain cause and an 
 element in the succession of after-effecte shall be regular and cer- 
 tain, and, if possible, reversible. Any element, in the total ' after- 
 effect,' that is so cormected with the cause can in the truest scientific 
 sense be called an effect of the cause. As a matter of fact it is 
 usually very close to the cause itself. 
 
 \\'here the direct causal action lasts an appreciable time, it is 
 customary to distinguish in the causal process three clearly deGned 
 time-stages : 
 
 *o^ 
 
 (a) The time prior to tlie causal action (antecedent) ; 
 
 (6) The time when the cause is actually jjroducing an effect 
 
 (synclironous) ; 
 (c) The time following on the direct causal action (subsequent). 
 
 Example. — When A stabs B, the direct causal action, strictly 
 so called, is limited to the time in which the dagger is penetrating 
 the body and tearing the tissues. From this should be distinguished 
 that which precedes and that which follows the actual womiding : 
 the .seizing of the dagger, the movement of the arm, etc., on the 
 one hand ; the opening of the bloodvessels, the loss of blood, and 
 bLill remoter effects, possibly death itself, on the other.* 
 
 • Dr. Chri.stoph Sigwart, ' Logik,' vol. ii., pait iii., ch. i., § 73 : English Transla- 
 tion by Ifelf-n Dendy, vol. ii., p. 100.
 
 Chap. XLIII.] CAUSATION 3S3 
 
 We have already drawn attention to the logical relation between 
 the ' immediate ' and the ' unconditional '; and we have also seen 
 that the ' unconditional ' antecedent must be identified with the 
 ' total ' antecedent — i.e., with the sum of the relevant conditions. 
 The practical consciousness attaches as little value to the causal 
 requisite of totality as it does to the requisite of immediacy. It is 
 ' characteristic of the popular conception of Causation that it is 
 content to identify any single condition with the whole causal 
 antecedent, provided that condition is a sufficiently striking one. 
 In ordinary life we look out for causes, not primarily to gratify 
 intellectual curiosity, but chiefl}'' in order to produce some par- 
 ticular desired effect. We are anxious to pr«)duce ratlier than to 
 explain. Hence the discovery of any obvious antecedent con- 
 dition suffices us. As against this view. Science, as wo have seen, 
 insists on the identification of the Cause with the totahty of the 
 relevant conditions. 
 
 We have now to point out how the further insistence on the part 
 of Science that the effect can be justly estimated onJy in so far as 
 it is adequately analysed enables it to keep clear, at least theo- 
 retically, of the popular pitfall known as ' Plurality of Causes.' 
 
 Plurality of Causes. 
 
 By * Plurality of Causes ' we do rwt mean that a number of con- 
 ditions or partial causes cons'pire to produce a certain effect. We 
 mean tliat the same effect may at different times be produced by 
 different total causes. Plurality of causes means plurality of 
 possible causes, of which only one is actual in any given instance. 
 
 The doctrine of a plurality of causes asserts tliat the relation 
 between Cause and Effect is not necessarily reciprocal. It requires 
 invariability of connexion between Cause and Effect only in the 
 one sense, not in the other. The same cause may invariably pro- 
 duce the same effects, but, given the effect, we camiot with cer- 
 tainty argue back to the cause. 
 
 It has been maintained, and with justice, that this doctrine is 
 consistent with the popular, though not with the scientific, view of 
 Cause and Effect. It is quite true that ' death ' is an effect that 
 may be due to droAvning, strangling, overeating, etc. ; but this 
 ' death ' is a general and not a particularized effect. The effect is, 
 in fact, loosely defined, in accordance with the popular view, by 
 some one or two salient characters — characters common to the 
 eff'ects respectively produced by a large number of possible causes ; 
 Drowning, strangling, overeating may all cause death ; but (and 
 this is what the practical consciousness commonly forgets to notice) 
 the death wliich they cause in common is marked by differeni
 
 3S4 THE PEOBLEM OF LOGIC [XIII. i. 
 
 symptoms, according as it is this cause or that which lias been 
 operative, the syniptora being just part of the effect. ' There are 
 many causes of deatli,' as Dr. Mellone concisely puts it, 'only because 
 there are many kinds of death.'* 
 
 Professor Minto gives the following illustration : ' A man's body 
 is foimd dead in water. It may be a question whether death came 
 by drowning or by pre\'ious violence. He may have been suffo- 
 cated and afterwards thro\\Ti into the water. But the circum- 
 stances will tell the true storj\ Death by drowning has distinctive 
 symptoms. If dro^\Tling was the cause, water will be found in the 
 stomach and froth in the trachea.'f 
 
 We see, then, that, given a certain very general effect and nothing 
 further, it is impossible to tell to which of a given number of pos- 
 sible causes it is due ; but that, if we study the effect more closely, 
 and observe its more specific symptoms or marks, we are able to 
 point out the cause (c/. Minto, ibid., p. 343). 
 
 We conclude, then, that, if only the effect is given in all its detail, 
 there will never be more than one possible cause to account for it. 
 The relation between Cause and Effect is thus, ideally, reciprocal 
 or reversible. Each additional Ime in the delineation of an effect 
 may strike out one or more of the various conceivable causes, until, 
 when the whole effect is analysed out, all possible causes except 
 one are eliminated. It may happen that all known possible 
 causes are thus eliminated. In that case the cause remains 
 unkno^\^l. 
 
 But however exhaustively an effect be analysed and its remotest 
 ascertainable sj'mptoms studied, it cannot be traced back to non- 
 essential antecedents. If these were to be reckoned as part of the 
 antecedent of a causal sequence, then the causal relation would no 
 longer be reciprocal, and the plurality of causes would still sub- 
 sist. It is only when the total cause is regarded as consisting cx- 
 clu.sively of essential or effective conditions that it is possible to 
 specify the cause by diagnosis of the effect. E.g., a nail is driven 
 one inch into a piece of wood by a blow from a blue-coloured 
 hammer made of steel. Now, it will be driven in just as far and in 
 exactly the same way by a red-coloured hammer made of iron, 
 provided the weight of the two hammers is the same, and each 
 offers the same surface-resistance to the nail. Whenever the blue- 
 coloured steel hammer strikes with a certain momentum a nail of 
 a certain kind placed in a certain position, the nail will be driven j 
 one inch into the wood ; but the colour, and, in a more restricted ' 
 degree, the material, are non-effective conditions, and, as such, ^ 
 cannot enter into any reciprocal connexion with the effect, 
 
 * Dr. Sydney Herbert Mellone, ' An Introductory Text- Book of Logic,' first 
 e<lition. eh. viii., p. 200. 
 
 t • I^gic Inductive and Deductive,' p. 342. Cf. also Professor Carveth Read, 
 'Logic Deductive and Inductive,' third edition, ch. xiv., pp. 179, 180.
 
 CuAP. XLIII.] CAUSATION 385 
 
 The preceding discussion on the Plurahty of Causes has an im- 
 portant bearing on the theory of Deductive Inference in its liypo- 
 thetical form. For it shows that in proportion as our knowledge of 
 RcaHty becomes more accurate and comprehensive, the less do tlie 
 fallacies of affirming the consequent and denying the antecedent 
 imply any real error. We are, indeed, led to see that when hotlt 
 Consequent and Antecedent are precisely defined — i.e., defined with 
 a precision that is relevantly ideal — they mutually iniply each 
 other, so that when we deny the antecedent or affirm the conse- 
 quent no fallacy is committed. Thus, to quote from Professor 
 Stout : ' I can legitimately infer from the proposition " If a closed 
 figure is three-sided, it has angles " that if anything has not angles, 
 it is not a closed figure with three sides. The denial of the con- 
 sequent involves denial of the antecedent. But I cannot infer that 
 " If anything is not a closed figure with three sides, it has not 
 angles." I cannot do so because I have not precisely defined 
 the consequent. But when I do this in the proposition " If a closed 
 figure has three sides, it has three angles," the fallacy of denying the 
 antecedent is no longer a fallacy. 
 
 ' Take a different example. " If all vegetables were cabbages, 
 the vegetable I am eating would be a cabbage." Even if some 
 vegetables arc not cabbages, the one I am eating may still be a 
 cabbage ; but this is only because I have included more in the 
 antecedent than is necessary for it qua antecedent of the assigned 
 consequent. It is presupposed in the unity and identity of the 
 universe that the fact of the vegetable I am eating being a cabbage 
 has some precise antecedent statable in the form " If vegetables 
 with certain specific characters are cabbages." Given tliis precise 
 antecedent, I can argue from the denial of the antecedent to that of 
 the consequent.' 
 
 Quantitative Aspect of the Causal Relation. 
 
 For comprehensive generahzations, such as the principles of the 
 Conservation of Energy and of Mass, Science has the greatest 
 respect. But what is typical of scientific procedure is not that it 
 bears towards breadth of generalization — so, alas ! does ignorance — 
 but that it consistently refuses to reach comprehensiveness at the cost 
 of accuracy. Hence the acliieved reconcihation of the ' generiil ' 
 and the ' accurate ' which constitutes the essential triumph of 
 Science. As Science has become more comprehensive both in the 
 array of its data and in the scope of its generalizations, it has 
 pari passu advanced in accuracy and precision. But it is not its 
 comprehensiveness of spirit that has ensured the precision. On the 
 contrary, the precision, and that slowness of advance which pre- 
 cision demands, have alone made possible the vast airl fruitful 
 comprehensiveness. 
 
 25
 
 3SG THE PROBLEM OF LOGIC [XIII. i. 
 
 Precision is doubtless a fundamental requisite of all scientific 
 investigation. It is as essential to the success of the biologist or 
 the arcb»ologist as it is to that of the physicist or the mathematician. 
 But it is only witliin the strictly physical and mathematical sciences 
 that the numerical precision required for the quantitative treat- 
 ment of Causation can be said to be obhgatory. This is frequently 
 forgotten by the votaries of these abstracter sciences. Thus Sir 
 John Herschel, the astronomer, writes as follows : ' Numerical 
 precision ... is the very soul of science ; and its attainment affords 
 the only criterion, or at least the best, of the truth of theories, and 
 the correctness of experiments. . . . Indeed, it is a character of all 
 the higher laws of nature to assume the form of precise quantitaiive 
 statement.'* This statement is true of the physical sciences 
 only. 
 
 The need for numerical accuracy, where such is obtainable, is 
 excellently brought out by Dr. Sigwart in the following passage of 
 his ' Logic ' : ' If we are to speak accurately,' he says, ' we cannot 
 saj- that eating stiUs hunger and drinking thirst, for a mouthful or a 
 sip is no good ; nor can we say that arsenic kills or quinine reduces 
 fever, for it depends upon the dose ; it is inaccurate, again, to say 
 that common salt is dissolved hj water, for it is not true that any 
 quantity of salt is dissolved by any quantity of water.' j 
 
 The quantitative aspect of Causation is regulated by certain 
 Principles of Conservation, notably by the Peen'ciple of the Con- 
 SEEVATiox OF ExEEGY. This principle has been defined by Clerk- 
 Maxwell in the following terms : 
 
 ' The total ?nergy of any material system is a quantity which can 
 neither be increased nor diminished by any action between the parts 
 of the system, though it mav be transformed into anv of the forms 
 of which energj' is susceptible.' J 
 
 It is often stated as an axiom that the amount of energy in the 
 material universe is constant. This statement is far from being 
 the record of an ascertained fact. We may safely aver that no 
 phj'sicist has ever estabhshed an equation between the wliole energy 
 of the Universe at any time, including the energies of all the stars of 
 heaven and all the cells of all living bodies, and its energy at a 
 subsequent moment of time. The equation of constancy — of the 
 constancy, that is, of the energj^ of the L'niverse as a whole — is, in 
 fact, a most imjustifiable extension in indefinxtum of the empirically 
 established equation of equivalence in a hmited and closed system. 
 The fallacy involved in this extension is picturesquely expressed 
 by Dr. James Ward : ' Those who insist that the quantity of tliis 
 energy in the universe must be constant seem to me,' says Dr. Ward, 
 
 * Discourse on Natural Philosophy, §§ 11.5, 116, quoted by Professor Wehon, 
 ' A Manual of Lojric,' second impression, vol. ii., ch. vi., p. 161. 
 
 t ' Ix.gik.' Part III., vol. ii., ch. v.. § 95 : English Translation, pp. 346, 347. 
 
 ♦ Di. J. Oerk-Masrvrell, 'Matter and Motir.n.'^ch. v., p. 60.
 
 Chap. XLIIL] CAUSATION 3S7 
 
 * in the same position as one who should maintain tliat the quantity 
 of water in a vast lake must be constant merely because the surface 
 was always level, though he could never reach its shores nor fathom 
 its depth.'* 
 
 Moreover, this so-called principle of the constancy of the energy 
 of the Universe has not even the limited relative necessity of a 
 regulative principle of Physics. What guides the physicist in 
 forming his energy-equations is not the idea of the constancy of 
 energy within the universe, but that of the balance of energy alDout 
 any given change as fulcrum. The energy-level must remain con- 
 stantly the same ; or, in other words, the energy distributed within 
 the given system must remain, through all its redistributions, con- 
 stant in amount. The ' constancy of energy,' as a working Idea of 
 Physics, comes, indeed, to nothing more than tliis : Given a finite, 
 known quantity of energy, then, if that energy be measured after 
 any transformation, it must be precisely equivalent in amount to 
 the original quantity. In other words, lost energy can always be 
 found again, provided the precise amount lost is known. There is 
 no attempt to deal with the whole amount of energy in the universe 
 at any time, a perfectly indefinite, incalculable quantum. Just as 
 the ' postulate ' of the indestructibility of matter is really nothing 
 more than the balance of weights before and after a chemical 
 change, so that of the indestructibility of energy is nothing more 
 than the mathematical balance between the capacity for work 
 within a certain closed system, before a certain amount of actual 
 work is done, and the capacity for work witliin the same closed 
 system after the transformation has taken place, f 
 
 The numerical equivalency between Cause and Effect required 
 by the principle of the Conservation of Energy can be ascertained 
 only when the amount of each causal agency and the total effect can 
 be measured in terms of one and the same unit. In the case of the 
 physical sciences such a unit has been found. It has been found 
 possible to state the amomit contributed by each physical agent 
 in an act of causation in the form of its mechanical equivalent 
 expressed in terms of the amount of work done in lifting against 
 gravity, from the sea-level, a certain specified weight to a certain 
 specified height. 
 
 Let E be a certain effect measured in terms of the unit U, and 
 represented, let us say, by 47 U. Let C^, Cg, Cg . . . be the causes 
 or agents known to be involved in producing E. Let the amount 
 of work contributed by each be measured, as before, in terms of U. 
 Let Ci= -hS U; C2= -f 20 U; C3= -i- 15 U. Then Ci-f-Co-hCg^ 
 -I- 43 U. We infer, therefore, on the ground of the principle of the 
 Conservation of Energy, that we have not j^et found all the causes ; 
 
 ♦'Naturalism and Agnosticism' (Gifford Lectures, 1S9G-189S), vol. ii., 
 Lecture XIII.. pp. 76, 77. 
 
 t Cf. • Personal Idealism : Philosophical Essays,' edited by H. Sturt, pp. 151-155. 
 
 25—2
 
 38S THE PROBLEM OF LOGIC [XIII. i. 
 
 but that causes whose combined capacity for work is equivalent to 
 + 4 L' still remain to be discovered. 
 
 With regard to the principles of the Transformation and Conser- 
 vation of Energy in their application to Causal Explanation, we may 
 note the following considerations : 
 
 L Our knowledge that energy is not lost, but transformed or 
 redistributed, helps us in discovering the factors that make up 
 cause or effect, as the case may be. It used to be thought — by 
 Newton, for instance — that energy lost in friction was lost for 
 good. Xow it is known to be transformed into heat and sound. 
 Moreover, as the following examples will show, a causal explanation 
 will often consist in noting what transformations actually do take 
 place. 
 
 Examples : 
 
 (i.) Question. — Why can a man endure the heat of an oven in 
 wliich a beefsteak is cooked ? 
 Answer. — Because he perspires freely, and thus a large part 
 of the heat which enters his body is used in 
 the performance of a certain saving me- 
 chanical work, while only a fraction of the 
 total amount is effective in raising the 
 temperature. ' The excess of heat, instead 
 of being applied to increase the temperature 
 of the body, is applied to change its aggrega- 
 tion ; the heat prepares persjiiration, forces 
 it through the pores, and vaporizes it. Heat 
 is thus consumed in work.'* This is the 
 fact that makes a Turkish bath endurable. 
 
 (ii.) Question. — Why does the Safety-lamp used in coal-mines 
 prevent explosions ? 
 Answer. — Because, though the explosive gas may ignite 
 within the lamp, yet, when the combustion 
 reaches the close-knit wire gauze, a large 
 part of the energy of the burning gas is 
 transformed into the vibrational energy of 
 the gauze, and thence into the vibrational 
 energy of the heavy metal of the lamp. The 
 conducting power of the gauze lowers the 
 temperature of the gas, thereby stopping 
 the progress of combustion. Hence the 
 flame is kept within the lamp, and does not 
 come into contact with the explosive gas 
 outside, f 
 
 • John Tyndall, ' Heat a Mode of Motion,' ninth edition, Lecture IX., p. 243. 
 t John Tyndall, ibid.. Lecture IX., pp. 2G1-263.
 
 Chap. XLIIL] CAUSATIOX 3S9 
 
 2. When once tlie factors in an effect are discovered in accordance 
 with our knowledge of the transformation of energy, the quantum 
 of effect contributed by each factor can be estimated in accordance 
 with the theory of mechanical equivalence. In this way all the 
 various quota are measured in terms of one and the same unit. 
 
 3. Finally, by applying the principle of the Conservation of 
 Energy we are able to test the adequacy of our explanations. The 
 capacity for work contained in the effect must be the same as the 
 capacity for work contained in the cause, the work done consisting 
 in every case of some transformation or redistribution of energy. 
 
 Transition to the Method of Causal Explanation. 
 
 In the present chapter, relying mainly on ^lill's guidance, we have 
 attempted to reach a concept of ' Cause ' that would provide a 
 secure basis for the application of methods of Causal Explanation. 
 The outstanding conclusion is that the ideal after which we should 
 aim in Causal Explanation is to reach reversible relations between 
 Cause and Effect. It would, therefore, seem to follow that the 
 method which the interest in Causal Explanation requires is one 
 that will adequately fulfil tliis fundamental requirement. Further, 
 since the ideal of Reversibihty — or of Unideterminism, as it is 
 sometimes called — remains unrealized so long as the defect of a 
 plurality of causes still subsists, it must be the primary aim of a 
 Causal Method through the precision of its procedure to eliminate 
 this radical defect. 
 
 CHAPTER XLIV. 
 
 XIII. (ii.) THE PROCESS OF SCIENTIFIC OBSERVATION. 
 
 The goal of Observation, we may say, is to see a fact as a fact-under- 
 law. Xow, a closer analysis of what is involved in such Observa- 
 tion brings into relief its essentially experimental character, and 
 convinces us that, if the facts are to exercise their full natural effect 
 upon the mind, they must be subjected to an analysis of an essen- 
 tially experimental kind. Through the very pressure of its o\vn 
 interest Observation develops into a process of experimental 
 Anatysis that can rest satisfied only when its experimental tests 
 have brought out the law wliich interprets the fact. 
 
 Observation, therefore, cannot be adequately conceived as a mere 
 preliminary stage in inductive procedure ; for the principle of 
 Fidelity to Relevant Fact, which actuates and inspires all scientific 
 observation, requires that the whole induction, from first to last.
 
 390 THE PROBLEM OF LOGIC [XIII. ii. 
 
 shall be under the control of the relevant facts. When Induction 
 is conceived as explicitly dominated by this Inductive Idea, we see 
 the observational interest predominant all through. Observation 
 becomes a pcnnaneJit factor in inductive inquir\', and its interest — 
 the interest in deahn^ faithfullv with fact — the controllins; interest 
 in the whole process. 
 
 This view, of course, implies a deepened view of the meaning of 
 Observation. It imphes that Observation is the expression of the 
 observational Interest and Purpose. Thus it is no mere inspection 
 of the senses, but, as Descartes clearly saw, an inspection of the 
 mind ; it is not a process of mere sensation, but a process of pur- 
 posive perception, in whose service are engaged all the constructive 
 powers of scientific Thought and Imagination. 
 
 We are only expressing the same conviction in another form when 
 we add that Scientific Observation is essentially an active process. 
 It is not a mere passive submission of the mind to its object, but an 
 active attempt to see that object in the Ught of some dominant 
 interest, and to bring it into touch with the body of our previously 
 acquired knowledge. Were we content to allow the object to stamp 
 itself in upon our minds, we, as a writer has tersely put it, should 
 simply be stamped out. 
 
 The characteristics of this activity of Scientific Observation are 
 (1) its purposiveness, and (2) its experimental character. 
 
 (1) True Scientific Observation is essentially Purposive, 
 
 For Observation to be effective, we must first know what is the 
 aim or purport of our observation, and we must steadfastly adhere 
 to the lines marked out by this purpose. It is of no use to observe 
 ' in general ' or at random. We must select M^hat is relevant to 
 our interest. In a book entitled ' A Chapter of Science ' (p. 19) 
 3Ir. Stuart has an illustration in point. ' Many people,' he says, 
 ' took observations of the meteor display on November 13, 1866. 
 Some thought that their colour was the important point, and 
 noted the colour of each they could see. Others thought that the 
 rate they seemed to move at was the important point, and noted 
 that. Others thought that the points they disappeared at was the 
 important point to note, and noted that. Others thought that the 
 direction of their motion was most important, and noted that ; 
 and others — and these the best informed — thought that the most 
 important thing to note was the points at which they made their 
 appearance, and noted that.' The writer then goes on to remark 
 that, though the observations of the radiant point were of greatest 
 scientific value, seeing that they served to fix the position of the 
 swarm — the result most especially desired — yet all the other sets 
 of observation, restricted as each was to some one aspect of the 
 whole phenomenon, were serviceable in a subordinate way. On
 
 Chap. XLIV.] OBSERVATION 391 
 
 the other hand, he adds : ' It is very improbable that anything 
 could be learned from the observations of a man wlio noted of one 
 meteor what was its colour, of another the point at which it appeared, 
 of a third that at which it disappeared, of a fourth, its size, and so 
 on ; from such a man's observations there is very little chance that 
 we should discover anytliing common to all the meteors.' 
 
 In this connexion it is extremely important to distinguish between 
 the relevant and the favourable. True selective interest fastens 
 faithfully on the relevant instances, whether these be favourable 
 to preconceived ideas or not. The failure to give the relevant 
 priority over the favourable constitutes what has been called the 
 fallacy of ' Non-observation,' or of ' overlooking what ought to be 
 observed.' ' It is exceedingly rare,' remarks Jevons,* ' to find 
 persons who can with perfect fairness estimate and register facts 
 for and against their own peculiar views and theories. Among 
 uncultivated observers the tendency to remark favourable and 
 forget unfavourable events is so great that no reliance can be 
 placed upon tlieir supposed observations. Thence arises the enduring 
 fallacy that the changes of the weather coincide in some way or other 
 with the changes of the moon, although exact and impartial registers 
 give no countenance to the fact. The whole race of prophets and 
 quacks live upon the overwhelming effect of one success, compared 
 witli hundreds of failures which are unmentioned and forgotten. 
 As Bacon says, " Men mark when they liit, and never mark wlien 
 they miss." We should do well to bear in mind the ancient storj", 
 quoted by Bacon, of one who in Pagan times was sho^^ni a temple 
 with a picture of all the persons who had been saved from sliip- 
 wreck, after paying their vows. When asked whether he did not 
 now acknowledge the power of the gods, " Aye," he answered, 
 " but where are they painted that were drowned after their 
 vows ?" ' 
 
 Among the rare persons of strict impartiality to whom Jevons 
 here alludes must be included Charles Darwin. ' The success of 
 the " Origin," ' he writes, f ' may, I think, be attributed in large 
 part to my having long before written two condensed sketches, 
 and to my having finally abstracted a much larger manuscript, 
 which was itself an abstract. By this means I was enabled to 
 select the more striking facts and conclusions. I had also, during 
 many years, followed a golden rule — namely, whenever a published 
 fact, a new observation or thought, came across me, wliich was 
 opposed to my general results, to make a memorandum of it without 
 fail and at once ; for I had found by experience that such facts 
 and thoughts were far more apt to escape from the memory than 
 favourable ones. Owing to this habit, very few objections were 
 raised against my views which I had not at least noticed and 
 attempted to answer.' 
 
 * 'The Principles of Science,' vol. ii., p. 5. t 'Autobiography,' p. 87.
 
 392 THE PROBLEM OF LOGIC [XIII. ii. 
 
 It need not be deliberate bias which causes us to neglect these 
 instances. Our non-observation of them may be due simply to 
 the psychological law according to which we tend to perceive only 
 such objects as fit in with the idea that is filling our mind. ' The 
 only things,' saj's Professor James, ' which we commonly see are 
 those which we pre-perceive ;'* and it is a well-known fact that to 
 have an idea of the object we wish to observe is the surest guarantee 
 of our seeino; it. 
 
 o 
 
 Example. — (1) Think of a square lamp-shade viewed from above, 
 and then look at the accompanying figure. 
 
 (2) Think of looking down a long picture-gallery or ball-room, 
 and then look again at the given figure. 
 
 It will probably be found that having idea (1) in mind prevents 
 us from regarding the figure as a representation of (2), and that 
 having idea (2) in mind prevents our regarding it as representing (1). 
 
 Purposive observation must be appropriately and adequately 
 amilytic. ' The observer,' saye Mill, " is not he who merely sees 
 the thing which is before his eyes, but he who sees what parts that 
 thing is composed of.'f So Lord Avebury reminds us that we have 
 not really seen a mountain until we have seen through it. How 
 an observed fact is thus mentally split up into parts or aspects will 
 depend on the purpose of the investigation. 
 
 Observation inc-ludes not only perceptions, but also the interpre- 
 tation of perceptions. But the observer must very carefully guard 
 against confusing these two elements, more especially if the iuter- 
 jiretation is merely conjectural. 
 
 An appeal to tlie testimony of the senses has logical value in 
 proportion as, in observation, we distinguish between mere state- 
 ment of fact as observed and conjectures drawn from the observed 
 facts — i.e., in proportion as we accurately and pointedly describe 
 what actually appeared, keeping to essentials, and avoiding any 
 mental confusion between what did happen, on the one hand, and 
 what we feel ourjlit to have happened, or what we should like to have 
 happened, on the other. In reporting an experience we tend to 
 relate not what we have actually seen or heard, but rather the 
 impression wliich the events experienr.ed have made upon us. We 
 confuse fact, not with opinion only, but also with feeling. 
 
 * 'The Principles of Psychology,' vol. i., ch. xi., p. 444. 
 t J. S. Mill, 'A System of Logic,' Book III., ch. vii.. § 1.
 
 Chap. XLIV.] OBSERVATION 393 
 
 When the conjecture or interpretation is false, the fallacy takes 
 the form of Mal-ohservatioii, the fallacy of wrongly interpreting 
 what we immediately perceive through our senses. 
 
 In Mal-observation it is not the senses that are wrong. Thus, 
 suppose we plunge our hands, one into ice-cold water, the other 
 into very hot water, and then dip both into water that is tepid. 
 Tile one hand will feel a sensation of heat, the other a sensation of 
 cold. There is no mistaking the sensations. I\Ial-observation 
 would come in were we to interpret these sensations by the state- 
 ment that the water must be warmer at the one spot than at the 
 other. Again, when our eyes seem to tell us that the sun rises, 
 passes to its highest point, and sets, while our motor-sensations 
 seem to assure us that the earth is at rest beneath us, we make a 
 mal-observation if we interpret our sensations by saying : ' The 
 earth is at rest, and the sun goes round it ;' for it does not follow 
 from oxir being at rest relatively to the earth that the earth itself 
 is at rest. Hence by the fact that we feel no motion we are not 
 justified in recording that the earth is at rest. It may be either at 
 rest or in motion, for aught that observation can tell us. In order 
 to avoid mal-observation in tliis particular instance, we must 
 record our observations as follows : ' I feel that the earth is at 
 rest, and I can see the sun in daily movement from East to West ; 
 but whether these perceptions are illusions or not onlj^ systematic 
 Science can decide.' For interpretation of our perceptions we must 
 appeal from Sense to Reason.* 
 
 (2) True Scientific Observation is essentially Experimental. 
 
 To see a fact scientifically is to see it in the light of an idea. 
 Observation expresses a form of scientific endeavour which can be 
 satisfied only when the fact is observed as fulfilling some law. The 
 merely passive assimilation of fact is only the first stage in the com- 
 plete process of Observation. Through such absorption of sense- 
 data the mind develops suggestions which enable it to observe with 
 new interest and efficacy. The facts, we say, are seen in a new 
 light. We aim now at ascertaining whether the susrfrestion fits the 
 fact. The centre of observational interest is transferred from 
 passive watcliing to active testing, and the observation inevitably 
 takes an experimental form. 
 
 Observation, we repeat, as animated by scientific purpose, is a 
 process whose natural end remains unattained so long as the fact 
 is not clearly seen as obej'ing law. Hence the experiments to which 
 we subject facts enter as an integral 'part into every purposive 
 observation. It is, of course, not mere manipulation of the object 
 
 * On Mal-observation, cf. Berkeley, the ' Third Dialogue between Hylas and 
 Philonous,' pp. 455. 456 : and for a good illustration of the same fallacy, see Pro- 
 fessor Royce'a ' Psychology,' p. 27.
 
 394 THE PROBLEM OF LOGIC [XIII. ii. 
 
 that marks the difference between the passive and exerimpental 
 stages in Observation. We may observe just as passively with our 
 fingers as we can with our eyes or ears. Observation is passive just 
 so long as it is purely assimilative. But once the manipulation 
 passes into the service of an idea, and we proceed to put our object 
 under such conditions as the idea requires for t-esting and verifying 
 itself, our observation is no longer merely passive, but experimental. 
 A definite, purposive interference of some sort with the production 
 of the phenomenon to be noted is the desideratum of the experi- 
 mental or testing stage. 
 
 Where this interference is impossible — i.e., where we are unable 
 to place our object in contexts that answer to the requirements of 
 our idea — experimenting takes place under difficult conditions. 
 We have to wait till Nature presents us our object under the desired 
 conditions. But even then it is almost certain that Nature will 
 only very roughly satisfy our requirements, and we may have to 
 wait a long time. Nature, again, may present the right kind of 
 instance, but in too strong a degree. For delicate electric investiga- 
 tions the electricity in a thunder-cloud will not do. Or it may be 
 in too weak a degree, as in the production of heat by gentle friction, 
 when a furnace-heat is required. 
 
 Control over the conditions under which the object shall 
 present itself for study is the great desideratum, and the factor in 
 experimental observation which most influences the 'precision and 
 certainty of the results obtained. Such control enables us to vary 
 the circumstances, and to repeat the observation as often as we 
 please by simply reproducing the concurrence of conditions requisite 
 for bringing into play the desired effect. 
 
 In certain sciences, as in Astronomy, such control is out of the 
 question. The Astronomer must test his theories concerning the 
 ' canals ' in Mars or the belts of Jupiter whenever the conditions 
 happen to be favourable, the planet in opposition, the weather 
 fine. Occasionally, however, Naturo contrives the very condi- 
 tions that the idea requires for its verification. We then have a 
 ' Natural Experiment.' As an excellent illustration of this, we 
 may take that noticed by Dr. Thomas Fowler * — the observation 
 of prominences during a total solar eclipse. Here, under normal 
 conditions, the observer is dazzled by the excess of light, and the 
 question as to the nature of the prominences cannot be answered ; 
 but when the moon appropriately obscures the sun's disc, we have 
 ' experimental ' conditions under which the answer is obtainable. 
 
 * ' The Elements of Inductive Logic,' sixth edition, pp. 50, 51.
 
 Chap. XLV.] CAUSAL METHOD 395 
 
 CHAPTER XLV. 
 
 XIII. (iii.) THE METHOD OF CAUSAL EXPLANATION. 
 
 Where explanation is causal, fact, as we have seen, means efject, 
 and the problem lies (1) in discovering the probable causes or con- 
 ditions upon which the effect depends, and the laws according to 
 which they are operative ; (2) in testing the actual efficiency of these 
 causes or conditions. We must be able to trace effects back to their 
 determining conditions, and justify our surmises as to the nature 
 of these conditions by working back with explanatory power from 
 causes to effects. 
 
 The procedure in Causal Explanation is, in its general outline, 
 dictated in advance by its inevitably inductive character. The 
 Principle of Fidelity to Relevant Fact requires that, in our ex- 
 planation, we shall pass from fact back again to fact ; that we 
 shall not be content tentatively to trace effects to causes, but that 
 we shall also argue back from causes to effects. Preliminary 
 observation of effects requires as its inductive complement experi- 
 mentation upon causes. Hence, though it is customary to speak 
 of Methods of Causal Enquiry, and to divide these into Methods of 
 Observation and Methods of Experiment, yet, when we look at the 
 matter synthetically, in the light of our inductive principle, we see 
 that the r\Iethods in question cannot be other than co-operative in 
 the task of a complete Explanation, and that the so-called ' Obser- 
 vational Methods ' cannot be more than preliminary to the Methods 
 which test the hypotheses built upon these observations. Our 
 main endeavour, therefore, will be to bring out the unity of the 
 one Method of Causal Explanation and the interdependence of its 
 various component stages. 
 
 A Method of Pure or Simple Observation, in so far as it con- 
 cerns the problem of Causal Explanation, may be diagnostically 
 defined as a method which proceeds from effects to causes ; and a 
 Method of Experiment may be similarly defined as a method which 
 proceeds from causes to effects. 
 
 I. Purely Observational IMethods. 
 
 The typical form of Purely Observational ^Method has been 
 kno^vn since Mill's day as the I\Iethod of Agreement. He enunciates 
 it somewhat formally, as follows : 
 
 ' // two or more instances of the phenomenon under investigation 
 have only one circumstance in common, the circumstance in which 
 alone all the instances agree is the cau^e {or efject) of the given phe- 
 nomenon.''* 
 
 * ' A System of Logic,' Book III., ch. viii., § I.
 
 396 THE PROBLEM OF LOGIC [XIII. iii. 
 
 In practical application, the term ' circumstance,' occurring in 
 tliis enunciation of the Canon of the Method of Agreement, as given 
 by !Mill. must be replaced by the expression ' relevant and important 
 circumstance.' 
 
 The requirement that the instances shall have only one relevant 
 and important circumstance in common makes it imperative to 
 ascertain that all the other relevant circumstances are different. 
 Hence the method has been called the Method of Single Agreement, 
 A method of mere agreement could be founded on simple enumera- 
 tion only ; a method of Single Agreement must be founded on 
 elimination as well, for the singleness is established onlj^ through 
 eliminating all the circumstances that arc not common to all the 
 instances in question. 
 
 By way of illustrating this purely Observational Method, it may 
 be sufficient to quote the following example from Professor Read, 
 who has it from Professor Bain : 
 
 ' Let me begin by borrowing an example from Professor Bain 
 ("Logic," Book III., ch. vi.). The North-East wind is generally 
 detested in tliis country : as long as it blows, few people feel at their 
 best. Occasional well-known causes of a wind being injurious are 
 violence, excessive heat or cold, excessive dryness or moisture, 
 electrical condition, the being laden with dust or exhalations. 
 Let the hj^jothesis be that the last is the cause of the North-East 
 wind's unwholesome quality ; since we know it is a ground current 
 setting from the pole toward the equator and bent westward 
 by the rotation of the earth ; so that, reaching us over thousands 
 of miles of land, it may well be fraught with dust, effluvia, and 
 microbes. Now, examining many cases of North-East wind, we 
 find tliat this is the only circumstance in which all the instances 
 agree : for it is sometimes cold, sometimes hot ; generally dry, but 
 sometimes wet ; sometimes light, sometimes violent, and of all 
 electrical conditions. Each of the other circumstances, then, can 
 be omitted without the North-East wind ceasing to be noxious ; 
 but one circumstance is never absent — namely, tliat it is a ground 
 current. That circumstance, therefore, is probably the Cause of 
 its injuriousness.'* 
 
 The fact that this method is essentially a method of elimination 
 explains the fact that the force of the method depends on the 
 number and variety of the instances ; for the greater the number 
 of varied instances, the greater will be the chance of successfully 
 eliminating the circumstances that are not common to all instances 
 of the phenomenon under investigation. Thus Dr. Mellone (follow- 
 ing the lead of Professor Henry Laurie,t) states Mill's First Canon 
 as follows : 
 
 * Carveth Read, ' Logic Deductive and Inductive,' third edition, ch. xvi., 
 p. 198. 
 
 t 'Methods of Inductive Inquiry,' Mind, New Series, vol. ii. (1893), p. 321.
 
 Chap. XLV.] CAUSAL METHOD 397 
 
 ' When observation shows that two events accompany one another 
 (either simultaneously or in succession), it is probable that they are 
 causally connected ; and the probabihty increases with the number 
 and variety of the instances.'* 
 
 Now, as Dr. Thomas Fowler has pointed out.f it is through this 
 elimination (with its accompanying proccs.ses of analysis) that the 
 Method of Single Agreement — which itself proceeds by enumeration 
 of instances — essentially transcends the limitations of the un- 
 analytic Method of Simple Enumeration. But this is not the only 
 essential difference between the two methods as we regard them. 
 The Method of Agreement is frankly and confessedly a merely 
 heuristic method, proceeding tentatively from effects to causes, 
 and is in no sense a Method of Verification, whereas the Method of 
 Enumerative Induction (as we have formulated it) claims to be a 
 complete Inductive Method which aims at the verification-ideal of 
 a Complete Enumeration. 
 
 The essential limitation of the Method of Agreement, as Mill 
 himself has clearly shown, J is the uncertainty due to the Plurality 
 of Causes. We call it a limitation, and not a defect, since the 
 method, being purely observational, cannot claim more than a 
 heuristic function. It is frankly ' a mode of reconnoitring,' sug- 
 gesting well-grounded starting-points for experimental inquiry, 
 but making no pretence at causal explanation. 
 
 To illustrate the limitation in question, let us suppose that we 
 have two instances which we may somewhat artificially, but still 
 conveniently, symbohze as follows : 
 
 Antecedents.-! .' -p^ ^„ J '/ Consequents. 
 
 [A, ±> , b X, y , z .j 
 
 As Mill points out, it is not certain that A, the circumstance 
 in which the antecedents agree, is the cause of the phenomenon 
 X. For X — so long as the limitation due to Plurality of Causes 
 holds good — is not necessarily produced on each occasion by the 
 same cause ; for all we can say, it may be due to B in the first 
 instance, and to B' in the second. Thus, if we were to give to each 
 of half a dozen persons a glass of water to drink, having poured 
 into each glass a different poison, it could hardly be said to be the 
 common element — namely, the water — which caused the death of 
 those six persons. 
 
 It is true that the uncertainty of tlie method due to this limitation 
 is greatly reduced by increasing the number and variety of tlie 
 instances ; but the limitation is intrinsic to the method, being rooted 
 m its merely observational and heuristic character. 
 
 If it is objected that the difficulties due to PluraUty of Causes 
 
 * S. H. Mellono, 'An Introductory Text-Book of Ix>gic,' ch. is., p. 271. 
 t ' The Elements of Inductive Logic.' sixth edition, ch. iv., p. i^J. 
 : ' A System of Logic,' Book III., ch. x.. § 2.
 
 39S THE PROBLEM OF LOGIC [XIIL iii. 
 
 may be met by making the observations less enumerative and more 
 analytic in character, with a view to describing the nature of an 
 effect with the greatest possible precision {vide p. 384), the reply 
 must be that the results of such precise analysis can be utilized 
 for the purpose of eliminating plurality of causes only when an 
 adequate knowledge of causal connexions has already been gaineji 
 through precise applications of Experimental Methods. Suppose 
 that E is the effect in question, and that it is described with all 
 desirable precision. The suggestion is then made that E may be the 
 effect either of Cause C^, or of Cause C,, or of some other cause. This 
 suggestion that E may have a plurality of causes can be met only 
 in so far as we are able to show that C^ cannot be the cause of E, 
 because C^ has a known constant effect, E^ ; and, again, that Cj 
 cannot be the cause of E, because Cg has a known constant effect, 
 E^, and so forth. But these assertions presuppose an accurate 
 knowledge of the causal action of C^, C2, etc. It may be perfectly 
 true that a Method of Agreement in conjunction with Experimental 
 Methods may, by a sufficiently precise delineation of the effects 
 from which it starts, get beyond any practical annoyance from 
 plurality of causes ; but the Method of Single Agreement, by itself, 
 as a Method of Pure Observation, remains permanently at the 
 mercy of a possible plurality of causes. For the overcoming of 
 this limitation we must look to the experimental stage of the 
 complete Causal Inquiry. 
 
 But before we pass on to the consideration of the methods dis- 
 tinctive of this later stage, we must notice an important modifica- 
 tion of the Method of Agreement, which, while it strengthens the 
 Method, does not deprive it of its purely observational character. 
 The Method in question has had various names given to it, but the 
 ' Method of Double Agreement ' expresses its real nature most 
 simpl}^ and aptly ; for, as Mill liimself points out, its use involves a 
 double employment of the Method of Agreement. We start, in this 
 method, from two sets of instances : (1) from instances in Avliich 
 the phenomenon E, to be causally explained, is present ; (2) from 
 instances in which tlie same phenomenon is absent. Applying the 
 ]\Iethod of Agreement to the first set of instances, we look out for 
 that relevant circumstance, C, which alone is present in all the 
 instances we start from ; and then, applying the Method once again 
 to the instances in which the phenomenon in question is absent, we 
 examine whether, in all these cases of the absence of E, C also is 
 absent, and is the only relevant circumstance invariably absent. 
 If we find that it is so, the suggestion furnished by the first applica- 
 tion of the Method, that C is the cause of E, is very consider- 
 ably strengthened. 
 
 In illustration of the Method of Double Agreement, we adapt the 
 following from Professor Carveth Read's ' Logic ' :* 
 * Third edition, ch. xvi., pp. 204, 205.
 
 Chap. XLV.] CAUSAL METHOD 399 
 
 Elaborating an illustration suggested by Dr. Fowler,* Professor 
 Read supposes the case of a man who, having a taste for cucumber, 
 attributes his chronic indigestion now to the salmon, now to the 
 cheese, now to the pastry — no one of which is an invariable feature 
 of his evening meal — but never to the cucumber, which he takes 
 every evening. However, after having dined without cucumber 
 on several consecutive evenings, whilst taking salmon one evening, 
 pastry the next, and so on through the whole list of suspicious 
 dishes, he cannot but notice that on no one of these evenings did 
 any indigestion occur, and is thus brought to confess that the cucum- 
 ber, after all, must have been the offending cause. 
 
 Let us briefly analyse this example, so as to see precisely how 
 this double apjDlication of the method excels the single application 
 in logical strength. We see that, so far as the single application is 
 concerned, the man cannot logically be silenced when, having 
 declared that on the evening when he partook of salmon as well 
 as cucumber the cause of his indigestion was ths salmon, he goes 
 on to affirm that, wore he to take salmon every evening, he would 
 always get indigestion from so doing. There is, in fact, nothing to 
 disprove the suggestion that salmon, pastry, etc., are each and all 
 causes Mhich, when present in a certain organism, invariably 
 produce indigestion. It is this suggestion that they are constant 
 causes of the phenomenon in question which is mere or less con- 
 vincingly refuted by the second application of the method. The 
 salmon is taken without the cucumber ; so is the pastry ; so also 
 is the cheese ; and in none of these three eases does indigestion occur. 
 It is true that on these three occasions the tendenc}^ of the sus- 
 pected dishes to produce indigestion may have been diverted into 
 some other form of harmful influence, or counteracted by some other 
 article of diet, such as pineapple, which is supposed to aid digestion ; 
 the conjecture that salmon, pastry, and cheese constantly tend, 
 when eaten, to cause indigestion may still remain unrefuted. But 
 it has been clearly shown that there are occasions on which they 
 do not actually cause it ; and in this way, if the instances of absence 
 are sufficiently varied, the presumption that, in the case of the 
 patient in question, the only constant cause of indigestion is the 
 eating of cucumber may become very strong indeed. 
 
 It is, however, important to notice what is not proved through 
 this second aj^plication of the metliod. It is not proved that the 
 salmon maj- not really have at least helped to cause the indigestion 
 on the first evening, nor that the pastry may not have caused, or 
 helped to cause, it on the second. The indigestion still remains 
 an effect that on any given occasion may be due to any one or more 
 of a plurality of causes. Hence, in principle, the defect of Plurality 
 of Causes is not overcome through this double application of the 
 Method of Agreement. And the reason is that the effect to be 
 
 ♦ 'The Elements of Inductive Logic,' cli. iii., pp. 13S, 1G4.
 
 400 THE PROBLEM OF LOGIC [XIII. iii. 
 
 accounted for is as unprecisely conceived in the second application 
 as in the first. Xothing can obviate this defect, in our supposed 
 inquiry, except an adequate specification of the varietj- of indiges- 
 tion suffered on each successive evening — a specification sufficiently 
 detailed to cancel all suggested causes save one. But this analytic 
 process, as we have already seen, presupposes, for its success, the 
 use of strictly experimental methods. 
 
 IL The Experimextal Stage in Caus.vl Ixquiey. 
 A. Positive and Xejative Instances. 
 
 Let E stand for an effect for the explanation of which a cause, C, 
 is suggested. On inquiry, the suggestion, let us say, is found to be 
 justified. The supposition of C's operation can be shown adequately 
 to account for E. We have, then, shown that, given C, E will be 
 its effect. But we have not yet shown that when E is the given 
 effect, C must have been the cause. There may possibly be other 
 causes which also have E as their natural effect. There may be a 
 ' plurality of causes.' 
 
 But the interests of scientific precision cannot be satisfied by the 
 conclusion that a given effect is due either to this cause or to that, 
 or to some third or fourth cause. The ideal of Causal Explanation 
 remains unrealized so long as the causal connexion between C and 
 E is not proved to be reciprocally unideterminate. We must find 
 a cause which satisfies the two conditions : 
 
 Whenever C, then E ; 
 ^Vhenever E, th.^n C. 
 
 Expressing the latter condition in the equivalent ' contrapositive' 
 form 
 
 ^^^leneve^ not C, then not E, 
 
 we have the requirement that C shall satisf}^ the two conditions : 
 
 (1) ^^^lenever C, then E ; 
 
 (2) Whenever not C, then not E. 
 
 Experimental instances which illustrate (1) are named positive 
 instances. We might also call them instances of the co-presence of 
 cause and effect. Experimental instances which illustrate (2) are 
 named negative instances. These we might call instances of the 
 co-absence of cause and effect. For the adequate verification of 
 a causal hypothesis instances of both types are essential. 
 
 Negative Listances should not be confused with Exceptions. 
 The former are instances which corroborate a suggested causal
 
 Chap. XLV.] CAUSAL METHOD 401 
 
 connexion ; tlie latter are instances which disprove it. The form 
 of argument from a negative instance may be stated as follows : 
 
 Here C is absent, and so also is E. (Negative instance.) 
 .-., so far as this evidence goes, C may be causally connected 
 with E. 
 
 The form of argument in which an exception is being urged may 
 take the following form : 
 
 Here C is present, but E is absent. (Exception.) 
 .-., so far as this evidence goes, C is not causally connected 
 with E. 
 
 Again, under exceptions — or what appear to be exceptions — 
 we should distinguish such results as are merely negative {mde 
 pp. 334, 335) from such as are positively incompatible with the 
 hypothesis we are seeking to prove. Negative results, in par. 
 ticular, should be carefully distinguished from negative instances. 
 
 In the light of the foregoing, we may state in a more specific form 
 the distinction between Purely Observational Methods on the one 
 hand, and Experimental Methods on the other. On p. 395 we 
 described a Purely Observational Method as a Method that proceeds 
 from effects to causes ; an Experimental Method as one that proceeds 
 from causes to effects. We may now state the distinction more 
 precisely : 
 
 Where the start is made from effects — i.e., from facts or events 
 viewed in the light of a causal interest — and causal explanations are 
 tentatively formulated, without any attempt to verify them by 
 appeal to positive a-nd negative instances, the method is Purely 
 Observational. 
 
 Where, on the other hand, both positive and negative instances 
 are used, with a view to the verification of suggested causal con- 
 nexions, the method is Experimental. Wliere the control over 
 positive and negative instances is adequately exhaustive, the 
 connexion between cause and effect can, by this method, be shown 
 to be reciprocal, and the ideal of Causal Explanation is realized. 
 
 B. The Method of Scientific Experiment. 
 
 The differentiae of a strictly experimental method are (1) that 
 the conditions of observation are under the observer's control ; 
 (2) that there is a well-controlled interference with the object to be 
 observed. Thus, mere dissection of an animal or plant is not 
 experiment. In dissection our aim is precisely not to interfere with 
 the nerve, or the member of a flower-whorl, or any other structure 
 that we desire to examine — except, indeed, so far as interference 
 
 26
 
 402 THE PROBLEM OF LOGIC [XIII. iii. 
 
 is necessary as a mere aid to observation. Absolute non-inter- 
 ference would preclude the possibility of all but the most super- 
 ficial observation. In order to make our object accessible to 
 simple observation and study, we must often in one sense interfere 
 with it. We must inflate collapsible structures, or inject with some 
 highly coloured stain the finer ramifications of a vascular system ; 
 or we must cut sections with the microtome and make microscopic 
 preparations, rendering differences of tissue conspicuous by means 
 of careful staining. But all such interferences as these are aids to 
 observation only. Wlien we go further than this — when, for 
 instance, instead of merely laying a nerve bare, and perhaps treating 
 it with acetic acid to make it more conspicuous and easy to trace, 
 we apph' an electric stimulus to a living nerve and watch the effect — 
 then we have at once a genuine experiment. In vivisection we have 
 a constant application of the experimental method. For example, 
 a carefully localized cortical point in tlie brain of an anaesthetized 
 animal is stimulated, and the ensuing movements are accurately 
 observed. In both dissection and vivisection there is control over 
 the conditions ; but in the latter there is also a modifying or con- 
 structive interference with the phenomenon to be observed, while 
 in the former there is only so much interference as is necessary to 
 aid observation of the object as it already exists. 
 
 Further, in a scientific experiment, properly so called, there is 
 made a deliberate attempt to obtain precise and unambiguous 
 results. This implies that, before experimentation, the conditions 
 have been carefully analysed. Thus, the Method of Scientific 
 Experiment, in the strict sense of the term, must be a method 
 which presupposes control over conditions, implies a directly 
 modifying or constructive interference with the phenomenon to be 
 observed, and aims at precision both in procedure and in result. 
 
 The typical Method of Scientific Experiment, so understood, is 
 discussed by Mill under the title of ' the Method of Difference.' 
 Its essential import is very simply and concisely presented by 
 Dr. H. S. Mellone in the following words : 
 
 ' Wh/^n (he addition of an agent is followed by the appearance, or 
 its subtraction by the disappearance, of a certain event, other circum- 
 stances remaining the same, that agent is the cause of the event.' 
 ' When the suspected agent is present,' continues Dr. Mellone, ' we 
 have the positive instance ; when it is absent, the negative instance. 
 What cannot be eliminated without doing away with the event is 
 causally connected with it.'* 
 
 It will be gathered from this passage, taken together with the rest 
 of Dr. Mellone's exposition, that his definitions of ' positive and 
 negative instances ' are different from ours. The ' positive instance,' 
 as we have defined it, implies not only the presence of the suggested 
 cause, but also the co-presence of the effect. It is an instance which 
 
 * 'An Introductory Text-Book of Logic,' ch. ix., p. 274.
 
 Chap. XLV.] CAUSAL METHOD 403 
 
 is always corroborative of a suggested causal connexion. Similarly 
 the ' negative instance,' as we understand it, implies not only the 
 absence of the suggested cause, but also the co-absence of the effect, 
 and, like the positive instance, is always corroborative. According 
 to Dr. Mellone's definitions, the positive and negative instances do 
 not, as such, necessarily confirm some suggested causal connexion ; 
 they simply test it. They are not necessarily instances of a causal 
 law, but they are instances by means of which the claim of a sug- 
 gested connexion to the title of causal law is challenged and sifted. 
 In order to distinguish these ' positive and negative instances ' 
 (in Dr. Mellone's sense of the Avords) from the positive and negative 
 instances of our own definition (which are all instances of the law, 
 and serve to establish it), we propose to call the former ' test- 
 instances.' Thus, when the suspected agent is present, we have 
 the positive test-instance ; when it is absent, the negative test- 
 instance, whether the expected effect is present or not. 
 
 Following Dr. Mellone, we may illustrate the application of the 
 Method of Difference by the familiar experiment of ' the coin and 
 feather.' The question to be decided is, Why is it that, when a coin 
 and a feather are simultaneously let fall from the same height, 
 the feather does not reach the ground so soon as the coin ? 
 
 We will suppose that Analytic Observation suggests that the re- 
 tardation of the motion of the feather (the phenomenon or effect 
 to be explained) is caused by the resistance of the air (the suspected 
 agent). 
 
 We proceed to test this hypothesis by a systematic application of 
 the Method of Difference. 
 
 (a) Positive Test-instance — i.e., the instance in which, under 
 experimental conditions, the suspected agent is present : 
 
 The two objects ' are dropped simultaneously in the receiver of 
 an air-pump, the air being left in.' 
 
 Result. — ' The feather flutters to the ground after the coin.' 
 
 (6) Negative Test-instance — i.e., the instance in which, under 
 experimental conditions, the suspected agent is absent. 
 
 The air is, as far as practicable, pumped out of the receiver, and 
 the coin and feather are simultaneously dro])ped in the receiver 
 as before. 
 
 Result. — They reach the bottom approximately at the same 
 instant. 
 
 For the ideal success of the Method of Difference it is essential 
 that the positive and negative test-instances should differ only in 
 one relevant and important circumstance. As in the coin and 
 feather experiment, so in other applications of the Method, the 
 difficulty of adequately ensuring this requisite is, as a rule, over- 
 come by using in both cases what is practically one and the same 
 instance, modified, in the second case, in some one single point. 
 
 26—2
 
 404 THE PROBLEM OF LOGIC [XIII. iii. 
 
 This is explained by Mill in the ' Logic ' (Book III., ch. viii., § 3), 
 and is emphasized by Dr. Mellone's above-quoted restatement of 
 Mill's Second Canon. 
 
 Precautions to he Observed in Applying the Method of Difference 
 
 • 
 
 The fundamental requirement for the successful application of 
 the Method of Difference is that only one condition shall be varied 
 at a time. Upon the observance of this rule depends the scientific 
 precision of the method. There are two main reasons for observ- 
 ing it : 
 
 (i.) If vre vary two conditions at a time, and find some effect, 
 we cannot tell whether this effect is due to one, or to the other, or 
 jointly to both. 
 
 (ii.) If no effect ensues, we cannot safely conclude that either of 
 the two changes was ineffective ; for the effect of one may have 
 neutralized the effect of the other. 
 
 Mill accordingly points out that, in using tliis method, the change 
 must be introduced as rapidly as possible, and the whole process 
 must not last a long time. For, whenever the introduced change 
 takes a considerable time to make itself felt, and opportunity is 
 thereby given for its effect to be fused with other unintended and 
 unobserved changes, no satisfactory conclusion can be drawn from 
 the experiment. We must beware of Intermixture of Effects. 
 
 The impossibility of complying with tliis requirement in the case 
 of investigations belonging to the province of such a science as 
 Greology is the main reason why in a science of this kind, which 
 concerns itself with slow and age-long processes, the Method of 
 Difference is inapplicable. In the investigation of geological 
 causation it is inconceivable that we should ever have control over 
 all the relevant conditions ; and even if we had, and could proceed 
 to introduce among those conditions that change which we have 
 previously called C, there would still remain the difficulty that the 
 causal change thereby initiated might take many centuries to 
 produce any effect of geological significance. During this time 
 there would creep in countless other agencies, and the ' effect,' 
 when at length perceived by the geologists of a later generation, 
 would probably be in great measure due not to our own cause, C, 
 but to those other agencies at work without our sanction. 
 
 Limitations of the Method of Difference. 
 
 I. -Mill* points out that the Method of Difference cannot cope 
 with the difficulty of permanent causes — causes, that is, whose 
 agency can never be excluded ; — for the very essence of the method 
 lies in its producing an instance in wliich the cause in question is 
 
 * ' A Syatem of Logic,' Book III., ch. viii., § G.
 
 CuAP. XLV.] CAUSAL METHOD 405 
 
 absent. But he goes on to show that, though the permanent cause 
 can never be other than a ' coexisting fact,' it may be prevented 
 from operating as an ' influencing agent.' In illustration of this, 
 he supposes that, in experimenting with a pendulum, we find that 
 its oscillation is afiFected by the vicinity of a mountain. Now, 
 though we cannot possibly remove the mountain, so as to apply 
 the method of Difference directly, yet we can remove the pendulum 
 to such a distance from the mountain that the disturbing attractive 
 influence becomes practically inappreciable ; and this amounts, 
 indirectly, to a genuine application of the Method of Difiference. 
 But, again, suppose we wish to estimate experimentally the effect 
 of the attractiv^e power of the earth on the motion of the pendulum. 
 Here, as Mill says, ' we cannot take away the earth from the pen- 
 dulum, nor the pendulum from the earth.' The Method of Dififer- 
 ence, strictly so called, is here unavailable. All we can do is, as 
 Mill expresses it, to modify what we cannot exclude, the modifica- 
 tion of a condition meaning ' change in it not amounting to its 
 total removal.' It is a methodical modification of this kind wliich 
 constitutes the Method of Concomitant Variations ; and the most 
 striking applications of this method, as Mill points out, take place 
 in the cases in which the Method of Difference is quite inapplicable 
 — e.g., in estabhsliing laws of heat, gravitation, friction. We here 
 make a series of partial experiments, in which we proceed by a 
 gradual quantitative modification of the constituent wliich cannot 
 be wholly witlidrawn. The following is an abbreviation of Mill's 
 Canon of the Method of Concomitant Variations : Whatever pheno- 
 menon varies in any manner whenever another phenomenon varies 
 in some particular manner is causally connected with that phenomenon. 
 Thus, if in a tropical country, as the rainfall of each year is more 
 or less, the rice-crop is observed to be greater or less, or to fail, 
 correspondingly, the two phenomena are at once cormected together, 
 and we proceed to determine according to what quantitative law 
 the one varies with the other. In the same way, as Mr. Carveth 
 Read points out,* the use of the thermometer illustrates this 
 method. The rise and fall of the mercury is connected by the 
 observer with the increase and decrease of the amount of heat, 
 per unit of volume, in the atmospheric air in contact with the 
 glass, and at once suggests a quantitative comparison of these two 
 varying facts. 
 
 As a simple and typical example of this Method of Concomitant 
 Variations, let us suppose that an electric bell is placed ringing 
 under the receiver of an air-pump. The air is now gradually 
 exhausted, and it is noticed that, pari passu with the exhaustion 
 of the air, the sound of the bell grows fainter and fainter, until a 
 point is reached at which it is no longer heard at all. The air is 
 now allowed to pass back gradually into the receiver, and, as it does 
 
 * ' Ijogic Deductive and Inductive,' third edition, ch. ivi., p. 216.
 
 40G THE PROBLEM OF LOGIC [XIIL iii. 
 
 so, tlie sound of the bell is heard, at first faintly, then more and more 
 loudly, until the clearness of the original note is reproduced.* 
 
 2. The Method of DiiTerence cannot establish between cause and 
 effect any precise qtiantitative relation. 
 
 Here again, in order to supplement the Method of Difference, 
 we must have recourse to the Method of Concomitant Variations. 
 When, by the Method of Difference, it has first been ascertained 
 that a certain cause produces a certain effect, the Method of Con- 
 comitant Variations may usefully be called in to determine accord- 
 ing to what quantitative law the effect follows the cause. 
 
 But the application of the Method of Concomitant Variations 
 is not without its dangers. When, within a limited range of varia- 
 tion, a continuous change of a phenomenon, C, in one direction is 
 found to be always accompanied by a continuous change of another 
 phenomenon, E, also in one direction, there is a tendency to take 
 for granted that this correspondence will always hold even beyond 
 the limits within which our investigation has been conducted, and 
 we are inclined forthwith to frame a universal law to that effect. 
 But there is such a thing as discontinuous variation. Change 
 produced by a varying cause is often continuous only between 
 two critical points, of which each marks a sudden change in the 
 law of variation. Thus, as the temperature of water at the sea- 
 level continuously increases from 0^ C. to 100° C, its density in- 
 creases between the temperatures of 0° C. and 4° C, but after 
 4° C. is reached it decreases. In order to meet the requirements 
 of such cases as this the experiments by which concomitance of 
 variations is established should extend over a wide range. 
 
 Another danger which peculiarly, though not exclusively, affects 
 the application of this method is that of supposing, whenever we 
 find two series of phenomena varying concomitantly, that one of 
 the series is causally responsible for the other. But this by no 
 means follows. It is quite as likely that the two series are co-effects 
 of one and the same cause. By the Method of Concomitant Varia- 
 tions alone it is, in fact, never possible to ascertain which of the two 
 suppositions is true. 
 
 3. A single application of the Method of Difference is not, as a 
 rule, sufficient to verify the reciprocal relation between a suggested 
 cause and its supposed effect. To reach this ideal of Causal Ex- 
 planation we must have exhausted the series of the relevant repre- 
 sentative negative test-instances, f and in each case established 
 afresh the truth of the proposition 
 
 If not C, then not E. 
 
 * lor further illustration of this method, see Dr. W. Stanley Jevons, ' Elemen- 
 tary Jyessons in IvOgic' Lesson xxix., pp. 249-2.51. 
 
 t We could not atU^mpt to deal •with more than a series of typical or representa- 
 tive instances, and even that series would not be exhaustively representative, but 
 would be constituted by a relevant selection.
 
 Chap. XLV.] CAUSAL METHOD 407 
 
 To this development of the Method of Single Difference — a 
 development first suggested by Hermann Lotze — Professor Laurie 
 gave the name of ' The Double Method of Difference,' and he 
 formulated it, after Mill's manner, in an Inductive Canon. The 
 title does not seem to be very appropriate, as a multiple applica- 
 tion of a method is hardly the same as a double method ; but the 
 enunciation of this developed method in an additional canon is a 
 useful step to have taken. Professor Laurie's enunciation runs as 
 follows : 
 
 ' When, by the Method of Difference, we have established a 
 causal law connecting certain conditions with the production of a 
 phenomenon, and when, further, we have failed to discover any 
 case in which the phenomenon occurs without these conditions, 
 there is a probability, increasing with the extent and variety of 
 our negative instances, that the phenomenon can be produced in 
 no other way.'* 
 
 We have now discussed at sufficient length the limitations of the 
 typical method of Scientific Experiment, the Method of Difference. 
 We have shown, with the single exception of the limitation due to 
 the presence of an intermixture of effects, that remedies may be 
 found in the use of supplementary methods, such as the Method of 
 Concomitant Variations, or else in a repeated and varied applica- 
 tion of the Method of Difference itself — an application so contrived 
 as to exhaust the relevant and representative negative test-instances. 
 There still remains the question whether the difficulties due to 
 Intermixture of Effects may not be met in similar ways. 
 
 They may, in fact, be met by strengthening the application of 
 the method by means of certain purposive analj-ses and deductions. 
 
 If the effect to be accounted for is ' intermixed ' — e.g., a heat 
 effect — Analysis is required in m'der to break up the effect into as 
 many constituents as there are causes which contributed to produce 
 it, and to separate out the various lines of agency — e.g., combus- 
 tion, compression, electric action — which converge to produce the 
 effect. Having thus passed, through Analysis, from effects to 
 causal agencies, we may discover the laws according to which 
 these agencies operate, through the application of Experimental 
 Methods. But when the Experimental Methods have done their 
 utmost, there will still remain the task of gathering up the various 
 threads which these methods have singly and separately laid bare, 
 and this can be done only through Deduction or Deductive Synthesis. 
 The whole process represented by the methods hitherto discussed, 
 comprising the purely observational and the experimental stages, 
 is thus made preliminary to processes of deductive synthesis through 
 which the final Verification is alone made possible. And yet not 
 wholly preliminary, for a further appeal to the Experimental 
 
 ♦ 'Methods of Inductive Inquirj',' Mind. New Series, vol. ii. (ISD3), p. 333.
 
 40S THE PROBLEM OF LOGIC [XIII. iii. 
 
 Methods is needed in order to carry out satisfactorily these final 
 processes of Verification. Referring to this very difficulty of Later- 
 mixture of Effects, in connexion with the application of his ' four 
 Experimental Methods,' Mill writes : ' The instrument of Deduction 
 alone is adequate to unravel the complexities proceeding from this 
 source ; and the four methods have little more in their power than 
 to supply premisses for, and a verification of, our deductions ' 
 (Book III., ch. X., § 3). 
 
 In the light of this conviction. Mill proceeds to elaborate a 
 strengthened Method of Causal Explanation, a Method especially 
 devised to deal with the difficulties arising from Intermixture of 
 Effects : and he gives it the name of the Deductive Method (see 
 Book III., ch. xi. ; also Book III., ch. xiv., § 4). 
 
 The Deductive Method, as Mill conceives it, is a Method in three 
 stages : 
 
 (i.) Induciion* — Starting with the given complex phenomenon, 
 we aim first at discovering, through Analysis and Ex- 
 periment, the simple antecedents to whose combined 
 operation it is due, and the laws according to which 
 these antecedents several!}- act. 
 (ii.) Batiocination (or Deduction). — We calculate what would be 
 the joint effect of the operation of these antecedents, 
 each acting according to its own law. 
 (iii.) Verification. — ' To warrant reliance on the general con- 
 clusions arrived at by deduction, these conclusions 
 must be found, on careful comparison, to accord with 
 the results of direct observation wherever it can be 
 had' (Book III., ch. xi., § 3j. 
 
 In illustration of the application of tliis Deductive Method, we 
 can hardly do better than quote the instance given by ]\Iill himself — 
 ' the deduction,' namely, ' which proves the identity of gravity 
 with the central force of the solar system.' 
 
 ' First, it is proved from the moon's motions that the earth 
 attracts her with a force var\-ing as the inverse square of the dis- 
 tance. This (though partly dependent on prior deductions) corre- 
 sponds to the first or purely inductive step, the ascertainment of 
 the law of the cause. Secondly, from this law, and from the know- 
 ledge previously obtained of the moon's mean distance from the 
 earth, and of the actual amount of her deflection from the tangent, 
 it is ascertained with what rapidity the earth's attraction would 
 cause the moon to fall, if she were no farther off and no more acted 
 upon by extraneous forces than terrestrial bodies are. That is the 
 second step, the ratiocination. Finally, this calculated velocity 
 
 • 'In many particular investigations,' says Mill (Book III., ch. xi., § 1), 'the 
 7<larp of the inriu'-tion miy be sup7>lie<l b}' a prior deduction : but the premises 
 of this prior deduction must have been derived from induction.*
 
 Chap. XLV.] CAUSAL METHOD 409 
 
 being compared with the observed velocity with which all heavy 
 bodies fall, by mere gravity, towards the surface of the earth 
 (sixteen feet in the first second, forty-eight in the second, and so 
 forth, in the ratio of the odd numbers, 1, 3, 5, etc.), the two quantities 
 are found to agree ' (Book III., ch. xiv., § 4). 
 
 To the Deductive Method, with its three stages of Induction, 
 Ratiocination, and Verification, Mill attaches the greatest import- 
 ance. ' The human mind is indebted to it,' he writes, ' for its most 
 conspicuous triumphs in the investigation of nature. To it we 
 owe all the theories by which vast and complicated phenomena 
 are embraced under a few simple laws, which, considered as the 
 laws of those great phenomena, could never have been detected 
 by their direct study ' (Book III., ch. xi., § 3). It is, moreover, of 
 supreme value in Sociological inquiry. It is, indeed, only when 
 Mill comes to treat of the Logic of the Moral Sciences that he 
 develops the full resources of the Method. In the first place, a 
 distinction is drawn between an abstract and a concrete applica- 
 tion of the Method. Where the simple procedure of Geometry is 
 taken as the model, the application — so far, at least, as the require- 
 ments of Sociology are concerned — is abstract ; where the deduc- 
 tions, and the inductions which they presuppose, are of the more 
 complex kind proper to Astronomy, the application is concrete. 
 In the former case Mill speaks of the Geometrical or Abstract 
 Method, in the latter of the Physical or Concrete Deductive Method. 
 ' The Social Science, therefore (which, by a convenient barbarism, 
 has been termed Sociology), is a deductive science ; not, indeed, 
 after the model of geometry, but after that of the more complex 
 physical sciences. It infers the law of each effect from the laws 
 of causation on which that effect depends ; not, however, from the 
 law merely of one cause, as in the geometrical method, but by 
 considering all the causes which conjunctly influence the effect, 
 and compounding their la\\s with one another. Its method, in 
 short, is the Concrete Deductive Method — that of which astronomy 
 furnishes the most perfect, natural philosophy a somewhat less 
 perfect example, and the employment of which, with the adapta- 
 tions and precautions required by the subject, is beginning to 
 regenerate physiology ' (Book VI., ch. ix., § 1). 
 
 But not only is there a distinction drawn between an abstract 
 and a concrete application of the method ; the concrete application 
 itself maj'- be either direct or inverse. Thus, we ma}' start with 
 laws of human nature, deduce conclusions from these tlirougli a 
 process of ratiocination, and then verify these by comparing them 
 with observed facts. 
 
 This is the Direct Method. But, as Mill goes on to add. ' there 
 is a kind of sociological inquiries to which, from their prodigious 
 complication, the method of direct deduction is altogether inap- 
 plicable ' {ibid.). Resort must then be had to the Inverse Method.
 
 410 THE PROBLEM OF LOGIC [XIII. iii. 
 
 Here we start by direct consideration of the facts of social life and 
 of history, and obtain thence empirical laws or generalizations 
 which we then proceed to verify, not by comparison with further 
 facts, but by deduction from the laws of human nature. It is ' an 
 imperative rule ' of the Historical Method ' never to introduce any 
 generalization from history into the social science unless sutficient 
 grounds can be pointed out for it in human nature ' (Book VL, 
 ch. X.. § 4). Verification here takes the new form of successful 
 deduction, and we have a striking inversion of the ordinary pro- 
 cedure. ' Wheroas in the " method of direct deduction" we com- 
 pare the results of deduction with observed facts, we here begin 
 by provisionally formulating empirical laws gathered from facts 
 of observation, and then verify these laws by deducing them from 
 '■ the principles of human nature." '* 
 
 In the Deductive Method, with its inverse development, we seem 
 to reach what is, perhaps, the concretest form which Scientific 
 Method can take. The march of Mill's exposition from the opening 
 of the third Book onwards may be described as a progressive 
 development from the abstract to the concrete. Beginning with 
 abstract views of the universe, and with the partial methods that 
 correspond to them, he gradually takes in more and more elements 
 of Reality, and shows at each step how the partial methods are in 
 themselves inadequate to cope with these newly recognized aspects 
 of the world. Every step towards a concreter view of the world 
 is shown to neces,sitate a development of Method, until he has 
 welded all his partial Methods together, and comes out facing a 
 concrete (material) universe with weapons fitted at length to deal 
 with its complexity.f 
 
 But there is a sense in which even the Deductive Method is not his 
 final .standpoint, for in his paragraphs on ' Hypotheses ' Mill gathers 
 up a further and a most important thread into the tissue of logical 
 method. He enriches the conception of Method by an explicit 
 reference to the part played in it by Hypothesis. In Book III., 
 ch. xiv., § 5, he writes : ' The function ... of hypotheses is one 
 which must be reckoned absolutely indispensable in science. . . . 
 Without such assumptions, science could never have attained its 
 present state. . . . Nearly everything which is now theory was 
 once hypothesis.' 
 
 This ' Hypothetical Method,' to use Mill's own expression, is 
 regarded by Mill as a modification of the Deductive Method. It 
 differs from the latter in one important particular. The law, or 
 laws, which are developed through ratiocination are, in the case of 
 the Hypothetical Method, not proved by experimental methods, 
 but assumed. The Hypothetical Method, as Mill conceives it, is 
 
 * I bclievo this to be a quotation, but cannot trace it to its source, 
 t For a comparison between Mill's ' expository ' development of the Theory 
 of ScientiGc Method and our own exposition of Induction, vide pp. 419, 420.
 
 Chap. XLW] CAUSAL METHOD 411 
 
 therefore a simplificaticm of the Deductive Method. It consists of 
 two stages only — that of Ratiocination, or the deductive develop- 
 ment of Hypothesis, and that of Verification. 
 
 This abridgment, however, does not imply any loss of inductive 
 soundness. For, as Mill expressly points out, this abridged method 
 is legitimate ' on one supposition ' only — * namely, if the nature of 
 the case be such that tlie final step, the verification, shall amount to 
 and fulfil the conditions of a complete induction ' (Book III., 
 chap, xiv., § 4). The testing through experimental methods, 
 though not here present in the first stage of the inquiry, is in the 
 third stage reinforced and applied with additional vigour. The 
 Method of Difference exercises its verificatory function in its most 
 conclusive form. As Mill expresses it in an earlier chapter of his 
 work, ' The great generalizations which begin as Hypotheses must 
 end by being proved, and are in reality (as will be shown hereafter) 
 proved, by the Four Methods ' (Book III., ch. ix., § 6). 
 
 The Hypothetical Method, when thus concretely conceived, is 
 distinguisliable from the Inductive Method, as we have interpreted 
 it, only in its exclusion of the preliminary stages of Observation and 
 Formulation of Hypothesis. As Mill understands Induction, these 
 are not parts of Induction itself, but operations subsidiary to it. 
 
 It is in connexion with the application of the Deductive Method 
 that JMill's Method of Residues finds its natural place. But, as 
 formulated by Mill himself, the method at first appears under the 
 head of Induction, and as preliminary to Deduction and Verifica- 
 tion. Mill's enunciation runs as follows : 
 
 ' Subduct from any phenomenon such part as is known by previous 
 inductions to be the effect of certain antecedents, and the residue 
 of the phenomenon is the effect of the remaining antecedents.'* 
 The distinction between two quite different uses of this method 
 we owe to Professor Laurie. f In one application of the method we 
 are acquainted with all the antecedents concerned. Professor 
 Laurie (followed by Dr. Mellone) believes this to be the application 
 which Mill has in mind. But this is surely a misinterpretation, 
 and is in direct opposition to the following statement of the 
 method made by Mill himself : 
 
 ' Subducting from any given phenomenon all the portions which, 
 by virtue of preceding inductions, can be assigned to known causes, 
 the remainder will be the effect of the antecedents ivhich had been 
 overlooked, or of which the effect was as yet an unknown quantity.' 
 
 The expressions we have ' underlined ' seem to sliow that Mill's 
 Method of Residues is not really different from that wliich Dr. Mel- 
 lone cliaracterizes as regulated by ' a distinct rule,' which he thus 
 formulates : 
 
 ' When any part of a complex phenomenon is still unexplained 
 
 * ' A System of Ixjgic' Book III., ch. viii., § 5. 
 
 •f ' Methods of Inductive Liquu-y,' Mind, New Series, vol. ii. (1S93), pp. 335, 336.
 
 412 THE PROBLEM OF LOGIC [XIII. iii. 
 
 by the causes which have been assigned, a further cause for this 
 remainder must be sought.'* 
 
 Here the antecedents are not all ' known causes.' Some of them 
 (one or more) are causes that have hitlierto been overlooked. This 
 is the form of the Method of Residues contemplated by Mill, and 
 Dr. Mellone rightly regards it as the more important of the two. 
 
 Suppose the total cause in any given instance to contain the 
 known conditions, A, B, C ; and suppose the total effect to be 
 represented by a, f^, 7, S- Singling out the kno\^'Ti causes A, B, C, 
 we can deduce from their known modes of operation the effect that 
 they will produce when acting together under the given conditions. 
 Suppose this effect to be represented by a, /3, 7 ; we are tlien left 
 with a residual phenomenon 8, to which Ave must assign the unknown 
 cause X. 
 
 Example. — The Perturbation of the Planet Uranus. 
 Before the discovery of Xeptune in 1846 this astronomical j)roblem 
 miglit have been rousrlilv stated thus : 
 
 Cause ABC . . . gives aftjB as effect. 
 
 I.e., Tangential momentum of Uranus at a given moment (A) 
 
 + Attraction of the Sun (B) 
 
 + Perturbing influence of known planets (C) 
 
 + Perturbing influence of unknown cause 
 
 gives, as the observed effect, the known orbit of Uranus (0^78). 
 Taking the known causes A, B, C, astronomers were able, by mathe- 
 matical deduction, to ascertain the effect in the motion of Uranus 
 due to these causes acting in combination. This effect was repre- 
 sented by a&y, and the investigators were left with the problem of 
 a residual effect 8, which represented the discrepancy between the 
 observed effect aPy8 and the effect afty already deduced from the 
 laws of all the known causes. John Couch Adams in England and 
 Urbain J. J. Leverrier in France (working quite independently of 
 one another) then argued that this effect 8 must be due to the per- 
 turbing influence of some unknown planet, which we may call x.f 
 In this way an application of the Method of Residues pointed to the 
 fact of the planet Neptune's existence. 
 
 From the foregoing it will be seen that the essential function of 
 the Method of Residues is to ensure the adequacy of a causal explana- 
 tion. The ascribed cause must be qualitatively adequate to account 
 for the ob.served effect — i.e., all the requisite conditions must be 
 considered, and the cause established as a total cause. 
 
 When the Method of Residues is further refined through insistence 
 on quantitative adequacy, we reach a conception of Causal Explana- 
 
 * ' An Intrfxiuctory Text- Book of Logic,' first edition, ch. ix., p. 287. 
 f For a brief authoritative account of the history of the discovery of Xeptune, 
 c/. Simon Newcomb's ' Astronomy for Everybody,' pp. 232-235.
 
 Chap. XLV.] CAUSAL METHOD 413 
 
 tion which can be satisfactorily realizrd only through the application 
 of quantitative methods. It was througli precise quantitative in- 
 vestigation of residual errors that James Bradley discovered the 
 nutation of the earth's axis and the aberration of light, and that 
 Leverrier and Adams ascertained not only the existence of Xeptune, 
 but also the mass and the orbit of the yet unseen planet. It was 
 also through this same method that Argon, Helium, and other con- 
 stituents of our atmosphere were recently brought to light. 
 
 As characteristic of these more exact modes of scientific pro- 
 cedure, we may note the Elimination of Error in Observation. 
 The following extract from a popular treatise on Astronomy* may 
 serve to illustrate the method : ' Observations made with the transit 
 circle must ... be corrected for errors in the instrument itself. 
 The astronomer will see to it that his instrument is made and is 
 set up as perfectly as possible. The pivots on which it turns must 
 be exactly on the same level ; they must point exactly east and west, 
 and the axis of the telescope must be exactly at right angles to the 
 lines joining the pivots in all positions of the instrument. These 
 conditions are never absolutely fulfilled. Day by day, therefore, 
 the astronomer has to ascertain just how much his instrument is in 
 error in each of these three matters. Were his instrument absolutely 
 without error to-day, he could not assume that it would remain so. 
 The astronomer finds that his own presence near the instrument is 
 sufficient to disturb it. . . . The great interest attaching to transit 
 circle work is this striving after ever greater and greater precision, 
 with the result of bringing out fresh little discordances, which, at 
 first sight, appear purelj'^ accidental, but wliich, under further 
 scrutiny, show themselves to be subject to some law. Then comes 
 the hunt for this new unknown law. Its discovery follows. It 
 explains much, but when it is allowed for, though the observations 
 now come much closer together, little deviations still remain to 
 form the subject of a fresh inquiry.' 
 
 Where errors cannot be individually eliminated, we have to resort 
 to the best argument whereby we can temper our ignorance — the 
 argument from probability. Our reasoning is here based on the 
 reduction of errors to average and probable errors, for which due 
 allowance is made. 
 
 III. The Essence of Inductive Method as an Instrument of 
 
 Causal Inquiry. 
 
 ]\Iill maintains (Book III., cli. viii., § 1) that his methods involve 
 two simple principles : 
 
 (1) That of the Method of Agreement ; 
 
 (2) That of the Method of Difference. 
 
 * I regret that I am unable to state the reference more precisely.
 
 414 THE PROBLEM OF LOGIC [XITI. iii. 
 
 1. Tlie Method of Agreement consists in comparing together 
 dilTerent instances in which a phenomenon occurs. 
 
 2. The Method of Difference consists in comparing instances in 
 wliich a phenomenon does occur with instances in other respects 
 simihar in which it does not. 
 
 Both metliods, according to Mill, agree in this — that they are 
 methods of Elimination. ' The Method of Agreement,' he says, 
 ' stands on the ground that whatever can be eliminated is not con- 
 nected with the phenomenon by any law. The Method of Differ- 
 ence has for its foundation that whatever cannot be eliminated is 
 connected with the phenomenon by a law ' (Book III., ch. viii., § 3). 
 
 It would appear from the foregoing that Mill regards Inductive 
 Method as fundamentally a weapon of Elimination. Whether we 
 make use of the Method of Agreement or of the Method of Difference, 
 we proceed, according to Mill, by Elimination. But by Elimina- 
 tion Mill means, not a logical process, but a physical exclusion. 
 The term, he says, ' is well suited to express the operation . . , 
 wliich has been understood since the time of Bacon to be the founda- 
 tion of experimental inquiry — namely, the successive exclusion of 
 the various circumstances wliich are found to accompany a phe- 
 nomenon in a given instance, in order to ascertain what are those 
 among them which can be absent consistently with the existence of 
 the phenomenon ' (ibid.). 
 
 In thus reducing his Inductive Methods to Methods of so-called 
 Ehmination, Mill here explicitly attaches his view of Induction 
 to that adopted by Francis Bacon. 
 
 To Bacon the data of observation presented themselves as com- 
 plex groups of properties, each property having as its essence some 
 corresponding simple cause. These single causes he called forms; 
 and the aim of Induction, in his opinion, is the discovery of these 
 forms. The Method of Induction which Bacon adopted he called 
 the Method of Exclusion. In order to discover the form or cause 
 of a certain phenomenon — for instance, that of heat — we must, 
 according to this method, take three preliminary steps : 
 
 1. We must collect all the known cases in which the phenomenon 
 of heat is present. These constitute a tabula essentice et prcesentice. 
 
 2. We must make a collection of instances, cognate or similar 
 to the former, in which heat is absent. These constitute a tabula 
 absenlicB in prozimo parallel to the first table. 
 
 3. We must compile a tabula graduum, in which the varying in- 
 tensity with which the phenomenon appears may be compared with 
 the varying intensities of other phenomena that accompany it. 
 
 We then seek to discover the form of heat by applying the 
 principle that this form can be nothing but that which is always 
 present where heat is found, which is not present where heat is 
 lacking, and which perpetually decreases as heat decreases, and 
 increa.ses as heat increases.
 
 CiMP. XLV.] CAUSAL MIvTHOD 415 
 
 This invcstic^ation, wo are told, must be conducted by means of 
 progressive exclusion. Thus we argue that tiie form of lieat is not 
 tenuity ; for the form of heat is always present where heat is, and 
 tenuity is not always where heat is. Gold and other metals, 
 though they may be heated, are still of great density throughout. 
 For this reason tenuity must be rejected. Again, the form of heat 
 is not present where heat is lacking ; but tenuity is where heat is not, 
 for the air is often cold, and yet remains uncondensed. Thus, for 
 this reason also, tenuity must be rejected. And so it is with many 
 other phenomena that in some cases accompany heat.* 
 
 It will be observed that, whether Bacon makes use of the positive 
 or of the negative test-instance, he makes use of it in the interest of 
 logical exclusion. His aim is to include all possible conjectures 
 as to the nature of the form under investigation in one disjunctive 
 judgment, and then to reach the true nature by excluding all con- 
 jectures which fail to satisfy the two Canons of Causation — viz. : 
 
 1. The cause or form must always be present where the effect 
 is found. 
 
 2. The cause or form must never be present where the effect is 
 lacking. 
 
 Now, when we compare Bacon's procedure with that of Mill, it 
 becomes evident that the latter is not throughout a Method of 
 Exclusion in the same rigorous and logical sense as is the former. 
 Mill's Elimination is not logical but physical exclusion. In the 
 logical and proper sense of the word, the Method of Agreement only, 
 and not that of Difference, can be called a Method of Elimination. 
 So far as the Method of Agreement is concerned. Mill remains faith- 
 ful to Bacon's Method of Exclusion ; but the Method of Difference 
 is emphatically not a method of logical Exclusion or Elimination. 
 Its essence is not the elimination of the non-cause, but the establish- 
 ment of the cause. ' The Method of Difference has for its founda- 
 tion, that whatever cannot be eliminated is cormected with the 
 phenomenon by a law.'f 
 
 Here the word ' eliminated ' is not well chosen, but its meaning 
 is not doubtful. IVIill here uses the word ' elimination ' consistently 
 (though, as we think, unadvisedly) in the sense of physical, not 
 logical, exclusion. It is quite clear that in the enunciation of the 
 Canon of the Method of Difference the idea of logical Ehmination is 
 nowhere implied. ' If an instance,' says Mill, ' in wliich the 
 phenomenon under investigation occurs, and an instance in wliich 
 it does not occur, have every circumstance in common save one, 
 that one occurring only in the former ; the circumstance in which 
 alone the two instances differ is the effect, or the cause, or an indis- 
 pensable part of the cause, of the phenomenon ' (Book III., cli. viii., 
 
 * Vide 'Novum Organum,' lib. ii., c. .xi.-xviii. 
 t ' A Sj'stem of Logic,' Book III., ch. viii., § 3.
 
 416 THE PROBLEM OF LOGIC [XIII. iii. 
 
 § 2). Now, we saw, in the case of tlie Method of Single Agreement, 
 that the singleness of the agreement could be established only through 
 a process of logical elimination ; but the singleness of the difference 
 between two instances is establislied, not by any such elimination, 
 but by making a single change in a certain given instance. We do 
 not pass from the positive to the negative test-instance, in experi- 
 ment, by elimination, except in this sense — that we ' eliminate ' a 
 circumstance in the positive instance in order to secure the negative 
 instance. But this, of course, is not a logical process of elimination, 
 but a physical process of subtraction. It would therefore have been 
 better if Mill had avoided the use of the term ' eliminated ' in his 
 statement of tlie foundation of the Method of Difference. He 
 would have expressed his own meaning more unambiguously if he 
 had said : ' What cannot be absent without doing away with the 
 phenomenon is causally connected with it.' 
 
 The view that Induction is essentially a Method of Elimination 
 or Exclusion has recently received the powerful support of Mr. 
 H. W. B. Joseph, Mr. Joseph is convinced, not only that the 
 Method of Exclusion was Bacon'sMethod, but that it was also Mill's ; 
 and he has uncompromisingly adopted it as his own. ' Inductive 
 conclusions,' he roundly asserts, ' are established disjunctively by 
 the disproof of alternatives.'* And again : ' The inductive proof of 
 a conclusion rests on excluding alternative explanations.'! 
 
 It will be useful to consider Mr. Joseph's position more closely. 
 He has formulated four ' grounds ' of elimination, each of which 
 ' points to some particular requirement of the causal relation, 
 failure to satisfy winch disproves that relation as between two given 
 phenomena. 'J 
 
 These grounds of Elimination are enumerated as follows : § 
 
 ' L Nothing is the cause of a phenomenon in the absence of 
 
 which it nevertheless occurs. 
 ' 2. Nothing is the cause of a phenomenon in the presence of 
 
 which it nevertheless fails to occur. 
 ' 3. Nothing is the cause of a phenomenon which varies when it 
 
 is constant, or is constant when it varies, or varies in 
 
 no proportionate manner with it. 
 ' 4. Nothing is the cause of one phenomenon which is known to 
 
 be the cause of a different j)henomenon.' 
 
 Tlie application of these four canons of Elimination enables us to 
 pass to a knowledge of the cause by a successive exclusion of all non- 
 causes, and inductive reasoning takes the form of a disjunctive 
 argument of the Modus Tollendo Ponens. 
 
 Mr. Joseph takes care to point out that ' the character of the 
 reasoning is unaffected either by the completeness of the elimina- 
 
 • ' An Introduction to Logic,' ch. xx., p. 408. t Ibid., p. 415. 
 
 X Ibid., p. 403. § Ibid., pp. 403, 404.
 
 Chap. XLV.] CAUSAL METHOD 417 
 
 tion ... or by the ground of elimination used.'* Whatever fails 
 to satisfy any single one of these grounds cannot be a cause. Again, 
 if I am unable to state my disjunctive basis exiiaustively, or to elimi- 
 nate all alternatives save one, this failure on my part in no way 
 alters the character of my argument. The method of reasoning 
 remains precisely the same whether I am able to conclude that the 
 cause of E is C, or only that the cause of E must be C^ or Cg or C3, 
 or some cause hitherto undetected. ' The getting of a positive con- 
 clusion, but not the inductive character of the argument, depends 
 on the completeness of the elimination.'! 
 
 In a footnote J Mr. Joseph connects these grounds of elimination 
 with Mill's Inductive Canons. ' On these grounds of elimination,' 
 he writes, ' Mill's " Inductive Methods " severally repose. Tlie 
 first is the foundation of his " Method of Agreement," the second of 
 his " Method of Difference," the first and second jointly of his 
 " Joint Method of Agreement and Difference," the third of his 
 " Method of Concomitant Variations," and the fourth of his 
 " Method of Residues." ' JMill's statement that the Methods of 
 Agreement and Ditlerence are both Methods of (physical) Elimina- 
 tion is thus developed into the surprising consequence that Inductive 
 Method generally is a Method of Logical Elimination. ' It would be 
 well to recognize that IMill has not formulated four (or five) but one 
 " Method of Experimental Inquiry " — as indeed Bacon might 
 have shown him — of which the essence is that you establish a par- 
 ticular hypothesis about the cause of a phenomenon by showing 
 that, consistently with the nature of the relation of cause and effect, 
 the facts do not permit you to regard it as the effect of anything 
 else (and mutatis mutandis if you are inquiring into the effect of 
 any thing ).'§ 
 
 Now, it is true that MiU has laid himself open to this interpretation 
 of liis position by his statement that not only the Method of Agree- 
 ment, but the Method of Difference also, is a Method of ' Elimina- 
 tion.' But, as we have already said, that statement does not refer 
 to logical Elimination at all, nor would such an interpretation of it 
 tally with Mill's actual handling of the Method of Difference. Xor 
 is there any hint of the Methodus Exclusiva in his formulation and 
 treatment of the Method of Concomitant Variations. Where ^Ir. 
 Joseph saj's, ' Nothing is the cause of a phenomenon wliich varies 
 when it is constant, or is constant when it varies, or varies in no 
 proportionate manner with it,' Mill says : ' Whatever phenomenon 
 varies in any manner whenever anotlier phenomenon varies 
 in some particular manner, is either a cause or an effect of that 
 phenomenon, or is connected with it through some fact of causa- 
 tion. '|| It is impossible not to be struck at once by the opposition 
 between the negative character of the third ground of Elimination 
 
 * 'An Introductioa to Logic' ch. xx., p. 39G. t Ibid., p. 417. 
 
 t Ibid., p. 4U4. § Ibid., p. 39'J. II Book III., ch. viii.. § 6. 
 
 27
 
 418 THE PROBLEM OF LOGIC [XIII. iii 
 
 and the positive character of Mill's fifth Canon. Nor is there any- 
 thing to suggest that the latter reposes on the former, in the sense 
 that the application of the former paves the way for tlie statement 
 of the latter. If we consider, for instance. Mill's illustration of the 
 experimental corroboration of the first law of motion through the 
 Method of Concomitant Variations (Book III., ch. viii., § 7), we 
 find that the application is perfectly positive and direct. Nor can I 
 find any evidence for supposing that the fourth ground of Elimina- 
 tion is the foundation upon wliich the Method of Residues rests. 
 
 As the gist of our criticism of Mr. Joseph's view consists in the 
 contention that the Method of Difference is not really a Method of 
 logical Elimination, it may be useful to make our meaning clearer 
 by reference to a particular example. Mr. Joseph illustrates the way 
 in which inductive conclusions are estabhshed through the disproof 
 of alternatives by reference to the problem as to the power possessed 
 by chameleons and frogs of changing colour according to the colour 
 of their surroundings. ' Granting,' he says, ' in the absence of ariy 
 other alternative, that it depends on the colour as such, we may ask in 
 what way the differently coloured rays affect the animal. Lord 
 Lister showed that they affected it through the eyes ; for a specimen 
 of Rana temporaria whose e3'es had been removed was no longer 
 affected by any change in the colour of the surroundings in which 
 it was placed ; thus the alternative, otherwise not unreasonable, is 
 excluded, that the reaction is somehow determined through the 
 skin, the principle applied being that no circumstance in the 
 presence of which the phenomenon fails to occur is its cause.'* 
 But, we ask, did not the experiment, on Mr. Joseph's own explicit 
 .showing, prove the eye-theory rather than disprove the skin-theory ? 
 Did not Lord Lister's experiment show that the rays affected the 
 frog through the eyes ? And was not the principle applied just this 
 — that the circumstance in the absence of wliich the phenomenon 
 fails to occur is causally related to that phenomenon ? 
 
 The pivot upon which Mr. Joseph's argument in favour of the 
 Method of Elimination appears to turn is the conviction that to 
 disprove through a ground of elimination is a far easier and surer 
 process than to prove through a ground of verification. Referring 
 to Bacon's Methodus Exdusiva, he writes : ' We must proceed, then, 
 by exclusions. Where a hundred instances will not prove an uni- 
 versal connexion, one will disprove it.'f 
 
 This is, no doubt, true of causal methods in wliich stress is laid 
 on the number of positive instances ; but in the apphcation of the 
 Method of Difference the verification is as easy and as convincing as 
 the di.sproof. The positive verification-method of Mill is in no sense 
 weaker or more unwieldy than the negative elimination-method of 
 Mr. Jo.seph. Even when the verification of C as causally connected 
 with E does not give the total cause on wliich the phenomenon E 
 
 • 'An Introduction to Logic,' cli. xx., p. 409. t ■^^*'^-' ^h. xviii., p. 365.
 
 Chap. XLV.] CAUSAL METHOD 419 
 
 depends, still, at the very least, we have ascertained (as Mill puts it 
 in discussing the force of the Method of Agrcomr'nt) ' one invariable 
 antecedent . . . however many other invariable antecedents . . . 
 may still remain unascertained.'* This result is surely as definite 
 and as valuable for the purposes of causal elaboration as the disproof 
 of the causal connexion in question for those of causal elimination. 
 But there is, I believe, a deeper reason for the discrepancy 
 between Mr. Joseph's standpoint and our own than any which the 
 preceding argument has yet suggested. It appears to me that there 
 is involved in the issue a conflict between logical and methodological 
 ideals. The strictly logical tendency is to presuppose the ideal ; 
 the methodological tendency is to aim at progressively realizing it. 
 Where the ideal is presupposed, we tend to frame our rules of pro- 
 cedure on the assumption that it is realized. Thus, in Mill's case, 
 the start is made, not methodologically, from the actual conditions 
 of sense-experience, but from a simplified and idealized conception 
 of such experience. Mill, as his own words and the testimony of 
 his own artificial symbolism assure us, assumes as the basis for the 
 application of his canons a causal situation in which there is neither 
 any plurality of causes nor any intermixture of effects.! 0^^ this 
 simplified basis a conception of Induction is developed which can 
 proceed without the aid of Deduction, and, as we might perhaps add, 
 without the aid of Hypothesis, or even of Verification. Unfortunately 
 these presuppositions are not explicitly embodied in the enuncia- 
 tions of the canons, so that the latter wear an air of unreality which 
 haunts the student even when he fails to suspect the abstract formal 
 character of the methods in question. It is only when these formal 
 limitations are abandoned, when the Method of Agreement is shown 
 to be at the mercy of plurality of causes, and forthwith depressed to 
 the level of a mere preliminary or adjunct to the Method of Differ- 
 ence, and the latter, again, shown to be unable to cope with inter- 
 mixture of effects, and, in its turn, transformed into a preliminary 
 to Deduction, that we approach the true conception of a working 
 Inductive Method. The stages of Observation, Hypothesis, Deduc- 
 tive development and A]'>plication of Hvpothesis, Progressive Veri- 
 fication — stages which, from the strictly methodological point of 
 view, impose themselves from the outset as the first essentials of 
 inductive procedure, are thus gradually made more and more 
 familiar to the reader, until, in the concrete development of the 
 
 • Book III., ch. viii., §3. 
 
 ■\ ' In the preceding exposition of the four methods of observation and experi- 
 ment, by which we contrive to distinguish among a mass of co-existent phenomena 
 the particular effect due to a given cause, or the particular cause which gave birth 
 to a given effect, it has been necessary to suppose, in the first instance, for the 
 sake of simplification, that this analytical operation is encumbered by no other 
 difficulties than what are essentially inherent in its nature ; and to represent to 
 ourselves, therefore, every effect, on the one hand, as connected exclusively with 
 a single cause, and on the other hand as incapable of being mixed and confounded 
 with any other co-existent effect ' (Book III., ch. x., § 1). 
 
 o- 
 — / -
 
 4-0 THE PROBLEM OF LOGIC [XIII. iii. 
 
 ' Deductive ' and ' Hypothetical ' Methods, we reach at last a true 
 methodological conception of Inductive Inquiry. 
 
 It is an interesting question whether Mr. Joseph has not been 
 developing his Methodology on a similar plan. We would readily 
 admit that the ideal of a reversible or reciprocal causal connexion 
 is common alike to the ' logical ' and the ' methodological ' points of 
 view ; but the presupposing of the ideal as already realized seems to 
 liave the effect of diverting the energies of the logician from the task 
 of seeking how progressively to realize it, and concentrating them 
 on methods for the elimination of whatever fails to satisfy certain 
 standard rules or canons. The logical presupposition supplements 
 the methodological goal, though presupposition and goal embody, 
 each in its own way, the same inductive ideal. The difference is one 
 of Means rather than of Ends, though it is a difference which touches 
 very closely on fundamental principles. 
 
 Tlie balance between terminus a quo and terminus ad quern — for 
 both are essential — might, perhaps, be adjusted by a purposive 
 change in the character of the former. For the logical JPresupposition 
 we might substitute the methodological Postulate ; or, in less tech- 
 nical language, we might place at the forefront of Inductive Inquiry 
 not an ideal assumed to be realized, but a condition which must 
 be satisfied in order to render Inductive Inquiry, not indeed ideal, 
 but 'possible. The methodological Postulate would state the con- 
 dition on which a legitimate scientific explanation depends, and the 
 Unideterminate Ideal — the ideal of a system of reversible causal 
 connexions — would then become the methodological guiding idea. 
 It is in the light of this ideal that Science presses forward to the 
 terminus ad quern of all Inductive Inquiry — namely, complete fidelity 
 to fact, in so far as fact is relevant to the demands of the Inductive 
 Postulate. 
 
 If we are asked to define the method which we propose to substi- 
 tute for the Method of Elimination, we can hardiy do better than 
 point to Professor Bosanquet's Method of Analysis, to his view of 
 Induction as the progressive ' moulding of Hypothesis ' (' Logic,* 
 vol. ii., ch. v., pp. 166, 167) through a process which he elsewhere 
 describes as a purifying by exceptions and a limiting by negations 
 (vol. ii., ch. iv., p. 117). In this process we have central importance 
 attached to the positive, progressive elaboration of causal con- 
 nexions, with a clear recognition of the part played by negation, and, 
 in particular, by elimination (ch. iv.). The view of Induction 
 adopted in the present work has been largely influenced byProfessor 
 Bosanquet's treatment of the Inductive Problem in this second 
 volume of his ' Logic,' though the inspiration has been tempered 
 by pragmatic leanings for which tlie author of the ' Logic ' might 
 have little sympathy. 
 
 In Professor Bosanquet's view, the primary and essential business 
 of an inductive investigator is not to attack all rival theories,
 
 Chap. XLV.] CAUSAL METHOD 421 
 
 supporting the truth of his own theory by showing that no other 
 can explain tlic facts so well : his main endeavour should be to 
 strengthen his own theory in a positive and constructive spirit. 
 This he must do by showing that all objections advanced against 
 it are ungrounded, or, if not wholly ungrounded, still, by limiting 
 the theory in determinate ways, support rather than refute it. A 
 theory can establish itself only by successfully meeting all relevant 
 objections. These objections will be brought forward as exceptions 
 to the truth of the statement which expresses the theory. If the 
 exception proves to be genuine, the theory must be correspondingly 
 modified by the introduction of a limiting condition ; if only ap- 
 parent, it is refuted by a negative instance confirmatory of the 
 theory. A genuine exception modifies the rule by restricting its 
 universality ; an apparent exception is but a mistaken interpretation 
 of the facts which the negative instance interprets rightly. 
 
 As exemplifying this typical form of inductive procedure, we may 
 cite an investigation discussed by Professor Welton,* who also is 
 confessedly indebted to Professor Bosanquet for his general line of 
 treatment. Professor Welton — from the point of view of the 
 inductive interest — is analysing a part of Charles Darwin's inquiry 
 into the formation of vegetable mould through the action of earth- 
 worms, f 
 
 A number of careful observations had gone to support the state- 
 ment that the ' formation of vegetable mould is due to the action of 
 earth-worms.' The sign by which the fresh production of vegetable 
 mould was tested was the sinking of objects strewed on the ground, 
 and the process through which these objects were gradually buried 
 deeper and deeper in the soil. 
 
 But here the investigator was met by an apparent exception to 
 the universal activity of earthworms, which he desired to establish. 
 Large boulders do not sink. Darwin, however, showed that tliis 
 fact was no genuine exception, but that it furnished confirmatory 
 negative instances. ' If a boulder is of such huge dimensions,' he 
 writes, ' that the earth beneath is kept dry, such earth will not be 
 inhabited by worms, and the boulder will not sink into the ground.' J 
 Thus the positive statement, ' Where there are worms there is vege- 
 table mould,' was corroborated and defined by the establishment, 
 through negative instances, of the negative statement, ' Where 
 there are no worms there is no vegetable mould.' 
 
 Tliis same example furnishes also a simple instance of the way 
 in wliich a genuine exception necessitates a modification of the 
 original statement of a hypothesis — a restriction of its universality. 
 It might be urged that soil is brought from beneath the surface by 
 moles and other burrowing creatures no less than by worms. Tlie 
 
 * ' A Manual of Logic,' vol. ii.. Book V.. ch. v., pp. 124-127. 
 t Vide 'The Formation of Vegetable Mould through the Action of Worms," 
 ch. ii.. pp. 131-177. X Ibid., p. 152.
 
 422 THE PROBLEM OF LOGIC [XIII. iii 
 
 fact cannot be denied ; but it can be shown that their contribution 
 to the surface layer of soil is far less than that of worms. The 
 original statement should, then, be modified by the introduction 
 of the word ' mainly,' and should be enunciated thus : ' The forma- 
 tion of vegetable mould is mainly due to the action of earthworms.' 
 A series of accurate comparative measurements of the castings of 
 earthworms, on the one hand, and of the soil thrown up by moles, 
 etc.. on the otlier, would complete the causal value of the proposition 
 by giving ' mainly ' its proper fractional values (expressed in average 
 form) for different soils and climates. This illustrates the fact that, 
 before exact results can be obtained, the data upon which the judg- 
 ment is based must be of a quantitative kind, and that corresponding 
 numerical limits must be introduced into our general statement. 
 
 In conclusion, we would add that, as we conceive the matter, 
 the part played by Elimination in the complete Method of Analysis 
 is essential!}^ heuristic, as in the case of the Method of Single Agree- 
 ment. Elimination has a further function to fulfil in connexion 
 with the necessary though secondary work of meeting and refuting 
 rival hypotheses, especially in the devising of crucial instances and 
 experiments which shall approve one hypothesis by disproving its 
 rivals. But even here it seems gratuitous to bring these rival 
 theories into prehminary relation with each other as co-alternatives 
 of a disjunctive proposition. The disjunction has force only in so 
 far as the verification of the theory we are defending rests upon the 
 disproof of the rival theories. But the verification rests essentially 
 on the po.sitive value of the theory as a means for systematically 
 explaining the relevant facts, and the disproof of rival theories — a 
 disproof which, if undertaken in an impartial spirit, may be almost 
 as arduous a piece of work as the justification of the theory itself — 
 is an operation subsidiary and supplementary to this. 
 
 CHAPTER XLVI. 
 
 XIII. (iv.) ILLUSTRATIONS OF THE APPLICATION OF INDUCTIVE 
 
 METHOD. 
 
 As examples of the ways in which Inductive Method is apphed, we 
 may take the following : 
 
 \. The Problem of Fermentation.* 
 2. The Case of Algol, the Demon Star. 
 The Rigidity of the Earth. 
 
 •J. 
 
 * This illustration is given by Dr. McUonc as an example of the application of 
 the Method of Single Agreement and the 'Double Method of Difference' ('An 
 Introductory Text-Book of Logic,' first edition, ch. ix., pp. 272, 273, 282, 283). 
 Our ovTO account of this investigation is mainly an adaptation of Professor W. 
 Dittmar's article on ' Fermentation ' in the ' Encyclopeedia Britarmica,' ninth 
 edition, vol. ix.
 
 Chap. XLVI.] ILLUSTRATIONS 423 
 
 1. The Problem of Fermentation. 
 
 By the word ' Fermentation ' we designate, as Professor Dittmar 
 tells us, ' a peculiar class of metamorphoses which certain complex 
 organic materials are liable to, and of which the well-known change 
 which grape-juice undergoes when it " ferments " into wine, the 
 souring of wine or milk, and the putrefaction of animal or vegetable 
 matter, may be cited as familiar examples.'* 
 
 Chemically, the processes of fermentation admit, as a rule, of 
 fairly simple statement. Thus, the change through which milk 
 turns sour consists chemically in the passing of the milk-sugar in 
 the milk into lactic acid by a rearrangement of atoms symbolized 
 by the chemical equation : 
 
 C12H22O11.H2O = 4C3Hg03. 
 
 (Milk-sugar) (Lactic acid) 
 
 So, again, in vinous fermentation we find that the formation of 
 alcohol can be chemically explained as the result of the decomposi- 
 tion of the sugar contained in grape-juice, which breaks up into 
 alcohol, carbon dioxide gas, and small quantities of other compound 
 substances. The effervescence wliich is so noticeable a feature in 
 the fermentation of must is caused by the rapid evolution of the 
 carbon dioxide gas which rises in bubbles to the surface. 
 
 There remains, however, the question, What is the ' exciting 
 cause ' of this chemical change ? 
 
 Let us, for simpUcity's sake, follow the investigation of the 
 alcoholic fermentation of grape- juice only. The facts are these : 
 Grape-juice, clarified by filtration into a perfectly Umpid and trans- 
 parent liquid, may remain unchanged for an indefinite time. But 
 when the smallest quantity of unfiltered juice is introduced into 
 this pure liquid, it becomes turbid, the turbidity being due partly 
 to the evolution of carbon dioxide gas, partly to the presence of a 
 finely diAided solid material, some of which rises to the surface and 
 forms a scum. This solid material is known as yeast. These 
 changes include a growing effervescence of the must, wliich may 
 reach the stage of Wolent ebullition ; and as the effervescence grows, 
 so does the yeast. A climax is reached, and the effervescence at 
 last dies away. The yeast then settles down as a slimy deposit, 
 above wliich there is left a clear yellow alcohohc liquid, characterized 
 no longer by its original sweetness, but by a Nanous taste and other 
 properties of ' fermented liquors.' 
 
 Thus, in all ordinary cases of the phenomenon we are investigat- 
 ing, the formation of alcohol is found to be accompanied by the 
 presence of yeast. Here, then, is an opportunity for the applica- 
 tion of the Method of Agreement. It had long been noticed that 
 
 * Loc. cit., p. 91.
 
 424 THE PROBLEM OF LOGIC [XIII. iv. 
 
 yeast was apparently an invariable concomitant of vinous fermenta- 
 tion. Accordingly, a causal hypothesis was formulated ; 
 
 First Hypothesis : The substance known as yeast is a causal 
 antecedent of the alcoholic fermentation of grape-juice. 
 
 Tliis earliest form of the hypothesis could not easily be developed 
 or applied. Wliile microscopic research was still in its infancy, and 
 the microscope itself hardly yet worthy of its name, such a hypo- 
 thesis was of necessity comparatively barren. But about the year 
 1840 it was discovered by Schwann and Cagniard-Latour that the 
 yeast wliich is found in fermenting grape-juice consists of millions 
 of globules possessing the morphological characteristics of vegetable 
 cells. This, coupled wdtli the fact that these cells multiply as the 
 fermentation proceeds, convinced the investigators of the fact that 
 yeast is a plant. Thus, they were led to put forward an important 
 modification of the original hypothesis. 
 
 Second Hypothesis : The chemical changes known as the alcoholic 
 fermentation of grape-juice are caused by the physiological 
 activity of the living yeast-cell. 
 
 Thus, we see how advancing knowledge, due to a deeper and 
 more precise analysis of the data, results in the moulding of hypo- 
 theses. Historically, the h3rpothesis, thus modified, was consider- 
 ably strengthened by the close analogies observed between the 
 processes of putrefaction and alcoholic fermentation, tending to 
 show that in both processes living agency is involved. Moreover, 
 we notice that in its second form the hypothesis is no longer barren. 
 Firmly rooted in the systems of morphological and physiological 
 Science, it can be logically developed into forms which are capable 
 of experimental verification. The recognition of the yeast-cell as 
 a micro-organism possessing the properties of vegetable cells in 
 general enables us to draw upon botanical Science for premisses 
 with which to combine our hypothesis. Such premisses are the 
 following : 
 
 (i.) Vegetable cells, including the yeast-plant, cannot hve at very 
 high or at very low temperatures. 
 
 (ii.) They are immediately killed by being treated with certain 
 substances known as ' antiseptics ' — e.g., corrosive sublimate, sul- 
 phuric acid, carbolic acid. 
 
 (iii.) Micro-organisms cannot (so far as we know) arise ' spon- 
 taneou.sly.' Thus, a liquid which by prolonged boiling or other 
 means lias been freed from all living micro-organisms will remain 
 so if kept from all contact with free atmospheric air and otherwise 
 uninfected. 
 
 Combining these statements in turn with our modified hypothesis, 
 we obtain, as ' developed ' forms of that same hypothesis, the state-
 
 CHAt. XLVI.] ILLUSTRATIONS 425 
 
 ments that, in the process wliich we are investigating, the exciting 
 cause o' the chemical change (and therefore in this instance the 
 chemical change itself) is (i.) something that cannot exist at ex- 
 tremely high or extremely low temperatures ; (ii.) something that 
 cannot exist in the immediate presence of certain antiseptics ; 
 (iii.) something that, if not antecedently present, cannot arise 
 spontaneously in any substance kept under aseptic conditions. 
 
 These, and other developed forms of the hypothesis, were duly 
 applied, and tested by means of a long series of experiments, of 
 wliich these are some of the results 
 
 (i.) Grape-juice ferments mtliin a certain range of temperature 
 only. It does not ferment at any temperature much higher than 
 60° C. If the fermenting liquid is boiled, the fermentation invari- 
 ably ceases ; nor does it recommence on cooling unless after the 
 liquid has been left for some time in contact with atmospheric air. 
 
 (ii.) The fermentation may always be arrested by treating the 
 liquid with .sulphuric acid, bisulphide of carbon, carbolic acid, or any 
 other of a number of antiseptics. By an experiment in antiseptic 
 treatment Schwann verified that part of the modified hypothesis 
 wliich assigned a vegetable as distinguished from an animal organism 
 as the exciting cause in alcoholic fermentation, and at the same 
 time experimentally discovered the limit of the analogy between 
 this kind of fermentation, on tlie one hand, and putrefaction on the 
 other. ' He found that white arsenic and corrosive sublimate, 
 being poisonous to both plants and animals, stop both putrefaction 
 and fermentation, while extract of nux vomica, being destructive 
 of animal but not of vegetable life, prevents putrefaction, but does 
 not interfere with vinous fermentation.'* 
 
 (iii.) ' Perfectly pure grape-juice does not ferment, unless the 
 process has been started by at least temporary contact Avith ordinary 
 air. This cardinal fact was observed by Gay-Lussac in a now 
 classical series of experiments. He caused clean grapes to ascend 
 through the mercury of a large barometer into the Torricellian 
 vacuum, where he crushed them by means of the mercurial column. 
 The juice thus produced and preserved remained unchanged, but 
 the addition to it of ever so small an air-bell (as a rule) induced 
 fermentation, wliich, when once started, was always found to take 
 care of itself. 'f 
 
 Such experiments (and others of wliich we have still to speak) 
 were the means of a prolonged and varied application of the Method 
 of Difference — an apphcation in which it is noticeable that the 
 positive is in many cases prior to the negative instance. Instead 
 of introducing the suspected causal antecedent, and so obtaining 
 the positive instance from the negative, the investigators more 
 often subtracted the suggested agent, and so obtained the negative 
 instance from the positive — though, in the case of the air-bubble 
 
 * • Encyclopaedia Britannica,' loc. cit., p. 95. t l^id., p. 94.
 
 420 THE PROBLEM OF LOGIC [XIII. iv. 
 
 admitted to the Torricellian vacuum, and the atmospheric air left 
 in contact vrith the boiled and coohng must, the positive instance 
 was re-obtained from the negative. 
 
 The obvious danger in each case was that of subtracting too much. 
 For instance, when air was excluded from the germless liquid, was 
 it not possible that the micro-organisms in atmospheric air were 
 not the only relevant and important circumstance thereby sub- 
 tracted ? j\Iight not the exciting cause of the fermentation be the 
 air itself, or some constituent of it ? This point was decided by 
 means of a crucial instance devised by Schwann. He boiled grape- 
 juice, thus subtracting all organic life, and then admitted air to it, 
 but only through a red-hot glass tube, which, wliile allowing the 
 ])assage of the air, must destroy any living germs that it might 
 contain. The grape-juice so treated did not ferment, and thus, 
 by a striking application of the Method of Single Difference, it was 
 proved that air, qtui air, is not responsible for the fermentation of 
 grape- juice. 
 
 Meanwhile the rapid advance of Science was making possible a 
 fuller analysis of the data and a further development of the hypo- 
 thesis. Rooted now in the system of chemical as well as in that 
 of biological Science, that hypothesis could be combined with such 
 premisses as — 
 
 (iv.) Living vegetable cells cannot, as such, constitute a soluble 
 substance ; while advancing knowledge of micro-organisms and 
 their movements had furnished the premiss — 
 
 (v.) A certain thickness of cotton-wool is an impassable barrier 
 to micro-organisms [wliich become enmeshed in it, and cannot slip 
 through]. 
 
 Combined in turn with each of these premisses, the hypothesis 
 develops into the statements that the causal antecedent for which 
 we are seeking is (iv.) not soluble, and (v.) something of which the 
 passage may be prevented by the interposition of cotton-wool. 
 
 From the statement that the required agent is not soluble Helm- 
 holz deduced the still further developed hypothesis that it is unable 
 to pass through the wall of a bladder. This form he was able to 
 verify experimentally by suspending a sealed bladder containing 
 grape- juice in a quantity of fermenting must. The liquid contained 
 in the bladder remained unchanged. 
 
 Many years after this Hoffmann ' took a test-tube full of sugar- 
 water, and by a plug of cotton-wool inserted within it divided the 
 liquid into two parts. To the upper part he added yeast, which, 
 of course, induced fermentation there ; but the change did not 
 propagate itself through the cotton-wool to the lower portion.'* 
 Thus, again, a developed form of the hypothesis was conclusively 
 verified. 
 
 All these investigations, taken together, pointed irresistibly to 
 
 * ' Encyclopaedia Britannica,' he. cit., p. 95,
 
 Chap. XLVL] ILLUSTRATIONS 427 
 
 the truth of the Second Hypothesis — i.e., the orif^inal hypothesis 
 as modified by Soliwann and Cagniard-Latour. That the cliemical 
 changes known as the alcohoUc fermentation of grape- juice are 
 caused by the physiological activity of the living yeast-cell could 
 no longer be doubted. It had been triumphantly established as a 
 causal law. The ideal of ' reversibility,' indeed, had not been, and 
 possibly will never be, reached in this case. In view of the facts 
 (I) that a number of fermentations have been proved to be inde- 
 pendent of physiological activity, and (2) that even alcoholic 
 fermentation of minute quantities of sugar seems in some cases 
 to have occurred without the intervention of micro-organisms, we 
 cannot yet venture to assert that aseptic fermentation of minute 
 quantities of grape-juice has never taken place. But that the 
 ordinary process of vinous fermentation is in all cases due to yeast- 
 plants has long been an established fact. 
 
 But the Second Hypothesis could not be regarded as final. In 
 several points it is lacking in scientific precision. What is meant 
 by the word ' caused ' ? Is the yeast-cell to be regarded as an im- 
 mediate or only as a remote causal antecedent ? Is the presence 
 of the living cell in all cases necessary, or is it only a separable 
 product of the living cell that is indispensable ? What precisely 
 is meant by the use of the word ' physiological '? Does it imply 
 that the process of fermentation is in itself anything but a purely 
 chemical reaction ? And, finally, what is meant by ' the yeast- 
 cell '? Is it one species or many ? If there are different species 
 of the yeast-plant, are there corresponding differences in the fer- 
 mentations which they severally induce ? 
 
 We have not space in which to follow out in detail the investiga- 
 tions to which these questions have given rise. With regard to the 
 interesting botanical researches into the nature of the yeast-plant, 
 we can only sa.y that they have resulted in the establishment of the 
 fact that vinous fermentation results from an activity (probably to 
 be regarded as pathological) of the genus of fungus known as Sac- 
 charomyces. This genus includes a considerable number of species, 
 of which the most important is S. cerevisice. Each plant of this 
 species is a single cell of globular form measuring about yg-jy milli- 
 metre in diameter. 
 
 From the logical point of view, the most interesting question 
 suggested is that of remote or immediate causation. This problem 
 gave rise to two rival hypotheses — that is, the hj'pothesis was, by 
 different investigators, moulded in two (apparently) opposite ways ; 
 its progressive modification advanced along divergent lines. 
 
 Third Hypothesis : The changes knoivn as the alcoholic fermenta- 
 tion of grape-juice are of a purely chemical character. The 
 living organism is a remote causal antecedent. 
 
 We may call this the Hypothesis of Liebig.
 
 42S THE PROBLEM OF LOGIC [Xm. iv. 
 
 Fourth Hypothesis : The changes known as the alcoholic fer- 
 mentation of grape-juice represent a physiological process. 
 They take place within the organism of the yeast-plant, and 
 are immediately caused thereby in this sense — that the products 
 of the fermentation are to he regarded as actual products of 
 the metabolism of the cell. 
 
 This is the Hypothesis of Pasteur. 
 
 The question between these rival hj'pothcses cannot yet be said 
 to have been finally decided. In forms modified b}" recent research 
 they still confront one another. But for a time Pasteur's theory 
 was completely triumphant, and until the year 1897 that of Liebig 
 was regarded as finally discredited. Both hypotheses were duly 
 developed, applied, and tested. The liistory of the (temporary) 
 estabhshment of Pasteur's theory is logically the more interesting. 
 
 Pasteur's work was at first of a purely revisionary and critical 
 kind. ' He did the whole of the work of Schwann and the rest of 
 his predecessors over again, modifying and perfecting the experi- 
 mental methods, so as to silence any objection or doubt that might 
 possibly be raised, repeating and multiplying his experiments until 
 every proposition was firmly established.'* 
 
 Pasteur's positive work consisted essentially in generalizing the 
 theor}^ of the connexion of alcoholic fermentation with organic 
 agency. He supported his hypothesis not as an isolated thesis, 
 but as part of a larger theoretical system. To express the matter 
 in a way less adequate but more logically precise, he supported his 
 hj'pothesis by strong analogical arguments which his extended 
 researches had made possible. 
 
 It is well knoviTi that alcoholic or ' vinous fermentation is only 
 one of a number of fermentative changes to which sugar is liable. 
 The samef substance — sugar — which, when placed under certain 
 conditions, breaks up into alcohol and carbonic acid, under certain 
 other sets of conditions ferments into lactic acid, and through lactic 
 into butyric acid, or into gum plus mannite. , . . What Pasteur 
 showed is that each of these changes is the exclusive function of a 
 certain species (or at least genus) of organism. What the yeast- 
 plant is for vinous a certain other organism is for lactic, a third for 
 mannitic, a fourth for butyric fermentation. No two of these 
 species, even if they belong to the same genus, will ever pass into 
 each other. Pasteur arrived at this great generahzation by means 
 of his invention of an ingenious method for cultivating pure growths 
 of the several species, so that each of them could be examined 
 separately for its chemical functions.'^ More recently this method 
 has been superseded by that of Hansen, a Danish chemist and 
 
 * ' Kncyclopaedia Britannica,' loc cit., p. 9.5. 
 
 t This is not quite accurate. Different sugars are variously constituted. 
 
 X ' Encyclopaedia Britannica,' loc. cit., p. 95.
 
 Chap. XL^^.] ILLUSTRATIONS 429 
 
 botanist, who showed that Pasteur's so-called pure cultures did 
 not really deserve the name, and that, in order to obtain a culture 
 that is certainly and strictly pure, it is necessary to begin with a 
 single cell ; but Pasteur must still be regarded as having done 
 inestimable service to Science in the investigation of yeast-fermenta- 
 tion, however imperfect some of his methods may have been. His 
 researches were accepted by the scientific world as having conclu- 
 sively established the theory that yeast-fermentation is a ' vital 
 phenomenon ' taking place within the living cell. 
 
 But the Chemical as opposed to the Physiological Hypothesis had 
 not yet been finally disproved. In 1897 an important discovery 
 was made by Buchner. By means which Professor Ray Lankester 
 has truly characterized as ' heroic mechanical methods,' he suc- 
 ceeded in obtaining from brewing-yeast what he regards as an un- 
 organized substance, which certainly is able, quite apart from the 
 living yeast-cell, to induce alcoholic fermentation. By prolonged 
 grinding with quartz sand, and by then subjecting the mass 
 of disintegrated cells to high pressure, he at length produced the 
 liquid ' zymase ' which he regards as an enzyme that by a purely 
 chemical process effects alcoholic fermentation. This substance 
 appears to be soluble — at least, the liquid may be made to pass 
 through a filter of porous porcelain without losing its property of 
 exciting fermentation. Thus, the Chemical Hvpothesis of Liebig, 
 so long discredited, may be said to have been revived, though in 
 a modified form, in the Enzyme Hjrpothesis of Buchner. The 
 Ph3'siological Hvpothesis of Pasteur can no longer stand as he 
 formulated it. At present it is struggling to save its life by sub- 
 mitting to an important modification. The fermentation (says the 
 physiological faction) induced by Buchner's wrongly termed enzyme 
 is due not to a ferment separable from the substance of the cell, but 
 to minute particles of living protoplasm, wliich are not, indeed, 
 soluble, but are sufficiently miscible with water to be able to pass 
 through the porcelain filter. 
 
 Thus, the Chemical and the Physiological hj'potheses both (in 
 modified forms) still survive. Will some investigator of the future 
 devise at last some crucial instance which will lead to the final 
 triumph of one, the final defeat of the other ? Or will some recon- 
 ciliation between them be brought about by a deeper analysis of 
 the meaning of that difference between chemical and physiological 
 process of which we speak so readily, but which yet is not witiiout 
 that suspicion of vagueness which may be the indication of the 
 need of prcciser definition ? Is it possible that the two divergent 
 forms of the hj^othesis may be so modified as to converge again, 
 and finally to coincide ? As Organic Chemistry advances, may it 
 prove that our distinction betAveen physiological and ' purely 
 chemical ' process was, after all, iu tliis case a distinction without 
 a difference ?
 
 430 THE PROBLEM OF LOGIC [XIII. iv. 
 
 2. The Case of Algol, the Demon Star* 
 
 Algol is a variable star in the constellation Perseus. It goes 
 through a regular cycle of changes that are visible to the naked 
 eye, and have therefore been known for centuries. Algol is the 
 brightest of those variable stars which, instead of varying con- 
 tinuously, remain constantly at their maximum brightness during 
 the greater part of their period, and then temporarily lose a part 
 of their light, soon to regain their usual brilliancy. Stars which 
 vary in this peculiar way are relatively very few, only twelve being 
 as yet known to us. 
 
 The problem of causally accounting for the variability of Algol 
 is of long standing, and some of the earlier hypotheses were crude 
 and fanciful. The star was at one time believed to be the eye of a 
 demon. The development and application of this attractive hypo- 
 thesis may safely be left to the ingenuity of the reader, who will 
 find no difficulty in combining it with such premisses concerning 
 the blinking of demons as Mythology or Folk-lore may supply. 
 We need hardly say that, considered from the Astronomical stand- 
 point, tliis prcscientific hypothesis is unfortunately barren. 
 
 From the later stages of this long-continued inquiry we select a 
 few salient points, ^^^thout aiming at strictly chronological order. 
 
 First Hypothesis : The variability of Algol is due to a series of 
 explosions. 
 
 This hj'pothesis did not prove itself fruitful in any precise or 
 satisfactory sense. Too vague to take firm root in any scientific 
 system of Astronomy, it put forth only feeble shoots of indeter- 
 minate probabihty. Yet, sUght as its vitality was, the hypothesis 
 developed far enough to be quite sufficiently disproved. Into 
 whatever specific forms it ramified, the feebly swaying branches 
 all pointed away from the facts. Explosive action is sudden, 
 seldom rcjgular, likely to result in a maximum brilliancy of short 
 duration and long intervals of diminished light. The variation 
 of Algol is steady and regularly recurring, and consists in the 
 occurrence of a relatively short interval of diminished brilliancy 
 between much longer times of maximum brightness. Even the 
 idea of a venerable demon indulging every three days in one 
 solemn and deliberate wink was in some respects less incredible 
 than the Explosion Hypothesis. Such a suggestion had not 
 strength enough even to bear modification. It was simply dis- 
 carded. 
 
 The next hypothesis that we select was based on an ingenious 
 argument from Analogy. 
 
 • Our ar-count of this investigation is partly based upon that given by Sir 
 Kobert Ball in the volume entitled ' In the High Heavens,' pp. 179-190.
 
 Chap. XLVI.] ILLUSTRATIONS 431 
 
 Second FltjpotJiesis : The case of Algol is analogous to that of 
 our SU71. The periodic increase and decrease of sun-spots 
 constitutes our sun a variable star. The periodic variation 
 of Algol is due to similar causes. 
 
 Here was a hypothesis that was not wanting in vitality. It 
 could be developed in various ways, but all of them pointed to its 
 own disproof. If we were as far from our Sun as we are from 
 Algol, even were the sun-spots much larger and more frequent 
 than they are, the variation in brightness would be altogether 
 imperceptible. And when the two periods had been precisely ascer- 
 tained, the analogy entirely broke down. That of the sun-spots 
 is a period of eleven years, that of Algol's variation less than tliree 
 days ; and 610 days is the very longest period that we know as that 
 of any variable star. We need not follow out in detail the process 
 which led to the rejection of tliis attractive but untenable con- 
 jecture. 
 
 Third Hypothesis : The variation of Algol is due to a rotatory 
 movement of the star, of which one aspect is brighter than 
 another. 
 
 Here we stand on much firmer ground. The laws of rotatory 
 motion are well established, and furnish undoubted premisses 
 with which to combine our hj'pothesis so as to cause it to develop 
 in more than one direction. Moreover, we are now considering a 
 stage of the investigation at which the original problem has been 
 more precisely stated. The telescope and the camera have both 
 been brought to bear upon the problem, and fuller data have thus 
 been acquired for scientific analysis. The results of observation as 
 they stand at the present time are as follows : 
 
 For about two days, ten hours, the star remains constant in 
 brilhancy. Then its brightness declines so rapidly that in a few 
 hours it has lost three-fifths of its brightness. In tliis stage of 
 lowest brilliancy it remains about twenty minutes, and then begins 
 to grow brighter again, so that in a few hours more Algol is as 
 brilliant as ever. 
 
 Days. Hours. Minutes. Seconds. 
 
 Time of maximum brilliancy . . 2 10 — — 
 
 (appro.ximately) 
 Time of decline and rise of brilliancy — 10 48 52 
 
 Time of wliole period . . . . 2 20 48 52 
 
 (approximately) 
 
 This period is liable to fluctuations of a few seconds, but other- 
 wise it is wonderfully uniform. 
 
 Now, when the rotation-hypothesis is developed into forms which 
 can be precisely applied, and compared with these data, verifica- 
 tion is found to be impossible. No form of the hyjiotliesis can be 
 obtained wliich does not require tlie time of maximum brightness
 
 432 THE PROBLEM OF LOGIC [XIII. iv. 
 
 to be much more nearly equal to that of diminished lustre than is 
 actually the ease. If the hj^othesis is not to be entirely discarded, 
 it must at least submit to a radical reconstruction. 
 
 Fourth Hypothesis : The apparent variation of Algol is due to 
 Us being periodically eclipsed by a darker companion re- 
 volving round it. 
 
 This h^'pothesis was not chronologically the latest, but it was 
 the last to be so developed that a conclusive verification could be 
 attempted. Even our present telescopes enable us to see Algol 
 only as a pin-point of light, and the star is at an immeasurable 
 distance from us. When first suggested, and for long afterwards, 
 the h3^othesis was comparatively barren. Even the modem tele- 
 scopic camera, with its films so much more sensitive than the 
 human eye, could alone give no clue for its profitable development. 
 It could, indeed, be combined with premisses furnished by the Law 
 of Gravitation and the Laws of Motion, and so developed into a 
 wide range of possible results, some of which agreed with the 
 observed data ; but such a test as this, though it sufficed to dis- 
 prove the rotation-h3^pothesis, did not afford for the eclipse-hypo- 
 thesis an adequate verification. To disprove an erroneous hypo- 
 thesis is often easy ; to establish a true one is usually very difficult. 
 
 But at length the spectroscope gave the required clue. It was 
 observed that during that half of Algol's period which follows the 
 first half of the time of minimum brilliancy the dark lines in the 
 spectrum of the star shifted slightly towards the violet end, and 
 that during the other half of the period they shifted towards the 
 red end of the spectrum. 
 
 Now let us again consider our Fourth Hypothesis, to see whether 
 it can in any way be developed, so that this observed fact may be 
 u.scd for its verification. 
 
 For brevity's sake, we formulated our hypothesis inaccurately. 
 What the Law of Gravitation requires it to mean is not that Algol 
 stands still and his dark companion revolves round him, but that 
 the Algol system consists of two bodies (of which one is luminous) 
 revolving about their common centre of gravity. Now, a little 
 consideration will show that if that be really the case, then the 
 luminous body vnW always be approaching us for some time after 
 the eclipse, and retreating from us for an equal time before it, these 
 two times, together with the time of the eclipse itself, constituting 
 the time of the whole period. As the luminous body moves not 
 in a straight line, but in an elliptical orbit, the velocity of the 
 luminous body in the line of sight will, of course, not be uniform. 
 Now, it can be deductively shown tliat, assuming the truth of the 
 now well-established Undulatory Theory of Light, the dark lines 
 in the spectrum of a source of light which is moving towards the 
 observer will shift slightly towards the violet end ; and that if the
 
 Chap. XL\I.] ILLUSTRATIONS 433 
 
 source of li;,'iit be moving away from the observer, they will shift 
 sli,i?litly towards the red end of tlie spectrum. 
 
 Thus we see that, by combining our Fourtli Hypothesis Avith 
 premisses furnished by the Law of Gravitation and the Laws of 
 Motion, it has been possible to develop it into a form in which it 
 can be ai)])licd by means of deductions from an accejjted theory 
 of physical Science. Thus the eclipse-hypothesis was at length 
 established by a convincing verification. Further accurate ob- 
 servation of the j^hotographic spectrum of Algol showed the exact 
 length of the times during which the dark lines moved towards 
 the violet and the red end respectively, and it was shown that during 
 just one-half of its period Algol must be moving towards us with 
 a maximum velocity of twenty-six miles per second in the line of 
 sight, and that during the other half it must be receding from us 
 Avith the same maximum speed of twenty-six miles per second. 
 These spectrosco23ic observations confirmed the hj^pothesis not only 
 in regard to the motion of Algol, but also in the statement that the 
 companion-star must be dark relatively to Algol, since, were both 
 stars brigiit, two sets of dark lines would have appeared instead of 
 only one. Moreover, from these observations, combined with pre- 
 misses furnished by the Law of Gravitation and the Laws of Motion, 
 it was found possible to deduce a form of motion for Algol which, 
 while perfectly agreeing u-ith the undeveloped hypothesis, could 
 now be more jjreciseh^ stated. Of the dark companion and its 
 movements the spectrum analysis could, of course, say nothing. 
 
 The previously observed facts as to the variability of Algol could 
 now be interpreted in the light of the established hj^pothesis, and 
 of the results of deduction, themselves based upon the analysis of 
 the spectral phenomena. Each detail of the original observations 
 can now be interpreted in terms of the newly established theory. 
 Thus : 
 
 {n) The period of Algol's variation means the period of Algol's 
 revolution in its orbit. Therefore, the time of this j^eriod 
 is two days, twenty hours, forty-eight minutes, fifty-two 
 seconds (approximately). 
 
 (b) The periodic fall and rise of brilliancy in Algol means the 
 
 regularly recurring eclipse of Algol by its dark companion. 
 The brillianej^ begins to diminish as Algol begins to pass 
 behind the dark body ; it begins to increase as the dark 
 body begins to pass from the disc of Algol. The whole 
 time of eclipse is, therefore, about eleven hours. 
 
 (c) The time of minimum brilliancy 7)icans the time during which 
 
 the ivhole of the dark body is eclipsing Algol. Therefore 
 tliis time of the transit of the dark body over Algol is 
 twenty minutes. 
 
 28
 
 434 THE PROBLEM OF LOGIC [XIII. iv. 
 
 {(I) Algol's losing three-fifths of its brightness during this time of 
 greatest eclipse means that the dark companion is then 
 covering three-fifths of its disc. 
 
 All these are not deductions in the true sense of the word, but 
 interpretations of one set of symbols in terms of another. 
 
 We come now to a stage in the investigation which we may call 
 that of qiiantitative deduction. Given (a), (6), (c), {d) (as above) in 
 conjunction with what we know about the velocity of Algol in its 
 nearly circular orbit, we are able to deduce the following results : 
 
 (1) The magnitude of Algol's orbit about the common centre of 
 
 gravity. 
 
 (2) Tlie distance between the two globes. 
 
 (3) The masses of the two globes, and also their dimensions, if we 
 
 are allowed to assume what we have reason for believing 
 is more or less true — that the two bodies are of the same 
 density. 
 
 (4) The nature of the dark body's orbit, and its velocity in that 
 
 orbit, provided that we are allowed to assume that this 
 dark body, as well as Algol itself, obeys the Law of Gravita- 
 tion — i.e., a law of attraction varying directly with the 
 mass of two attracting bodies, and inversely as the square 
 of their distance. 
 
 The following are some of the results thus obtained : 
 
 Algol is twice as large as our sun, having a diameter of more than 
 a million miles. Its weight is only one-half that of our sun, its 
 average density being less than that of water, a little greater than 
 that of cork. 
 
 Algol's dark companion is of the same size as our sun, and has 
 one-quarter of our sun's mass. Its orbit, like that of Algol, is 
 nearly circular, and its velocity in that orbit almost uniform — 
 about fifty miles per second. 
 
 The distance between the two globes is about three million miles. 
 
 So ends for the present the story of Algol, the Demon Star. The 
 results may be regarded as perfectly trustworthy, provided we have 
 been right in assuming — 
 
 (i.) The truth of the Law of Gravitation as applicable to Algol 
 
 and its companion, 
 (ii.) The truth of the Laws of Motion, 
 (iii.) The truth of the Undulatory Theory of Light, 
 (iv.) The equal densities of the two stars. 
 
 Of these, (i.), (ii.), and (iii.) are fundamental, and about as certain 
 as any fact of knowledge whatsoever, especially (i.) and (ii.). The
 
 Chap. XLVL] ILLUSTRATIONS 435 
 
 only elomont of real uncertainty lies in (iv.). Cut even if the 
 densities should be found not to be equal, the only consequence 
 would be a comparatively small modification of the numerical 
 results. Substantially our account of the Algol s\'.stem would 
 remain the same. 
 
 If the stor}' of Algol is to close %nth a logical moral, we would 
 point out that the great lesson it teaches is that of the importance 
 of Analysis. But for the minute and careful analysis of the spectro- 
 scopic data the causal hypothesis could never have been established 
 or the quantitative deductions made. We must not, indeed, confuse 
 logical with physical anah'sis. The so-called spectrum analysis — 
 the spreading out of the light of Algol into its elements — was not a 
 logical but a physical process. The logical analysis came after- 
 wards. It was the careful distinction between tho.se spectro.scopic 
 phenomena which concerned the qualitative properties of the light 
 and those which pointed to movements of the source of light itself. 
 It is noticeable that tliis analysis was exercised upon the results 
 not of direct, but of what we may call indirect observation. When 
 the direct observation of Algol's luminous changes had been carried 
 as far as the case admitted, it gave place to the observation of the 
 phenomena of Algol's spectrum. Thus, it often happens in scientific 
 investigation that the analysis of some obscure phenomenon enables 
 us sooner or later to discover some hitherto uninterpreted indica- 
 tions which, though not obviously or directly bearing upon the 
 problem before us, yet enable us to apply to its elucidation some 
 part of our previously acquired knowledge. Analysis is that 
 essential process in scientific investigation whereby it is possible 
 to bring old knowledge to bear effectively on that which is yet 
 unknowTi. 
 
 3. The Bigiditij of the Earth.'' 
 
 The high temperature of the interior of the Earth — i.e., of the 
 whole bulk of our planet M-ith the exception of a relatively very 
 thin crust — is a well-established fact. From early times such 
 phenomena as those of hot springs and the lava-streams of molten 
 rock ejected by volcanoes had attracted attention and called for 
 explanation. More recently, the fact that the temperature in the 
 lower parts of deep mines is higher than that at the earth's surface 
 was recognized as pointing in the same direction, and experimental 
 investigations have now established the fact that, from a point 
 about 100 feet beneath the surface, the temperature of the Earth 
 increases at the rate of 1° F. for every 66 feet of descent — i.e.. 80° 
 for the first mile. 
 
 The causal investigation of such phenomena as pointed to the 
 
 * Our account of this investigation is based mainly upon Sir Robert Ball's 
 discussions in ' The Earth's Beginning,' oh. ix., and ' In the High Heavens,' ch. iii. 
 
 2s 2
 
 436 THE PROBLEM OF LOGIC [XIII. iv. 
 
 existence of extremely high tempei-atures beneath the Earth's crust, 
 when viewed from the standpoint suggested by the initial stages 
 of the Nebular Theory, gained immensely in interest and import- 
 ance, and necessitated the formulation of some definite hy2:»othesis 
 as to the conditions upon whicli these j^henomena depend. The 
 earlier forms in which tlie hj^othesis took shape were founded on 
 a consideration of the high-temperature conditions of the Earth's 
 interior, while the high-pressure conditions, which are C[uite as 
 important, were comjiaratively ignored. If we may be allowed to 
 use the word ' fluid ' in that jiopularly accepted sense in which it is 
 opposed to ' rigid,' we may briefly formulate the initial hyiDothesis 
 thus : 
 
 First Hypothesis : The interior of the Earth is fluid. 
 
 We have not space in which to follow out the various processes 
 whereby this hyi^othesis was developed, applied, and disjjroved. 
 Speaking quite briefly, and aiming at logical rather than chrono- 
 logical arrangement, we may say that this early form of the hypo- 
 thesis was developed in two main directions : 
 
 (i.) Combined vrii\\ i:)remisses furnished by the Laws of Motion, 
 it Avas developed into conseciuences which could be com- 
 2oared with facts to which Sir Robert Ball somcAvhat 
 vaguely refers as ' certain astronomical phenomena con- 
 nected wdtli the way in which the earth turns round on 
 its axis.' 
 (ii.) Combined with the Law of Gravitation and certain well- 
 known laws of Hydrostatics, it was developed into con- 
 sequences which could be compared with established facts 
 regarding the ebb and flow of the tides. 
 
 It was found that in both these developed forms the hypothesis 
 broke down under the test of attempted verification. 
 
 Meanwhile, Astronomers had been implicitly building many of 
 their theories and investigations upon an hypothesis directly op- 
 po.sed to tliat which we have been considering. It is true that 
 absolute rigidity was not supposed to have any actual existence, 
 but it was in many investigations taken for granted that the rigidity 
 of the Earth is so great that without practical risk of error the 
 Earth may be considered as behaving like an absolutely rigid body. 
 We must now consider the fortunes of this form of hypothesis. 
 
 Second Hypothesis : The Earth may he taken as behaving like 
 an absolutely rigid body. 
 
 Passing over many interesting developments of this hypothesis, 
 and the partial verifications which in many cases justified its use
 
 CiTAP. XLVT.] ILLUSTRATIONS 437 
 
 as a simplification resulting in no practical error, we must give some 
 account of one development which is of very special interest, since 
 it ultimately led to the disproof of the hypothesis itself. Moreover, 
 it is specially interesting as being an unintentional development in 
 this sense — that it was not carried out with any intention of testing 
 or verifying the h^'pothesis of which the truth was implicitly 
 assumed. The development took the form of an investigation 
 respecting a supposed movement of the North Pole of the Earth 
 relative to the Earth's surface. 
 
 Most of us can identify the Pole Star in the heavens. Observa- 
 tion shows that its apjDarent position remains practically the same 
 all through the night, while the other stars appear to revolve round 
 it. As a matter of fact, it does every twenty-four hours apparently 
 describe a small circle about another point in the heavens. This 
 point, which is entirely stationary relatively to the diurnal apparent 
 motion of the stars, is called the Celestial Pole. 
 
 Let the reader imagine that his eye is placed at the centre of the 
 Earth, and that a long, slender tube passes from that centre to the 
 surface. If this tube be so placed that, when looking through it 
 from the centre of the Earth, the eye is directed exactly to the 
 celestial pole, then that spot at wliich the end of the tube passes 
 out through the surface of the Earth is the North Pole. 
 
 We have now to consider whether this imaginary tube or axis, 
 about which the Earth daily rotates, always cuts the surface of the 
 Earth (north of the Equator) at exactly the same point of that 
 surface. We must remember that it is only an imaginary axis, or 
 we shall tend to think that this must necessarily be the case. If 
 I make an orange spin round a steel knitting-needle that pierces 
 its centre, that needle, however it moves, \vill always cut the 
 surface of the orange at the same point of the jDecl. But there is 
 no such axis as this to exert a physical compulsion on the Earth ; 
 and the question arises : ' Assuming that the Celestial and North 
 Poles remain in fixed positions, does the Earth move at all in 
 such a way as to make it necessary to identify in succession 
 different points of its surface with the North Pole V Or, to put 
 it differently, is there a movement of the -pole over the surface of the 
 Earth I 
 
 Now, the great mathematician Euler, assuming the absolute 
 rigidity of the Earth, and combining that assumption vdih well- 
 established laws of Astronomical Science, was able to show that a 
 rotatory movement of the North Pole is physically possible, and 
 that, if it takes place, the period must be completed in ten 
 months. 
 
 This statement about a ten months' period, ha\ing been 
 deduced from the assumed absolute rigidity of the Earth, we 
 may regard as a developed form of our Second Hypothesis. 
 That Euler did not call it so, or regard it as a means of testing
 
 43S THE PROBLEM OF LOGIC [XIII. iv. 
 
 that hypothesis, need not disturb us. We are regarding the 
 investigation from the logical, and not at all from the liistorical 
 point of view. 
 
 Euler's calculation remained unchallenged for more than a 
 centurv. At length it occurred to a certain Mr. Chandler that 
 Euler's deductive result should be tested by comjDarison with the 
 results of observations made during the past century as to the 
 movement of the North Pole. A careful scrutiny of these recorded 
 observations enabled liim to discover that the rotatory movement 
 does indeed take place, that the pole describes a circle not more 
 than a dozen yards in radius, but that the period is completed, not 
 in ten mouths, but in fourteen. 
 
 Xow, Chandler was not consciously testing the absolute rigidity 
 assumption any more than Euler had been consciously developing 
 it, and the true import of the discrepancy between the results of 
 deduction and the results of observation was not at first perceived 
 by anyone. Chandler's observational results were received with 
 incredulity by the theorists, who maintained that Euler's reasoning 
 was faultless. The outcome was a veritable dead-lock between 
 theory- and observation. 
 
 At length Simon Xewcomb reasoned thus witliin himself : Euler, 
 in making liis calculations, assumed that the Earth is a perfectly 
 rigid body. His assigning of a ten months' period for the rotatory 
 movement may depend entirely on that assumption. It is possible 
 that, if the Earth is not taken as being perfectly rigid, there may 
 still be deducible a rotatory movement of the pole, but that tliis 
 movement may be shown to have a period of fourteen months 
 instead of ten. These ideas were worked out, and theory and 
 observation, after many hitches, were finally reconciled. Thus 
 Xewcomb showed the true meaning of the discrepancy discovered 
 by Chandler. He showed that it amounted to a disproof of the 
 absolute-rigidity assumption, to the establishment of the theory 
 that the Earth is far from being ideally rigid. More recent researches 
 have confirmed this conclusion, and have shown that the measure 
 of the Earth's plasticity is exactly given by the observations which 
 Chandler adduced. 
 
 We .see, then, how, by a long and intermittent process, ca.rried 
 out at different periods by different investigators, and by means 
 of a method of which the application was to a great extent un- 
 conscious, the assumption which we have called the Second Hypo- 
 thesis was developed, applied, and in the end disproved. 
 
 This Second Hypothesis concerned the behaviour of the Earth 
 considered as a whole. Its disproof and the establishment of the 
 plasticity of the Earth as measured by Chandler's results bear 
 only indirectly on the original question as to the conditions of the 
 interior of the Earth as distinguislied from its outer crust. We
 
 Chap. XLVL] ILLUSTRATIONS 439 
 
 have now to consider the recent investigations by which the rigidity 
 of the Earth's interior relative to that of its outer crust has been 
 accurately ascertained. We have to consider the development, 
 verification, precise formulation, and final establishment of the 
 theory which, for the sake of simplicity and clearness, we may call 
 the Third H\^othesis, it being, of course, understood that our 
 numbers have no precise chronological significance. 
 
 Third Hypothesis : The interior of the Earth is at least as rigid 
 as its outer crust. 
 
 In order to understand the possible developments of this hj-po- 
 thesis, we must make sure that our conception of rigiditj' is suffi- 
 ciently precise. The question to be decided is not exactlj' the 
 question whether the matter of which the interior of the Earth 
 consists is solid rather than hquid. Not only is that matter in- 
 tensel}' hot ; it is also subjected to a pressure of manj^ thousands 
 of tons on the square inch. We have no experience of matter under 
 such conditions. Whether it is solid or liquid we cannot tell. It 
 is quite possible that, as Sir Robert Ball suggests, the words ' liquid ' 
 and ' solid ' may be equally inapplicable to it in the senses in wliich 
 we understand them. As ' many, if not all, solids may be made to 
 flow like liquids if only adequate pressure be applied,'* so it may 
 be that (for instance) water, if subjected to the enormous pressure 
 under which the materials of the bulk of the Earth exist, would in 
 respect of rigiditj' behave like cast-iron. Our hj'pothesis, then, 
 does not assert that the materials of which the interior of the earth 
 consists are solid as opposed to liquid. It asserts only their com- 
 parative rigidity. Now, the rigidity of any substance is measured 
 by the amount of force required to make its particles change their 
 positions relatively to one another. The greater the force required 
 to bring about a certain given amount of such change, the more 
 rigid is the body in question. 
 
 What, then, do we know about rigid bodies so defined ? What 
 statements about them does Physical Science supply which may 
 serve as premisses with which to combine our hypothesis with a view 
 to its fruitful development ? There are two such statements which, 
 representing well-established laws, enable the li3i)othesis to develop 
 in two important directions. These statements of fact are the fol- 
 lowing : 
 
 (i.) When a body receives a blow, the disturbance, jsropagated in 
 undulatory tremors through the substance of the body, travels at 
 greater speed the greater is the rigidity" of the body struck. 
 
 (ii.) When such tremors are propagated through a body, the 
 greater the body's rigidity, the less ^\ill be the consequent displace- 
 ment of its particles. 
 
 * ' The Earth's Beginning.' ch. ix., p. 1(51.
 
 440 THE PROBLE:\r OF LOGIC [XIII. iv. 
 
 These well-kno^ni facts of Physical Science enable the In^othesis 
 to be developed into two forms capable of being experimentally 
 tested : 
 
 (i.) Tremors propagated through the interior of the Earth 
 travel w-ith at least as great a speed as those propagated 
 through the outer crust. 
 
 (ii.) Tremors propagated through the interior of the Earth 
 cause no more, and possibly less, displacement of the 
 particles of the substances traversed than do tremors 
 propagated through the Earth's outer crust. 
 
 The development of the hypothesis in these two directions was no 
 difficult matter. But what of the experimental verification which 
 was necessary for its establishment ? What fabled giant of the 
 ^vildest fairy-tale could make or "^^'ield a sledge-hammer heavy 
 enough to give such a blow to the Earth as might result in the pro- 
 pagation of tremors throughout the globe ? The developed forms 
 of the h;v'pothesis might have waited long for their experimental 
 testing had not the Earth itself come to the rescue and supplied 
 (as, indeed, it had supplied during countless ages) a series of phe- 
 nomena which were now for the first time recognized as natural 
 experiments of the very kind that the hypothesis needed for its 
 convincing verification. 
 
 We have spoken above of the enormous amount of heat that 
 exists in the interior of the Earth. Now this heat is very slowly 
 indeed, but still unceasingly, passing away from the Earth. It is 
 continually rising by conduction to the surface, and thence is lost 
 by radiation into space. Professor J. D. Everett has estimated 
 that, were the whole globe covered with a shell of ice one-fifth of an 
 inch in thickness, that shell could be entirely melted by the amount 
 of heat which thus escapes annually from the Earth. 
 
 As a consequence of this continual loss of heat, the Eartli con- 
 tracts. Furtlier, the Earth-crust has to accommodate itself to this 
 perpetual shrinkage, and this adjustment causes violent shocks, 
 which reveal themselves at the Earth's surface in the form of earth- 
 quakes. Suppose an adjustment of this kind to take place among 
 the rocks at a depth of ten miles, where the pressure is thirty-five 
 tons on the square inch. Under such pressure as this even a slight 
 adju.stment must produce an exceedingly violent shock, of which the 
 effect is propagated in the form of undulatory tremors throughout 
 the globe. If the shock is sufficiently intense, the surface of the 
 Earth above the centre of disturbance will shake and rend as the 
 vibrations reach it. The wave-commotion spreads in all directions, 
 decreasing in violence as the distance from the centre of disturbance 
 increases, until it is no longer directly perceptible. Yet thousands 
 of miles away from the disturbed area, though we may not feel the 
 tremor, delicate instruments can. A seismometer, set up at any
 
 Chap. XLVL] ILLUSTRATIOXS 441 
 
 point on the Earth's surface, will be sensitive to earthquake tremors 
 which have become far too faint to be directly perceived bv our 
 senses, and faithfully records on its revolving paper drum the 
 particulars of all the earthquakes that take place even in the most 
 distant countries. These seismograms vary in character. Those 
 which represent Earth-tremors originating in any specific area have 
 a family resemblance. 
 
 Tiius, Professor Milne, in his laboratory at Shide, in the Isle of 
 Wight, has accurate news of every earthquake a very short time 
 after its occurrence. He looks at the seismogram, and observes 
 the nature of the tracing. If it is of a certain kind — cr^ — he is able 
 to say : ' This is the tracing proper to the Japanese group of earth- 
 quakes. Therefore an earthquake has been taking place in Japan 
 during the last half-hour, and the tracing tells me the magnitude of 
 the shock.' If the seismogram is of anotlier kind — o-g — he will say : 
 ' This is the tracing proper to the West Indian earthquakes ; I .see 
 that an exceptionally violent earthquake has just now taken place 
 in that region.' 
 
 Japan is the scene of very frequent earthquakes ; about a 
 thousand take place there every year. Let us suppose that a 
 vigorous earthquake has occurred in the neiglibourhood of Tokio. 
 The earth-tremors are propagated thence over the surface and 
 through the interior of the globe in all directions. Speaking 
 roughly, and putting the matter in the simplest way possible, we 
 may say that they reach the Isle of Wight by three main routes : 
 
 (1) Tlie direct route through the interior of the Earth ; 
 
 (2) The shorter superficial route through the Earth's crust ; 
 
 (3) The longer superficial route through the Earth's crust. 
 
 <ve> Si ^ 
 
 Shide / i'l''£<=i_IOi>ie_ --\ Tokio 
 
 (Isle of WIghty through Earth's interior. VjapaiO 
 
 i u p c r t ici*
 
 442 THE PROBLEAI OF LOGIC [XIII. iv. 
 
 The tremors propagated along these different routes reach Shide 
 at different times : 
 
 (1) About a quarter of an hour after an earthquake-shock has 
 
 been felt in Japan the pencil of the seismometer at Shide 
 begins to record the tremors arriving by the direct route 
 tlirough the interior of the Earth. 
 
 (2) Three-quarters of an hour after this the pencil makes 
 
 another record of precisely- the same form, but on a much 
 larger scale. Tliis represents the tremors arriving by the 
 shorter superficial route. 
 
 (3) About half an hour later still a precisely similar record is 
 
 made on a larger scale than the first, but on a smaller 
 scale than the second. 
 
 We are chiefly interested in two deductions that have been made 
 from these observations : 
 
 (i.) By comparing times of arrival and distances travelled, it has 
 been shown that the tremors propagated through the interior of the 
 Earth travel at greater speed than do those propagated by way of the 
 Earth's crust. Indeed, it has been shown that the velocity varies 
 with the square root of the depth beneath the surface. When the 
 tremors are traversing the Earth's centre the velocity of propagation 
 is more than ten miles per second ; near the surface it is not two 
 miles per second. 
 
 (ii.) Measurements of the amplitudes of the tremors, as repre- 
 sented by their respective seismograms, show that the tremors which 
 have travelled by the direct route have much smaller ampUtudes 
 than those which have travelled by either of the superficial routes. 
 Hence is deduced the fact that the particles of which the materials 
 in the Earth's iyiterior are composed are less displaced by any given 
 earthquake-shock than are those of the Earth's crust. 
 
 Thu.s the two developed forms of our M^pothesis have been 
 experimentally tested and established ; and so precise are the 
 results obtained that they not only verify the hypothesis itself, 
 but enable us to state it in a far preciser form than that in which 
 it was first proposed. It is now an established fact of knowledge 
 that the materials of which the interior of the Earth are composed 
 are more rigid than solid steel as we know it at ordinary tempera- 
 tures and under atmospheric pressure at the Earth's surface. 
 Thus we have reached the solution of the problem of the Earth's 
 riindity. 
 
 The investigation of this problem, when studied from the 
 methodological standpoint, offers several points of interest and 
 instruction. 
 
 In the first place, the long acceptance and the ultimate fate of 
 the assumption which we have called the Second Hypothesis teaches
 
 Chap. XLVI.] ILLUSTRATIONS 443 
 
 us the imijortant lesson that, when deductive theory on the one 
 hand, and the results of observation on the other, come to a dead- 
 lock, we must fall back upon such questions as these : Are the 
 assumptions adopted by Theory for simplicity's sake really trust- 
 worthy ? May not these simplifications have ignored some element 
 which, if taken into account, might bring about a reconciliation 
 between the results of Deduction and the data of Observation ? 
 Instead of discrediting the results of Observation, may not the dis- 
 crepancy be interpreted as pointing to the disproof of some hypo- 
 thetical assumption ? 
 
 A second important feature of the investigation is the fact that, 
 in the testing of the Second and the Third Hypotheses respectively, 
 the problem was api^roached from two different and independent 
 standpoints. We have here two separate groups of observed facts : 
 (1) Observations respecting the movements of the jjole ; (2) observa- 
 tions respecting earth-tremors. From each of these groups are 
 deduced certain conclusions respecting the rigidity of the Earth. 
 The first set of observations tends to emphasize its plasticity, the 
 second set its rigidity. In such a case it sometimes happens that 
 the conclusions drawn from the one point of view simply serve to 
 give greater definiteness to those drawn from the other standpoint, 
 in the present case it is possible that the amount of rigidity required 
 by the seismic observations may be found to be incompatible 
 with the amount of plasticity required by the observations regarding 
 the movements of the pole. If this is so, the whole argument will 
 have to be revised on both sides until the two groups of observa- 
 tions can be shown to be concordant. If such complete harmony 
 can be reached, it will involve the assigning of a narrowly restricted 
 degree of rigidity. The interior of the Earth must be shown to be 
 just rigid enough to please Mr. ]\Iilne, and just plastic enough to 
 content Mr. Chandler. The hope of an accurate solution of the 
 problem is thus greatly increased by the fact that the investi- 
 gation has been carried on from two points of view which are 
 apparently antagonistic in their requirements. If solutions of 
 a problem offered from two opposite sides of the question can 
 be shown to be coincident, the probability that the common result 
 is true is very great indeed. Generally speaking, there is no more 
 convincing \'indication of the truth of a questioned fact than the 
 proof that various independent lines of evidence all converge in 
 maintaining it. 
 
 A third point which tliis investigation illustrates is the intricate 
 complexity %\ith which phenomena cohere in Nature. Whether we 
 consider the ocean tides, or the rotatory movements of the Earth 
 and of its poles, or its internal tremors, the question of the Earth's 
 rigidity is seen to be equally involved. The methodologically 
 important conclusion that we draw from such instances as this is 
 that, whenever we reason from any grou^) of observed facts, we
 
 444 THE PROBLEM OF LOGIC [XIII. iv. 
 
 are drawing conclusions to which the remaining facts of Xature are 
 not by any means indifferent. Our conclusions are always liable 
 to be called to account by reasonings based on other groups of facts, 
 perhaps apparently remote from those on which we reasoned, but 
 really together with them belonging to one and the same har- 
 monious svstem.
 
 XIV. 
 THE INDUCTIVE POSTULATE.
 
 CHAPTER XLVII. 
 
 THE INDUCTIVE POSTULATE : THE POSTULATE OF CAUSAL 
 
 EXPLANATION. 
 
 The Meaning of the Term ' Postulate.'' 
 
 In Scientific Theory we have four important terms which are too 
 often confused vnih one another, though they are by no means 
 synonymous. These are the terms ' presupposition,' ' working 
 idea,' ' hypothesis,' and ' postulate.' 
 
 The 'presuppositions of an abstract .science are the estabhshed 
 results of the simpler sciences, results which it accepts uncritically, 
 or takes for granted. Thus, Geometry presupposes the funda- 
 mental results of the Science of Number, Physics presupposes an 
 already elaborated science of quantity in general, Chemistry pre- 
 supposes the results both of Mathematics and of Physics — i.e., it 
 assumes, as already established, the properties of Quantity and 
 Motion, dealing mainly with the atomic and molecular composition 
 of bodies. Biology presupposes the results of Chemistrj% Physics, 
 and Mathematics — the properties of Quantity, Motion, and mole- 
 cular composition — as already established, dealing directly with 
 organic relations only. 
 
 The working ideas of a Science are certain fundamental assumja- 
 tions which, in the attempt to elucidate its own subject-matter, it 
 finds itself called upon to make. They differ from presuppositions 
 in this — that they concern the subject-matter of the science itself, 
 and not that of other more elementary sciences ; and they differ 
 from hypotheses in this — that they represent the relatively per- 
 manent, whilst hypothesis represents the more fluctuating, element 
 in the inquiry. As in military strategj^ so in scientific research, it 
 is necessary to have a basis of operations which remains fixed, 
 however the plan of the inquiry be modified. A science of Com- 
 parative Biology may posit Evolution as its fundamental working 
 idea, while holding itself free to change or modify its hvpotlieses 
 as to the nature of the evolutionary process — Aristotelian, La- 
 marckian. Darwinian, and the moi'c specific hypotheses that depend 
 on these. Such an inquiry consists in the constant remodelling of 
 a system of hypotheses so as to adjust it to the facts, the working 
 
 447
 
 44S THE PROBLEM OF LOGIC [XIV. 
 
 idea, in close relation to which the hypotheses are framed, remain- 
 ing stead}'- throughout the inquiry. If any discrepancy occurs 
 between calculated results and facts, the hypothesis must first be 
 modified or changed so as to fit the facts, and only in last resort 
 must the Idea be brought in question. 
 
 Postulate and Working Idea. 
 
 The term ' postulate ' ma}' conveniently be used in one or the 
 other of two main senses. It may stand either for an a priori 
 necessity of the reason, or for a 'methodological guiding principle.' 
 The Principle of Logical Consistency would be an a priori postulate 
 — that is. a postulate which, if denied, would leave the Reason 
 irrational. An a priori postulate is the Reason's demand to have 
 its own intrinsic nature respected as an indispensable precondition 
 to its functioning at all. Such a postulate cannot be denied with- 
 out self-contradiction. For what is to guarantee the rational 
 character of tlic denial if the requirement of logical consistency be 
 not respected ? What is to hinder the denial from being explained 
 away as a corroborative affirmation in disguise ? 
 
 Under the second of the two headings we have the Inductive 
 Postulate of Determinism. This deterministic postulate is not 
 a priori in the sense in which this term has just been defined. It 
 is not a law of Thought. The Reason and this fundamental stipu- 
 lation of Inductive Metliod do not stand and fall together. Tlie 
 Method, however, of which the postulate is the jorinciple, does 
 stand or fall with the postulate, and it is on this account that the 
 postulate is called ' methodological.' A methodological postulate 
 lies at the root of rational inquiry in this sense — that it defines the 
 type of explanation that such inquiry must reach after, and the 
 type of viethod appropriate to sucli explanation. ' Inductive Prin- 
 ciple,' ' Inductive Postulate,' ' Inductive Method,' ' Inductive Ideal 
 of Explanation,' ' Inductive Conception of Fact,' are but different 
 expressions of one and the same dominating determinant : the 
 scientific point of view, the point of view of the external observer. 
 
 We propose, then, to use the term ' postulate ' in the sense of 
 methodological postulate. Hence we shall be free to discuss the 
 limits of the Inductive Postulate, for these will not necessarily 
 coincide with the limits of the human understanding. 
 
 In a certain sense, a methodological j)ostulate may be described 
 as a working idea. It is a working idea of the Reason, all Reason's 
 methods being working methods to be appraised by their explana- 
 tory power. But it is not a working idea of the scientific point of 
 view, for its limitations coincide with the .scientific horizon. It is 
 constitutive of the scientific outlook, and not tentatively regulative 
 of it. Relative to the scientific point of view, the Inductive Postulate 
 is an a priori postulate, a vital requirement which cannot be severed
 
 Chap. XLVIL] THE IXDUCTIVE POSTULATE 449 
 
 from tlie method whicli it informs wathout leaving that method 
 incapable of development and condemned to barrenness. 
 
 The deterministic postulate should be distinguished from the 
 wwideterministic ideal of Natural Science. If the former defines 
 the legitimacy of a liypotlietical explanation, the latter defines the 
 essential condition on which its conclusiveness depends. An ex- 
 planation may be said to be conclusive when accompanied by the 
 demonstration that no other is possible. Thus, the proof that a 
 given causal relation is necessarily reciprocal — a proof, in other 
 words, which eliminates the possibility of a plurality of causes — 
 would satisfy the unideterministic ideal of Scientific Explanation. 
 
 The Inductive Postulate and the Inductive Principle. 
 
 The Inductive Principle of Fidelity to Relevant Fact can be fully 
 understood only in the light of the Inductive Postulate. For it is 
 only through the latter that we can clearly perceive the meaning 
 of the word ' relevant.' Fact is relevant to inductive inquiry only 
 in so far as it is conceived as the expression of Natural Law — only 
 in so far, that is, as it is studied under the limitations of the Induc- 
 tive Postulate. The Inductive Postulate simply specifies the 
 enunciation of the Inductive Principle ; an inquiry carried out 
 under the general inspiration of the latter has its direction specifi- 
 cally determined by the meaning put upon the former. 
 
 Postulate and Necessary Truth. 
 
 Dr. Whewell defined a necessary truth as a proposition the nega- 
 tion of which is not only false, but inconceivable ;* and Herbert 
 Spencer's definition is practically the same as Whewell's. 
 
 Mill denies that this is an adequate account. He objects, and 
 rightly, to measuring the possibility of things by our human cajDacity 
 of conceiving them. He explains that ' when we have often seen 
 and thought of two things together, and have never in any one 
 instance cither seen or thought of them separatelj^ there is by the 
 primary law of association an increasing difficulty, which may in 
 the end become insuperable, of conceiving the two things apart.' 
 Thus, even eminent persons — e.g., Comte, when he urged (at the 
 very time when the principles of Spectroscopy were being dis- 
 covered) that it was inconceivable that we should ever discover 
 what the stars were made of — have seemed unable to conceive 
 what was afterwards found to be not only quite conceivable, but 
 quite true. 
 
 ^lill has done good ser\-ice by clearly distinguishing between the 
 inconceivable and the unbelievable ;1[ but a further distinction is 
 
 * Vide J. S. Mill. ' A System of Logic," Book II., ch. v., § 6. 
 t Hid., Book II., ch. vii., §3. 
 
 29
 
 450 THE PROBLEM OF LOGIC [XIV. 
 
 imperatively called for — that between the inconceivable and the 
 unimoginable. These two words ^lill seems to regard as identical 
 in meaning. For our part, we hold that the only genuine incon- 
 ceivable is the strictly irrational, the self-contradictorj^ the mean- 
 ingless. Where it is still conceivable that greater knowledge may 
 shed light upon a mvsterj'', that mystery cannot logicallj'- be termed 
 ' inconceivable.' We may feel very sceptical about it, but that is 
 another matter. 
 
 Mill's own xiew of necessary truth is far from satisfactory. He 
 holds that what we call necessary truths are experimental truths, 
 generalizations from observation. ' The proposition, Two straight 
 lines cannot enclose a space ... is an induction from the evidence 
 of our senses.'* He argues that it is unreasonable to attribute to 
 these truths an origin different from that of all the rest of our 
 knowledge of Xature when their existence is joerfectly accounted 
 for by supposing their origin to be the same. Speaking still of the 
 axiom, ' Two straight lines cannot enclose a space,' he writes : 
 ' Experimental proof crowds in upon us in such endless profusion, 
 and T^ithout one instance in which there can be even a suspicion 
 of an excejDtion to the rule, that we should soon have stronger 
 ground for beheving the axiom, even as an experimental truth, 
 than we have for almost any of the general truths which we con- 
 fessedly learn from the evidence of our senses.' 
 
 We shall presently discuss the weakness of this position of Mill's 
 — i.e., of the attempt to treat a necessary truth [e.g., an axiom of 
 Geometry or the Law of Causation) as a mere hypothesis that is 
 more than usually well grounded. Meanwhile we have already 
 committed ourselves to the con\nction that the demands or postu- 
 lates which the Reason is obliged to make as a condition of its 
 effective exercise are necessary truths, and that these are of two 
 kinds : 
 
 (a) A priori postulates — the absolutely necessary truths {e.g., 
 
 the Law of Xon-Contradiction). 
 
 (b) Methodological postulates — truths necessary, not, indeed, 
 
 for the reason generally, but for the reason as self- 
 limited to some particular universe or aspect of Reality. 
 
 The Inductive Postulate as the Postulate of a Mechanical or 
 Deterministic Explanation of Nature. 
 
 The Inductive Postulate — the Postulate of Causal Explanation — 
 may be enunciated as follows : Fact is inductively explained only in 
 80 far as it has been determinately brought under Causal Law. This 
 may be more expressively named the Deterministic Postulate. 
 
 * Vide J. S. :Mill, 'A System of Logic.' Book II., ch. v., § 4.
 
 Chap. XLVIL] THE INDUCTIVE POSTULATE 451 
 
 The demand that the explanation of Xature shall be f^iven in 
 terms of Causal Law is far from being a mere abstract formula, of 
 interest to logicians only. It represents the hardest-won victory 
 of Science, and is a demand over which the keenest controversies 
 are waged even at the present day. It represents the victory of 
 the mechanical \'iew of Nature over the magical. Indeed, the true, 
 living value of the deterministic conception of Explanation is seen 
 only in contrast with the magical, anthropomorphic conception 
 which it supplanted. 
 
 To the savage all things appear to have psychical life, and he 
 interprets the movements of inanimate objects as though they had 
 their source in psychological motives. He thinks of plants and 
 animals as quasi-persons, and seeks to determine their beha\iour 
 by prayers, sacrifices, etc. This is his anthropomorphism. 
 
 Again, the savage sees a sympathetic connexion between objects 
 either because they are like one another (resemblance), or because 
 they have previously been connected together (association). Thus 
 the cut hair of a man, his shadow, image, or picture are conceived 
 of as so closely related to him that it is possible by injuring them 
 to injure the man liimself. If a waxen image of a man be set in 
 the sun to melt, the man himself will waste away (c/. D. G. Ros- 
 setti's ' Sister Helen '). Hence the custom of burying hair or nail- 
 parings, the dislike of being sketched or photographed, and the 
 care with which savages often keep their names, or the names of 
 their gods, a profound secret. 
 
 Here are other instances : A savage ^vill wear a ring of iron in 
 order that it maj' impart its quality of hardness to his body, or 
 when bargaining for a cow, or asking a woman for wife, he will 
 chew a piece of wood to soften the heart of the person he is dealing 
 with. So, again, having discovered in the lion the quality of 
 courage, or in the deer that of swiftness, he eats the former that he 
 may become bold, and the latter that he may run well. So he will 
 eat an enemy to acquire liis boldness, or a kinsman to prevent his 
 virtues from going out of the family. 
 
 These instances might be indefinitely multiplied. Whether 
 antliroiiomorphic or magical, the explanations are all equally non- 
 mechanical. The postulate of Induction is the protest of Science 
 against Anthropomorphism and Magic. It requires that natural 
 effects shall follow from natural conditions, and vice versa that 
 natural conditions shall give rise to natural effects. It is now 
 recognized as the specific regulative principle of all the natural 
 sciences — a principle, that is, which determines the general method 
 or direction of inquiry, however hypotheses may be modified or 
 working ideas displaced. 
 
 Towards the middle of the nineteenth century, certain puzzling 
 irregularities were observed in the movements of the planet L'ranus. 
 
 There were two conceivable ways of hypothetically explaining 
 
 on -t
 
 452 THE rROBLE:M OF LOGIC [XIV. 
 
 these irregularities. The orbit of Uranus must have been modi- 
 fied either mechanicall}', by the influence of matter in motion 
 elsewhere, or else by the immediate operation of some volitional 
 agency of more than human power. This second alternative was 
 not in itself inconceivable. It was inadmissible onl}^ as a scientific 
 explanation. It was an alternative that Astronomy could not 
 possibl}' have admitted without admitting that it had transgressed 
 its own limitations — i.e., without ceasing to be Astronomy. Sup- 
 posing, now, that some incalculable demon had really been respon- 
 sible for the perturbations, could Astronomy, we ask, ever have 
 found it out ? By no means. It would still be puzzling its mighty 
 intellect for a mechanical solution, and meanwhile blaming its 
 telescopes, or the irreflective nature of the surface of the disturbing 
 body, or its extraordinary density that resulted in its being too 
 small for visibility, etc. ; and so it would go puzzling on for ever, 
 readjusting its hypotheses and working conceptions — even, perhaps, 
 that of gravitation itself — in order to render the phenomenon 
 mechanically intelligible. It would, in fact, simplj^ rejDcat over 
 again, in its improved modern way, those processes of adjusting 
 epicycles and excentrics which were forced by the same respect 
 for regulative Ideas upon the bewildered observers of the Middle 
 Ages. The postulate or Supreme Idea is a principle for working 
 with, and not for discussing. 
 
 The same great regulative principle — the principle of Deter- 
 minism — is sometimes assumed even in Psychology, and by the 
 most modern writers. ' Psychology, like every other science,' 
 writes Hoffding, ' must be deterministic — i.e., it must start from 
 the assumption that the causal law holds good even in the life of 
 the will, just as this law is assumed to be valid for the remaining 
 conscious life and for material nature. If there are limits to this 
 assumption, they will coincide with the limits to Psychology.'* 
 Professor James speaks in a precisely similar manner : ' Psychology, 
 as a would-be Science, must, like every other Science, postulate 
 complete determinism in its facts, and abstract consequently from 
 the effects of free-will even if such a force exists. 'f 
 
 The identification of the Inductive with the Deterministic Postu- 
 late may appear to some to be arbitrary, and to involve a gratuitous 
 restriction of the Inductive Method. Tliis, however, is not the case. 
 Tliere is inlierent in the very attitude of Science towards its facts 
 a restriction which compels the identification in question. An 
 intrin.sic requirement of scientific method is that the theorist shall 
 approach his facts from a standjooint external to the facts them- 
 selves, and this external attitude is responsible for the form which 
 Inductive Method necessarily takes. There are two main ways in 
 
 * ' Outlines of Psychology,' English translation, by Mary Lowndo3, p. 345. 
 t 'Text-book of Psychology,' p. 456. For a criticism of these views, see 'Per- 
 sonal Idealism, Philosophicaf Essays,' edited by Henry Sturt, pp. IGG et seq.
 
 Chap. XLVII.] THE IXDUCTIVE POSTULATE 453 
 
 which we can study facts. We may study them in relation either 
 to a scientific or to a philosophic interest. In the former case, we 
 are concerned solely uith the relations in which the facts stand 
 to each other. In the second case we are concerned with their 
 relation to us, who know them and observe them, Mith their function 
 as factors in a concrete spiritual experience. The dominant ques- 
 tion here is : ' How do the facts wo are studying express spiritual 
 purjDOse V Such teleological investigation, however, is possible 
 only when the inner standpoint of personal experience is adopted. 
 When, by the nature of the case, the object is approached from a 
 standpoint external to it, it is only the external, sense-perceivable 
 behaviour of the object that admits of being studied. Thus, in 
 this case the only question which we can legitimately ask about 
 our object is : ' How does this object embody natural law V Now, 
 the Sciences of Xature are obliged to approach their object from 
 the outside. ' Whatever life or mind may constitute the inner 
 being of so-called inanimate Nature, the scientist cannot share it 
 in such a way as to make any knowledge of its procedure as a pur- 
 poseful agent a basis for liis investigations.'* But just in so far as 
 we fail to regard an object from the inner point of view of the end 
 or purpose that object may be tending to realize, a procedure of 
 tentative explanation becomes imperative. Laws of beha\'iour 
 must be hj^pothetically superinduced upon the object, and be left 
 entirely dependent on verification for their objective accej:)tance. 
 
 If the question should be asked, ' Why do we need to make postu- 
 lates at all V the reply is simply this : that the postulate is needed 
 to define what we mean by a legitimate explanation. It is of no 
 use to begin an inquiry without a test which shall enable us to 
 decide whether an alleged explanation is legitimate or not — whether, 
 in fact, it can be accepted as a possible explanation. The postulate 
 of scientific method is a test of this kind. If a suggested explana- 
 tion \aolates the postulate of mechanical connexion, Inductive 
 Science ^nll have none of it. The true interests — indeed, the neces- 
 sities — of Inductive Science demand or postulate that no explana- 
 tion of a magical or otherwise indeterministic character be enter- 
 tained for a moment, even as a possible explanation. 
 
 In a famous chapter of his 'Logic ' (Book III., ch. xxi.) Mill has 
 endeavoured to present the Inductive Postulate, or, as he calls it, 
 ' the Ground of Induction,' as a generalization from experience, and 
 as standing on precisely the same inductive footing as a well- 
 established hypothesis or law of Nature. He enunciates tliis Law 
 of Causation, or Ground of Induction, as follows : 
 
 ' Ev^ery event, or the beginning of everv phenomenon, must have 
 some cause, some antecedent, on the existence of wliich it is in- 
 variably and unconditionally consequent.'! 
 
 * From ' A Philosophical Introduction to Ethics,' p. 54. 
 t * A System of Logic,' Book III., ch. xxi.. § 1.
 
 454 THE PROBLEM OF LOGIC [XIV. 
 
 Tliis liiw, he maintains, can be inductively proved, and that by 
 a process of Simple Enumeration. In certain cases, he says, we 
 may have the completest proof based on simple enumeration of 
 in-stances — namely, in those cases in which our survey over instances 
 is so extensive as to leave us convinced that, had there been any 
 instance contrary to the law, we should have met "\\itli it. 
 
 In order to be sure of the truth of an Induction by Simple 
 Enumeration, we must, according to Mill, be able to affirm two 
 things : ( I) that ' we have never known an instance to the contrary '; 
 and (2) ' that if there were in Nature any instances to the contrary, 
 we should have known of them.'* In most cases of this kind of 
 Induction the first affirmation has to be made without the second. 
 But the peculiarity of Induction by Simple Enumeration, argues 
 Mill ingeniously, is that it ' is delusive and insufficient exactly in 
 proportion as the subject-matter of the observation is special and 
 limited in extent. As the sphere widens, this unscientific method 
 becomes less and less liable to mislead ; and the most universal class 
 of truths — the law of causation, for instance, and the principles of 
 number and of geometry — are duly and satisfactorily proved by 
 that method alone, nor are they susceptible of any other proof. 'f 
 
 Mill sustains tliis point by the following argument : An Induc- 
 tion by Simple Enumeration can be affirmed as true only ' within 
 certain limits of time, place, and circumstance,' the reason for not 
 extending its apj)lication beyond those limits being ' that the fact 
 of its holding true within them may be a consequence of colloca- 
 tions, which cannot be concluded to exist in one place because 
 they exist in another ; or may be dependent on the accidental 
 absence of counteracting agencies, which any variation of time or 
 the smallest change of circumstances may possibly bring into 
 play.' 
 
 Xow, argues Mill, ' if we suppose . . . the subject-matter of any 
 generalization to be so A^ndely diffused that there is no time, no 
 place, and no combination of circumstances, but must afford an 
 example either of its truth or of its falsity, and if it be never found 
 otherwise than true, its truth cannot be contingent on any colloca- 
 tions, unless sucli as exist at all times and places ' — i.e., it must 
 hold good for all collocations ; J ' nor can it be frustrated by any 
 counteracting agencies, unless by such as never actually occur. It 
 
 * ' A System of Logic,' Book III., ch. iii., § 2. 
 t Ifjtd., ch. xxi., § :i. 
 
 * By ' collocation ' (an expression borrowed from Dr. Chalmers) Mill means 
 the coexistence of causes or causal tendencies in certain relative positions and 
 relations. For example, a lady is troubled, in cycling, by a constant noisy rattle, 
 and succeeds in tracing this result to an interference between gear-case and loose 
 chain. The gear-case is seen to be slightly displaced, but this of itself could not 
 have caused the rattle provided the chain had been sufficiently taut, nor would 
 the mere looseness of the chain have caused it if the gear-case had been in order. 
 The fact of the two circumstances being present together — their .' collocation,' as 
 Mill would put it — is indispensable for the ijroduction of the effect.
 
 Chap. XLVIL] THE IXDUCTIVE POSTULATE 455 
 
 is, therefore, an empirical law coextensive with all human expcri- 
 ence, at which point the distinction between empirical laws and 
 laws of Xature vanishes.' In other words, the oriirinal limitations 
 characteristic of an enumerative induction (in Mill's sense of the 
 term) are then entirely cancelled. The law established by such an 
 induction, says Mill, ' takes its place among the most firmly estab- 
 lished as well as largest truths accessible to science.' 
 
 Criticism of MilVs Attempted Proof of the Law of Causation. 
 
 Our criticism of this justly celebrated proof may conveniently 
 be arranged under three heads : 
 
 (i.) Even if the Law of Causation were a generalization from ex- 
 perience* it could not he established by the simple enumeration of 
 instances. 
 
 Mill's proof supposes tliat the uniform working of the Law of 
 Causation is much more obvious than is, in fact, the case. He 
 does, indeed, clearly distinguish the Uniformity of Nature (the 
 ' universality of the Law of Causation 't) from any obvious uni- 
 formity in the succession of ' physical facts ' ; J and we must also 
 remember that ]\Iill regards the Law of Causation as a generaliza- 
 tion not from individual instances of causal sequence, but as 
 obtained ' by generalization from many laws of inferior generality, '§ 
 as ' obtained by induction from particular laws of causation.'|| 
 But the process by which these particular laws are in the first 
 instance established is ' the loose and uncertain mode of induction 
 per enumerationem simplicem '; nor are we given to understand that 
 the liigher generalization is reached by means of any other less 
 superficial method. Now, this method is essentially unanalytic, 
 and therefore powerless to elicit from the apparent confusion of 
 natural phenomena their underlying regularity and order. As Mill 
 truly says, ' The course of Nature ... is not only uniform ; it is 
 also infinitely various.'^ But if this is so, then investigation by 
 means of Simple Enumeration alone would as often oppose the 
 hypothesis of Causal LaAV as it would confirm it. As Dr. Sigw-'^rt 
 has shown, all we can attain to by this method is a number of 
 empirical rules accompanied by a large and equally imposing 
 number of exceptions. The process leads naturally to the con- 
 viction ' that law and disorder bear sway in wild alternation.'** 
 To this rudimentary form of observation ' the universe becomes 
 
 * The term ' experience ' is here naturally, indeed inevitably, used in the 
 restricted sense of ' experience as relevant to a pre-scientidc or scientific 
 interest.' 
 
 t ' A System of Logic.' Book III., ch. xxi., § 1. 
 
 X Cf. Ibid., §2, second footnote. § Ibid., §2. 
 
 II Ibid., §3. H Ibid., ch. iii.. §2. 
 
 *♦ ' Logik,' vol. ii., ch. v., § 93, 12. English translation, pp. 306, 307.
 
 456 THE PROBLEi\r OF LOGIC [XIV. 
 
 divided into one sphere, in which we feel at home, and are accus- 
 tomed to expect results with certainty, and another composed of 
 phenomena which are changing, variable, and fortuitous.'* 
 
 (ii.) The Law of Causation, as the Ground of Induction, cannot he 
 a mere generalization from experience ; it must have a methodological 
 significance. 
 
 Let us, then, suppose that this proof by Enumeration is aban- 
 doned, and that tlie attempt is made to prove the Law of Causation 
 by a scientific induction. We will suppose that the surface-view 
 of fact, inseparable from the Method of Simple Enumeration, is 
 given up. and that a thorough-going anal3^sis takes its place. We 
 will further suppose that, as the result of such analysis, Nature 
 has never failed to reveal the uniformity which the hypothesis 
 requires for its verification. There are, indeed, indications that 
 some such procedure as tliis is what Mill really has in mind, and 
 that his Enumeration process is not so unanalytic as its name 
 imi^lies : for he says that it is as phenomena ' become better known 
 to us ' that they are found to obey the law of uniformity of succes- 
 sion ; that it is ' after due examination,' and when we know a 
 phenomenon ' sufficiently well,' that we are able to perceive its 
 obedience to causal law.f In any case, it might be thought that, 
 if the suppositions suggested above were granted. Mill's main con- 
 tention that the Law of Causation is a generalization from experi- 
 ence would be justified, though we might be dissatisfied with his 
 method of supporting his thesis. Let us consider this point. 
 
 We must admit that, if Mill were prepared to accejjt the Law of 
 Causation as a simple hypothesis of precisely the same standing as 
 the hj^othesis of gravitation or any other well-grounded hypo- 
 thesis, and did not attempt to erect the law into a Ground of In- 
 duction on which ' the validity of all the Inductive Methods 
 depends,':!: the proof might be accepted as amounting to a very 
 satisfactory verification. And, indeed. Mill does not regard the 
 Uniformity of Nature as anything more than the most general and 
 the most extensively verified of causal laws, as witness the following 
 famous passage : 
 
 ' Tlie uniformity in the succession of events, otherwise called the 
 Law of Cau.sation, must be received not as a law of the universe, 
 but of that portion of it only which is within the range of our means 
 of sure observation, with a reasonable degree of extension to 
 adjacent ca.ses.'§ 
 
 But if the Law of Causation is itself an induction, it cannot 
 be at the same time the Criterion of Induction. The Ground of 
 Induction, as Mill understands it, cannot be the Standard of Induc- 
 
 * ' Logik,' vol. ii., cli. v., § 93, 12. English translation, p. 300. 
 
 t ' A .System of Logic,' Book III., ch. xxi., § 4. X Ibid., § 1. 
 
 § Ibid.. § 4.
 
 CuAP. XLVII.] THE INDUCTIVE POSTULATE 457 
 
 tion. It cannot tell us what we are to understand by an inductive 
 explanation, and so enable us to distinp;uish between an explana- 
 tion that is inductively legitimate and one that is inductively 
 illegitimate. In so far, then, as the methodological significance of 
 a ground of Induction depends on its ability to supply a criterion 
 for the legitimate application of Inductive Method, the Law of 
 Causation, as interpreted by Mill, cannot be said to possess any 
 methodological significance. 
 
 The question then remains : What significance other than this 
 methodological significance can a ground of Induction possess ? 
 Or, to state the question in the narrower form which alone is rele- 
 vant to the present criticism : What significance other than this 
 methodological significance can MilVs Ground of Induction lay 
 claim to ? The function of the Ground of Induction, according 
 to Mill, is to serve as the ultimate major premiss for every specific 
 induction, ' not contributing at all to prove it, but being a necessary 
 condition of its being proved ' (Book III., ch. iii., § 1). Thus, the 
 statement, ' The course of Nature is uniform ' — Mill's favourite 
 expression for the Ground of Induction — is a necessary condition 
 for proving, for instance, that heat causes evaporation. For if we 
 had no warrant for assuming the uniformity of Nature, we should 
 not feel safe in concluding from our observations on heat that it 
 tends to cause — i.e., uniformly tends to cause — the evaporation of 
 liquids. Now, our sole guarantee, according to ]\Iill, for assuming 
 that Nature is uniform is that Ex-perience shows with convincing 
 consistency that ' it is a law that there is a law for everything ' 
 (Book III., ch. v., § 1). The word ' Experience,' however, is am- 
 biguously vague. So far as the term has any relevancy to IMill's 
 argument, it should mean ' Experience as relevant to the inductive 
 interest.' It is, then, ' Experience as relevant to the inductive 
 interest ' which estabhshes on a firm inductive basis the uniformity 
 of Nature. Hence, since tlie inductive interest itself defines the 
 very exi:)erience which establishes tliis uniformity, we are driven 
 to ask whether, apart from that interest, the uniformity could ever 
 be established. The query, so formulated, leads us at once to the 
 crux of the indictment. We shall see that this inevitable reference 
 to the inductive interest requires that our confidence in the uni- 
 formity of Nature as the Ground of Induction shall itself rest on 
 the postulate that Nature is uniform. It is only as depending on 
 this methodological demand that the Ground of Induction can 
 guarantee specific inductions in the sense indicated by Mill. 
 
 (iii.) 31 ill's proof of the Law of Causation involves, either directly 
 or indirectly, a Petitio Principii. 
 
 That the Law of Causation, the ' ultimate major premise ' (as 
 Mill calls it) of every scientific induction, should itself have been 
 obtained by ' Induction ' does not, from Mill's point of view,
 
 45S THE PROBLEM OP LOGIC [XIV. 
 
 involve any peiitio principii ; for he maintains that the major 
 premiss is never ' the proof of the conclusion, but is itself proved, 
 along with the conclusion, from the same evidence.'* But we hold 
 that the fallacy of petitio principii vitiates the very process by 
 which, according to Mill, the ' ultimate major premise ' has been 
 obtained. A successful proof of the Law of Causation, even within 
 the limits of a restricted range of experience, must necessarily be 
 based on the assumption that there is more uniformity in Nature 
 than at first meets the eye. As wo have already seen, the facts do 
 not thrust the idea of Causal Law upon the impressionable investi- 
 gator. Apparent exceptions to the reign of law will inevitably be 
 met with. But if the scientist is not assuming that a fact must 
 exemplify law, there is no reason why he should not accept these 
 exceptions as final. Why should he suspect that the exceptions 
 are but disguised exemplifications of law ? 
 
 The simple truth is that we are logically justified in treating 
 apparent exceptions to order as merely disguised instances of it 
 only on condition that we deliberately make it a postulate of the 
 search after knowledge that we shall look for order even where 
 order is not palpably manifested in the facts. The mind must 
 make its own demand for Causal Uniformity, or it will never find 
 it realized. So far as Mill finds order beneath the surface of natural 
 phenomena, he does so by implicitly assuming the fundamental 
 postulate of all inductive inquiry ; he assumes that the intelligi- 
 bility of fact for Science depends on its being conceived in the light 
 of a law. 
 
 Limits of the Inductive or Deterministic Postulate.'^ 
 
 Provided that the Postulate of Determinism be treated simj^ly 
 as a methodological guiding principle — as limiting the sphere in 
 which Inductive Method is applicable, and as having no relevance 
 to facts that refuse to be explained on the ground of it — this jDrin- 
 ciple needs no other limitation than that imposed upon it by the 
 facts themselves. 
 
 From our present point of view facts may be roughly classed 
 under three heads : 
 
 (1) Inorganic. 
 
 (2) Organic. 
 
 (3) Self-conscious. 
 
 In reference to this rough division of facts, we may say that the 
 postulate works well within the first two realms of fact (though the 
 
 * 'A System of Logic' Book III., ch. xxi., §4. 
 
 t In what follows I am much indebted to Dr. Sigwart's section on ' Explanation 
 by the Nature of Substances ' in the fifth chapter of his already classical Method- 
 ology (' Lofik,' vol. ii., ch. v., § 100. Engli.sh translation by Helen Dendy, 
 pp. 460-480).
 
 Chap. XLVIL] THE IXDUCTIVE POSTULATE 459 
 
 nature of the determinism has to be understood differently in the 
 two cases), Avhereas with what is central and essential in the third 
 group of facts it is entirely inadequate to deal. 
 
 Whether Determinism is an effective postulate or not, and how, 
 when adequate and fruitful, it is exactly to be interpreted, depends 
 on the ultimate nature of the substance whose movements or 
 actinties it is concerned with interpreting. We require Ideas or 
 working concepts as to the nature of the substances between which 
 causal interaction takes place, as well as Ideas concerning the 
 nature of that causal interaction itself. 
 
 Thus, in dealing with inorganic phenomena, Science has adopted 
 the concejjts of the molecule, the atom, and the electron. In 
 attempting to deal with organic phenomena on the basis of the 
 same suppositions as to the nature of substance, it has found itself 
 unable to give complete explanation of the facts, and it has been 
 obliged to adopt the further concepts of iiulividuality and develop- 
 ment. Finalh', in attempting to explain mental activity, the con- 
 cept of freedom, which has been found indispensable for giving any 
 meaning to human action, has sprung up in direct antagonism to 
 the deterministic postulate. 
 
 Let us consider these points more closely : 
 
 1. The concept of the atom. The atom (or the electron, which 
 seems likely to take its place as representing the ultimate nature 
 of substance for physical Science) is conceived as an indinsible, 
 invariable force-centre, inherently possessing certain fundamental 
 force-attributes ; and upon this view of the ultimate nature of 
 (material) substance is based ' the conception of a mechanism of 
 the universe wliich attempts to rej)resent all perceptible events as 
 the motion of invariable atoms according to invariable laws.'* 
 
 The mechanism of the heavens, of which this mechanism of the 
 universe is the imaginative extension, furnishes the best example 
 of what such a conception involves. It is needless to say that the 
 extended conception of a thorough-going mechanical (atomistic) 
 explanation of the universe has not the same justification as that 
 of the mechanical explanation of the planetary system. Tliat 
 could be the case only were this explanation to interpret the whole 
 course of the universe as intelligibly as the motions of the planets 
 and their satellites are interpreted by the laws of gravitation and 
 inertia. In the one case the Given is completely explained on the 
 ground of the mechanical hj^othesis ; in the other it is far indeed 
 from admitting of any such complete explanation. 
 
 Thus, the attempt to explain the facts of develoi^ment by the 
 interactions of atoms that mutually attract or repel each other is 
 far from being satisfactory. We need here other concepts or 
 categories than those of ' atom ' and ' inherent force. ' The thorough- 
 
 * Dr. Christoph Sigv.-art, ' Logik.' vol. ii., ch. v., § 100. 11. English traaslatioQ 
 by Helen Dendy, p. 469.
 
 ^60 THE PROBLEM OF LOGIC [XIV. 
 
 going atomist tells us that each individuality is only a collective 
 aercrecate of atoms which interact bv virtue of their inherent forces, 
 and that the development of these indiWdualities, their disposition 
 to pass through successive stages, is already pre-established in the 
 original configuration of the atoms relative to one another. It 
 need hardly be pointed out that upon this view ' life ' and ' mind ' 
 are by-products, the world's course being already mechanically 
 predetermined. Thus, consistent atomists regard Consciousness as 
 epiphenomenal, as a mere spectator of its own predetermined 
 changes. 
 
 2. Truer conceptions of development and of indiWduality are 
 ob\nousl3^ needed when we come to deal with organic life. Let us 
 first consider the meaning of ' development,' as this meaning has 
 itself gradually developed. In the meaning of the term ' organic 
 process ' or ' development ' we can, as Dr. Sigwart says, distinguish 
 several stages of growth. 
 
 First, we have the original meaning of the word, a mere unfold- 
 ing, as in the opening of a rolled-up scroll or the expansion of a 
 bud into the full-blown flower. This conception is then enlarged, 
 so as to take in at the same time the idea of growth, a growth not 
 only in volume, but also in differentiation. This meaning also is 
 illustrated by the development of a flower-bud. Not only do its 
 parts unfold, but they also change in size and shape, and their 
 tissues become continually less and less homogeneous. A still 
 fuller meaning is gained when all the particular stages of the process 
 are explicitly referred to one developing individual, and an anti- 
 thesis is drawn between the beginning jrom which and the end 
 towards which the subject develops. The end of the development 
 is then conceived as reveaHng what the beginning contained, as 
 the oak reveals the true nature of the acorn. Finally, when the 
 concept of development is made to extend ' beyond particular 
 individuals to the whole range of the organic universe,' it has 
 reached its deepest meaning ; but in estabhshing this we find a 
 difiiculty which lies in our inability to fix definitely upon ' the 
 subject to which this universal development is to be ascribed.' 
 
 Now, from the point of view of our logical inquiry, the main 
 thing to note is that, though we may still profitably adhere to our 
 deterministic postulate in the investigation of organic as in that of 
 inorganic forms, we have to introduce into the former investigation 
 a new conception of causal explanation. The earlier stages of the 
 growth of an organism cannot be said to account for the later 
 stages in the development in the same sense as that in which the 
 original distribution of atoms in space can be said to account for 
 the later distribution of these atoms. For the change we call 
 ' development ' is always qualitative as well as quantitative, and 
 atomistic explanation practically ignores all qualitative changes. 
 
 Thus, in the case of organic development, we cannot account for
 
 Chap. XLVII.] THE IXDUCTIVE POSTULATE 4G1 
 
 any one phase by merely pointing out the phases antecedent to it. 
 All we can truly say is tliat the phenomena of the beginning of a life 
 are imperfect manifestations of ai^rinciple which is more completely 
 manifested in the later stages. Hence, the study of antecedents 
 gives us the least distinct clue to this inner principle. The forest- 
 tree, with its deep-striking, ramifying root, its massively towering 
 trunk, and its far-spreading system of branches, more trulv ex- 
 presses the nature of its species than did the embryonic infant- 
 plant contained in the seed from which it sprang. The strongly 
 developed root-system that is able to withstand the mighty strain 
 and leverage of the storm-wind was but feebly represented by tlie 
 minute and unbranched radicle ; the giant branches upholding their 
 dense cloud of luxuriant foliage are more explicitly significant than 
 the tiny and delicate plumule ; the fully differentiated tissues of 
 leaf and t\\ig, of pith and bast and woody fibre, were but dimly 
 foreshadowed in the soft, rudimentary features of the small white 
 folded embryo. Most significant of all is the mature tree in its 
 flowering and fruiting seasons, for an organism is never so truly or 
 so explicitly itself as in that process of reproduction which is the 
 culminating-point of its development. In the physical, no less 
 than in the moral, world does the saying hold good : ' By their 
 fruits ye shall know them.' 
 
 3. That the end explains the beginning in a profounder and 
 completer sense than that in which the beginning explains the end 
 is a principle that applies to all developing life as such, whether 
 the life, like that of a tree, is unconscious of its own development, 
 or is conscious, or at least partially conscious, of it, as in the case 
 of the life of Mind. This type of explanation, wliich is known as 
 teleological, is indeed characteristic of all our attempts to under- 
 stand conscious experience. Just as a finished essay explains a 
 writer's idea far more truly than do the first rough, incoherent 
 jottings — as these jottings, though they do explain something, 
 though they tell us how the thoughts struggled into being, and 
 give us the early history of the idea, yet do not give us its true 
 meaning — so Man is explained, indeed, both by his past and by his 
 future, but more truly by his future — by his destiny — than by 
 his past. The teleological explanation reaches deejier than the 
 genetic. 
 
 The attempt, when dealing with mental development, to explain 
 the subsequent completely by means of its antecedents inevitably 
 issues in fallacy. Thus, even if we were to grant that the religious 
 sentiment was at first no other than a belief in ghosts, we could 
 not reasonably go on to argue that, since the religious sentiment of 
 to-day is but a development of the religious sentiment of j^rimitive 
 man, it must therefore still be essentially a mere transfigured belief 
 in ghosts. We might just as profitably argue that, since the first
 
 462 THE PROBLEM OF LOGIC [XIV. 
 
 efforts of Science produced notliing but fanciful conjecture, there- 
 fore Science must be at bottom a mere collection of fancies. 
 
 But it is when we come to deal with the self-conscious activities 
 of moral beings that we most clearly realize the limits of all deter- 
 ministic — i.e., of all inductive — explanation. For anj^ attempt to 
 explain morality on the basis of determinism completely stultifies 
 that wliich it seeks to exj)lain. Where there is no freedom, there 
 is no responsibility, no dutj-, no ideal to be striven after — for why 
 should we strive against the inevitable ? — so that morality is a 
 question no longer of character or even of conduct, but merely of 
 customary behaviour ; and ethics a science no longer of what ought 
 to be, but of what has been and must be. 
 
 With the conception of Freedom, and still more so "udth the 
 further concepts of ImmortaUty and God — concepts needed, we 
 believe, for explaining the deepest rooted of our moral difficulties — 
 we leave the deterministic postulate far behind. The category of 
 mechanical causation must be transcended. What categories of 
 explanation there ma}^ be which can transcend it — this is a ques- 
 tion which, when systematically conceived, forms the main problem 
 of a Pliilosopliical Logic.
 
 INDEX, VERBAL AND ANALYTIC. 
 
 • A ' AS mark of undistributedness : 
 Pages 158, 159. 
 
 ' A few ' : 155. 104, 169, 170. 
 
 A fortiori SvUogism : 250, 251. 
 
 ASIrmo : 147. 
 
 ' A is A ' : 21, 90, 97. Sec ' Identity.' 
 
 ' A is either B or C ' : 132. 
 
 ' A is not non-A ' : 96. 
 
 A priori Postulate : 448, 450. 
 
 A Proposition : 147-152, 154, 156, 159- 
 162, 166-172, 239, 282. Its Con- 
 version : 194. Its Definition : 150. 
 Its Diagram : 151. 
 
 Aberration of Light : 339, 413. 
 
 Abscissio Infiniti : 48. 
 
 Absolute Being: 42, 75-77. 79. A. 
 Identity : 122 (footnote). A. Sub- 
 ject : 119. 
 
 Absolutely Necessary Truth : 450. See 
 ' A priori . . .' 
 
 Abstract character of Formal Logic : 
 157. A. Concept, Meaning, Term : 
 22 (ftn.), 85-89, 161. A. Deduc- 
 tive Method : 409. A. Identity : 
 97 (see ' Identity '). A. Nature of 
 the Term, HI. A. Object: 85, 86, 
 161. A. Proposition : 161. A. 
 Science : 386, 447. A. Statement : 
 119. A. Study of Inference or Proposi- 
 tion : 145, 146. A. Validity : 0, 7, 9, 
 10, 148, 326 (see 'V.'). A. View of 
 Propositional Import : 94, 95. See 
 ' Statement-import.' 
 
 Abstraction : 21, 22, 51, 75, 76, 85, 
 87, 88. 118, 127, 129, 340, 341, 369, 
 371, 410, 452. 
 
 Acceptance (or Affirmation) of the Ante- 
 cedent : 264, 265, 271, 285. Of the 
 Consequent : 264, 265, 269, 271. 337. 
 A. of Contradictories : 104, 105 (sec 
 ' Self-contradiction '). Of Ends or 
 Means : 128. A. (or granting) of 
 Premiss, Statement, etc. : 7, 8, 98, 
 99. 130, 140-142, 145, 149, 150, 157, 
 161 (ftn.), 166. 172-178, 180. 182. 
 183. 187, 189. 194-196, 201-207, 214- 
 219. 223, 236, 263, 265, 266, 269. 286. 
 
 290-292, 304, 307, 309, 310, 321-325, 
 328 (ftn.), 401 (see ' Premiss '). 
 A. of Theory or Law : 433, 453. 
 
 Accident: 22, 25-29, 37, 120. See 
 ' Fallacy of A.' ' Converse F. of A.' 
 
 Accidental Elements in the Ante- 
 cedent : 380. 
 
 Accompaniment : 4See ' Concomitance.' 
 
 Accuracy : See ' Precision.' 
 
 Achilles and the Tortoise : 291, 292. 
 
 Actualization : 131, 137. 
 
 Adams, John Couch : 412, 413. 
 
 Added Determinants : 208, 209. 
 
 Adequacy of Explanation : 389, 412. 
 
 Admission : See ' Acceptance.' 
 
 .EquipoUence : 190 (ftn.). 
 
 Aflirmation : 94, 98 (ftn.), 122-124, 146, 
 166, 193, 194. 235, 324. 448. 454 
 (see ' Affirmative '). Its relation to 
 Negation : 123, 124. A. of the Ante- 
 cedent : 137, 142 (see ' Acceptance 
 of the A.'). A. of tho Consequent : 
 137. See ' Acceptance of the C 
 
 Affirmative Assertion or Statement : 
 Sec 'Affirmation.' A. Conclusion: 
 217. 220, 221, 224, 228, 230, 232. A. 
 Force or Meaning : 168-170. A. Pre- 
 miss : 217, 220, 221, 223. 224, 228, 
 230, 237, 238. 243, 258, 264. A. 
 Proposition: 123, 128, 147-150, 161. 
 163 (see ' Affirmation,' ' A, I, Y, U 
 Proposition '). A. Quality : 123, 124 
 (see ' Q. of Categorical . . .'). A. 
 Synthesis : 123 (sec ' S.'). A. Quality- 
 mark : 163. A. use of ' Any ' : 
 164. 
 
 After effect : 381, 382. Sec ' Effect.' 
 
 Aggregate : 353, 360. 
 
 Agreement : 18-22, 58, 59, 75, 341, 350, 
 353, 395-400, 413-417, 422. 423. -See 
 ' Determinate A,' ' Indeterminate 
 A," ' Method of A,' ' Resemblance.' 
 
 Aim : See ' Purpose,' ' Teleology.' 
 A. of Logic : 157. 
 
 Alcoholic Fermentation : 422-429. 
 
 Algebra : 14, 362. 
 
 422. 430-435. 
 
 Algol 
 
 463
 
 464 
 
 THE PROBLEM OF LOGIC 
 
 All: See 'Possibly AIL' 'All' as 
 Qimntitv-mark : 4S (ftn.). 152-lGO, 
 102. UU" ICS, -241-243 (see 'Categorical 
 Judgment,' ' Collective Meaning,' 
 'Conjunctive use . . .,' 'Distribu- 
 tive Meaning '). ' All A's are either 
 B or C: 132. 'AH P is S ' : 130, 
 100. ' All S is all P ' : 159 {see 
 'U . . .'). 'AH S is-not P' : 242. 
 ' All S is P ' : 130, 160. ' All S is 
 some P ' : 159. ' All (the) S's arc 
 all (the) P's ' : 149, 153, 158, 182, 198 
 {see ' U . . .'). 'AH S's are-not 
 P's • : 147, 149, 164. ' AH (the) S's 
 are P's ' : 147, 149, 153, 159, 162, 
 166, 239. All ' S's are some P's ' : 
 149, 158. 
 
 Alphabetical Key : 65, 66. 
 
 Alternative Clauses: 112 (see 'Dis- 
 junctive Judgment '). A. Deter- 
 minations or Possibilities : 112, 113, 
 131-138, 236, 262, 271, 274, 292, 293, 
 324 {see ' Disjunctive . . .'). A. 
 Explanations : 416-418, 422, 452 
 {see 'Rival Hypotheses'). A. 
 Species : 21 (see ' Disjunctive Di- 
 vision '). A. Subjects: 110, 117. 
 
 Ambiguity : 14, 16-18, 26, 29, 50, 62, 
 74, 76," 78, 102, 105, 121, 135, 154, 
 159, 164, 208, 246, 281, 282, 285, 
 286. 327 (ftn.). 457 (see 'Non- 
 Ambiguity '). A. of the Middle 
 Term': 218. 
 
 Ambiguous Structure : 285, 286. 
 
 Amphibole : 285, 286. 
 
 Ampliative Judgment or Proposition : 
 120-122. 
 
 .Analogical Argument : See ' Analogy.' 
 ' A. Inference ' : 327. 
 
 Analogy (Biological) : 60. 
 
 ^Vnalogy (Logical) : 321, 327, 347, 355, 
 358-364, 369, 424, 425, 428, 430, 
 431. -See '"False A.'" 'Illegitimate 
 A.,' 'Sound A.' 
 
 Analysis : 29-31, 40, 47, 61, 85, 80, 
 117, 121, 122, 157, .303, 305, .325. 
 329, 341. 343, 344, 353, 355, 357, 
 358, 362, 364, 368, 380, 383, 3,S4, 
 389, 397-400, 402, 407, 408, 419- 
 422, 424, 426, 429, 431, 433, 435, 
 450 {see ' Experimental A.'). A. of 
 Categorical Proposition: 115-120. 
 Of Complex Epicheirema : 201. Of 
 Exponiblos : 166. Of Judgment : 
 122. Of Logical Proposition : 109- 
 142. Of Sorites : 2.55-200, 270, 271. 
 Of Syllogisms: 1.30-1.35. 
 
 ' Analytic ' Judgment or Proposition : 
 120-122. A. Method or Inquiry : 
 See ' Analysis.' A. Observation : 
 392, 403. 
 
 Analytical Key : 63-00. 
 
 Au^Ujiay, 57, 74, 75. 
 
 ' And ' in Division : 49, 53. In Ex- 
 tension-formula : 72, 154, 158. 
 
 Animal: 22, 40, 42, 47, 52, 57, 00, 73-75, 
 88, 89, 152, 314, 359, 300, 374, 401, 
 402, 418, 425, 451. ^ee 'Zoology.' 
 
 Animism : 368. 
 
 Answer: 90, 112-114. 
 
 Antecedent: 112, 137, 138, 141, 142, 
 145, 203-274, 320, 325, 309-372, 
 377-385, 397, 408, 411, 412, 419, 
 424-427, 453, 458, 401. See 'Ac- 
 ceptance of the A.,' ' Hypothetical 
 Proposition.' 
 
 Anthropology : 46. 
 
 Anthropomori)hism, 368, 451. 
 
 Antinomy : 105. 
 
 •Any ' : 149, 150, 158, 159, 104, 168. 
 
 Apodcictic Hypothetical : 139-142. Its 
 connexion Avith Inference : 145, 146. 
 Its Opposition : 178, 179. 
 
 Apparatus (of Experiment or Obser- 
 vation): 145, 334, 335. See 'Transit 
 Circle.' 
 
 Apparent Contradiction : 100-106, 282. 
 A. Exception: 357, 40i, 421, 458. 
 
 Applicability of Inductive Method : 
 458-402. <S'cc ' In. Postulate.' 
 
 Application of Causal Methods : 389, 
 390, 399-413, 418-420, 454. Of 
 Hypothesis : 313, 318, 320-322, 327, 
 419, 424, 425, 428, 430, 431, 433, 
 436, 438. Of Inductive Method : 422- 
 444. Of Meanings, Terms, Words : 
 18, 21 (ftn.), 71, 72, 83, 84, 124, 146, 
 209 (see 'Indication ...'). Of a 
 Universal to Particulars : 244-247. 
 
 Apprehension : ;S'ee ' Immediate A.' 
 
 Approximation : 62, 63, 331. 
 
 Archaeology : 386. 
 
 ' Are,' ' Are not,' ' Are-not ' : 163, 104. 
 See ' Copula-mark.' 
 
 Arguing in a Circle : 289. See ' Pe- 
 titio . . .' 
 
 Argument from Analogy : See ' An- 
 alogy.' 
 
 Arguments reduced to Syllogistic 
 Form : 230-235. 
 
 Argumentum a dicto secundum quid . . . 
 etc. : 284, 285. A. a dicto siuipli- 
 citcr . . ., etc. : 284, 285. A. a 
 fortiori : 250. A. ad Hominem : 
 
 287, 288, 307. A. ad Ignorantiam : 
 289. A. ad Judicium : 287. A. ad 
 Passiones : 288. A. ad Populum : 
 
 288. A. ad Rem : 288, 307. A. ad 
 Verecundiam : 289. 
 
 Aristotelian Evolution : 447. A. 
 Logic : v/ii, 307, 353 (see ' Aris- 
 totle '). A. Sorites : 2.55-200, 319. 
 
 Aristotle : 79, 233, 310. A. and 
 Classification : 341. A. and Enu- 
 meration : 3.53. A. and the Fourth 
 Figure : 236. A. and the Postulate
 
 INDEX, VERBAL AND ANALYTIC 
 
 465 
 
 of Mediation : 213, 21G. A. and 
 Scholastic Logic : 307. A. and the 
 Sorites : 255. A. on Achilles and 
 tho Tortoise : 292. A. on Defini- 
 tion : 38 (ftn.). On the Indefinite 
 Term : 49 (ftn.)- On Indirect Re- 
 duction : 32-L On Subalternation : 
 172 (ftn.). 
 
 Aristotle's Categories : 31, 32, 47 (ftn.), 
 78. A.'s Organon : 116, 117. A.'s 
 Scheme of Predicables : 22, 23, 28, 32. 
 
 Arrangement : See ' Spatial grouping,' 
 ' Classilication.' 
 
 Artificial Classification : 03-66. 
 
 Ascertainability : 140, 141, 369. 
 
 Assertion : 93," 94, 96, 97, 104, 113, 
 
 116. 119, 122-128, 133-139, 141, 142, 
 149-152, 166, 167, 291. 
 
 Assertive character of Propositions : 
 123, 124. 
 
 Assertorial Hypothetical : 139-141. Its 
 Opposition : 179. 
 
 Association : 16, 17, 28, 29, 66, 81, 
 105, 303, 449, 451. 
 
 Assumption: 113, 142, 145, 289, 290- 
 292, 308, 324, 325, 330, 337, 338, 
 341, 410, 419, 432, 434-438, 442, 
 443, 447, 457, 4.58 (see ' Presuppo- 
 sition '). A. of Science : 368, 452. 
 
 'Assuredly ' : 130, 132 (ftn.), 157, 175, 
 176. 
 
 Astronomical . . . : See ' Astronomy.' 
 
 Astronomy : 38, 69, 70, 129, 134, 245, 
 329-333, 338, 339, 361, 362, 377, 386, 
 394, 409, 412, 413, 430-438. 452. 
 See 'Aberration . . .,' 'Copernicus,' 
 ' Eclipse,' ' Moon,' ' Planet,' ' Pto- 
 lemaic A.,' 'Star,' 'Sun,' 'Transit 
 Circle.' 
 
 Atom : 423, 459, 460. 
 
 Atomism : 459, 460. 
 
 Attribute : 22, 25, 26, 40, 43, 65, 70. 
 75, 81, 86. 147, 159, 161, 299, 340, 
 341. 348. See ' Character,' ' Con- 
 notation,' ' Indeterminate Ele- 
 ments ...,'' Property.' ' Quality.' 
 
 Attributive view of Import : 147 (ftn.). 
 
 Augmentative Judgment or Proposi- 
 tion : 120-122. 
 
 Avebury, Lord : 392. 
 
 Average : 352, 422. 434. A. Error : 413. 
 
 Axiom : 251, 306, 322. 386. 450. 
 
 Axiomatic Ideas or Propositions : 329 
 (ftn.). 330. A. Premisses : 318. 322. 
 
 Bacon. Francis, Lord Verulam : IKi, 
 
 117, 313, 351, 391, 414-418. 
 Baconian Method : viii. 313-317, 414- 
 
 418. 
 Bain, Alexander (Prof.) : 372. 396. 
 Balfour, the Rt. Hon. Arthur J. : 81. 
 Ball. Sir Robert: 374, 375, 430 (ftn.), 
 
 435 (ftn.). 436, 439. 
 
 Barbara : 215, 226, 230, 237. 241, 247. 
 
 250, 256. 
 Bare Denial : Sec ' Pure Negation.' 
 Baroco : 226, 230, 235, 242, 248, 250. 
 
 Its Indirect Reduction : 324, 325. 
 
 Its Ostensive Reduction : 248-250. 
 Barren Hypothesis : 313, 337, 338, 424, 
 
 430, 432. 
 Begging the Question : 289. See ' Pe- 
 
 titio . . .' 
 Being: 31, 42, 47. 75-77, 79, 86 (ftn.). 
 
 122 (ftn.), 131. 
 Belief: 93, 112, 122, 123. 
 Bentham, Jeremy : 301. 
 Bergson, Prof. : 74 (ftn.). 
 Berlteley, George (Bishop) : 370, 393 
 
 (ftn.). 
 Bernoulli : 333. 
 Berry, Arthur: 339 (ftn.). 
 Biology : 2, 20, 374, 386, 426, 447. 
 
 See ' Botany,' ' Zoohjgy.' 
 Bocardo : 226, 227, 234, 243, 248, 250. 
 
 Its Indirect Reduction : 324. Its 
 
 Ostensive Reduction : 248-250. 
 Bosanquet, Bernard (Prof.) : viii, 15, 
 
 107 (ftn.), 119, 358 (ftn.), 420, 421. 
 Bosanquet (Mrs.) : 120 (ftn.). See 
 
 ' Sigwart.' 
 Boscovich, Roger Joseph : 371. 
 Botany: 28, 43, 51 (ftn.). 59-08. 73- 
 
 75, 87, 129, 236, 237. 342, 343, 368. 
 Bowen. F. (Prof.) : 196. 
 Bradley. James (The Rev. Dr. ) : 339, 413. 
 BramarUip : 226, 227, 229, 237. 
 Breach of Method : 281, 295. 
 Buchner, Prof. : 429. 
 
 Cagniard-Latour, Charles (Baron) : 424, 
 427. 
 
 Caird, Edward: 63 (ftn.). 
 
 Calculus of Relations : 251. 
 
 Camenes : 226, 232, 237. 
 
 Camcstres : 226, 232. 236, 242, 249, 269. 
 
 ' Can ' : 127-129. 
 
 Canons of Causal Methods : 395-397, 
 402, 404, 405, 407, 411. 412. 415. 
 417-420. Of Causation : 415. Of 
 Elimination : 416-418. 
 
 Capability : See ' Capacity.' 
 
 Capacity : 128. 129, 131 (see ' Potenti- 
 ality ■). C. for Work : 387-389 (see 
 'Energy'). 
 
 Cartesian Vortex Theory : 332, 333, 
 336. 
 
 Catalogue : 63, 65, 69. 
 
 Catchpool, Edmund : 374. 
 
 Categorical Assertion, Judgment, Pro- 
 position. Statement : viii, 111-126. 
 131-133, 130-139, 239. Its Formal 
 Import : 140-161. Its Reduction to 
 strict Logical Form : 161-171. Its 
 Relation to the Hvpothetical : 138, 
 139, 141, 268 (ftn.). C. Basis of 
 
 30
 
 466 
 
 THE PROBLEM OF LOGIC 
 
 Disjunction : 113, lU. 131, 133, 136. 
 ' C. Equivalents ' of Hvpotheticals : 
 13S, 139. 2GS (ftn.). C. Form : 138. 
 139, 141. '2(iS (ftn.). C. Form of 
 Sorites : ::.55-'260, 319. C. Inference, 
 C. Syllogism: 111-261. 269, 323. 
 C. Meaning : 139, 26S (ftn.). 
 
 Category : ix. 31, 32, 47 (ftn.), 63 (ftn.), 
 75-78, 4.".9. 462. 
 
 Causal Antoce.lent : 370 (ftn.), 380, 
 
 381, 383. 424-427 {see 'A.'). C. 
 Chain : 378, 381, 382. C. Con- 
 nexion : 357-361, 364, 368, 371. 377- 
 384, 397-407. 414-420, 428, 4.52. C. 
 Consequence, C. Sequence : 377, 381, 
 
 382, 384, 455 {see ' Causation,' ' Con- 
 sequence ')• C. Elaboration: 419- 
 421. C. Elimination : 419 (.see ' E '). 
 C. Explanation, C. luquirv, C. In- 
 vestigation : 333. 342, 365-462 (see 
 'E'). C. Hypothesis : (see ' H.'). 
 C. Interaction : 459. C. Interest : 
 401. C. Law: 138, 303, 341, 342, 
 357, 367-389, 403, 407-409, 412, 427, 
 450, 451, 455. C. Method : (see 
 ' M. of C. Explanation '). C. 
 Methods : (see ' Inductive M's '). 
 C. Nexus : 361. C. Process [or 
 Action] : (see ' Causation '). C. Re- 
 lation : 341, 379. 384-389, 406, 416- 
 418,449. C. Tendency : 454 (ftn.). 
 {see 'T.'). C. Theory: 378. C. 
 Uniformity : 458. See ' U.' 
 
 Causality: 369. See 'Causal . . .,' 
 ' Causation,' ' Cause.' 
 
 Causation : 309, 341, 367-389, 398, 399, 
 403-409, 415, 417, 419, 422, 427-458, 
 462. iSee 'Canons ofC.,' "Causal . . .,' 
 'Law of C 
 
 Cause: 35, 333, 337, 342, 367-389, 
 395-426, 4.30, 449, 453, 4.54 (ftn.). 
 C. as the Sum Total of the Condi- 
 tions : 370-378. 
 
 CdarerU : 220, 237, 242, 247, 249, 256, 
 260. 
 
 Central Definition : *See ' D. by Type.' 
 
 C«rtainty : 75, 134, 139, 155, 157, 158, 
 381-383, 394, 397, 456. 
 
 Cesare : 226. 
 
 ' C'-aaante caufd . . ..' etc. : 382. 
 
 Chain of Causation : 378, 381, 382. C. 
 of Divisions : 44. C. of Reasoning : 
 255, 260, 261 (see ' Epicheirema,' 
 ' Sorites'). 
 
 Chalmers, Thomas (Dr.) : 454 (ftn.). 
 
 Chandler : 438, 443. 
 
 Character, Characteristic Attribute: 43, 
 57-64, 68, 70, 74, 75, 83, 87, 236, 341, 
 359, 360, 364. 371, 383, 385, 386, 
 424. See ' Common A.,' ' Pro- 
 Tierty.' 
 
 C 'iritic : (See 'Character.' C. 
 
 y : 26, 27, 37, 165. 
 
 Characteristics of the Four Syllogistic 
 Figures : 235-238. 
 
 Characterization : 59-64, 67, 93, 136, 
 167, 357. 
 
 Chemistry: 28, 30, 61 (ftn.), 68, 69, 
 119, 361, 368, 380-382, 387, 423-429, 
 447. 
 
 Choice : *See ' Free Choice.' 
 
 Circle: See 'Circular . . .,' ' Cir- 
 cuius. . . .' 
 
 Circles : See ' Diagrams.' 
 
 Circular Reasoning : 289. See ' Pe- 
 titio. . . .' 
 
 Circulus in Definiendo : 32, 34-36, 38, 
 77, 83, 84. C. in Proponendo : 103, 
 121 (see 'Tautology'). 
 
 Circumstance : 394-39S, 402, 403, 414- 
 416, 418, 426, 454. 
 
 Clarke, R. F. (Father) : 33, 102, 261, 
 282, 290. 
 
 Class. Class-term, etc. : 18-24, 32, 33, 
 39, 42, 43, 52, 54, 55, 70-72, 80, 
 88, 89, 131, 148-155, 159-162, 166, 
 219, 220, 239-242, 327, 340, 341, 348, 
 351, 352, 356-359, 368, 454. See 
 ' Classification,' ' Xomenclature,' 
 ' Species.' 
 
 Class-concept : 22. See ' Class.' 
 
 Class-designation : 18, 43. 
 
 Class-distinction : 357. 
 
 Class-extension : 151. See ' E.,' ' Ex- 
 tensive. . . .' 
 
 Class-generalization : 352 (Note). See 
 ' G. proper.' 
 
 Classification : 30, 42, 46, 54, 56-74, 
 70-79, 88, 89, 95, 136, 137, 244, 245, 
 341, 368, 369. C. distinguished from 
 Real Division : 56-57 (ftn.). C. by 
 Series: 61 (ftn.). C. of Fallacies: 
 281. Of Logical Principles : 281. 
 
 Class-inclusion : 20-21 (ftn.), 159 (ftn.), 
 239-245. 
 
 Classing : 18, 19, 42, 43, 57. 58, 60, 
 64, 66, 340. 
 
 Class-mark : 352. See ' Definition- 
 mark.' 
 
 Class-relations : 152, 1.53. 
 
 Clause: 112, 138, 216. See 'Alter- 
 native C.'s,' ' Antecedent,' ' Con- 
 sequent.' 
 
 Clearness : 66, 112, 121. 
 
 Clerk-Maxwell, James (Dr.) : 335, 386. 
 
 Closed Conclusion : 309. 
 
 Co-absence of Cause and Effect : 400, 
 403. 
 
 Co-alternative : 422. See ' A. . . .' 
 
 Co-division : 43, 44 (ftn.), 53, 55, 56. 
 
 Co-effect : 309, 406. 
 
 Coexistence : 342, 357, 368, 369, 419 
 (ftn.), 454 (ftn.). 
 
 Coexisting Fact : 405. 
 
 Coextensiveness : 149, 151, 153, 455. 
 
 Cognate Membra Dividentia ; 52.
 
 IXDEX, VERBAL AXD AXALYTIC 
 
 4G7 
 
 Coherence of Natural Phenomena : 443 
 (see ' S3'6tematic Conception . . .'). 
 C. of Thought : 7, 8, 98, 106 {see 
 'Consistency,' 'T. and Identity,' 
 ' Validity '). 
 
 Coherency : 343. 
 
 Coincidence : 14G, 147, 149, 150, 152, 
 153, 158, 159, 161, 240, 391. C. of 
 Antecedents or Consequents in Di- 
 lemma : 272, 273. 
 
 Coincidence-import : 159 (ftn.). 
 
 Coincident Explanations : 443. 
 
 Collateral Condition : 372, 375. See 
 ' Predisposing. . . .' 
 
 Collection of Facts : 313-315. 
 
 Collective Aggregate : 460. C. Con- 
 ception : 349. C. Expression or 
 Term : 162, 163. C. Meaning : 154, 
 158, 160, 162, 163, 171, 239, 241, 
 283, 284. C. Statement or Propo- 
 sition : 154. 
 
 Colligation : 347, 349-351, 
 
 Collocation : 372, 454. 
 
 Command : 93, 94. See ' Imperative.' 
 
 Commensurate Terms : 55. 
 
 Common Attribute, Mark, Property, 
 etc. : 18-21, 38, 57, 61, 62, 65. 236, 
 
 340, 350, 356, 358, 359, 361, 383 (see 
 'Agreement,' 'Resemblance'). C. 
 Circumstance or Element : 395-398, 
 
 415. C. Nature : 22, 24, 81, 85, 
 114, 115, 121, 123, 131. C. Sense: 
 4, 5, 9, 97, 347. C. Term : 213, 217, 
 222 {see ' Middle T.'). 
 
 Comparison : 18-20, 29-33, 37-39, 57, 
 75, 340, 341, 350, 353, 355, 408- 
 410, 414. 422, 431, 436. 438, 442. 
 
 Compatibility : 8, 133, 156. See ' Inc.' 
 
 Complementary Terms : 48. 
 
 Complete Act of Judgment : 122. C. 
 Definition : 39. C. Division : 44 {see 
 ' Continued D.'). C. Elimination : 
 
 416, 417. C. Enumeration : 43, 356, 
 357, 397. C. Explanation : 368- 
 370, 395, 459, 461. C. Induction, 
 C. Causal Inquiry : 326, 327 (ftn.), 
 
 341, 361, 398, 411 {see 'Ind.'). 
 C. Inductive Method : 397. C. 
 Method of Analysis : 422. C. Pro- 
 cess of Observation : 393. C. Proof : 
 454. 
 
 Complex Categorical Syllogism : 255- 
 261. C. Conception : 209. C. Con- 
 structive Dilemma : 272-277. C. 
 Destructive Dilemma : 272, 273. C. 
 Epicheirema : 261. 
 
 Composition (Fallacy) : 283, 284. C. 
 of Causes : 375-377. Of Forces : 
 376. Of Laws : 409. 
 
 Compound Hypothetical Syllogism : 
 271 {see ' Dilemma '). C. Proposi- 
 tion : 132. 153, 158, 100, 274 ; its 
 Contradictorv : 182, 262. 
 
 Comprehensiveness : 157, 385. 
 
 Compulsion : 127. See ' Necessity.' 
 
 Comtc, I. Augusto M.F.X. : 449. 
 
 Conccivability : 72, 130, 449, 4.50. 
 
 Concept, Conception, etc. : ix, 22, 24, 
 25, 29, 31, 34, 40-43, 49, 50, 57 (ftn.), 
 62. 63, 71-85, 88, 89, 95, 96, 97, 112, 
 116, 121, 125-127, 329, 335, 340, 341, 
 349, 353, 449, 459-402 {see ' Abstract 
 C' 'Fundamental C.,' 'General 
 C' ' Idea,' ' Individual C.,' ' Logical 
 C' 'Singular C' ' Subject-c.'). C. 
 in relation to Logical Interest : 78. 
 
 Conception of Fact : 448, 449. 
 
 Conceptual Being : 77. C. Definition : 
 24, 25, 93. C. Determinacy: 80. 
 C. Ordering of Fact, C. System, etc. : 
 43, 73, 78-80, 82, 88, 89, 95 {see 
 'Classification,' 'Division,' 'Por- 
 phyry's Tree '). 
 
 Conceptualistic Definition : 24, 25. 
 
 Conclusion : 8, 105, 129, 137, 145, 194, 
 208, 213-238, 242, 243, 246-249, 
 255-263, 268, 270, 273, 275, 282, 
 289-291. 300-306. 309, 319-328, 337. 
 349, 355, 359-364, 389, 391, 400, 404, 
 408, 409, 416-418, 438, 443, 444, 
 454, 457, 458. 
 
 Conclusive Explanation : 449. 
 
 Concomitance : 165, 342, 397, 414, 415, 
 423, 424. 
 
 Concomitant Variations : 405-407, 417, 
 418. 
 
 Concrete Conceptualism : 24, 25. C. 
 Context : 85. C. Experience : 453. 
 C. form of Scientific Method : 410. 
 C. (Deductive) Method : 409, 410, 
 419, 420. C. Object : 85, 86, 340. 
 C. Science : 136. C. Terms : 85-89. 
 C. Thought : 6. 
 
 Concurrence of Antecedents or Causes : 
 375-383. C. of Conditions : 394. 
 
 Condition : 130, 246, 305, 306, 309, 319, 
 334, 336, 344, 360-363. 309-381, 
 394-404, 407, 411-413, 419-421. 425, 
 428, 436, 439. 448-451. 457, 458. 
 
 Conditional Uniformities : 377, 378. 
 C. Validity : 141, 142. 
 
 Conduct. See ' Morality.' 
 
 Configuration : 460. 
 
 Conic Sections (or Analvtical Geo- 
 metrv) : 29, 52, 362. 
 
 Conjecture : 130, 327, 348. 351. 357, 
 359, 392, 393, 399, 415, 431, 462. 
 
 Conjunctive Division : 49, 53. C. 
 interpretation of Extension : 72 
 154, 284. C. use of ' All ' and 
 ' Some ' : 154. 
 
 Connexion : .S'ec ' Causal C.,' ' Rela- 
 tion,' 'Necessary C C. of Possi- 
 bilities : 138-141, 180. 
 
 Conno-denotation : 21 (ftn.), 146. See 
 ' Intension.' 
 
 30—2
 
 4GS 
 
 THE proble:\i of logic 
 
 Conno-denotative Terms : 73, SO. 
 Connotation : 70-S,"). 89, 147, 105, 170, 
 245 (sec ' Connotative . . .'). C. as 
 formal or real : 70. 71. Its relation 
 to Denotation : 71-73. To Exten- 
 sion : 71, 72. 
 Connotative Name or Term : 73, 74, 
 
 89. 
 Coriscioiisness : 77, 97, 155, 381, 383, 
 452. 4U0. 461 {sec ' Immediate Ap- 
 prehension,' ' Psj-chical . . ..' 'Psy- 
 chology '). C. of a Limit : 79. 
 Consequence : 13G, 304, 319, 325, 328, 
 335-337. 377-383, 435, 436, 440, 
 451-454. 
 
 Consequent: 112. 137, 138, 141, 142, 
 145, 264. 265. 269-274, 320, 325, 337, 
 369, 370, 37S-385, 397, 439. See 
 ' Acceptance of the C.,' ' Rejection 
 of the C' 'Hypothetical Judgment.' 
 
 Conservation of Energy : 333, 385-389. 
 C. of Mass : 385. 
 
 Consistency : 1-9, 16, 98, 99, 102, 103, 
 106, 112, 145, 157, 172, 180, 214, 
 282, 286, 448, 457. Distinguished 
 from I\Iaterial Compatibility : 8. 
 From Validity : 8. C. in Hypo- 
 thesis : 318. 
 
 Consistency-Logic : 1. 
 
 Consistency-principle : 99. See ' P. of 
 (Logical) C' 'P. of Non-Contra- 
 diction.' 
 
 Consistent Statement : 145. See ' Con- 
 sistency.' 
 
 Constancy of Energy : 386, 387. 
 
 Constant Cause : 399. C. Effect : 398. 
 
 Construction : 26, 29, 65, 246, 321, 
 322, 328, 390, 402, 421. 
 
 Constructive Dilemma : 271-277. C. 
 Hypothetical Syllogism : 263, 264. 
 
 Content: 23, 24, 27, 28, 78, 118, 120, 
 125, 126, 358. 
 
 Context: 15-17, 85, 106, 116-120, 282, 
 394. See ' formal C.,' ' Logical 
 C' ' Real C' ' Ultimate C 
 
 Contingence : 379, 454. See ' Possi- 
 bility.' 
 
 Continued Division : 41-52, 72-80. 
 
 Continuity : 106, 124 («ee ' Coherence,' 
 ' Unity '). C. of Cause and Effect : 
 381 {see ' Immediacy in Causal Con- 
 nexion '). C. in Division : 44-40. 
 
 Contradiction (or Contradictory Oppo- 
 sition) : 48, 94, 98-107, 122 (ftn.), 
 124, 128, 130, 150, 164, 172-174, 
 188, 208, 264, 324, 325, 448, 450. 
 See ' Apparent C.,' ' Non-C.,' ' Self- 
 C 
 
 Contradictory Stat<!ment3 or Propo- 
 Bition.s : 98-106, 128, 1.30, 264-266, 
 268, 291 , 324, 325. ' C. Terms ' : 48. 
 
 Contraposition : 189, 197, 198. ' C. of 
 Hypothcticala ' : 265, 266. 
 
 Contrapositive (Converse) : 197-202, 
 206, 248, 249, 400. 
 
 Contrariety (or Contrary Opposition) : 
 128, 172, 176. 
 
 Contrary Propositions : 128, 130, 
 268, 291 (see ' Contrariety '). ' C. 
 Terms ' : 48. 
 
 Control of Conditions : 371-373. 394, 
 401, 402, 404. C, or Limitation, 
 through Relevance to Purpose : See 
 ' Purpose.' 
 
 Conventional : See ' formal.' C. Clas- 
 sification : 64-66. C. Meaning : 16. 
 C. Signs or Svmbols : 25, 93, 115 
 (see 'Sign,' 'Words'). 
 
 Convergence of Evidence : 443. 
 
 Converse : 193-201, 206, 229, 232, 237, 
 249 {see ' Weak C). C. Fallacy of 
 Accident : 284, 285. 
 
 Conversion : 189, 193-198, 200, 202. 
 260, 265, 324, 325. C. bv Limitation 
 or per accidens : 194, 197, 248, 249. 
 C. by Negation : 197 {see ' Contra- 
 position,' 'Contrapositive'). ' C. of 
 Hypothcticals ' : 265. C. of : 54. 
 
 Convertcnd : 193-195, 201. See ' Con- 
 version.' 
 
 Conviction : 123, 377. 
 
 Co-operating Causes : 375, 376. 
 
 Co-ordinate Possibilities : 136 {see 
 ' Disjunctive . . .'). C. Species (or 
 Co-species) : 32, 33, 39, 75, 136 {sec 
 ' Division '). 
 
 Copernicus, Copernican System : 316, 
 330, 331, 362. 
 
 Co-presence of Cause and Effect : 400, 
 402. 
 
 Copula : 124-126, 251. 
 
 Copula-mark : 124 (ftn.), 126, 163, 168, 
 250, 251. 
 
 Corpuscular Theory of Light : 336, 338, 
 339. 
 
 Correct Thinking : Sec ' Right T.' 
 
 Correction of Observations : 413. 
 
 Correlate : 159. 
 
 Correlated Differences : 58, 59. C. 
 Properties : 368, 369. 
 
 Correlation : 344, 368. 
 
 Correlative Terms : 30, 35. 
 
 Corroborative Instance : 403. See 
 ' Negative I.,' ' Positive I.' 
 
 Co-species : See ' Co-ordinate S.' 
 
 Countable Objects : 146. 
 
 Counteraction : 371, 374-377, 381, 399, 
 404, 454. 
 
 Counter-dilemma : 275-277, 293, 294. 
 
 Counter-extreme : 123. 
 
 Counting : ;S'ee ' Enumeration.' 
 
 Creighton, James Edwin (Prof.) : 313, 
 337. 
 
 Criteria of Legitimate Hypothesis : 
 328. Of Validity in Dilemma : 
 274.
 
 INDEX, VERBAL AXD ANALYTIC 
 
 4G9 
 
 Criterion of Consistency : 98, 99. C. 
 (or Test) of Explanation : 453. C. 
 of Induction : See ' Inductive C 
 C. of Intuitive Certainty : 75. Of 
 Trutii : 38G. Of Valid Syllogism : 
 
 241, 243. 
 
 Criticism : 119 (ftn.). C. of a Dis- 
 junction : 137, 13H. Of Definitions : 
 31, 3G-39. Of Dichotomy: 50-52. 
 Of the distinction between Analytic 
 and Synthetic Judgment : 120-122. 
 Of the Hamiltonian Scheme of Cate- 
 gorical Propositions : 159-161. Of 
 Mr. Joseph's view of Induction : 
 416-420. Of Mill : 81, 82, 301-304, 
 455-4.'J8. 
 
 Cross-division : 45, 52, 53, 55. 
 
 Crucial E.xperiment : 336, 338, 422. 
 
 C. Instance : 336. 338, 355, 422, 426, 
 429. 
 
 Cumulative Effect : 373, 374. 
 
 Darapd : 222, 226, 227, 231, 232, 236, 
 
 242, 243. 
 
 Darii : 226, 237, 242, 247, 249, 256. 
 
 Darwin, Charles : 270, 314, 315, 353, 
 354, 391, 421, 447. 
 
 Darwin, Francis : 314 (ftn.), 315. 
 
 Datisi: 226, 227. 
 
 De Morgan, Augustus (Prof.) : 49 (ftn.). 
 
 Death as an Effect : 379, 381-384, 397. 
 
 Declaratory Sentence : 94. 
 
 ' Decrease ' in Denotation : 72. 
 
 Deducible Property : 26, 369. 
 
 Deduction : 246, 321, 322. 326, 327 
 (ftn.), 337, 342, 343, 407-412, 419, 
 426, 432-438, 442, 443 (see ' De- 
 ductive ...,*' Quantitative D.'). 
 Its relation to the Dictum de Omni : 
 244-247. 
 
 Deductive Development of Hypothesis : 
 352, 356, 411, 419, 426, 438 (see 
 ' Dev. of H.'). Ded. Inference: 
 246. 247, 318-328, 348, 385. D. 
 Method: 244, 246, 408-411, 420. 
 
 D. Process : See ' Deduction.' D. 
 Reasoning : 251, 304, 309. D. 
 Science : 245. 409. D. Theory : 
 443. D. S}-nthesis : 407. D. Vi- 
 tality : 318, 430. 431 {see ' Fruitful 
 Hypothesis '). 
 
 Defective Argument : 230-232. 
 
 DeGnability : 25, 73. D. of Experi- 
 ence-concepts : 82-85. Of the In- 
 dividual : 81. Of Infima Species: 
 74. Of Particular Existences : 83. 
 Of Proper Names : 74, 79-85. Of 
 Subaltern Genus : 74. Of Summum 
 Genus : 73-79. Of Symbols : 25. 
 
 Detiniendum : 23-25, 121. See ' Defi- 
 nition.' 
 
 Defining Terms : 33, 34. See ' Defi- 
 nition-mark.' 
 
 Definite Answer: 113. D. Interest: 
 118. D. Limitation of Knowledge: 
 157. D. Possibility: 112, 132. D. 
 Species or Term : 47-52. D. Sub- 
 ject-term : 116. 
 Definiteness: 157. »See 'Determinacy.' 
 Definition : ix, 3, 16-39, 66, 67. 71-81, 
 93, 102, 115, 118, 120, 121, 125, 138, 
 
 146, 147, 165, 214, 281, 282, 300, 
 309, 316 (ftn.), 322, 340, .341, 353, 
 302. 308, 383, .385, 421, 429 [see 
 ' Circultis in Dcfiniendo,' ' Complete 
 D.,' ' Conceptualistic D.,' 'Defin- 
 ability,' ' Definiendum,' ' Diagnostic 
 D.,' 'Dictionary-definition,' 'Etymo- 
 logical D.,' ' "Fitting" D.,' ' formal 
 D.,' 'Genetic D.,' 'Homogeneous 
 D.,' ' Metaphoric D.,' ' Nomiualistic 
 D.,' 'Object of D.,' 'Partial D.,' 
 'Peripheral D.,' 'Pragmatic D.,' 
 ' Realistic D.,' ' Relational D.,' 
 ' Verbal D.'). D. as Progressive : 
 63, 79. D. by Characterization : 
 61-63. By Ends : 62, 63. By Type 
 (Central D.) : 62, 63, 71, 136. D. in 
 Formal Logic : 214. D. in relation 
 to Classification: 57 (ftn.). To 
 Division : 3, 39, 41, 43, 45. 52, 70, 
 71. 89. D. of Correlatives: 30, 35. 
 Of Semi-correlatives : 35, 30. Of 
 Words: 17-20, 23-25, 115. D. per 
 genus ct diffcrentiam : 17-22, 25, 32- 
 39, 57. 68, 69, 73-77, 82, 89. 
 
 Definition-mark: 62-04, 70. 81, 121, 
 105, 351. 352. See 'Differentia,' 
 ' Genus,' ' Negative Def.-m.' 
 
 Definition-rules : 32-30. 
 
 Definitions as Simply Convertible : 33, 
 197. Ds. tested : 32-39. 
 
 Degree of Deterrainacy : 89. Of Gen- 
 erality : 45, 88. 
 
 Demand : See ' Postulate.' 
 
 Demonstrandum : 323, 325. 
 
 Demonstration : See ' Proof.' 
 
 Demonstrational Fallacies : 286-295. 
 
 Denial : 94, 98, 123, 124, 126, 134, 139, 
 140, 166, 173, 174, 193, 235, 236, 
 324, 422. D. of Being : 75. 76. Of 
 Postulate : 448. Of the Antecedent : 
 See ' Rejection of the A.' 
 
 Denotation : 70-73, 147 (see ' Denota- 
 tive ' ' formal Denotation,' ' Real 
 
 D.'). D. in relation to Connotation : 
 71-73, 89. To Extension: 71, 72, 
 
 147. D. of General Collectives : 
 162 (ftn.). 
 
 Denotative Names or Terms : 73, 74, 
 
 89. ' D. Reading ' : 147. 
 Derivative Objects : 85. 
 Descartes, Rene : 318, 332, 333, 362, 
 
 390. 
 Description : 17, IS. 24, 62, 342, 349. 
 
 350, 369, 392, 3JS.
 
 470 
 
 THE PROBLEM OF LOGIC 
 
 Descriptive Law : 342 {see ' Empirical 
 L.'). D. Terminology : 66, 07. 
 
 Designation : SO (ftn.). D. by Class- 
 names : IS, 19. 
 
 Destructive Dilemma : 271-273, 277. 
 D. Hypothetical Svllogism : 263, 
 264. 
 
 Determinacv : 16, 17, SO, 88, 89, 112, 
 113. See ' Determinate ..." 
 
 Determinant : 20S. 200, 448. 
 
 Determinate Agreement : 20. D. 
 Answer: 112, 113. D. Categorical: 
 113. D. Connotation: 72, 73. D. 
 Content : 78. D. Exhaustiveness : 
 46. D. Form: 112, 113. D. Mark 
 or Meaning : 18-22, 41, 43, 88, 95, 
 
 121. 105 (ftn.), 285. 
 Determination : Sec ' Definition,' ' De- 
 
 terminacy.' D. of Import by Iden- 
 tity : 146. D. of Meaning through 
 Disjunction : 131 {see ' Disjunc- 
 tive . . .'). Through Division : 41, 
 127. Through Interest or Purpose : 
 51. D. of Possibilities : 131. 
 
 Determining Cause : See ' Exciting 
 C D. Condition : 395 {see ' C). 
 
 Determinism : 367, 448, 449, 450-455, 
 458-462. See 'Inductive Postulate,' 
 ' Law of Causation,' ' Uniformity.' 
 
 Deterministic Explanation, D. Postu- 
 late : See ' Determinism.' 
 
 Development: 58-63. 129, 341, 447, 459- 
 462. D. of a Classificatory System : 
 74 (ftn.), 76 {see 'Classification'). 
 Of a Possibility : 138. Of a Topic : 
 88, 89, 119. D. of Hypothesis: 
 136, 318-320, .326-328, 335-339, 3.12, 
 356, 361, 411, 419, 424-433, 436- 
 442. Of Implications : 229. Of 
 Interest : 51, 88. Of Laws : 343, 
 367, 368. D. of Meaning, etc. : 3, 
 22. 26, 41, 4.3-45, 50, 70, 71. 73, 74, 
 76, 80, 88, S9, 95-98, 146, 148, 262 
 (Xote), 321 {see 'Variation in M.'). 
 D. of M. in Proposition : 119, 121, 
 
 122, 124. D. of M. through Classi- 
 fication : See 'C Through Defi- 
 nition : See ' Def.' Through Disjunc- 
 tion : 112-114, 131, 133. Through 
 Division : See ' Div.' Dev. of 
 McthofJ : 449. Of Purpose : 57. 58. 
 Of the Category of Substance : 47 
 (ftn.). Of the F. D. : See ' Unity of 
 F. D.' Of the Genus : 20-22. Of 
 the ' That ' : 82. Of Thouglit : 31, 
 112-114, 138. 
 
 Developmental : ,See ' Genetic' 
 Dewey, John (Prof.) : viii. 
 Diagnosis : 03, 64. D. of Effect : 384. 
 Diagnostic Classification : 58, 63-66. 
 D. Definition : 63, 64, 341, 395. 
 Diagrammatic Representation of A, E, 
 i7,I, O: 150-153. 160. 
 
 Diagrams : 20-21 (ftn.), 150-153. 160, 
 195, 196, 208, 209, 240, 244, 323, 
 392, 441. 
 
 Dialectic : 307. D. of Hegel : 75, 76. 
 
 Dialysis : 123. 125, 126. 
 
 Dialytic Copula or C.-mark : 124 (ftn.). 
 126. 
 
 Dichotomy: 46-52, 79. 
 
 Dicta : 239-247, 250. 
 
 Dictionary-definition : 19, 20. 
 
 Dictum de Diver so : 240, 242, 243. D. 
 de Exemplo et de Ezcepto : 241-243. 
 D. de Omni et Nullo : 240-247, 2.50, 
 251 ; its relation to Deduction : 
 244-247, 324. D. de Reciproco : 
 241. 
 
 Difference : 43, 57-59, 75, 122-124, 146, 
 245, 358-360. 362-364, 396, 402-407. 
 413-419, 422 (ftn.), 425, 426 {see 
 ' Differentia,' ' Double Method of 
 D.,' ' Essential D.,' ' Method of D.'). 
 D. in relation to Identity : 96-98, 
 121-124. 
 
 Differencing : 122. See ' Differentia 
 tion.' 
 
 Differentia : 17-22, 25-29, 32-40. 54, 
 57, 08, 09, 72-77, 82, 89, 120, 121. 
 159. Its relation to the Genus: 50. 
 
 Differentiation : 400, 461. D. (or 
 Specification) of Meanings : 96, 123, 
 124, 131 (see 'Differentia,' 'Dis- 
 crimination '). D. of M. in Classifi- 
 cation : See ' C. ' ; in Division : See 
 ' D.' Diff. of S from P : 122. 
 
 Dilemma: 271-277, 289, 290, 292-295. 
 
 Dilemmatic Fallacies : 276, 289, 292- 
 295. 
 
 Dimaris : 226, 227, 237. 
 
 Direct Apprehension, D. Cognition : 
 See ' Immediate A.' D. Causal 
 Action : 382. D. Deductive Method : 
 409. D. Effect: 382 {see 'Imme- 
 diacy in Causal Connexion '). D. 
 Observation : 341, 408, 435 {see 
 'O.'). D. Perception: 440, 441 
 {see 'P.'). D. Reduction: 247-250, 
 324. 
 
 Disamis : 226, 227, 233. 
 
 Disbelief: 123. 
 
 Discontinuity in Division : 44, 45. D. 
 of Thought : 106. 
 
 Discontinuous Variation : 406. 
 
 Discourse (or Conversation) : 17, 22, 
 97, 116, 154, 155. See ' Subject of 
 D.,' ' Universe of D.' 
 
 Discovery: 331, .336, 338, 339, 348, 
 350, 359, 361, 362, 369, 383, 388, 
 389, 391, 395, 407, 408, 412-414, 424, 
 425, 429. 435, 438, 449. D. of a 
 Middle Term: 320. D. of Pre- 
 misses : 325. 
 
 Discrimination of Meanings : 56. Of 
 Qualities, Relations, etc. : 85, 86.
 
 I^sDEX, VERBAL AND AN.ILYTIC 
 
 471 
 
 DiscusBion : 5, 102, 307, 309, 313. 340. 
 
 Dirtiniplication : 188, -210, 246, 301, 
 303, 306, 307, 327 (ftn.). -S'ee ' Ex- 
 plication.' 
 
 Disjunction: 113. 132-134, 136, 137, 
 190, 213, 422 (see ' Disjunctive . . .'). 
 D. in Dilemma : 271-274, 293. 
 
 Disjunctive Argument, D. Inference : 
 190. D. Denotation : 1G2 (ftn.). 
 D. Development of Meaning : 113, 
 114, 131, 132. D. Dichotomy, D. 
 Division : 39-47, 49-53, 70-73, 75, 
 77, 89. D. Exhaustiveness : Sec 
 ' Exhaustive Division.' D. Interpre- 
 tation of Intension : 154. D. Judg- 
 ment, D. Proposition, D. Statement : 
 vii, 111-114, 131-138, 178, 262, 2G6, 
 274, 415, 417, 422. Its Definition : 
 
 137. Its Opposition : 178. Its re- 
 lation to the Categorical : 112-114, 
 131 ; to the Hj'pothetical : 113, 114, 
 
 138. D. Meaning : 132, 284. D. 
 Moods : 262, 263. D. Obverse : 192. 
 D. Possibility : 138. D. Predicate : 
 
 132. D. Predications : -See ' Alter- 
 native . . .'I D. Specification : 21, 
 
 133. D. Syllogism: 236, 262, 263, 
 323, 416. 
 
 Disorder : 455. 
 
 Disposition : 129. 
 
 Dispositional Possibility : 129, 131 
 (ftn.). 
 
 Disproof : 94, 291, 328 (ftn.), 335, 336, 
 362-364, 401, 416, 430, 431, 436- 
 438, 443 {see 'Refutation'). D. of 
 Alternatives or of Rival Hypotheses : 
 422, 429, 432. 
 
 Disputation : See ' Discussion.' 
 
 Dissection : 315, 401, 402. 
 
 Distinction of Alternatives : 134-136. 
 D. of Meanings : 105 {see ' Differen- 
 tiation of M.'). D. of Subject and 
 Predicate : 96. D. of Summa 
 Genera : 74 (ftn.). 
 
 Distributed Term: 148-151, 153, 155 
 160, 161, 193, 194, 216-219, 221 
 223, 225, 228-230, 237, 238. 
 
 Distributedness : 158. 
 
 Distribution of Terms in Conversion : 
 194. In Proposition : 148-150. 158 
 {see ' Distributed ...,'' Undistri- 
 buted . . .'). In Syllogism: 216- 
 
 219, 225, 228, 230 {sec ' D.ed ', 
 
 ' Und. . . .'). D. Rules of the S. : 
 217-219, 223, 224. 
 
 Distributional SvUogism : See ' Simple 
 Categorical D." S.' 
 
 Distributive Form : 163. D. Meaning • 
 154, 158-160, 162, 163, 239, 283, 2S4! 
 D. Proposition : 154. 
 
 Dittmar, W. (Prof.) : 422 (ftn.), 423. 
 
 Dividendum : See ' Division,' ' Object 
 
 "^oi Division.' 
 
 Divisibility : 73. D. of the Infima 
 Species : 74, 79. 
 
 Divisio per saltum : 44, 4'). 
 
 Division : 3, 37, 39-57, 59, 61, 70-80. 
 136, 146, 159, 316 (ftn.), 368 {see 
 ' Class,' ' Continued D.,' ' Diciio- 
 tomy,' 'Disjunctive D.,' 'Exclusive 
 Reading ...,'' Exhaustive Division,' 
 'Fallacy of D..' 'formal D.,' 'In- 
 adequate D.,' ' Logical D.,' ' Meta- 
 physical D.,' 'Non-logical D.,' 'Object 
 of D.,' 'Physical D.,' 'Real D.,' 
 'Rules of Logical D.,' ' Verbal D.'). 
 D. Formula : 40, 72. D. in relation 
 to Definition : 3, 39, 41, 43, 45, 52, 
 57 (ftn.), 70, 71, S9, 131. D. of Plural 
 Terms : 53. 
 
 Divisional Basis : See ' Fundamentum 
 Divisionis.' 
 
 Divisions tested : 52-56. 
 
 Dogma : 313, 315, 316, 382. 
 
 Dogmatism : 291, 330. 
 
 Double Agreement : 398-400. D. 
 cheirema : 261. Double F. D. 
 
 Epi- 
 
 : See 
 of 
 
 ' Unity of F. D.' D. Method 
 Difference : 406, 407, 422 (ftn.). 
 
 Doubt: 75, 112, 113, 123. 
 
 Duncan, Prof. : 61 (ftn.). 
 
 Duty: 107, 127. 462. See ' Obli 
 gation.' 
 
 Dynamics : 318, 376. 
 
 E Proposition : 147-151, 156, 158-160. 
 162, 164, 166-172, 239, etc. It? 
 
 194, 195. Its Dia- 
 159-161. Its Dia- 
 
 Conversion : 
 
 gram : 151. 
 ■q Proposition 
 
 gram : 160. 
 ' Each ' : 154, 162, 168. ' E. of All ' : 
 
 146, 158, 159. 239. ' E. of Some ' . 
 
 149, 158, 159. 
 Earth : 38, 80, 81, 134, 238, 284, 316, 
 
 327, 329, 330. 334, 338, 342. 360, 
 
 372-377, 393, 396, 405, 413. 421, 422, 
 
 435-444. See ' Rigidity of the E.' 
 Earthquakes : 440-442. 
 Earth%vorms : 328, 374, 421, 422. 
 Eclipse : 335, 336. 394. 432-434. 
 Economics : 66. 
 Educt: 189, 192, 194, 200-202, 204. 
 
 See ' Contrapositive,' ' Converse,' 
 
 ' Inverse,' ' Obverse.' 
 Eduction : 7, 189-208, 248. .304. 
 Effect: 333, 334, 337. 307, 369-384, 
 
 387-389, 394. 395, 397-409, 411, 412, 
 
 415, 417-419, 451, 452, 454 'ftn.) 
 
 {see ' Total E.'). Its Definition^ 35. 
 Effective Condition : 384. 
 Efficiency : 68, 69, 395. 
 Efficient Cause : 378. E. Force : 370 
 
 {see ' F.'). 
 Eightfold Scheme of Categoricals : 159, 
 
 160.
 
 472 
 
 THE PROBLEM OF LOGIC 
 
 ' Either . . . or ' : 132-135, 137, 262, 263. 
 272. jSee ' Disjunction,' ' Disjunc- 
 tive . . .' 
 
 Ejaculation : 93, 116, 168. 
 
 Elaboration : 343. 
 
 Electricity : 335, 337, 372, 3S0, 394, 
 402, 407. 
 
 Electro-magnetic Theory of Light : 
 335. 
 
 Electro-magnetism : 334. 
 
 Electron : 459. 
 
 Elementary Propositions : 153. E. 
 Terms : '34, 36. 
 
 Elements of Spiritual Experience : 107. 
 E. of the Proposition : See ' P.al E.' 
 
 Elenchus : 291. 
 
 Elimination : 137, 343, 358, 384, 396- 
 398, 400, 402, 414-418, 420, 422, 449. 
 E. of Error : 413. 
 
 Elimination-method : See ' M. of E.' 
 
 ' Else ' : See' Or else.' 
 
 Emerson, Ralph Waldo : 282. 
 
 Emphasis : 116, 134, 156, 157. 
 
 Emphatic Term : 116. 
 
 Empirical Law, E. Rule : 341-343. 368. 
 410, 455. 
 
 Empiricism : 313-317. 
 
 End : 96, 128, 129, 283, 314, 429, 453, 
 460, 461. See 'Purpose,' 'Tele- 
 ology.' 
 
 Energy : 87. 333, 335, 342, 371. 385-389. 
 
 Enthvmeme : 230-232, 261, 305. 
 
 Enumeration : 42, 43, 71, 72, 327. 347. 
 351-361, .396-398. 454-456. See 
 ' Countable . . .' 
 
 Enumerative Argument : See ' Enu- 
 meration.' 
 
 Enumerative Universal (Judgment) : 
 147-148 (ftn.), 357. 
 
 Enumerative Induction (or Method of 
 Simple Enumeration) : 347, 351-358, 
 369, 397, 454-456 (see ' Enumera- 
 tion '). E. I. in the light of the In- 
 ductive Criterion : 355-357. 
 
 Epicheirema : 260, 261. 
 
 i:pimenide.s : 103, 182, 183. 
 
 P^piphenomenon : 460. 
 
 Epi.syllogism : 257. 258, 260. 
 
 Equivalence : 124, 147. 149. 150, 153- 
 160, 166, 177. 189, 190, 198, 199. 
 249, .324, 386-389, 400, 408, 409. 
 
 Equivalents of Modals : 130. 
 
 Equivocal Words : 37. See ' Ambi- 
 guity.' 
 
 Equivocation : 282-285. 
 
 Erroneous Hypothesis: See '"False" 
 H.' 
 
 Error: 295, 303, 331, 338. 385, 413. 
 4.36, 437. See 'False . . .,' 'Resi- 
 dual e: 
 
 Essence (or Essential Mcaninj^) : 23, 
 26-29, 38, 414. E. of Inductive 
 Method : 413-422. 
 
 Essential Characteristic or Property : 
 43, 369. E. Concepts : 63. E. 
 Condition : 372, 384. E. Difference : 
 28, 360, 363. E. Identity : 122 (ftn.). 
 E. Marks : 64. E. Meaning : See 
 ' Essence.' E. Resemblance : 28. 
 359, 360, 362. 
 
 Essentials of Induction : 316-325, 352. 
 419. 
 
 Ethics : See ' IMorality.' 
 
 Etymological Definition : 32. 
 
 Euckon. Rudolf (Prof.) : ix. 
 
 Euclid : 245, 246, 321-323. 
 
 Euler, Leonanl (Prof.) : 437, 438. 
 
 Event: 86, 114, 119 (ftn.), 130, 370, 
 392, 397, 401, 402, 453, 457, 459. 
 
 Everett, J. D. (Prof.): 440. 
 
 ' Every ' : 158, 164, 168. 
 
 Evidence : 130, 189, 246, 275, 286- 
 289, 291, 303-309, 313, 317, 326. 
 328, 337, 348, 350, 367, 401, 443, 
 458 (see ' Fallacies of Irrational E.,' 
 'formal E.,' 'Real E.,' ' Self-E.'). 
 E. of the Senses : 450. 
 
 Evolution : See ' Development.' 
 
 Exactitude : See ' Precision.' 
 
 Examples : See under subjects exem- 
 plifi'd. 
 
 Exception : 103. 315, 351, 357, 376. 
 377, 400. 401, 420, 421, 450, 455. 
 458. See ' Apparent E.' 
 
 Exceptive Proposition : 166, 167. 
 
 Exciting (or Determining or Initiating 
 or Precipitating) Cause or Condition : 
 371-373. 423, 425, 426. 
 
 Exclamation : 168. See ' Ejaculation,' 
 ' Exclamatory . . .' 
 
 p]xclamatory Judgment : 116. 
 
 Excluded Middle : See ' Law of E. 
 M.' 
 
 Exclusion : 323, 324, 414-418. E. of 
 Classes : 153, 339-244. E. of Pos- 
 sibilities : 113, 230. 
 
 Exclusive Proposition : 160, 167. 170, 
 198, 199. E. Reading of the Dis- 
 junctive : 131-137, 178, 263, 266, 
 274. E. Universal : 166. E. use of 
 ' Some ' : 155, 150, 175. 
 
 Exclusiveness of Alternatives in Dis- 
 junction : 131. 133. 135-137 (see 
 ' Exclusive Reading . . .'). E. of 
 Membra Dividentia : 44. 45. 54. 
 55. 
 
 Exclusivist : See ' Exclusive Read- 
 ing . . .' 
 
 Exhaustion : 323. 
 
 Exhaustive Di.sjunction : 417. E. Di- 
 vision : 44, 46, 49, 52-55. E. Read- 
 ing of the Di.sjunctive : 132-134, 136. 
 137, 263. 
 
 Exhaustiveness : See ' Exhaustive . . .' 
 
 Exhaustivist : See ' Exhaustive Read- 
 ing . . .'
 
 INDEX, VERBAL AND ANALYTIC 
 
 473 
 
 Existence : 75, 97, 131 {sec ' Being '). 
 E. in Thought. See 'T.-c.' 
 
 Existent : 24. 80, 115 (ftri.). 
 
 Existential Ground : 370 (ftn.)- E. 
 Import: 115 (ftn.), 18'J. 100. 
 
 Experience : viii, 5, G, 28, 29. 75, 70, 78. 
 82-85, 97, 107. 114, 105, 303, 326. 
 340, 351, 352, 377, 379. 392. 419. 439, 
 453, 4.55-458, 461. See ' Immediate 
 Apprehension,' ' Personal E.,' 
 ' Sense-e.,' ' Spiritual E.,' * Time-e.,' 
 ' Unorganized E.' 
 
 Experience-concept ; 82-85. 
 
 Experience-definition : 82, 84, 85. 
 
 Expcrient : 82. 
 
 Experiment, Experimentation : 2, 145, 
 334-330, 375, 386, 393-395, 397, 
 401-408, 414, 416. 41S, 419 (ftn.). 
 425, 426, 428, 435, 440. See ' Method 
 of E..' ' Natural E.' 
 
 Experimental Analysis : 94, 343, 357, 
 389. E. character of Scientific Ob- 
 servation : 389, 390, 394. E. Con- 
 ditions : 394, 403. E. Instance : 
 400 (.see 'Negative I.,' ' Positive I.'). 
 E. Inquiry, E. Investigation : See 
 ' Experiment.' E. Method : See ' M. 
 of Experiment.' E. Proof : 450. E. 
 Stage in Causal Inquirv : 398-408. 
 E. Test, E. Testing. E. Verification : 
 389, 424-426, 440, 442. E. Truth : 
 450. 
 
 Experimentum Crucis : 336, 338, 422. 
 
 Explanation : 3, 122, 137. 146, 245, 
 303, 304, 313, 317, 318, 321, 326. 
 328, 330-334, 336, 338, 339. 341- 
 344. 347. 348. 351, 355, 356, 305- 
 402. E. of Laws : 369. 
 
 Explanatory Law : 342, 357. 
 
 Explication of Meaning : 57 (ftn.), 70, 
 94. 95, 304. 322. 326. <Sfee ' Dis- 
 implication.' ' Implication.' 
 
 ' Explicative ' Judgment. ' E.' Propo- 
 sition : 120-122. 
 
 Exponible Proposition : 106, 170. In 
 relation to Eduction : 198, 199. 
 
 Expression of Ideas in Language : 16. 
 E. of Meaning in Words : 14-17. Of 
 Truth in Language : 16. 
 
 Expressive Signs : 14. 
 
 Extension, Kxtcnsity, Extent : 21 
 (ftn.), 80, 146-155. 158-161, 208, 219, 
 222. 239. Its relation to Connota- 
 tion : 71. 72. 
 
 Extensional Coincidence : 161 (ftn.) 
 (see ' Coincidence '). E. Correlate : 
 159. E. form of the Fourfold 
 Scheme : 159. E. Import : See 
 ' Extension-import.' 
 
 Extension-import : 57 (ftn.), 1.46-148. 
 150, 153. 1.58-160, 162. 190. 210, 239. 
 
 Extensity : See ' Extension.' 
 
 Extensity-rclations : 150. 
 
 Extensive Dcfmition : 21 (ftn.). E. 
 Import : <S'ee ' Extension-i.' E. in- 
 terpretation of Dividendum : 53. 
 E. Proposition : 147, 148 (ftn.), 101. 
 
 E. Reading : jS'ee ' Extension-im- 
 port.' E. Reference : 146-152, 155, 
 158, 239. 241. 
 
 Extent : See ' Extension.' 
 
 External Observation : 448. 452. 4.53. 
 
 Externality of the Pre-philosophical 
 
 Attitude : 4, 5, 80. See ' External 
 
 Observation.' 
 Extra-logical Elements in Deduction : 
 
 321, 322. 
 Extreme : 123. 
 Extremes in Division : 53. 
 
 Fact : ix, 114, 115, 140, 141, 246, 303- 
 309, 313-322, 326, 328, 334-344, 
 347-353, 356-358, 367-369, 377-380, 
 386, 389-395, 401. 405, 409-412. 417, 
 420-427, 430-444, 447-459. F. as 
 intelligible : 104 (.see ' Intelligibility 
 of Experience '). F. as Systematic : 
 See 'S. Conception . . .' F. in 
 relation to Analysis : 29 ; to Classi- 
 fication : 57 ; to Dichotomy : 52 ; 
 to Division : 43 ; to Enumeration : 
 43 ; to Experience : 6 ; to Inference : 
 290 ; to Interest or Purpose : 1-0, 
 453 [see ' Relevant F.') ; to Meaning : 
 35 ; to Proposition or Statement : 
 94, 96, 135 (see ' S. as bearing on 
 F.') ; to Thought: 1-6; to Time: 
 100-102 ; to Truth : 1-5, 97. F. of 
 Observation: 114, 115. F. under 
 (or expressing) Law : 114, 115, 138, 
 308, 307, 389, 393, 449, 453, 458. 
 
 Faithfulness : See ' Fidelity.' F. in 
 Thought : 107. 
 
 Fallacies : 279-295 (see ' Demonstra- 
 tional F.'). F. in Method : 295. 
 
 F. of Illicit Process : See ' I. P.' 
 Of I. Proof: 286, 289-295. Of Ir- 
 rational Evidence : 286-289. Of the 
 Mixed Hypothetical Syllogism : 264, 
 265. 
 
 Fallacious Inference by Added De- 
 terminant : 208, 209. F. I. by 
 Complex Conception : 209. 
 
 Fallacy : 106. 107, 228. 233. 376, 386. 
 391, 461 (see 'Converse F. of Acci- 
 dent,' ' Fallacies,' ' Formal F.'). F. 
 of Accident : 284. 285. Of Accept- 
 ing or Affirming the Consequent : 
 See ' Acceptance of the C Of Am- 
 biguous Structure or Amphibole . 
 285, 286. Of Composition : 283: 
 284. Of Denying or Rejecting the 
 Antecedent : 'See ' Rejection of the 
 A.' Of Division: 283, 284. Of 
 Dogmatism : 291. Of Equivocation : 
 282-285. Of False Parenthesis : 285.
 
 474 
 
 THE PROBLEM OF LOGIC 
 
 Of Flexion : 2S3. Of Ignoratio 
 Elcnchi : See ' /. E.' Of Irrelevant 
 Evidence : 2S8. Of Mal-observa- 
 tion : 393. ^ Of Mcntiem : 103. 182, 
 183. Of Negative Conclusion, Two 
 Atiirmative Premisses : 224. Of Xe- 
 
 • gative Premiss, Atiirmative Con- 
 clusion : 22-i. Of Non-obsorvation : 
 . 391. 392. Of One Particular Pr., 
 ^ Universal C. : 224. Of Petitio Prin- 
 cipii : See ' P. P.' Of ' Post hoc 
 ergo propter hoc ' : 377. Of Quatcrnio 
 Terminorum : See ' Q. T.' Of Two 
 Negative Premisses : 224, 233, 234, 
 258 (ftn.). Of Two Particular Pre- 
 misses : 224. Of Undistributed 
 Middle: See ' U. M.' Of Vicious 
 Circle : 32, 34, 35. 
 
 False . . .: See ' Falsehood . . .' ' F. 
 Analogy ' : 358, 3G2-3G4. ' F.' or 
 Erroneous Hvpothesis : 334, 336, 
 338, 339, 432.' F. Parenthesis : 285. 
 
 Falsehood, Falsity : 7, 93, 94, 99, 103, 
 104, 122, 141, 145, 306, 310, 393, 
 449, 454. Ignored by Formal Logic : 
 7-9, 157, 309. 310. F. in relation to 
 Hvpotheticals : 179, ISO. To Time : 
 100-102. F. of Conclusion: 324, 
 325. Of Premisses : 214, 292, 295, 
 324. 325. 
 
 Faradav, Michael : 335, 337. 
 
 Fatalistic Dilemma : 293. 
 
 Favourable Condition : 394. F. Fact, 
 F. Event, F. Instance : 391. 
 
 Feeling : 4, 82-85, 87. 392, 393. Right 
 F. : 118. 
 
 Feeling-immediacy : 85. See ' Imme- 
 diate Experience.' 
 
 Felaplon : 220. 227. 
 
 Ferio : 226, 235, 237, 242, 247. 248, 
 256. 
 
 Ferison : 226, 227. 
 
 Fermat, Pierre de : 333. 
 
 Fermentation : 422-429. 
 
 Ferrier, James Frederick (Prof.) : 77. 
 
 Fesapo : 226, 227, 237. 
 
 Festino : 226. 
 
 ' Few ' : 169. See ' A f.' 
 
 Fidelity to Fact : 313-318, 390. F. to 
 Pvclevant Fact : 3, 16, 314, 316 (ftn.), 
 317, 347, 356, 367, 389-391, 395, 
 420, 449. F. to Thought : 107. 
 
 Figurative Terms in Definition : 33. 
 
 Fidure : See ' Syllogistic F.' Fig. I. 
 215, 225, 220, 228-230, 234-237, 240 
 242-244, 240-249, 255-200, 324, 325 
 Fig. II. : 215, 225, 226, 228, 230 
 232-236, 238, 242, 243, 246, 258 
 260, 269. 324. Fig. III. : 215, 222 
 225-227, 233. 234, 236, 238, 241 
 243, 246, 2.58-260, 324. Fig. IV. 
 215, 225-227, 232, 236-238. 246. 2.55, 
 2.36. 324. 
 
 Final Cause : 370. 
 
 Fir.'^t Cause (or Ultimate C.) : 370. 
 378. 
 
 ' Fitting ' Definition : 32, 33, 36-39. 
 
 Fivefold Scheme of Class-relations : 
 152, 240. 
 
 Fixation of Attention by Words : 14. 
 F. of Meaning by Association : 16, 
 17 {see 'A.'); by Context: 15, 16. 
 93, 116-118; by Definition: 16-39. 
 81. 121. 309 (see ' D.') ; by Usage: 
 15 ; by Words : 13-lG, 23, 93. 
 
 Flexion : 283. 
 
 Force: 332, 333, 335, 337, .342, 343, 
 367, 370, 373, 375-377, 408, 409. 
 452, 459, 460. 
 
 Form : 125 (see ' Distributive F.,' 
 ' Logical F.'). ' F.' in Baconian In- 
 duction : 313, 414, 415. F. of the 
 Syllogism : 214, 215. 
 
 Formal Apodeictic Hypothetical : 140- 
 142, 145, 146. F. Argument: 290. 
 F. Discipline : 304-306. F. Fallacy : 
 106, 233, 286, 289 (see ' Fallacies . .'.,' 
 ' Fallacy ...'). F. Import of the 
 Categorical Proposition : 146-161. 
 F. Inference : 130, 135, 136, 187- 
 191, 194, 216 (ftn.), 250, 291. 321, 
 326, 327 (ftn.) {see 'Inference'). 
 F. Premisses : 326. F. Principle : 
 145, 146 (.see 'Law of Validity,' 
 ' P. of Identity,' 'P. of Non-Contra- 
 diction '). F. Process : 52. F. Proof : 
 '289. F. Syllogism : 214, 216 (ftn.) 
 {see 'S.'). F. treatment of Defi- 
 nition : 214. F. (t. of) Logic : ix, 
 3 (ftn.), 6-10, 21, 143-306, 309, 310. 
 321. F. t. of the Logical Propo- 
 sition : 143-183. F. Validity : 187. 
 191. 218, 223 (see 'V.'). 
 
 Formal Actuality, f. Fact : 138. f. 
 Argument : 288. f. Assertorial Hy- 
 pothetical : 140-142, 145, 146. f. 
 Classification : 57, 58, 64-66, 137. 
 f. Connotation : 70-72. f . Context : 
 120. f. Definition : 17-23. 29-31. 
 34, 71, 214, 309, 316 (ftn.). f. De- 
 notation : 71. f. Di.scus.sion : 313. 
 f. Division : 42, 316 (ftn.). f. Evi- 
 dence : 307-309. f. Induction : 316 
 (ftn.). f. Inference: 300-310. f. 
 Judgment, f. Proposition : 114, 115, 
 119. f. Method: 410. f. Process: 
 308. f. Reference to Reality : 4-7, 
 9, 114, 119, 140, .307-309. f. Re- 
 levance : 288. f. Singular Propo- 
 sition : 114, 115. f. Subject of 
 Discourse : 119. f. Truth and Fal- 
 sity : 141. 
 
 Formed Content : 125, 126. 
 
 Forms of Syllogism : 215-217, 224-227. 
 236, 237, 242, 243, 247-251, 253-277, 
 324, 325.
 
 INDEX, VERBAL AND ANALYTIC 
 
 473 
 
 Formula of Division : 40, 72. Of Ex- 
 tension : 7>. Of Identity : 21, Ofi- 
 98. Of Intension : 72. Of the Dis- 
 junctive : 132. 
 
 Formulation of Experience : 85. F. of 
 Hypothesis : 304, 326, 411, 432, 430, 
 439 {see ' H.'). Of the Law of Ex- 
 cluded Mitldlo : 99, 100. Of the 
 Principle of Non-Contradiction : 98. 
 
 Foucault, Jean Bernard Ldon : 336. 
 
 Foundation : See ' Ground.' 
 
 Four Terms : See ' Quaternio Ter- 
 minorum.' 
 
 Fourfold Scheme of Categorical Pro- 
 positions : 147, 150, 159, IfiO, 190. 
 
 Four-termed Arguments : 250, 251. 
 
 Fowler, Thomas (Dr.) : 274. 394, 397, 
 399. 
 
 Fragmentary Aspect of Reality : 4, 5, 
 23, 309. F. Thought : 106. 
 
 Fraser : 370 (ftn.). 
 
 Free Choice: 127, 140. See 'Free- 
 dom.' 
 
 Freedom : 4, 105, 120, 127, 140. 309, 
 459, 462. 
 
 Free-will : 105, 452. See ' Freedom.' 
 
 Fresuon : 226, 227, 237. 
 
 Fruitful Analogy : 30': 
 
 Hypo- 
 
 thesis : 313, 328, 337-339, 430, 431, 
 439. F. Postulate : 459. 
 
 Fundamental Concept : ix, 462 {see 
 ' Freedom,' ' Reality,' ' Truth,' ' Ul- 
 timate C). F. Forms of Syllogism : 
 227. F. Problems : 105. 
 
 Fundamentum Divisionis (or F. D.) : 
 43-46, 51-56, 73-75. 148. 
 
 Galen, Claudius : 230. 
 
 Galileo, G. G. : 213. 313, 316, 338, 302. 
 
 Garden, H. (the Rev.) : 68. 
 
 Gay-Lussac, Joseph Louis : 425. 
 
 General Categorical : 131. G. Col- 
 lective : 162 (ftn.). G. Class. Con- 
 cept. Idea. Meaning, Name, "Term : 
 19, 57 (ftn.), 72, 75, 80, 81, 88. 127 
 {see 'Class,' 'Genus'). G. Effect: 
 383, 384. G. Form of the Hypo- 
 thetical : 141. G. Law : See "' L.' 
 G. Maxim. G. Proposition. G. State- 
 ment : 250, 299-303, 348, 422 {see 
 ' Universal P.'). G. Truth : 351, 
 376. 450. G. Universal Judgment : 
 147 (ftn.). 
 
 Generality : 45, 52. 77, 88, 133, 385. 
 455. See 'General . . ..' 'Generali- 
 zation.' 
 
 Generalization: 21, 22, 88, 155, 299 
 302-304, 326, 339-344, 349-352, 355, 
 357, 379, 385. 410, 411, 428, 450, 
 453-456. G. Instances : 351. G. of 
 Concepts under Concepts : 341. Of 
 Facts under Concepts: 340. 341. 
 Of F.'s under Laws : 341-343. Of 
 
 Laws under Laws : 343. Of the 
 Meaning of a Word : 14, 15. G. 
 Proper : 351, 352 (Note). 
 
 Generic F. D. : See ' Unity of F. D.' 
 G. Idea: See 'Genus.' G. Mark: 
 18-22. G. Universal Judgment : 
 147-148 (ftn.). 
 
 Genetic Definition : 29, 30, 34. G. 
 Explanation : 461. G. Movement : 
 ix. G. Order : 60. 61, 65. G. (or 
 Developmental) Sciences : 58, 62 
 {see ' Science '). 
 
 Genuine Exception : 421. See ' E.' 
 
 Genus (Biological) : See ' Animal.' 
 ' Classification,' ' Plant.' G. (Log- 
 ical) : 17-22, 25. 27, 28, 32, 33. 
 35-40. 57, 61. 68. 79. 82, 89, 95, 
 120, 121, 127. 136, 159, 245 {see 
 'Division,' ' Summum G.'). Its 
 Definition : 30, 35. 
 
 Geocentric Theory : 329, 330, 338. 
 
 Geology: 51 (ftn.), 80, 314, 337, 374, 
 404. 
 
 Geometrical Method : 409. 
 
 Geometry : 20, 25, 26, 29, 33, 34, 36, 
 40, 41, 52, 54-56, 87-89, 120, 127. 
 131, 137, 138, 152, 169, 170, 197, 
 
 245, 246, 274. 284, 306, 321-323. 
 329, 331. 349, 362, 369, 409, 44'. 
 450, 454. 
 
 Gilbart. John William : 67. 231. 2SS. 
 
 Given : 145. See ' Acceptance of 
 Statement.' 
 
 Goal of Induction : 365-444. G. of 
 Observation : 389. 
 
 Goclenian Sorites : 255-260. 
 
 Goclenius. Rudolphus : 255. 
 
 Grades in Division : See ' Continued 
 D.' 
 
 Granted : See ' Acceptance of State- 
 ment.' 
 
 Grammar: 13, 93. See 'Grammatical.' 
 
 Grammatical Form : 382. 383. G. 
 Structure : 116, 285, 286. G. Sub- 
 ject : 116. 
 
 Gravitation, Gravitv : 29. 141. 245, 
 
 246, 318. 332-334, 337, 342, 343. 371, 
 373-377, 387. 405. 408, 409, 432- 
 434, 436, 452, 456, 459. 
 
 Ground : 127-130, 168, 269, 291. 303, 
 321, 323, 327, 349, 363, 364, 
 370 (ftn.), 387, 410, 450. 456. G. 
 of Elimination : 416-418. Of Ex- 
 planation or Induction : 453. 456- 
 459. Of the Joint Methud : 417. 
 Of the M. of Agreement : 414. 417. 
 Of the M. of Concomitant Varia- 
 tions : 417. 418. Of the M. of Dif- 
 ference : ^14-417. Of the M. of 
 Residues : 417. 418. Of Verifica- 
 tion : 418. 
 
 Group : 1.54. 15S. 162. See ' Class,' 
 ' Continued Division.'
 
 476 
 
 THE PROBLEM OF LOGIC 
 
 Growth : 4G0 {see ' Development '). G. 
 of Meanings : 14-17. 
 
 Guess : 313. 327 (ftn.), 361. See 
 ' Conjecture.' 
 
 Guiding Conception, G. Idea. G. Prin- 
 ciple : 329, 340, 420, 448, 458. 
 
 Ha : 178, 179. 
 
 Hamilton, Sir William : 159, 160. 
 
 Hamiltonian Scheme of Categorical 
 
 Propositions : 159-161. 
 Hansen : 42S. 
 Ht: 178, 179. 
 Heat: 334, 374, 380. 388, 393, 394, 
 
 396, 405, 407, 414, 415, 440. 457. 
 Hegel, G. W. F. : 2, 6, 14, 31, 75, 76, 
 
 96, 97, 107. 
 Hegelian School : ix. 
 Heliocentric Theory : 330, 331, 338. 
 Helmholz, Hcrmami L. F. von : 426. 
 ' Here ' : 82, 84, 100, 101. 
 Herschel, Sir John F. W. : 386. 
 Heterogeneous Effect, Heteropathic 
 
 E. : 375. 
 Heuristic function of Analysis : 301. 
 
 Of Elimination : 422. H" Method : 
 
 397. H. Supposition : 327. 
 Hi: 178, 179. 
 
 Highest Genus : See ' Summum G.' 
 
 Historical Methorl : 410. 
 
 History: 119, 410, 438. 
 
 Ho : 178, 179. 
 
 Hoffding, Harald : 452. 
 
 Hoffmann : 426. 
 
 Homogeneous Definition : 33, 34. II. 
 Effect : 375, 376. 
 
 Homology : 60. 
 
 Horns of a Dilemma : 274, 292. 293. 
 
 ' How ' : 342, 356, 357, 368, 369. 
 
 Huygens, Christian : 333. 
 
 Hydrostatics : 436. 
 
 Hypothesis : 99, 136, 204, 304, 313- 
 315, 317, 321, 326-339, 348, 351, 
 3.52, 3.55, 356, 361, 367, 395, 390, 
 400, 403, 410, 411, 417, 419-443, 
 447-453, 456, 459 (see ' Develop- 
 ment of H.,' 'Modification of H.,' 
 'Pvival H.es,' 'Verification'). H. 
 of Causal Law : 455, 456. H. of 
 Gravitation : Sec ' G.' 
 
 Hypothetical .Assumption : 443. H. 
 Explanation : See ' Hypothesis.' H. 
 Inference, H. Syllogism : 263-271, 
 323-325, 385 (see ' Dilemma '). H. 
 Judgment, H. Proposition, etc. : 
 vii, 111-114, 133, 136-142, 370 (ftn.) 
 (see ' Dilemma '). Its Opposition : 
 178-180. Its relation to Inference : 
 145, 146. To the Categorical : 138, 
 1.39, 141, 268 (ftn.). To the Dis- 
 junctive: 113, 114, 138. H. Method : 
 410, 411, 420. H. .Sorites : 270, 271, 
 31'J, 320, 325. H. status of lieality 
 
 and Truth in Formal Logic : 6-9. 
 H. treatment of Truth and Falsity : 
 
 145, 146. 
 
 'I': 82-84. 'I can.' 'l' cannot,' 'I 
 must,' ' I ought,' ' I will ' : 127. I 
 Proposition : 147-152, 155, 157, 159, 
 160, 162, 163, 169-172, 282. Its 
 Conversion : 194, 195. Its Dia- 
 gram : 151. Its relation to : 156 
 (see ' Subcontrarictv '). 
 
 Idea : 94, 393, 394 (see ' Class,' ' Con- 
 cept,' 'Guiding I.,' 'Ideas,' ' In- 
 ductive I.,' 'Principle,' 'Pvcgulative 
 I.,' 'Supreme L,' 'Working I.'). 
 I. and Fact: 6, 9, 313, 314, 317, 
 318 (ftn.), 339, 343. .393. 
 
 Ideal : 62, 63, 84, 127, 419, 420, 402 
 (see ' Inductive I.,' ' Logical I.,' 
 ' Unideterminism '). I. Construc- 
 tion : 26, 29, 61. 87, 88, 321, 322. L 
 of Consistency : See ' C I. of Ex- 
 planation : 448. 
 
 Idealism : ix, 6, 24, 25, 77, 107. 
 
 Idealization of Experience or of Fact : 
 0, 340, 419. 
 
 Ideas : See ' Meanings.' I. as more or 
 less elementary : 34. I. expressed 
 in Language : 16-18, 23-25, 32. 
 Fixed by Words : 13-16, 93. Not 
 true or false : 94. Suggested by 
 Words : 71. Their Association : 28, 
 29. Their Indefiniteness : 16. 
 
 Identification : 64, 69, 70, 97, 149, 
 158, 161, 218. I. of S and P : 122. 
 
 Identity : 16, 21, 60, 95-98, 104, 106. 
 122-124, 126, 145, 146, 149, 150, 
 152, 187, 188, 194, 219, 220, 223, 
 250, 290, 304, 325, 327 (see ' Agree- 
 ment,' ' Essential I.,' ' Material I.,' 
 ' Principle of I.'). I. as determining 
 Import : 146. I. in Difference : 
 96-98, 121-124, 128. I. of Cause 
 and Effect : 382. Of Pvcasoniug : 
 349. Of the Universe : 385. 
 
 Identity-import: 148.159(ftn.),239,245. 
 
 Identity-principle : 124, 223. See ' P. 
 of L' 
 
 Identity-relation between S and P : 120, 
 
 146, 219, 220. 
 
 'If in Hypothetical : 145 (see ' HI 
 Judgment '). ' If P is true, then Q is 
 true': 141, 145. 'If P, then Q ': 141. 
 
 If -clause : 112. See ' Antecedent.' 
 
 Ignava Ratio : 293. 
 
 Ignorance : 77, 79, 129, 157, 158, 385, 413. 
 
 Ignoratio Elenchi : 289, 291, 292. 
 
 Ignotum per ignotius : 34. 
 
 Illegitimate (or Unsound) Analogy : 
 359, 360, 363, 364. I. Explanation : 
 457. I. Hypothesis : 338 (see ' Bar- 
 ren H.'). I. Inference : 141. See 
 ' Invalid . . .'
 
 IXDEX, VERBAL AND ANALYTIC 
 
 477 
 
 Illicit Procf>ss : 219, 229 (ftn.). I- (P- 
 of the) Major : 210, 225, 237, 23,S, 
 257, 258, 2G9. (Of the) Minor : 219, 
 225. 22C. I. Proof : -S'ee ' Fallacies 
 of I. P.' 
 Illusion of Contradiction : 105, lOG 
 {see ' Apparent C). I. of Identity : 
 10(). 
 Illustrations : See under suhjccts illus- 
 trated. 
 Imagination : 71. 72, 246, 313, 317, 321. 
 
 328, 390, 459. 
 Immediacy : 85 {see ' Immediate ...'). 
 I. in Causal Connexion : 378, 381- 
 383, 427, 428. 
 Immediate Antecedent, I. Cause, I. 
 Effect : See ' Immediacy in Causal 
 Connexion.' I. Apprehension, I. 
 Experience, I. Perception : 4, 75, 
 76, 82, 85, 340, 393. I. Inference : 
 185-209, 306. ' I. I. by Privative 
 Conception': 190 (ftn.). 'I. Is' 
 from Hypotheticals : 265, 266. I. 
 Object : 85. I. Self-evidence : 84. 
 Immortality : 462. 
 Imperative : 93, 94, 127, 168. 
 Imperfect Disjunction : 137 {see ' Non- 
 exclusive . . .'). I. Enumeration : 
 347 (see ' Enumerative Induction '). 
 ' I.' Figures : 247, 248, 324 {see 
 'Fig. II.,' 'Fig. III.,' 'Fig. IV.') 
 I. Inductions : 347, 351-364 (see 
 ' Analogy,' ' Enumerative Induc- 
 tion'). 
 Impersonal conceptions of Truth and 
 
 Reality: 4, 5. I. Judgment: 116. 
 Implication : 35, 70, 71, 81, 94, 95, 112, 
 113, 123, 124, 148-151, 155-158, 
 165, 167, 168, 187-190, 229, 301, 304, 
 307, 321, 322, 326, 327 {see 'Im- 
 port . . .'). I. in Apodcictic Hypo- 
 thetical : 140, 142, 146. I. in Hypo- 
 theticals : 139-142, 385. I. in Syllo- 
 gism : 214, 223. 
 Implications of Disjunctives : See 
 
 ' Import of the Disjunctive.' 
 Implicit Assumption : 436-438, 458. 
 I. Exhaustiveness : 46. I. Self- 
 contradiction : 104. 
 Implied Common Nature : 115. I. 
 Objective Reference : 94. I. Pro- 
 perty : 26, 27. 
 Import : See ' Meaning,' ' Singular I.' 
 Determined bv Identity : 146. I. of 
 Propositions :" 94. 95, 99, 114, 115, 
 119, 123, 132 (ftn.), 145-148, 150, 
 155-159, 189, 190, 239-241, 244, 245 
 {see ' Attributive view of I.', 
 ' Class-inclusion,' ' Coincidence-im- 
 port,' ' "Denotative Reading," ' 'Ex- 
 istential I.,' ' Extension-i.,' ' Formal 
 I..' ' Identity-i..' 'Intensive Read- 
 ing,' ' Predicative view of I.,' 
 
 ' Rtatement-i..' 'Truth-i.'). I. of 
 ' Some ' : 157 {see ' S.'). I. of the 
 Categorical : 146-161. I. of the 
 Disjunctive : 132-137. 274. Of the 
 Hypothetical : 138-142. Of the Par- 
 ticular Proposition : 154-158. Of 
 the Undistributed Term : 157 («ee 
 ' Some.' 
 Important Attribute. I. Character, I. 
 Difference, I. Resoinblance : 58-60, 
 62, 65, 327, 3.58-360. I. Circum- 
 stance : 396, 403, 426. I. Condition : 
 370. 
 Impossibility : 131. 
 Inaccuracy : 66, 295, 386, 432. 
 Inadequate (or Incomplete) Division : 
 45, 46, 52-56. I. Knowlerige : 129, 
 130 {see ' Limitation of K.'). 
 Inclusion of Classes : ^ee ' Class-i.' I. 
 of Individuals in Classes : 43, 152 
 {see ' Class,' ' Indiviflual'). 
 Inclusive Idealism : 77. 
 Incompatibility : 156, 364, 401, 443. 
 
 See ' Compatibility.' 
 Incomplete Dichotomy : 47. I. Di- 
 vision : See ' Inarlequato D.' 
 I. Enumeration : 43. I. Explanation : 
 369. 
 Inconceivability : 449, 450, 452. 
 Inconclusiveness : 281. 
 Inconsistency : 8, 9, 17, 76, 98, 99, 
 
 102, 106, 107, 139, 140, 150, 164. 
 Increase of Determinate Connotation : 
 
 72. 
 Indefinability : 74, 76, 82, 84. See 
 
 ' Definability.' 
 Indefinite Categorical : 113. I. Char- 
 acteristic : See ' Indeterminate Ele- 
 ments . . .' I. Judgment : 165 (ftn.). 
 I. Proposition : 156. I. Species, I. 
 (or Negative) Term : 47-52. 124, 
 190-192, 250. Its Symbol : 201. I. 
 Subject: 116. I. Use of 'Some': 
 156-158, 175. 
 Indefiniteness : 16-18, 74, 157, 281. 
 
 See ' Indeterminacy.' 
 Independent Propositions : 166, 198, 
 
 199. 
 Indesignate Proposition : 128, 165, 147. 
 Indestructibility : 387. 
 Indeterminacy": 89, 112, 113. 116. See 
 
 ' Indeterminate . . .' 
 Indeterminate : See ' Indefinite . . .' 
 I. Agreement : 20. I. Categorical : 
 131. 133. I. Concept, I. Conception, 
 I. Meaning : 76-78. 88. 95, 121. 284, 
 285, 329. I. Elements in Definition : 
 20-22, 35, 41, 43, 44, 54, 71, 74. 75. 
 I. Knowledge: 113. I. Subject- 
 concept : 116, 121, 165 (ftn.). 
 Indoterminism : 453. See ' Non-me- 
 chanical . . .' 
 Index : 63-65.
 
 47S 
 
 THE PROBLEM OF LOGIC 
 
 Index-Classification : G4-C6. 
 
 Indication of Objects by Terms : 21 
 (ftn.), SO. SI, SG, 147, 161, 1G2 (ftn.), 
 •21S. 219. Sec ' Application of 
 Meanings,' ' Objective Reference.' 
 
 Indicative Sentence : 94, 1G7. 
 
 Indirect Mood- : 23G. I. Observation : 
 435. I. Proof : 237, 23S, 324, 325. 
 I. Reduction : See ' Reductio per 
 Impossibile.' 
 
 Individual : 22-24, 40, 42, 43, 57, G3, 
 64, 66. 71, 72. SO-82, 85-88, 114, 
 115, 14S. 149, 152, 157, 161-163, 165, 
 2SS, 299. 340, 352, 354, 356, 363, 
 455, 460 (see ' Objects . . .,' 
 ' Singular Term ' ). Its Definability : 
 iSee ' D. of Proper Names.' I. 
 Characteristic : 371. I. Concept : 
 80 {see 'Singular C). I. Elements 
 in Division : 47, 79, 80. I. (or Sub- 
 jective) Intension : 70, 71. I. Na- 
 ture : 81. I. Meaning : SO (see 
 ' Singular Meaning '). 
 
 Individuality : 82, 85, 459, 460. I. of 
 Reference : 80. See ' Singular Mean- 
 ing.' 
 
 Individualization : 118. 
 
 Indivisibility : 79. See ' Divisibility.' 
 
 Induction, Inductive Method, In- 
 ductive Procedure, etc. : 3, 94, 122, 
 295, 301-303, 306 (ftn.), 311-462 {see 
 ' Explanation,' ' Inductions,' ' Prin- 
 ciple of Induction,' ' Scientific I.'). 
 I. and ' Inductive Inference ' : 326, 
 327. I. and the Inductive Principle : 
 311-344. I. by Complete Enumera- 
 tion : 356, 357. I. by Simple 
 Enumeration: 347, 351-357, 454-456. 
 
 Inductions improperly so-called : 344- 
 3.50 {see ' Colligation,' ' Parity of 
 Reasoning,' ' " Perfect Induc- 
 tion " '). I. properly so-called : 347 
 (see ' Imperfect I.,' ' Scientific I.'). 
 
 Inductive Canons : See ' C. of Causal 
 Methods.' I. Conception of Fact : 
 448. I. Conclusion : 416, 418. I. 
 Criterion, I. Standard : 347, 348, 
 3.50, 3.55-357, 362, 456, 457. I. Ex- 
 planation : 343, 368-370 (see 'E.,' 
 ' Induction '). I. Hypothesis : 336 
 (sec 'H.'). I. Idea: 318 (ftn.), 
 3.52, 390. I. Ideal : 420, 448. ' I. 
 Inference ' : 299, .301, 303, 326, 327, 
 347-350. I. Inquiry, I. Method, 
 I. I'rocedure, I. Process : <See ' Induc- 
 tion.' 'I. In-stincf : 303. I. In- 
 terest : 381, 421, 457. I. Logic : 
 326 {see ' Induction '). I. Methods 
 326, 3.56, 395-422, 455, 456 (see 
 • Method . . .'). I. Movement : 313. 
 I. Postulate : 3, 338 (ftn.), 367, 368, 
 420, 445-402. I. P. and Deter- 
 minism : 450-455. I. P. and I. 
 
 Principle : 449. I. Principle : ix, 3, 
 311-364, 367, 395, 448, 449 (see 
 ' Fidelity to Relevant Fact '). I. 
 Proof : 416, 454. I. Scheme : 347. 
 I. Science : 313, 378, 379. 453. See 
 ' Induction,' ' Science.' 
 
 Inertia : 459. 
 
 Inference : 3 (ftn.), 7, 8, 114, 130, 141, 
 142, 146, 155, 167, 185-310, 318-327, 
 348, 349, 362, 368, 370 (ftn.), 385, 
 387, 409 (see ' Deductive I.,' ' For- 
 mal I.,' 'Immediate I.,' '"Induc- 
 tive I.",' ' Invalid ...,'' Svllogism,' 
 ' Validity '). I. as an Identity : 96. 
 I. by Added Determinants : 208, 209. 
 I. from Disjunctives : 133, 134 (see 
 ' D. Syllogism'). I. from Par- 
 ticulars to P. : 299, 300, 302. I. in 
 relation to Hypothetical Form : 145, 
 140. I. of Consequent from Ante- 
 cedent : 142 (see ' Apodeictic . . .'). 
 
 Inferential Fallacies : 286 (see ' Formal 
 Fallacy'). 'I. Possibility': 130. 
 
 hifima Species : 42-44, 72-74, 77, 80, 
 88, 89 (see ' Continued Division '). 
 Its Divisibility : 79, 89. 
 
 Infinite Regress : 103, 378. ' I. Term ' : 
 48, 49, 72 (.see ' Indefinite Species'). 
 
 Influencing Agent : 405. 
 
 Infra-scientific Viewpoint : 4, 5. 
 
 Inherent Force : 459, 460. 
 
 Initiating Cause, I. Condition : See 
 ' Exciting Cause.' 
 
 Inner Standpoint : 453. 
 
 Inorganic Fact, I. Phenomenon : 458 
 459. I. Nature : 61 (ftn.). I. Order : 
 33. 
 
 Inspection : 390. See ' Observation.' 
 
 Instance : 114, 131, 157, 302-304, 327, 
 340, 348-362, 371, 383, 391-408, 412- 
 418, 421, 422, 425, 426, 439, 443, 
 449-451, 454-458. ^See 'Negative 
 I.,' ' Positive I.' 
 
 Instantia Crucis : See ' Crucial In- 
 stance.' 
 
 Integrity of Thought : 98. See ' Con- 
 sistency.' 
 
 Intelligibility: 93, 104, 191, 452. I. 
 of Experience, of Fact, of Reality, 
 etc. : 78, 79, 99, 104, 139, 140, 458. 
 I. of Statement : 99. 
 
 Intelligible World : 105. 
 
 Intended Meaning : 97 (ftn.). 
 
 Intension : 21 (ftn.), 70-73, 80, 146, 
 147, 154. See ' Disjunctive De- 
 velopment . . .,' ' D. Interpretation 
 of I.,' ' Intensive . . .' 
 
 Intensive Reading: 147, 148 (ftn.). 
 
 Intent, Intention : 23, 24, 27, 28, 51 
 (ftn.), 97 (ftn.) 6'ee ' Purpo.se.' 
 
 Interaction : 29, 341, 459, 460. 
 
 Interconsistency : 7, 8, 9. See ' Con- 
 sistency.'
 
 INDEX, VERBAL AND ANALYTIC 
 
 479 
 
 Interest: 51, 77-79. 88, 93, 112, 114, 
 118-120, 135, 13G, 155, 247, 339, 
 3G7, 378, 3S1, 3S2, 389-393, 400, 401, 
 421, 430, 437, 442, 451 (see ' Causal 
 I.,' 'Inductive I.,' 'Logical I.,' 
 'Observational I.,' 'Purpose,' 
 'Scientific I.,' ' Truth-i.,' ' Validity- 
 i.'). I. in Development of Meaning : 
 22. I. in Non-ambiguity : 135 (see 
 ' X.-a.'). I. in Practical Life : 9, 23, 
 29. I. in Precision: 135 (see 'P.'). 
 I. in relation to Proposition : 94, 97, 
 112, 114, 123-125 (see 'Import of 
 P's '). In r. to Thought : 95, 90. 
 
 Interference : 375, 376, 394, 401, 402, 
 454 (ftn.). 
 
 Intermediate Links in Causation : 381, 
 382. 
 
 Intermixture of Effects : 375-377, 404, 
 407, 408, 419. 
 
 Interpretation : See ' Import,' ' Mean- 
 ing.') I. of Fact : 94. 
 
 Interrelatedncss, Interrelations : Sec 
 ' Relations.' 
 
 Interrogative : 5ee ' Question.' 
 
 Intimacy of S and P : 126. 
 
 Intrinsic Impossibility : 131. I. Ne- 
 cessity, I. Possibility : 127, 128. 
 
 Intuition : 4, 75, 322. 
 
 Invalid Argument : 8, 9, 263. I. Di- 
 lemma : 274. I. Forms of Syllo- 
 gism : 210, 224-220, 233-235, 243. I. 
 Hypothetical : 141. ' I. Ideas ' : 94. 
 Invalid Statement : 8, 9. 
 
 Invalidation : 141. 
 
 Invalidity : 281. -See ' Invalid . . .' 
 
 Invariable Antecedent : 379, 380, 419. 
 I. Concomitance, I. Connexion : 359, 
 380, 383, 424. I. Consequence, I. 
 Sequence, I. Succession : 370, 377- 
 383, 399, 453. I. Consequent : 380. 
 I. Law : 379, 459. 
 
 Inverse : 201, 202, 206. I. Deductive 
 Method : 409, 410. ' I. Relation ' of 
 Connotation and Denotation : 72, 73. 
 
 Inversion : 189. ' I. of Hypotheti- 
 cals ' : 265. 
 
 Inverted Sorites : 270. 
 
 Inviolability of the Laws of Thought : 
 104-107, 173, 191, 193, 327. 
 
 Involution of Meaning : 88. 
 
 Inward Conception of Truth : 3, 4, 
 6. I. Individuality : 82. 
 
 Irrational Evidence : Sec ' Fallacies of 
 I. E.' 
 
 Irrationality : 448, 450. 
 
 Irreducible Forms of Thought : 31. 
 
 Irrelevance: ix, 118, 123, 133, 214, 
 286-288, 310, 343. I. in Definition : 
 25-29, 38. In Division : 46, 51, 54, 
 79. 
 Irrelevant Evidence : See ' Fallacy of 
 I. E.' 
 
 'Is,' 'Is-not': 124 (ftn.), 126, 162. 
 163. iS'ec ' Copula-mark," ' Quality- 
 mark.' 
 
 I.solated Individual : 115. 
 
 Isolation of an Object : 145. 
 
 ' It ' as Subject: 116. 
 
 James, William (Prof.) : 392, 452. 
 
 Jevons, W. Stanley (Dr.) : 266, 283, 
 306, 334, 335, 448 (ftn.), 361, 362, 
 391, 406 (ftn.). 
 
 Joint Effect : 375, 376, 408. J. Method 
 of Agreement and Difference : 398- 
 400, 417. 
 
 ' Joints ' in Division : 40, 53. 
 
 Joseph. H. W. B. : viii, 38 (ftn.). Ill 
 (ftn.), 105 (ftn.), 194 (ftn.), 197 (ftn.), 
 229 (ftn.), 306 (ftn.). J. on ' Ana- 
 lytic ' and ' Sjmthetic ' Judgements : 
 122, 123 (ftn.). On Categories: 31, 
 32. On Designations : 80 (ftn.). On 
 Division : 46, 49-52. On Extensive 
 Import : 158, 159. On Fallacy : 
 295. On Fig. IV. : 236. On Genus 
 and Differentia : 20. On Induction 
 as a Method of Elimination : 416- 
 420. On Predicablcs : 22, 31, 32. 
 On Sorites : 255. On Subcon- 
 trarietv : 172 (ftn.). 
 
 Judgment : 91-97, 100. 112, 114, 116, 
 119, 120, 122-126, 129, 130, 133, 136, 
 137, 147 (ftn.), 370 (ftn.), 422 {see 
 ' Proposition '). J. in relation to 
 Proposition : 93. Suspended J. : 99, 
 123. 
 
 Jupiter : 245, 362, 394. 
 
 Justification of the ' Extensive ' View 
 of Import : 158, 159. J. of Piedic- 
 tions : 141. 
 
 Kant, Immanuel : 31, 78, 105, 120. 
 
 Karslake : 271. 
 
 Kelvin, Lord : 333. 
 
 Kepler, Johann : 331, 332, 338, 342, 
 343, 349, 350. 
 
 Key: See 'Analytical K.,' 'Index- 
 Classification.' 
 
 Ke^Ties, John Neville (M.A., Sc.D.) : 
 viii, 71, 72, 115 (ftn.), 179-182, 198- 
 200, 250, 251, 266. 
 
 Kind : 19, 22, 33. 38, 40. 45, 58, 60, 
 63-65, 67, 68. 71, 72, 89, 162 (ftn.). 
 384. -See 'Class,' 'Natural K.'s,' 
 ' Species.' 
 
 Knowable Being : 77-79. 
 
 Knowledge: 63, 77-79, 83, 123, 128- 
 130, 145, 152, 157, 158, 175. 176, 
 190, 208, 214, 246, 300, 321, 338, 
 348, 351, 353, 361, 377 (ftn ), 384, 385. 
 388-390, 398. 408, 411, 41 1. 410, 424. 
 420, 430, 434, 435, 440 442. 450, 
 453-458. K. and the /atecories : 
 31. K. in relation to Classification :
 
 480 
 
 THE PROBLEM 0? LOGIC 
 
 65. To Definition : G3. To Dicho- 
 tomy : o2. To Ignorance : 77, 79. 
 To Truth : 3, 4. K. of Classes : 340. 
 Of Meaning : 76. Of Nature : 29. 
 Of Self : 3. 4, 82. 
 Known Cause, K. Condition : 411, 412, 
 416. 
 
 Lamarck, J. B. P. A. de Monet : 447. 
 
 Lancaster, Ray (Prof.) : 429. 
 
 Language (or Speech) as expressing 
 Ideas : 16, 17. As ex. Truth : 16. 
 As the Instrument of Definition : 53, 
 84 {see ' D.'). L. in relation to 
 Locric : See under ' Logic' In r. to 
 Thought : 13-17, 24 {see ' Tech- 
 nology '). Its Ambiguity : 105 (see 
 'A.'). L. regulated by PJietoric : 13. 
 
 Latta, Robert (Prof.) : 322. 
 
 Laurie, Henry (Prof.) : 396, 407, 411. 
 
 Law of Association : 28, 81, 449. L. of 
 Causation, L. of Uniformity : 378, 
 379, 4.50, 452-458 {see 'Causal L.'). 
 L. of Contradiction, L. of Non-C. : 
 iSee ' Principle of Xon-C L. of Ex- 
 cluded Middle: ix, 98-100, 104, 146, 
 180 ; its relation to Obversion : 191 ; 
 to Time: 100-102; its Violation: 
 104. L. of Falling Bodies : 342, 
 409. L. of Formal Inference : See 
 ' Principle of F. I.' L. of Formal 
 Validity : See ' P. of F. V.' L. of 
 Gravitation : See ' G.' L. of Iden- 
 tity : <See ' Principle of I.' L. of 
 Inertia : 459. L. of Logical Validity: 
 See 'P. of L. V.' L. of Material 
 Identity : 146. L. of Nature, etc. : 
 5, 58, 114, 115, 138, 144-147, 295, 
 302, 303. 305, 307, 308, 313-315, 
 333, 341-344, 353-357, 367-389, 393, 
 395, 405-415, 437, 439, 449-459 {see 
 ' Empirical L.,' ' Mathematical L.'). 
 L. of Non-Ambiguity : See ' Prin- 
 ciple of N.-A.' L. of Perception : 
 392. L. of Self-Consistency : 98, 
 99 («ee ' Principle of Non-C). L. 
 of the Conservation of Energy : 333. 
 L. of the Land : 104, 140, 367. Peri- 
 odic L. : 61 (ftn.). 
 
 Laws of Causal Interaction : 341 {see 
 C. Law). L. of Contradictory Oppo- 
 sition : 173, 174. L. of Develop- 
 ment : 341. L. of Hydrostatics : 
 436. L. of Motion : 333, 335, 418, 
 431-4.34, 436. L. of Non-Contra- 
 diction and Excluded .Middle : ix, 
 98-104. L. of Planetary Motion : 
 332, 342, 343. L. of Science: See 
 'Scientiic Law.' L. of Thought: 
 vii. 3 (f n.), 6, 7, 9, 16, 95-107, 187, 
 203, 204, 448 ; their Inviolabili'ty : 
 104-107. 173, 191, 19.3, 327. Teleo- 
 logical 1 . -: 104. 
 
 Leading Category : 76 {see ' Being '). 
 L. Concept : 74. See ' Summum 
 Genus.' 
 
 Leaps in Division : See ' Saltus in 
 Divisione.' 
 
 Legitimate Explanation : 420, 449, 453, 
 457. L. Hypothesis : 328, 337, 338, 
 348. 
 
 Leibniz, Gottfried Wilhelm: 333. 
 
 Leslie, Sir John : 334. 
 
 Leverrier, Urbain J. J. : 412, 413. 
 
 Library-Classification : 64, 65. 
 
 Liebig, Justus (Prof.) : 427-429. 
 
 Life : 57, 425, 426, 4.53, 460, 461. See 
 ' Psychical L..' ' Vitality.' 
 
 Light: 3.35, 336, 338, 339, 394, 413, 
 430, 432-435. 
 
 Likeness : See ' Resemblance.' 
 
 Limit of Abstraction or of Generaliza- 
 tion : 75. L. of Determinacy : 88. 
 L. of Knowledge : 79. 
 
 Limitation by Negations : 420-422. L. 
 by Suppositio or Universe of Dis- 
 curse : See ^ S.' L. of Certainty : 
 157. L. of Content : 28. L. of 
 Extension by Connotation : 72. L. 
 of Knowledge: 128-130, 157, 158, 
 175, 176. L. of Meaning : 17, 158. 
 L. of Possibility : 131. L. of State- 
 ment : 156, 158, 176. L. (or Con- 
 trol) through Relevance (to Pur- 
 pose) : See ' P.' 
 
 Limitations of Enumerative Induc- 
 tion : 356, 357, 4.54, 455. L. (or 
 Limits) of the Inductive Postulate : 
 448, 449, 452. 458. L. of the 
 Method of Difference : 404-407. 
 
 Limited Universe: 49 (ftn.), 118, 138. 
 iS'ee ' Suppositio.' 
 
 Limiting Case of Interference : 376. 
 L. Condition : 421. L. Form of 
 Infima Species : 74, 80. Of the 
 Principle of Identity : 98. Of the 
 Universal Proposition : 161. 
 
 Limits of Definition and Division : 
 73-85. L. of Human Understand- 
 ing : 448. L. of the Inductive Pos- 
 tulate : See ' Limitations of the I. 
 P.' 
 
 Links in Causation : 381, 382. 
 
 Linnaeus, Carolus (Prof.) : 63, 64. 
 
 Lister, Lord : 418. 
 
 Liligiosus : 275, 293-295. 
 
 Living Being : 42. 
 
 Locke, John : 287, 289. 
 
 Logic and Truisms : 97. L. as a Con- 
 cept : 96. As a Science : 8, 13, 49, 
 94, 97, 111. As an Art : 13. As 
 appealing exclusively to the Reason : 
 13. As dependent on Language : 
 13. As Formal : ,S'ee ' F. . . . L.' As 
 formal : <S'ec ' f. Reference . . .' As 
 Material : See ' M. L.' As Philo-
 
 INDEX, VERB.AL AND ANALYTIC 
 
 481 
 
 Bophical : See ' P. L.' As Pre- 
 philosopliical : 3-5. As Real or 
 Scientilic : See ' R. L.' As Regu- 
 lative of I^ncjuage : 13, 16, 17. As 
 R. of Thout^ht : H, 9, 13, IG, 17, 27, 
 107. As tlic Medicine of the Mind : 
 IG. L. Defined: 1, 8, 33, 34. L. 
 in relation to Language : 11-89, 1G4. 
 In r. to Morality and Religion : 
 107. In r. to Words : 13, IG, 17. 
 93. Its Function : 1, 2. Its Theo- 
 retical Character : 13. L. of Ab- 
 stract Identity : 97. L. of Common 
 Sense: .5. L. of Discovery: 301. 
 L. of Evidence : 361. L. of K.xpori- 
 ence : viii [see 'Philosophical L.'). 
 L. of Form : See ' Formal . . . L.' 
 L. of Personality : 3-6, 114 (ftn.) (see 
 ' Philosophical L.'). L. of Relatives : 
 251. L. of the Moral Sciences : 409. 
 L. of Truth : 304. L. of Validity : 
 304 (see 'Formal . . . L.'). L. of 
 Verification : 361. 
 Logical Ambition : 2. L. Analysis : 
 435 {see 'A.'). L. Antecedent : 370 
 {see 'A.'). L. Concept : 95, 96. L. 
 Consistency : Sec ' C L. Context : 
 118-120. L. Criterion : 3. L. Dia- 
 grams : See ' D.' L. Division : 40- 
 56 (see ' D.'). L. EUmination, L. 
 Exclusion : 414-418 {see ' Elimina- 
 tion '). L. Equivalence : 121, 147, 
 166, 324 (see ' E.'). L. Form of 
 Arguments : See ' Reduction of 
 A. . . .' L. F. of Propositions : 116, 
 117, 139, 142, 154, 161-171, 193. L. 
 Ground: 370 (ftn.) (see ' G.'). L. 
 Ideal : 9, 124, 136, 137, 303, 419, 
 420. L. Identity : See 'I.' L. Im- 
 plication : See ' I.' L. Inference : 
 See ' I.' L. Interest : 77-79, 88, 93, 
 112, 114. 124, 134. 135 {see ' Pur- 
 pose ') ; its relation to the Concept : 
 78. L. Judgment : See 'J.' L. 
 Meaning of ' All ': 154 ; of ' Some ': 
 154-158. L. Method : 410. L. Ne- 
 cessity : 8, 9, 2G. 139, 141, 165, .306, 
 309, 321, 322, .324, 327 (ftn.) WIS, 
 336 (see ' Apodeictic . . .,' ' ^Vorr^- 
 sity '). L. Op])nsition: ^"JjS>t' 
 L. Order or Priurity of HfoJpiWonal 
 Forms: 112-U^l|g^B|bintv : I.;.!. 
 L. Pi°*^<;l^£^ttH^HP^' P-')- L. 
 
 ciplM^^^^^P^. . Pro- 
 
 pusiti^^^^T- 183 (see 'P.'). 
 L. Purpose : 93, 133 (see ' P.'). 
 L. Science : iSce ' Logic as a S.' 
 L. Status of the Dicta : 241-244. 
 L. Subject: 116 (see 'Subject Term'). 
 L. Theory : 305. L. Unit : 94, 96, 
 97. 111. L. Validitv: See 'V.' L. 
 Vitality: 157. L. Whole: 118. 
 
 Lotzc, Rudolph Hermann (Dr.) : 407. 
 Lowest Species : <S'ee ' Infima S.' 
 Lubbock, John (Lord Avebury) : 392. 
 Lyell, Sir Charles : 314, 337. 
 
 70. 
 
 M» : 130. 
 
 McTaggart, J. M. Ellis (Dr.) : 75, 
 
 Magic : 451, 453. 
 
 Magnetism : 97, 334, 335. 
 
 Major Premiss : 214, 215. 225, 226, 228- 
 231, 233, 234, 237, 238. 243, 257, 
 258, 261-264, 269, 271-274, 290, 300, 
 301, 324, 457, 458 (see ' Ultimate M. 
 P.'). M. Term : 213, 214, 219, 225, 
 228-230, 233, 234. 237, 269 (see 
 ' Illicit Process of the M.'). 
 
 Mal-observatioa : 393. 
 
 Malthus, the Rev. Thomas Robert : 314. 
 
 ' Many ' : 108. 
 
 Mark : 18-21, 27, 70. 81, 82, 327, 351, 
 352, 356, 357, 360. 362, 384. See 
 'Character.' ' Copula -m.,' 'Defini- 
 tion - m.,' ' Indeterminate Ele- 
 ments ...,'' Meaningless M.,' ' Qual- 
 ity-m.,' ' Quantitv-m.,' ' Tense-m.' 
 
 Mars : 327. 331. 349", 350, 360. 361, 394. 
 
 Marshall : 354. 
 
 Mass : 385, 434. 
 
 Material : See ' Object-matter.' ^L 
 Apodeictic Hypothetical : 140-142, 
 146. M. Compatibility : 8. M. Evi- 
 dence : 189, 275, 289. 308 (see ' E.'). 
 M. Fact : 306. M. Fallacy : 289. 
 M. Falsity : 310. M. Ground : 326. 
 M. Hypothetical : 140 - 142. M. 
 Identity : 146. M. Knowledge : 52. 
 M. (treatment of) Logic : ix, 7. M. 
 Nature: 452. M. Proposition: 115. 
 M. Reality, M. Truth : 8. 9, 10, 290, 
 304, 310. M. Substance : 459. :\1. 
 System : 386. M. Universe : 386, 
 387. 
 
 Mathematical Deduction : 369, 412. 
 M. Law : 333. M. Science : 386. 
 *See ' Mathematics.' 
 
 Mathematics: ix. S'.). ];ir,, 305, 306, 
 _34Ji)r--322, 3-':; i,tiu.j, WM), 333, fflJG, 
 :;:::, 36a, 3^i"., :n7, HT. .v. ■ Al- 
 gebra,' ' As' [■■ ':!■ MMV,' '(''ill..- .'^i-C- 
 
 tii'ri^,' 'Dynamics,' "Euclid,' ' Geo- 
 int uy,' ' Hydrostatics,' ' Number,' 
 ' Physics.' 
 
 Matter : 333, 387. M. of a Judgment : 
 125. 
 
 Maxim : 105. 382. M. of Identity : 
 97. 
 
 ' May ' : 128-130, 286. 
 
 M« : 130. 
 
 Mean : 99, 123. ]\L in Division : 53. 
 
 Meaning : 23-25, 51, 75, 77, 78, 80, 82, 
 8!^ 94, 97 (ftn.), 139, 461 {see ' Col- 
 lective M.,' ' Concept '). M. as 
 Abstract : iSee ' A. Conf^ept.' As 
 
 31
 
 482 
 
 THE PROBLEM OF LOGIC 
 
 Associative : See ' Association.' As 
 Conventional: IG. As Determinate: 
 See ' D. . . .' As Disjunctive : Sec 
 • D. . . .' As Essential : See ' Es- 
 sence.' As Indeterniinate : See ' I. 
 Concept,' ' I. Elements ..." As In- 
 dividual or Singular : See ' S. 'M.' 
 As Occasional : 15, 70. As Self- 
 consistent, True, or False : 93 {see 
 'Judgment'). As Stated: See 
 'Statement.' As the Object of 
 Definition : 17-23 (see ' D.') ; of Di- 
 vision : 3 (see 'D.'). Its Develop- 
 ment : See ' D. of if.' Its Involu- 
 tion : 8S. Its relation to Relevant 
 Fact: 3. M. of a Sentence : 116, 117, 
 168. Subjective M. or Intension : 
 70, 71, SI. Ultimate M. of a Propo- 
 sition : 118. 
 Meaning-identity : 95. 
 Meaningless JIark : 81, 82. 
 Meaninglessness : 27, 51, 77-79, 97, 
 98. 103-107, 122 (ftn.), 126, Ul, 149, 
 150, 188, 191, 317, 339, 450. 
 Meanings as Consistent or Inconsie- 
 tent : See ' Consistency,' ' Incon- 
 sistency.' As given in Dictionaries : 
 15, 19, 20. As Literrelated : 30, 31, 
 35, 89, 126 (see ' Conceptual Order- 
 ing ...'). M. expressed by Signs : 
 14-17, 24, 25, 32. M. of Immediate 
 Experiences : 82. Of Marks : 81. 
 Of Sound-complexes : 25, 81. Of 
 Svmbols : 25. Of Terms, of Words, 
 of Verbal Signs : 115, 214, 282 (see 
 ' Term,' ' Words,' ' V. Sign '). Their 
 Application : See ' A. of M.' Their 
 Classification : See ' C Their Com- 
 parison : See ' C Their Definition : 
 .S'ee ' D.,' ' Definability.' Their De- 
 scription : See ' D.' Their ' Dicho- 
 tomy ' : See 'D.' Their Differen- 
 tiation, Distinction, Specification : 
 See ' Differentiation of M.' Their 
 Division : See ' D.,' ' Divisibility.' 
 Their Fixation : See ' F. of Meaning.' 
 Their Generalization and Specializa- 
 tion : 14, 15. Their Indefiniteness : 
 16-18, 74, 281, 282. Their Vari- 
 ability : 14-18, 37 (ftn.), G6, 101, 102. 
 Ultimate M. : 82-85. 
 Means: 104, 128, 420. See ' Teleo- 
 
 logical . . .' 
 Mechanical Causation, M. Connexion : 
 370, 453, 462. M. Equivalent : 387, 
 389. M. Explanation : 450-455, 459, 
 iG(). M. Force: 333, 375. M. 
 Method : 429. M. Work : 388. 
 Mediate Inference : 189, 192, 194, 209, 
 216, 223, 244, 247, 251, 269. See 
 ' Syllogism.' 
 Mediation: 213, 216, 218, 223, 244, 
 251. 
 
 Mellono, Sydncv Herbert (Dr.) : viii, 
 52, 274, 3S4, 396, 397, 402-404, 411. 
 412, 422 (ftn.). 
 
 Members of a Class : 43, 61-63, 131, 166 
 (see 'C.'). Of a Genus: 20-22 (see 
 ' Species'). 
 
 Membra Dividentia : 45. See ' Di- 
 vision,' ' Species.' 
 
 Meudeleeff : 61 (ftn.). 
 
 Mental . . . : See ' Psychical . . .' M. 
 Development : 461. 
 
 Mentiens : 103, 182, 183. 
 
 Mere Possibility : 46, 157. 
 
 Metaphoric Definition : 33, 37. 
 
 Metaphysical Division : 40, 52, 56. 
 
 Metaphysics : 244. See ' Philosophy,' 
 ' Philosophical Logic' 
 
 Meteorology : 352, 391. 
 
 Method : 246, 281, 295, 316, 321, 347, 
 348, 351, 353, 357 (see ' Induction,' 
 'Methods . . .,' 'Scientific Method,' 
 'Unscientific M.'). M. of (Single) 
 Agreement : 395-399, 413-417, 419, 
 422, 423. M. of Analysis : 420-422. 
 M. of Causal Explanation or C. 
 Inquiry: 389, 395-422 (.see 'Induc- 
 tion '). M. of Concomitant Varia- 
 tions : 405-407, 417, 418. M. of 
 Deduction : See ' Deductive M.' 
 M. of Definition : 18-20, 29-33. M. 
 of (Single) Difference : 355, 402- 
 407, 411, 413-419, 425, 426 (see 
 'Double M. of D.'). M. of Double 
 Agreement : 398-400, 417. M. of 
 Elimination, M. of Exclusion : 396, 
 414-418, 420 (see ' Mcthodus Ex- 
 clusiva '). M. of Enumerative In- 
 duction : See ' E. I.' M. of (Scien- 
 tific) Experiment : 395, 398, 401- 
 413, 417, 419 (ftn.), 428. M. of 
 mere Agreement : 396. M. of Ob- 
 servation : See ' Obsjcrvational M.' 
 M. of Proof : ;Sfee ' P.' M. of Resi- 
 dues : 411-413, 417, 418. M. of 
 Simple Enumeration : See ' Enumer- 
 ative Induction.' M. of Statistics : 
 See ' S.' M. of Verification : 397, 
 418 (see 'V'). 
 
 Methodical Reference to Reality: 309. 
 See ' Real Reference . . .' 
 
 Methodological Fallacies : 295. M. 
 Goal : 420. M. Guiding Principle : 
 448, 458. M. Postulate : 420, 448, 
 450, 457 (see 'Inductive P.'). M. 
 Significance of the Law of Causation : 
 456 457. 
 Methodology: 3, 305, 419, 420, 442, 
 
 443. -S'ee ' Induction.' 
 Methods of Induction : See ' Induc- 
 tive M.'s.' M.'s of the Reason : 
 448. 
 
 Methodu8 Exclusiva : 414-418. 
 M': 130.
 
 INDEX, VERBAL AND ANALYTIC 
 
 483 
 
 Middle Terra : 213-218, 221, 227-229, 
 248. 251. 258. 284, 320 {see 'Cate- 
 gorical Syllogism '). Its Distribu- 
 tion : 217, 218. 222, 223, 228, 230, 
 237, 238, 209. 
 
 Mill, John Stuart : viii, 58. 321, 348, 
 407. His Estimate of the Syllogism : 
 250. 299-304. His Inductive Canons : 
 See ' C. of Causal Methods.' His In- 
 ductive Scheme : 347. His Proof of 
 the Law of Causatian : 453-458. His 
 View of the Nature of Inference : 
 302-304. M. on Analogy : 358. 3G1, 
 363. On Cause and Causation : 
 370, 371, 375-380, 382, 389. On 
 Colligation : 349, 350. On Con- 
 comitant Variations : 417, 418. On 
 Correlatives : 35. On Elimination : 
 414-417. On Enumerative Induc- 
 tion : 351. On Hypothesis: 328. 
 On Inductive Methods : 413, 414, 
 418. On Necessary Truth : 449, 
 
 450. On Non-connotative Names : 
 81, 82. On Observation : 392, 419. 
 On Parity of Reasoning : 349. On 
 the Deductive Method : 408, 419. 
 420. On the Hypothetical M. : 
 
 410, 411, 419, 420. On the Joint 
 M. : 398. On the M. of Agreement : 
 395, 397, 413. 415, 417, 419. On the 
 M. of Difference : 402, 404, 405, 414, 
 417, 419. On the M. of Residues : 
 
 411. On the SummumBonum: 283. 
 Milne. Prof. : 441, 443. 
 
 Milnes, Alfred : 323. 
 
 Mind : 95, 97, 112, 390, 392, 393. 409. 
 453. 458-461. 
 
 Minor Premiss : 214. 215, 221, 226, 
 228-231, 233, 237, 238. 242. 243, 248, 
 249, 257, 258. 262-264, 268-274, 294, 
 301, 324. M. Term: 213. 214, 219, 
 228, 233, 269 (see ' Illicit process 
 of the M'). 
 
 Minto, William (Prof.) : viii. 287, 290, 
 306 (ftn.) 307, 336, 384. 
 
 Mitchell, P. Chalmers (Dr.) : 62. 
 
 Mixed Hypothetical Syllogism : 263- 
 269. 
 
 Mnemonic Verses : 226, 229, 235, 249, 
 250. 
 
 M°: 130. 
 
 Modal ' May ' : 129, 130. Modal Pos- 
 sibility : 129-131, 138. M. Square 
 of Opposition : 130. 
 
 Modes of Differentiation : See ' D.' 
 
 Modification of Condition : 405. M. 
 of F. D. : See ' Unity of F. D.' M. 
 (or Moulding or Revision) of H^'po- 
 thesis or Theory : 334, 335, 420-422, 
 424. 425, 427-430, 432, 447. 448. 
 
 451, 452. M. of Statement : 285. 
 2Iodus Ponendo Tolleiis : 262, 263. 
 
 M. Ponem : 263, 264, 267. 268, 325. 
 
 J/. ToUendo Ponens : 202, 203, 410. 
 M. Tollens : 263, 264, 207-269, 324. 
 
 Molecule : 459. 
 
 Monism : 1, 2, 98. M. as a Postu- 
 late : 2. 
 
 Mood : -See ' Syllogistic M.' 
 
 Moon: 38, 129. 141, 329. .331, 3.32, 
 342, 355, 358. 361. 375-377, 391, 
 394. 408. 
 
 Moons of Jupiter : 362. 
 
 Moral Science : 409. 
 
 Morality : 61 (ftn.), 107, 283. 285. 340. 
 361, 462. 
 
 Morphology : 60, 62, 424. See ' Bo- 
 tany,' Structure.' ' Zoology.' 
 
 ' Most ': 165, 222. 
 
 Motion : 447. See ' Laws of M.' 
 
 Moulding of Hypothesis : See ' Modifi- 
 cation of H.' 
 
 ' Must ' : 127. 128, 378, 462. See 
 ' Necessity.' 
 
 Mutual Coincidence, M. Exclusion. M. 
 Exclusiveness, M. Inclusion, M. Non- 
 coincidence : See ' C' ' Exclusion.' 
 ' Exclusive ....'' Exclusiveness,' 
 ' Inclusion,' ' Non-c' 
 
 Mysticism : 2. 
 
 Name: See ' Connotative N.,' ' Deno- 
 tative N..' 'Nomenclature,' 'Proper 
 N.'s,' ' Term.' 
 
 Names defined : 24, 25 (see ' Defini- 
 tion '). N. of Classes or Kinds. See 
 ' Class,' ' General Concept.' Of Indi- 
 viduals: See 'Singular Meaning.' 
 
 Natural Classification : 57-66, 73, 368, 
 369. N. Condition : 451. N. Effect : 
 451. N. Experiment : 394, 440. N. 
 Fact, N. Phenomenon : 455, 458 {see 
 ' F.,' ' Nature,' ' P.'). N. Groups, N. 
 Kinds : 58, 60, 61. 63-65 {see ' Kind ') ; 
 their Phvlogenetic Relations : 60. 61. 
 N. History : 342. N. Law : See ' L. of 
 Nature.' N. Method of Definition : 
 32. N. Order : 60. N. Philosophy : 
 409. N. Science : See ' S.' N. Selec- 
 tion : 62, 314, 344. 
 
 Nature: 138, 307-309, 313. 314, 316- 
 318, 357. 368, 377, 379, 3S6, 394, 409, 
 443, 444. 450-458 {see ' Common N.,' 
 'Law of N..' 'Science'). N. as 
 Intelligible : See ' Intelligibility of Ex- 
 perience.' As Systematic: 119 {see 
 ' S. Conception . . .'). N. in relation 
 to Analysis : 29 ; to Knowledtre : 29 ; 
 to Thought : 2-5, 9, 23. N.' of the 
 Object : 54, 57, 5S, 64, 65. 
 
 Naville, Ernest: 316. 
 
 Nebular Theory : 436. 
 
 Necessary Conclusion : See ' Necessity 
 
 in Inference.' N. Condition : 457 
 
 {see 'C.'). N. Connexion between 
 
 Antecedent and Consojuent : 136, 
 
 31—2
 
 4S4 
 
 THE PROBLEM OF LOGIC 
 
 139. 14-2 {see ' Apodeictic Hypo- 
 thetical ' ) ; between Cause and Elfect : 
 3S2. X. Consequence : 319. 335, 336. 
 X. Deduction : 342. X'^. Implication : 
 149 {see 'I.'). X. Truth : 449, 450. 
 Xecessitv : 105, 127, 12S. 37S, 387, 448 
 {see ' Intrinsic X.,' ' Teleological X.'). 
 X. in Connexion of Possibilities : 139, 
 141. In Deduction : 26. In Infer- 
 ence : 8. 9. 130, 149. 190, 214, 216, 
 306. 309, 321-325, 327 (ftn.), 328 (see 
 ' Logical X.'). 
 ' Xeed ' : 12S. 
 X'egated Copula : 126. 
 Xegation: 49, 122-124. 193, 194, 420, 
 421, 449 {sec ' Xegative Proposition,' 
 ' Privative View of X.,' ' Pure X'.') 
 Its relation to Affirmation : 123, 124 
 Xegative aspect of Verification : 334 
 335. X. Assertion : 122 (.see ' Xega 
 tion ■). X. Conclusion : 217, 220, 223 
 224, 236-238, 257, 259, 260. X. Con 
 dition : 371, 379. X. Copula : 126 
 X". Definiteness : 157. X. Definition 
 mark: 36, 64. X. Evidence, X 
 Result : 334. 335, 401. X. Form and 
 X. Meaning: 169 (ftn.). X. Idea: 36 
 {see 'Indefinite Species'). X^. In- 
 stance : 400-403, 407, 416, 421, 425, 
 426. X. Interrogative: 168. X. 
 Limitation : 156. X'. Method : 418. 
 X'. ^Movement : 107. X'^. Premis-s : 
 217, 220, 221, 223, 224, 230, 232-234, 
 237, 238, 243, 264. X. P. in Sorites : 
 257, 2.58. X. Principle : 16. X. Pro- 
 position : 123, 124, 147, 149, 1.50, 158 
 (5ee 'E Proposition,' ' ?; P.,' ' X'ega- 
 tion.' 'O P.,' 'w P.'). X. Quality: 
 123, 124 {see ' Q.'). X. Quality-mark : 
 163. X. Statement : 122, 170 (see 
 ' Xegation '). X. Subject-term : 166. 
 X. Test-instance : 403, 406, 407, 414- 
 416. N. Universal: See ' U. N." 
 N. use of ' Any ' : 164. 
 
 X'egativity : 107. 
 
 X£gO: 147. 
 
 X'eo-Hegelians : 31. 
 
 Xeptune: 412, 413. 
 
 Xewcomb, Simon : 70, 412 (ftn.), 438. 
 
 Xewton, Sir Isaac : 318, 332, 333, 337- 
 339, 342, 343, 388. 
 
 X'ewtonian MethofJ : 315. 
 
 'Xo ' : 48 (ftn.), 164, 168. ' Xo A is 
 either B or C ' : 1.32. ' Xo S is any 
 P ' : 1.59. ' Xo S is P ' : 149. ' Xo S 
 is some P ' : 1.59, 160 {see ' t; . . .'). 
 
 Xomenclature : 61, 66-70, 281, 282. 
 Symbolic X. : 69, 70. 
 
 Xominalistic Definition : 24, 25. 
 
 Xon -acceptance of Statement : 269. 
 
 Xon-Ambiguitv: 3. 16. 20, 26, 27, 39, 
 l'r2, 103, 1.33, 1.35, 214. 
 
 X'on-causal Explanation : 369. 
 
 Xon-cause : 415, 41G. 
 Xon-coincidcnce : 150, 153. 
 Xon-coimotative Xames or Terms : 81. 
 
 89. 
 Xon-Contradiction : See ' Principle of 
 
 X.-C 
 Xon-cumulative Eflect : 373-375. 
 Xon-dcfining Predicables : 22, 25, 26. 
 Xon-denotative Terms : 89. See ' In- 
 
 fima Species.' 
 Xon-eflective Condition : 384. 
 X^n-essential Antecedents : 384. X.-e. 
 ^Mark: 64. X.-e. Meaning: 27-29. 
 Non-exclusive Reading of the Disjunc- 
 tive : 132-137, 178, 266, 274. 
 Xon-exclusivist : See ' Xon-exclu- 
 
 sive . . .' 
 X^'on-identity : 219, 220. 
 Xon-logical Division : ^ee ' Meta- 
 
 phjrsical D.,' 'Physical D.,' 'Verbal 
 
 D.' 
 Xon-mechanical Explanation : 451, 452. 
 Xon-observation : 391, 392. 
 Xon-periodic Causes : 373, 375. 
 X'on-propositional Sentences : 93, 94, 
 
 167, 168. 
 
 X^n-x : See ' Indefinite Species,' ' " In- 
 finite Term." ' 
 
 ' Xot any ': 164. ' N. assuredly ' : See 
 '"A."' 'X.neither': 133. Not(SisP): 
 98 (ftn.). 173, 174. ' N. statedly ' : 
 See ' S.' 
 
 Xotion : See ' Concept,' ' Idea.' 
 
 Xot-P, X^'ot-x : See ' Indefinite Species/ 
 ' " Infinite Term." ' 
 
 ' Xow ' : 82-85, 101. 
 
 Xumber : 447, 454. X. of Diilerences, 
 N. of Instances, X. of Resemblances : 
 43, 327, 353-355, 358-360, 396, 397, 
 418, 455 (see ' Enumeration '). N. 
 of Kinds denoted : 72. 
 
 Xunieration : See ' En.' 
 
 Xunierical Accuracy or Precision : 386 
 X. Coincidence : 146, 147. N 
 Equivalency : 387. N. Exceptive 
 (Proposition) : 167. X. Result : 435. 
 
 O Proposition : 147-151, 155, 157-161, 
 
 168, 171, 172, 182, etc. Its Con- 
 version : 54, 195, 196, 201. Its 
 Diagrams : 151, 160. Its relation to 
 1 : 156 {see ' Subcontrariety '). 
 
 w Proposition : 1.59-161. 
 
 Object : 78, 129, .341, 357, 359, 368, 451 
 (see 'Abstract O.,' 'Concrete O.,' 
 ' Countable O's.,' ' Immediate O.,' 
 ' Individual,' ' Objective Reference,' 
 ' Spatial O.,' ' Subject-0. Relation '). 
 0. of Definition : 23-26, 32, 51. O. of 
 Division : 51, 53, .54. O. of Enumera- 
 tion : 353. O. of Experience : 78. 
 O. of Experiment, O. of Study : 145, 
 453. 0. of Interest : 78, 118. O. of
 
 INDEX, VERBAL AND AN.AJLYTIC 
 
 485 
 
 Knowledge : 83, 208. O. of Observa- 
 tion : 340, 390, 303, 394, 401-403. O. 
 of Perception : 392. O. of Thought : 
 3-6, 9, 23, 104, 208. 
 
 Objective aspect of Meaning : 23-26. 30, 
 51, 77 ; of Reality or of Truth : 4, 5. 
 23, 309 (.«ce ' Real Reference . . .'). 
 O. Classification : 65 {see ' Real C). 
 O. Content: 23, 24. O. Control of 
 Thought : 3, 5. O. Nature : 23, 2G. 
 O. Possibility: 129 (see 'Real P.'). 
 O. Reality : 24, 118, 119. O. Refer- 
 ence : 85-89, 94, 103, 115, IGl (ftn.), 
 162 (ftn.). O. System: 318. 
 
 Object-matter : 57, 58, 65. 
 
 Objects classed : See ' Classing.' 0. 
 classified : 63 (see ' Classification '). 
 0. contained in Classes : See ' Class.' 
 O. defined : 23-25 (see ' Definition '). 
 O. ' denoted ' by Terms : 147. O. 
 indicated by Terms : See ' Indica- 
 tion . . .' O. named : 80-82. 
 
 Obligation : 107. 127, 293. 
 
 Observation: 114, 115, 136, 140, 166, 
 295, 299-303. 305, 313-315, 326, 327, 
 330-335, 338-341, 349-352, 355-360, 
 379, 389-398, 401-405, 408-414, 419, 
 421, 424, 425, 431-438, 441-443, 448- 
 457. 
 
 Observational Interest : 389-391. 393. 
 O. Method : 395-401, 419 (ftn.). O. 
 Purpose: 390-393. 
 
 Obverse: 190-193, 197-202,206,248,265. 
 
 Obversion : 189-193, 198, 200, 202, 221, 
 232, 234, 249, 250, 324. ' O. of 
 Hypotheticals ' : 265. 
 
 Obvertend : 191, 192, 201. See ' Ob- 
 version.' 
 
 Occasional : See ' formal.' 0. Interest : 
 114. O. Meaning : 15, 70. O. Pro- 
 perty : 27, 28, 37. 
 
 Odysseus: 118. 119. 
 
 (Ecology : 321. 
 
 <Ersted, Hans Christian: 334. 
 
 Omissions in Division : See ' Inadequate 
 D.' 
 
 ' One at least ' : 155-157, 161, 162. 
 
 Oneness : <See ' Unity.' 
 
 ' Only ' : 166, 167, 169-171, 198. 
 
 Ontological Cause : 378. See ' First C 
 
 Operations Subsidiary to Induction : 
 316 (ftn.), 349. 411, 422. See 'Col- 
 ligation,' ' Definition,' ' Division.' 
 ' Hypothesis,' ' Observation.' 
 
 Opinion : 145. 157, 392. 
 
 Opposite : 123. Sec ' Opposition.' 
 
 Opposition : 48, 123. 126, 304. O. of 
 Categorical Propositions : 128. 130, 
 156, 171-177, 181-183, 189. 203 (sec 
 'Modal Square of O.,' ' Teleological 
 Scheme of 0.'). O. of Disjunctives : 
 178. 0. of Hypotheticals : 178-180. 
 O. of Singular Propositions : 181. 
 
 Optative : 93. 94, 167, 168. 
 
 Optics : See ' Light.' 
 
 ' Or else ' in Disjunctives : 135. ' Or ' 
 in Division : See ' Disjunctive D.' 
 ' Or ' in Disjunctives : See ' Either . . . 
 or.' 
 
 Order. Ordering, Orderliness : 3, 5, 23, 
 29, 30, 43. 57, 58. 119 (ftn.). 337. 340. 
 368. 4.55. 458 (see ' Classification,' 
 ' Conceptual Ordering ...,'' Natural 
 Order,' 'System,' ' Systematization,' 
 ' Uniformity '). Order (or Degree) of 
 Generality : 45, 88. 0. of Propo- 
 sitional Forms : 112-114. 
 
 Ordered Connexion : See ' Order.' 
 
 Orders of Meaning : 33, 34. 
 
 Organic Development, O. Evolution : 
 See ' D.' O. Fact, O. Life. O. Pheno- 
 menon : 458-461. O. Order : 33. O. 
 Relations : 447. O. Unity of a 
 Natural Classification : 60 ; of Term 
 and Proposition : 125 ; of the Division 
 Process : 44, 45. O. Universe : 4G0. 
 
 Organization : See ' Conceptual Order- 
 ing ...,'' Systematization.' O. of 
 Thought: 96. 
 
 Organized Common Sense : 5, 347. 
 
 Ostensive Reduction : 247-250. 
 
 Otherness : See ' Difference.' 
 
 Over-exhaustive Division : 46. 
 
 Overlapping in Classification : 62. In 
 Division : 45, 52-55. 
 
 P as a Substitute Sign • 14. 
 
 Pairs of Terms : See ' " Contradictory 
 T.,'" '" Contrary T. '"' Correla- 
 tive T.' 
 
 Palaeontology : 61. 
 
 Paradox : See ' Zeno . . .' 
 
 Parallel interpretation of ' Eitlier . . . 
 or' and 'Some' : 132 (ftn.). 
 
 Parallelism of Division and Enumera- 
 tion : 43. 
 
 Parity of Reasoning : 347, 349, 359. - 
 
 Partial Antecedent, P. Cause : 378, 383. 
 P. Coincidence, P. Non-c. : 150, 152, 
 159. P. Definition : 74. P. Deter- 
 minacy : 112. P. Exclusion, P. In- 
 clusion : 153, 240, 241, 243. P. Ex- 
 periment : 405. P. Extension-refer- 
 ence : 147, 148, 219 (see ' Undis- 
 tributed . . .'). P. Ground: 129. P. 
 Knowledge : 129 (see ' Limitation of 
 K.'). P. Method : 410. P. Verifica- 
 tion : 436. See ' Progressive V.' 
 
 Particular: 83, 84, 87, 88, 129, 157, 
 244-247. 299. 300, 302, 303, 320, 326, 
 339, 349, 441, 460 (see • Individual.' 
 ' Inference from Ps . . .'). P. Aflir- 
 mative (Proposition) : 147 (see ' I 
 P.'). P. Conclusion: 217, 223. 226- 
 228, 236, 238, 259, 260. P. Exten- 
 sion-reference : 148 (see ' Undistri-
 
 4S6 
 
 THE PROBLEM OF LOGIC 
 
 390, 393, 394. 
 
 1G3, 104,371,461. 
 ) : 428, 429. 
 
 buted . . .'). P. Individual : 1,")" {see 
 'I.'). P. Judcinont, P. Proposition : 
 132. 147. 150. 153-158, 161, 163, 165, 
 166 (sec ' I P." O P.'). P. Law : 455. 
 P. Negative (Proposition) : 147 (see 
 'O P.'). P. Premiss: 217, 221-224, 
 228, 230, 232. 24S ; P. P. in Sorites : 
 257. P. Quantity-mark: 155, 158 
 [see * Some'). 
 
 Particularity : 165, 344. See ' Par- 
 ticular.' 
 
 Particularization : 133. See ' Specifica- 
 tion.' 
 
 Particularized Effect : 383, 384. 
 
 ' Parts ' in Division : 40, 45. Parts of 
 the Proposition : See ' Propositional 
 Elements.' 
 
 Passive Observation ; 
 
 Passivity : 31. 
 
 Past: 83,85, 101, 139, 
 
 Pasteur. Louis (Prof. 
 
 Pathology : 372, 427. 
 
 Peirce, C. S. : 63 (ftn.), 112. 
 
 Perception : 97, 338, 390, 392, 393, 440, 
 441, 453, 459. Sec ' Immediate Ap- 
 prehension,' ' Sensation,' ' Sense-ex- 
 perience.' 
 
 Perfect Disjunction : 136, 137 (see ' Ex- 
 clusive Reading . . .'). ' "P. Figure" ' : 
 257, 324, 325 {see 'Fig. I.'). ' P. In- 
 duction ' : 347, 348, 356. 
 
 Periodic Cause : 373-375. P. Law 
 (ftn.). 
 
 Peripheral Definition : 17, 18, 62. 
 
 Permanent (or Persistent) Cause 
 405. P. Conditions : 372, 373. 
 
 Permissible H3'pothesi8 : 
 ' Legitimate H.' 
 
 Permissive Possibility : 128, 
 
 Permutation : 190 (ftn.). 
 
 Persistent Cause, P. Conditions : 
 ' Permanent C 
 
 Person : 86, 87, 114, 128, 451. 
 
 Personal Agent : See ' Person.' P. 
 perience : 84, 453 (see ' Immediate 
 E.'). P. Xame : ,S'ee ' Proper N.'s.' 
 
 Personality: 3-6, 63 (ftn.), 114 (ftn.), 
 127. 
 
 P>-r8onification : 87. 
 
 Pctitio Princinii : 289-291, 300, 301. 
 P. P. in Mill's Proof of the Law of 
 Causation : 4.j7, 458. 
 "P Proposition : 156, 109. 
 
 Phenomenal World : 105. 
 
 Phenomenon: 313, 3.39. 34 T 349, 3.53, 
 355, 371, 375, 376, 378, 390, 394-411, 
 414-419, 423, 429, 433-436, 440, 443, 
 4.52-461. 
 Philosophic Definition : 62, 63. P. 
 Interest : 453. 
 
 Philo.sophical Hypothesis: 318. P. 
 Logic: Tiii, ix, 3-6, 10, 114 (ftn.), 
 402. 
 
 61 
 
 404, 
 
 337. See 
 
 129. 
 
 See 
 
 Ex- 
 
 Philosophv : ix, 97, 105, 316, 370 (see 
 ' Natural P.'). P. of Morals : 340. 
 
 Physical Agent : 387. P. Analysis : 435. 
 P. Cause : 378. P. Division : 40, 52. 
 56. P. Elimination, P. Exclusion, P. 
 Subtraction : 414-417. P. Fact : 455. 
 P. Force : 373. P. Method : 409. 
 P. Order: 33. P. Possibility: 437. 
 P. Science : 386 (see ' Astronomy,' 
 ' Chemistry,' ' Geology,' ' Physics '). 
 P. World : 461. 
 
 Physics : 2, 29, 82, 318, 374, 385-389, 
 409, 433, 439, 440, 447, 459. See 
 ' Dynamics,' ' Electricity,' ' Heat,' 
 ' Hydrostatics,' ' Light,' ' Magne- 
 tism,' ' Sound.' 
 
 Physiology: 82, 131, 364, 409, 424, 
 427-429. 
 
 Place : 31, 87, 100, 454. See ' Space.' 
 
 Planet, Planetary . . . : 80, 81, 97, 119, 
 134, 139, 230, 237, 245, 329-333, 338, 
 375, 376, 435, 439. See 'Earth,' 
 ' Jupiter,' ' Mars,' ' Neptune,' 
 ' Saturn,' ' Uranus.' 
 
 Plant : 42, 53, 59, 60, 64-68, 73-75, 87, 
 129, 231-234, 305, 314, 315, 320, 321, 
 342-344, 354, 355, 374, 401, 424-428, 
 451, 461. 
 
 Plato : 40 (ftn.). 47, 79, 80, 300, 331, 
 363 (ftn.). 
 
 Plural Term as Dividendum : 53. 
 
 Pluralism : 2. 
 
 Plurality of Causes : 379, 383-385, 389, 
 397-400, 419, 449. 
 
 Plurative Proposition : 222. 
 
 Political Economy : 66. 
 
 Polysyllogism : 255, 259. See ' Sorites.' 
 
 Popular Conception of Causation : 380, 
 381, 383. P. Use of ' Some ' : 155. 
 
 Porphyry : 22, 23, 47 (ftn.), 78 (ftn.). 
 
 Porphyry's Tree : 42, 47, 79, 80, 82. 
 
 Position : 14, 96, 123, 262, 263, 266 (see 
 'Affirmation'). P. in Space; 101 
 (.see ' Place '). P. in Time : 83. See 
 
 i rrt f 
 
 Positive Aspect of Verification : 335- 
 338. P. Conclusion : 417. P. Con- 
 dition : 371, 379. P. Definiteness : 
 157. P. Error : 339. P. Implication : 
 124 (see ' Indefinite Species '). P. 
 Incompatibility : 401. P. Instance : 
 400-403, 416, 41S, 425, 426. P. In- 
 terrogative : 168. P. Method : 418. 
 P. Species, P. Term : *S'ee ' Definite 
 S.' P. Statement: 421 (see 'Affir- 
 mation'). P. Test-instance: 403, 
 414-416. 
 
 Possibility: 112, 113, 122 (ftn.), 127- 
 141, 156-158, 437, 449, 454 (see 
 ' Alternative Determinations,' ' Con- 
 nexion of Ps,' ' Intrinsic P.,' 
 ' Modal P.,' ' Permissive P.,' ' " Pos- 
 sibly all,"' 'Purposive P..' 'Real
 
 IXDEX, VERBAL AXD ANALYTIC 
 
 487 
 
 P.,' ' Toloological P.'). P. in relation 
 to a Categorical Basis : 131. 
 
 Possible Cause : 383. P. Differentia- 
 tions : 46, 52 {see ' Alternative 
 Determinations'). P. Explanation: 
 449, 453 (.see ' Alternative Es,' 
 'Legitimate E.'). P. Moauings : 5G. 
 P. Properties : 27, 28. 
 
 * Possibly all ' : 150, 157. 
 
 ' Post hoc ergo propter hoc ' : 377. 
 
 Postulate: 2, 306, 322, 338 (ftn.), 367, 
 308, 387, 420, 445-462. P. and 
 Kocessary Truth : 449, 450. P. and 
 Working Idea : 448, 449. P. of Induc- 
 tion : ^ee ' Inductive P.' P. of Infer- 
 ence : 223. P. of Intelligibility : 78, 
 79, 99 (see ' I.'). P. of Mediate Infer- 
 ence, P. of Mediation : 213, 210, 218, 
 223, 244, 251. 
 
 Potential 'Can'': 127-129. P. Predi- 
 cato, P. Subject: 111, 112. 
 
 Potentiality : 129 (sec ' Capability,' 
 ' Possibility ']. P. of the Genus : 22, 
 127. 
 
 PQ: 135. 
 
 Practical aspect of Reality : 4, 5, 9 ; of 
 Logic : 13. P. Classification : See 
 ' Diagnostic C P. Consciousness : 
 155, 380, 381, 383. P. Definition : 
 See ' formal D.' P. Division : Sec 
 ' formal D.' P. Interest : 9, 23, 29. 
 P. Life : 58. P. Reasoning : 133. P. 
 Sense : 97. 
 
 Pragmatic Definition : 03 (ftn.). 
 
 Pragmatism : viii, ix, 420. 
 
 Precautions in applying Method of 
 Difference : 404. 
 
 Precipitating Cause : See ' Exciting C 
 
 Precision (or Accuracy) : 16, 66, 69, 79, 
 105, 135-137, 145, 282, 304, 330, 336, 
 353, 385, 386, 389, 392, 394, 398-400, 
 413, 422, 424, 427-435, 439-443. P. 
 in Hypotheticals : 137, 385. 
 
 Precondition : 448. See ' Condition.' 
 
 Predicables: 18-29, 31, 67, 120, 121, 
 159. 
 
 Predicate Concept : 320. P. Terra : 22, 
 28, 96, 103, 104, 100, 111, 112, 115- 
 120, 131, 132, 140-152, 150, 158-103, 
 105-168, 320 (see 'Distribution of 
 T.'s,' ' Quantification of the P.,' 
 ' Subject-P. Relation '). 
 
 Predication : 22-28, 31, 32, 34, 40, 96, 
 118, 120, 121, 124, 129, 154, 158, 159, 
 102 (ftn.), 105 (ftn.), 230. 
 
 Predicative view of Imjwrt : 147, 149, 
 101, 105. 
 
 Prediction : 141, 335, 336. 
 
 Pre-disjunctive Relation of Genus to 
 Species : 22, 50. 
 
 Predisposing Cause, P. Condition : 371, 
 372. 
 
 Pre-iudesignate Proposition : 163. 
 
 Preliminary Explanation : 369. P. 
 Observation: 351, 352, 356, 395. P. 
 Stages in Induction : 361, 395, 407, 
 411, 419. 
 
 Premiss: 8, 105, 1.30, 145, 157, 187-189, 
 194, 201, 208, 213-238, 241-243, 246- 
 249, 255-264, 208-275. 282, 286, 289- 
 295, 300-310, 318-327, 408, 424. 426. 
 430-433, 430, 439, 457, 4,58. See 
 ' Falsity of Ps,' ' Lifcrcnce,' ' Major 
 P.,' ' Minor P.,' ' Truth of Ps.' 
 
 Preperception : 392. 
 
 Pre-philosophical Logic : 3-5, 9, 10. 
 
 Prc-scientific Hypothesis : 430. P.-s. 
 Interest, P.-s. Thought : 4, 5, 455 
 (ftn.). 
 
 Present: 82-85, 101, 103. 
 
 Presumption : 399. 
 
 Presupposition: 370, 385, 400, 409, 
 447 (see 'Assumption'). P. of the 
 Ideal : 419, 420. 
 
 Preventine Cause : 371. 
 
 Primary F. D. : 74. -See ' Unity of 
 F. D.' 
 
 Principmm Individuationis : 82. 
 
 Principle : 193, 370, 399, 420, 448, 452 
 458, 461 (see 'Guiding P.,' 'Law') 
 P. of Classification : 57, 58, 61 (ftn.) 
 P. of (Logical) Consistency : 179, 180 
 213, 286, 448 (see ' C). P. of Deduc 
 tion : 245. P. of Deductive Infer 
 ence : 216, 219, 223, 244, 246, 281. P 
 of Definition : 39, 281. P. of Deter 
 minism : See ' D.,' ' Inductive Postu 
 late.' P. of Division : See ' Funda 
 mentiun divisionis.' P. of Empiri 
 eism : 313, 316 (see ' Fidelity to 
 Fact '). P. of Fidelity to Fact : See 
 ' F. to F.' P. of Fidelity to Relevant 
 Fact : See ' F. to R. F.? ' Inductive 
 Postulate.' P. (or Law) of Formal 
 Identity: 187, 188, 190, 194. Of 
 Formal" Inference: 187, 188, 191, 194, 
 286. P. of Formal Validity: 187,188, 
 191. 218, 223. P. of Gravitation : 
 -See 'G.' P. (or Law) of (Logical) 
 Identity: 16, 95-98, 104. 124, 145, 
 189, 193, 194, 213, 223, 250, 209, 290, 
 325, 327 ; Its relation to Inference : 
 90 ; to Judijment or Proposition : 96, 
 104, 121-124, 145. Its Violation : 104. 
 106. P. of Identitv-in-Difference : 
 90, 97, 121 (see ' I. "in D.'). P. of 
 Induction : See ' Inductive P.' P. of 
 Inductive Method : 2S1. P. of (Logi- 
 cal) Inference : 210, 219, 223, 244, 
 246, 281. P. of Judixment : 90. P. of 
 Logical Validity : 193, 243, 305, 326, 
 327. P. of Mediate Inference : 246 
 (see ' Postulate of M. I.'). P. of Non- 
 Ambiguity : 16, 26, 27, 39, 133 {see 
 'X.-A.'); Its relation to the P. of 
 Non-Contradiction : 102, 103, 286.
 
 488 
 
 THE PROBLEM OF LOGIC 
 
 P. (or Law) of Xon-Contradiction : 
 ix, 95, OS. 09, 102-104. 140, 145. 171, 
 172. 187-100, 103, 205. 250, 286, 200, 
 304. 327, 450; It.s Violation (are 
 ' Inviolabilitv ...'). P. of Proof : 281. 
 P. of Pure induction: 313, 310 (sec 
 'Fidelity to Fact"). P. of Statical 
 Identity: 00. 07. P. of Synthesis: 
 140. P". of Tautology: 97. P. of the 
 Conseryation of Eneriiy : Sec ' C. of 
 E." P. of the C. of Mass : -See ' C. of 
 M.' P. of the Constancy of Energy : 
 Sec 'C. of E.' P. of the Natural 
 Sciences: 451 (sec ' Inductiye Postu- 
 late'). P. of the Transformation of 
 Energy: -See ' T. of E.' 
 
 Principles of Conscience : 3G7. P. of 
 Consistent Thinking : 6, 16 (see 
 ' Principle of Consistency '). P. of 
 Cieometry : 454. P. of Human 
 Nature: 410. P. of Intelligibility: 
 104 (see 'Laws of Thought'). P. 
 of Mathematics : ix, 306. P. of 
 Mill's Inductiye Methods : 413 (sec 
 'Method of Agreement,' 'Method of 
 DiflFerence '). P. of Number: 454. 
 P. of Reason : 367. P. of Reasoning : 
 305 (see ' La\ys of Thought'). P. of 
 Science : 63, 305, 368. ' P.' of the 
 Syllogistic Figures : 240-247. P. of 
 Thought . 304 (see ' Laws of T.'). 
 
 Priority of Propositional Forms : See 
 ' Order of P. F.' 
 
 Priyatiye Conception : 190 (ftn.). P. 
 View of Negation: 103 (see' Pure N.'). 
 
 Probability : 359, 397, 407, 413, 430, 
 443. 
 
 Probable Cause: 395, 396. P. Con- 
 clusion : 363. P. Condition : 395. 
 P. Error : 413. P. Inference : 327. 
 
 Problem of Inference : 297-310. P. of 
 Reduction : 247-250. 
 
 Problematic ' May ' : 129. P. Pro- 
 perty : 27, 28, 37, 54, 165, 170, 359, 
 360. 
 
 Problems of Philosophy : 105. 
 
 Procedure in Proposition : 96. P. of 
 Analysis and Synthesis : 30, 31. P. 
 of Cou.sciou3nes8 : 97. P. of Induc- 
 tion : jS'ee ' I.' 
 
 Products (or Results) of Definition and 
 Division : 70, 146. 
 
 Progressive Definition : 63, 79. P. 
 Division : 79. P. Modification (or 
 Moulding) of Hypothesis : 420-422, 
 427 («ee ' Modification of H.'). P. 
 Realization of an Ideal : 419, 420. 
 P. Rea.soning: 260 (.tee 'Sorites'). 
 P. Simplification : 339. P. Sorites : 
 2.55-260. P. Verification : 349, 3.52, 
 3.56, 419. 
 
 Proof (or Demonstration) : 94, 235, 236, 
 246-248, 282, 286-291, 300, 301, 306, 
 
 321-325, 364, 399-401, 408, 411. 416, 
 418, 443, 449, 450, 4.54, 457, 458 (see 
 'Experimental P.,' 'Indirect P.'). P. 
 by Exclusion, P. by Exhaustion: 
 323, 324. P. of a Proposition from 
 Axiomatic Premisses : 318. P. of 
 Hypothesis : 336, 337. P. of the Law 
 of Causation : 453-458. 
 
 Proof-Fallacies : <S'ee ' Demonstrational 
 F.' P.-F. in Dilemma : 292-295. 
 
 Propaedeutic Character of Formal Logic: 
 157, 204. Of formal Logic : 5. 
 
 Proper Names : 69, 161. Their Defina- 
 bility : 74, 79-83. 
 
 Property (or Proprium) : 22, 25-29, 37, 
 40, 54, 81, 121, 159, 165, 170, 341, 
 351-353, 358, 359, 361-364, 368, 369, 
 414, 424, 447. See ' Characteristic 
 Property,' ' Essential Characteristic,' 
 ' Problematic P.' 
 
 Proposition (or Judgment) : 91-183, 
 422, 449 (see ' A Proposition,' 
 ' Affirmation,' ' Arapliative Judg- 
 ment,' '"Analytic J.,"' 'Asser- 
 tion,' 'Augmentative J.,' 'Cate- 
 gorical Assertion,' ' Collective P.,' 
 ' Disiunctive J.,' ' Distributive P.,' 
 ' E P.,' '17 P.,' ' Enumerative Uni- 
 versal,' ' Exceptive P.,' ' Exclusive 
 P.,' '"Explicative" J.,' ' Ex- 
 ponible P.,' ' Extensive P.,' ' Formal 
 treatment of the Logical P.,' ' formal 
 J.,' 'General Universal J.,' 'Generic 
 Universal J.,' ' Hypothetical J.,' 
 'I P.,' 'Import of Ps.,' 'Indefinite 
 P.,' ' Indesignate P.,' ' Negative P.,' 
 'OP.,' 'wP.,' 'Particular J.,' '*P.,' 
 ' Plurative P.,' ' Pre-indesignate P.,' 
 ' Propositional Forms,' ' Real J.,' 
 ' Singular P.,' ' Statement,' ' " Syn- 
 thetic J.," ■ ' U P.,' ' Universal J.,' 
 ' Y P.'). P. as the Logical Unit : 94, 
 97, 111. P. distinguished from Sen- 
 tence : 93, 94. P. in relation to Judg- 
 ment : 93. 
 
 Propo.sitional Elements : 98, 104, 111, 
 
 112, 118, 122, 125, 136, 137. 160, 216 
 (see ' Alternative Clauses,' ' Antece 
 dent,' 'Compound Proposition,' 'Con 
 sequent,' ' Term '). P. Form : 167 
 168 (see ' Proposition '). P. Forms 
 109-115, 1.36, 139 (see 'Proposition'). 
 P. Import : See ' I. of Propositions.' 
 
 Propositions as Interrelated : 122, 123. 
 As Self-contradictory : See under 
 ' Statement.' As Self-evident : 99, 
 
 113, .323 (ftn.). As Valid or Invalid : 
 8, 94, 140 (see ' Apodeictic . . .'). 
 Their Meaning: 102, 104, 117, 118, 
 135, 169 (ftn.). See ' Import of P.'). 
 
 Proprium : <S'ee ' Property.' 
 Prosyllogism : 257, 258, 260, 261. 
 Protagoras : See ' Liligiosus.'
 
 IXDEX, VERBAL AXD AXALYTTC 
 
 489 
 
 Provisional Definition : 63. P. Divi- 
 
 P. 
 145. 
 
 138, 
 
 122, 
 
 330, 
 
 . P. 
 12-t, 
 
 sion : 46. P. Explanation : 334 
 Isolation from Truth-interest : 
 P. Subject: 116. 
 
 Proximat.^ Cienus : 32, 33, 37. 
 
 Pseudo-rJileninia : 274. 
 
 Pseudo-hypothetical Proposition : 
 139. 
 
 Pseudo-Syllogism : 369. 
 
 Psychical (or Mental) Life : 83, 84, 105, 
 451, 459, 461. 462. See ' Conscious- 
 ness,' ' Mind.' 
 
 Psycholofrical Analysis of Jndrrment : 
 
 122. P. Law : 392. P. Meanings : 
 82. 
 
 Psychologism, ix. 
 
 Psychology : 71, 81, 106, 112, 113, 
 
 123, 451, 4.52. 
 Ptoleiua?us, Claudius: 329. 330. 
 Ptolemaic Astronomy : 38, 329, 
 
 332, 338, 362. 
 
 Pure Being: 75-77 (see 'Being'] 
 Denial, P. Negation: 49, 123, 
 193. P. Hypothetical Syllogism : 
 363, 369-37r(.»ce ' H. Sori'tcs'). P. 
 Inductive Method : 313-317. P. 
 Thatness: 82. 
 
 Purely Observational Method, P. O. 
 Stage in Causal Inquiry : 395-401, 407. 
 P. Problematic Properties : 27, 28. 
 
 ' Purifying by Exceptions ' : 420-422. 
 See ' Exception.' 
 
 Purport of Propositions : 138, 139, 145, 
 146. 
 
 Purpose (or Aim or Interest or Rele- 
 vancy to I. or to P.) : viii, ix, 77-80, 
 93, 95, 96, 115, 118-120, 132, 133, 
 137, 145, 146, 154, 155, 162, 295, 339, 
 
 • 340, 343, 347, 361. 370, 377, 380-382, 
 390, 302 (see 'Observational P.,' 
 'Logical P..' 'Scientific P.,' 
 * Spiritual P.'). P. (or Relevancy) in 
 Classification : See ' C In Collection 
 of Facts : 314. In Definition : See 
 ' D.,' ' Definability.' In Dichotomy : 
 Sec 'D.' In D'ivision : See ' D.,' 
 ' Divisibility.' In Induction : 316-318 
 (see 'Relevant Fact'). In Nomen- 
 clature : See ' N.' In Proposition : 
 97, 123 (see under ' Interest '). In 
 relation to Differentia : 20 ; to Mean- 
 ings of Wortls : 16, 17 ; to Reality and 
 Truth : 1-3. 5, 9. In Thought : 1-3, 5, 
 9, 95, 96. 
 
 Purposive Interference : 394. P. Per- 
 ception : 390. P. Possibility : 128, 
 129. 
 
 Purposiveness of Scientific Observa- 
 tion : 30O-:}03. 
 
 Pj'thagoreanism : 331. 
 
 Qualification ; 
 ' Quality.* 
 
 86, 87. 134, 168. See 
 
 Qualifying Mark : 18, 19, 80. See 
 'Mark,' 'Quality.' 
 
 Qualitative Adequacy : 412. Q. Change: 
 400. Q. Distinction : 124 (see 
 ' Quality '). Q. Property : 4.35. 
 
 Quality: 31, 85, 87, 168, 174, 3.33, .340, 
 3.J5, 361, 451 (see ' Attribute,' ' Char- 
 acter'). Q. of Categorical Proposi- 
 tions 122-124, 148 (see 'Affirma- 
 tion,' ' Negation '). Q. Rules of the 
 Syllogism : 217, 219-221. 
 
 Quality-difference : 123, 124. 
 
 (^i.ality-mark : 162-164, 168. 
 
 Quantification of the Predicate : 148 
 (ftn.), 149 (ftn.). 159-161. 
 
 Quantitative Adequacy : 412, 413. Q. 
 Aspect of Causal Relation : 385-389, 
 405, 406. Q. Change : 460. Q. Data : 
 422. Q. Deduction: 434, 435. Q. 
 Method : 413. Q. Reasoning : 306. 
 Q. Statement : 3S6. 
 
 Quantity: 31, 43. 146, 108, 174, 333, 
 386, 387. 389, 409. 447. Q. of Cate- 
 gorical Propositions : 148. Q. Rulea 
 of the Syllogism : 217, 221-224. 
 
 Quantity-mark : 147, 154, 155. 158-160, 
 162-164. See '"AH" as Q.-m.,' 
 ' Some.' 
 
 (Quantum : See ' Quantity.' 
 
 Qmilernio Terminorum : 218, 221, 232. 
 234, 235. 259. 285, 286. 
 
 Question: 95, 96, 112, 113, 164. 168, 
 170, 339, 340 (see ' Begging the Q.'). 
 Its relation to Reality : 114. 
 
 Range of Possibility : 131, 133. 
 
 Rank of Generality : 45, 52. 
 
 Ratiocination : 250, 408-411, 
 
 Rational Faculty: 104 (see 'Reason'). 
 R. Inquiry : 448. R. Obligation : 106. 
 
 Rationality : 30, 47, 76, 79, 448. 
 
 Rationalization : 369. 
 
 Ray, Lankester (Prof.) : 429. 
 
 Read, Carveth, M.A. (Prof.): viii, 36, 
 49 (ftn.), 235, 249 (ftn.), 273. 305, 337, 
 3S4 (ftn.), 396, 398, 399, 405. 
 
 Reading : See ' Import.' 
 
 Real Actuality, R. Fact : 138, 367. R. 
 Assertorial Hypothetical : 140, 141. 
 R. (or Scientific) Classification : 57- 
 64, 66, 82. 136, 341 (see ' C). R. (or 
 S.) Connotation : 70-72. R. Context : 
 120. R. (or S.) Definition : 23, 29-31, 
 34. 01-64, 214, 309, 316 (ftn.), 341. 
 R. Denotation: 71. R. Discussion: 
 307. R. Disposition : 129. R. (or 
 Scientific) Division : 30, 42, 56-57 
 (ftn.), 316 (ftn.). R. Evidence: 291, 
 3U3, 307-309. R. Ground : 321. R. 
 Induction: 316 (ftn.), (see 'I.'). R. 
 (or Scientific) Inference : 300-310, 
 321. R. Interest : 114. R. Judg- 
 ment, R. Proposition : 114, 115. 119,
 
 490 
 
 THE PROBLEM OF LOGIC 
 
 140. 141. R. Logic : 5-7, 9, 157, 304, 
 305. R. Objects : 72. R. Possibility : 
 1-27-129. 131 (ftn.). 13S. R. Proof: 
 291. R. Reference to Reality : 140, 
 307-309. R. Siuffular Proposition : 
 114. 115. R. World : 119 {see 
 'Reality'). 
 
 Realism : G. 77. 
 
 Realistic Dcftuition : 24, 25. 
 
 Reality : ix, 122 (ftn.). 145. 307, 30S, 
 3S5. 410. R. as a Whole : 119. R. as 
 couceiyed by Common Sense : 4, 5, 9. 
 R. as Intelligible: 99, 191 {see 
 • Intelligibility of Experience '). R. 
 ConyeationaUy (or formally) Con- 
 ceiyed : iSee ' formal Reference . . .' 
 R. in relation to Delinition : 23-25 ; 
 to Formal Logic: 0-10, 214 {see 
 ' F. L.') ; to Hypothesis : 99 ; to 
 Interest or Purpose : See ' P.' ; to 
 Personality : 4, 5 ; to Propositions : 
 123 (see ' Import ...'); to Question : 
 114; to Spiritual Experience: 114 
 (ftn.) ; to Thought : See tinder ' T.' ; 
 to Truth: 1, 2. 4, 5, 9. R. of 
 Nature: 114 (ftn.). R. Philosophi- 
 cally conceived: 5, 114 (ftn.). R. 
 Really, Scientifically, Systemati- 
 cally conceiyed : See ' Real Refer- 
 ence ...,'' Science,' ' Scientific Con- 
 ception ...,'' Systematic Concep- 
 tion . . .' 
 
 Reality-Principle : 191. See ' Law of 
 Excluded Middle.' 
 
 Realization of an End : 4.53. R. of Con- 
 cept : 353. R. of Contingency, R. of 
 Possibility : 4G, 379. R. of Ideal : 
 419, 420. R. of Problematic Pro- 
 perties : 27. 2S. 
 
 Reason : 6. 13. 75. 97. 104-106, 124, 127, 
 287, 28S, 3G7. 3G9, 393, 448, 450. 
 See ' Sufficient R.' 
 
 Reasoning : 7. 96, 133, 157, 2.50, 287- 
 289. 299-309, 322-327, 337, 347-349, 
 359, 301, 369, 413, 41G, 417, 438, 443, 
 444 {see ' Inference,' ' Parity of R.,' 
 ' Practical R.,' ' Ratiocination,' 
 'Scientific R.'). R. as Continuous: 
 106. R. from Particulars to P. : 157. 
 R. Tested by Rules : 107. 
 
 Rebuttal of Dilemma : 274-277, 293, 
 294. 
 
 Reciprocity {or Reversibility) of Causal 
 Connexion: 379-384, 389, 40f). 401, 
 406. 420, 427, 449. See ' Unideter- 
 minism.' 
 
 Recfjn.struction : (ij. 82, 119, 121. 343, 
 432 {see ' Modification of Hypo- 
 thesis '). R. of Definitions : 31, 
 36-39. 
 
 Reduciio nd Ahsurdum : 122 (ftn.), 247, 
 248. 324. 325. li. per Impoasibile : 
 247, '248, 2.30, 324, 325, 
 
 Reduction of Artruments into Syllo- 
 gistic Form : 230-235, 250, 251, 267, 
 2GS, 284, 286. R. of Categorical Pro- 
 positions to Strict Logical Form : 161- 
 171. R. of Certain Ambiguous Ex- 
 pressions : 164, 165. R. of Exceptive 
 and Exclusive Propositions : 166, 
 167. R. of Given Sentences to Strict 
 Logical Form : lGS-171. R. of Hypo- 
 theticals to Categorical Form, and 
 vice versa : 138, 139. R. of Non-propo- 
 sitional Sentences to Strictly Logical 
 Form : 1G7, 168. R. of Propositions 
 to L. F. : 116, 117, 139, 154. R. of 
 Syllogisms : 247-250. 
 
 Reference : See ' Application of Mean- 
 ings,' 'Extensive R.,' 'Indication 
 ...,'' Objective R.' R. to Fact : 94, 
 309. R. to Reality : See ' Reality.' 
 
 Reflection : 29, 82, 84, 105, 112. 
 
 Refutation: 276, 293, 399, 421, 422 
 (.sec ' Disproof '). R. of Hypothesis: 
 334. 
 
 Regress, Regressive Process : ^ee ' Infi- 
 nite Regress.' 
 
 Regressive Analysis : 325. R. Reason- 
 ing : 2G0, 261 {see ' Epicheirema '). 
 R. Sorites : 255-260. 
 
 Regularity : See ' Order,' 'Uniformity.' 
 
 Regulative Function of Logic : 13, 16, 
 17, 155. Of the Maxim of Identity : 
 97. Of the Principle of Non-Ambi- 
 guity : 39. R. Idea, R. Principle : 
 387, 451, 452. 
 
 Rejection : See ' Exclusion.' R. (of 
 Statement): 98, 99, 130, 172-178, 
 180, 182, 187, 188, 194, 195, 204, 205, 
 236, 262, 264-266, 269. R. of Hypo- 
 thesis : 334, 430-432. R. of the Ante- 
 cedent : 264, 265, 267-269, 271, 285. 
 R. of the Consequent : 264, 265, 268, 
 271, 385. R. of the Indcfimte Term : 
 48. 
 
 Reating Act, R. Activity : 125. See 
 Copula.' 
 
 Re 'lua: 31. 86, 125, 340, 341. R. 
 between Gravity and Electricity : 
 335, 337. R. b. Light and Magnet- 
 ism : 335. R. of Aflirmation and 
 Negation : 123, 124 {see ' A.,' ' N.'). 
 R. of Attribute to Substance : 40. 
 Pv. of Belief and Disbelief : 123. R. 
 of Cause and Effect : See ' Causal R,,' 
 'Causation.' R. of Concept to 
 Logical Interest : 78. R. of Connota- 
 tion to Denotation : 71-73, R. of 
 Disjunctive to Categorical: vii, 112- 
 114, 131. R. of Division to Defi- 
 nition : See under ' Division.' R. of 
 Ends and Means : 104, 128. R. of 
 Extension to Connotation : 71, 72. 
 R. of Formal to Verbal Fallacy : 2S6. 
 R. of Genus and Dillerentia : 50. H.
 
 INDEX, VERBAL AND ANALYTIC 
 
 491 
 
 of Genus and Species : 40-50. R. of 
 Hypothetical tu Categorical : vii, 138, 
 139, 141, 208 (ftn.). R. of Hypo- 
 thetical to Disjunctive: vii, 113, 114, 
 138. R. of I and O : 150 (see ' Sub- 
 contrarietv '). R. of Identity: See 
 ' I.' R ; of 'identity to Difference : 90- 
 98 {see ' I. in D.'). R. of Ignorance to 
 Knowledge : 77, 79. R. of Inference 
 to Hypothetical Form : 145, 140. R. 
 of Logic to Language : See under 
 'Logic' R. of Part to Whole: 40. 
 R. of Past, Present, and Future : 85. 
 R. of Possibilities to their Categorical 
 Basis : 131. R. of Proposition to 
 Judgment : 93. R. of Question to 
 Reality: 114. R. of 'Some' to 'All ': 
 155-158. R. of Statement to Fact : 99 
 {see ' S. as bearing on F.'). R. of Sub- 
 ject to Predicate: See 'Predicate,' 
 ' Subject,' ' S.-P. Relation.' R. of 
 Subject to Suppositio: 118-120. R. 
 of Term to Proposition : 111, 112. R. 
 of the Dictum de Omni et yuUo to 
 Deduction : 244-247. R. of the Ele- 
 ments {or Terms) of a Proposition {or 
 Statement): 94, 111, 112, 150 (see 
 ' Propositional Elements'). R. of 
 the Law of E.xcluded Middle to Time : 
 100-102. R. of the Principle of Iden- 
 tity to Inference : 90 ; to Judgment 
 or Proposition : 90, 104. R. of the 
 Principle of Non-Contradiction to 
 that of Non-Ambiguity : 102, 103. 
 R. of Thought to Interest: 95, 90 
 {see ' Purpose in T.'). R. of Truth to 
 Space : 100, 101 ; to Time : 100-102. 
 R. of Working Idea to W. Hypo- 
 thesis : 328. 
 
 Relational Definition : 30, 31, 35, 37, 
 83 
 
 Relations of Classes : 152, 153, 239, 240. 
 R. of Class-names : 08, 09. R. of 
 Concepts, R. of Meanings : 30, 31, 35, 
 89, 120 {sec ' Conceptual Ordering 
 .. .,' 'Opposition'). R. of Facts: 453. 
 R. of Natural Kinds : 00, 01. R. of 
 Opposition : 48 {see ' O.'). R. of Pro- 
 positions, R. of Statements : 94, 95, 
 123 {see ' Opposition . . .'). R. of 
 Uniformities : 309. 
 
 Relative Necessity : 387. 
 
 Relatives : 251. 
 
 Relativity to Purpose : See ' P.' 
 
 Relevance (to Purpose) : See ' P.,' 
 'Relevant ..." R. to Interest: See 
 ' I.' R. of Predicate to Subject : 28. 
 
 Relevant Circumstance : 390, 39S, 403, 
 420. R. Condition: 370, 380, 383, 
 404. R. Context : 118-120. R. Ex- 
 perience : 455 (ftn.). R. Fact: ix, 
 1-0. 9, 314, 310-31S, 320, 347, 350, 
 307. 389-391, 395, 420, 422, 449. 
 
 R. Instance : 391. R. Objection ; 421. 
 R. Test-instanco : 400, 407. 
 
 Religion : 84, 107, 401, 402. 
 
 Religious Idealism : ix. 
 
 Remote Causal Antecedent : 427 (see 
 ' Immediacy in C. Connexion '). R. 
 Effect : 381, 382. 
 
 Rendering : <S'cc ' Import.' 
 
 Representation by Diagrams : See ' D.' 
 
 Representative Test-instances : 400, 
 407. 
 
 Requirement of Relevance : See ' Rele- 
 vance.' 
 
 Resemblance {or Likeness) : 18, 19, 24, 
 28, 57-00, 04, 245, 299, 327, 358-304, 
 441, 451. See ' Agreement,' ' Es- 
 sential R.' 
 
 Residual Phenomena, Residues: 411- 
 413, 417, 418. 
 
 Resourceful Hypothesis : 32S. 
 
 Retort : 288. See ' Rebuttal . . .' 
 
 Reversibility : See ' Reciprocity . . .' 
 
 Reversible Premisses : 325. 
 
 Revision of Hypothesis : See ' Modifica- 
 tion of H.' R. of Working Idea : 
 337. 
 
 Rhetoric : 13. 
 
 Rhetorical Hypothetical : 180. R. 
 Question : 108, 170. 
 
 Right Feelmg: 118. R. {or Correct) 
 Thinking : 8, 9, 13, 10, 17, 27, 118, 
 
 204. R. Use of Words : 10-18. 
 Rigidity of the Earth : 422, 435-444. 
 Rival Hypotheses, R. Theories : 336, 
 
 420-422, 427-429. 
 
 Romanes, George John : 354. 
 
 Ross, G. R. T. : 133, 134. 
 
 Rossetti, Dante Gabriel : 451. 
 
 Rovce, Josiah, Pli.D. : 303 (ftn.). 
 
 Rule : 103, 107. R. for Inference : 350. 
 R. for reducing Exclusive Proposi- 
 tions to strict Logical Form : 100. 
 R. of Contraposition : 198. 
 
 Rules for Distribution of Terms : 150. 
 R. for reducing Categorical Propo- 
 sitions to strict Logical Form : 102. 
 R. for r. Indcsignate P. : 105. R. of 
 ' Aristotelian ' Sorites : 257. 258. R. 
 of Contradictory Opposition : 173, 
 174. R. of Contrary O. : 176. R. of 
 Conversion : 193-190, 205. R. of 
 Definition : 32-30. R. of Disjunction : 
 137, 138. R. of Goclenian Sorites : 
 258. R. of Logical Division : 44-40, 
 52-50. R. of Mixed H^TJOthetical 
 Syllogism : 204. R. of Opposition : 
 203. R. of Statistical Investigation : 
 353. R. of Subalternation : 170, 177, 
 188. R. of Subcontrariety : 174, 175, 
 
 205. R. of (S. C. D.) Syllogism : 107, 
 21()-225, 2:30, 233, 234, 237, 238, 243, 
 244, 247, 280, 305 {see ' Distribution 
 R ...,'' Quality R . . .,' ' Quantity
 
 492 
 
 THE PROBLEM OF LOGIC 
 
 R . . .," Structural R . . .'). R. of the 
 (Syllogistic) Figures : 229, 235, 237, 
 238, 243. 244. 
 
 S as a Substitute Sign : 14. S. C. D. 
 Svllogism : Sec ' Simple Catestorical 
 Distributional S.' ' S can be P ' : 12S. 
 120. ' S cannot be P," ' S may be P,' 
 ' S must be P,' ' S must not be P,' ' S 
 need not be P ' : 12S. ' S is a P : ' 1G2. 
 • S is assuredly P.' etc. : 130. ' S is 
 either P or non-P ' : 99 (ftn.). ' S is 
 not P ' : 49, etc. ' S is not-P ' : 49, 
 124, etc. ' S is P ' : 14, 22, 98-100, 
 1 11 12 0. 126, 130, 161, 173, 174, 320. 
 ' S is P ' : 98, 99. 173, 174. ' S is-not 
 P.' : 100. Ill, 124, 173, 174. 
 
 2. 2 : 9S, 99. 
 
 SaUus in Divisione : 44, 45, 52, 54-56. 
 
 SaP : 147, 201-203, 206, 207. 
 
 Saturn : 245, 331. 
 
 Savage Customs : 451. 
 
 Scheme of Enumerative Induction : 
 351, 352. S. of Extensive Proposi- 
 tions : 147, 148. S. of ' Inductions ' : 
 347. 
 
 Schemes of Categorical Propositions : 
 See ' Eightfold ...,'' Fourfold . . .,' 
 ' Threefold ..." S. of Opposition : 
 128, 130, 156, 178, 179. 
 
 SchiUer. F. C. S. (Dr.) : viii. 
 
 Scholasticism : 97, 123, 244, 248, 307, 
 315, 316, 3S2. 
 
 Schwann : 424-428. 
 
 Science : 2-6. 9, 16, 23, 29-31, 46, 62-67, 
 79. 114. 115, 119, 133, 136, 244, 274, 
 287, 288, 295, 304-309, 313-318. 321, 
 328, 331, 333, 334, 337-339. 342. 347, 
 3.52, 353, 3.55, 361, 362, 367-387, 393, 
 394. 404, 409, 410, 413, 420, 424, 426, 
 429, 433-440. 447-455, 458, 4.59, 462 
 (see ' Abstract S.,' ' Anatomy,' ' An- 
 thropology,' ' Archaeology,' ' Bio- 
 logy,' Botany,' 'Classification,' 
 ' Economics,' ' Logic as a S.,' ' Mathe- 
 matics.' ' Meteorology,' ' Moral S.,' 
 ' Morphology,' ' (Ecology,' ' Palajon- 
 tology.' ' Pathology,' ' Physical S.' 
 ' Physiology ' ' Scientific . . . ' 
 ' Seismometry,' ' Sociology,' ' Zoo- 
 logy '). S. and the Pre<Jicable3 : 22. 
 S. in relation to Common Sense : 4, 5, 
 347 ; to Philosophy : 4-0. 
 
 Scientific . . . : See ' Real . . .' S. 
 Analysis: 355, 431 (.see 'A.'). S. 
 Attitude, S. Intent, S. Interest, S. 
 Purpose, S. Standpoint : See ' Rele- 
 vant Fact,' ' Science.' S. Certainty : 
 381 (see ' C). S. Conception of 
 Truth : 3-6, 9, 141. S. Experiment : 
 See ' E.' S. Explanation : See ' E.' 
 S. Hj-pothesis : 328, 361, 302 (see 
 ' H.'). S. Imagination : See ' I.' S. 
 
 Induction : 316 (ftn.), 356, 357. 359. 
 361-364, 367, 456. 457 (see ' I.'). S. 
 Investigation, S. Knowledge, S. Pro- 
 cedure, S. Research : *See ' Science.' 
 S. Law : 357, 367, 368, 377 (see ' L. of 
 Nature '). S. Logic : 5, 6. S. Method : 
 119 (ftn.), 137. 304, 321, 347, 348. 
 410, 451, 453. S. Nomenclature : See 
 ' N.' S. Observation: 114, 136, 355, 
 389-394 (see ' O.'). S. O. as essentially 
 Experimental: 389, 390, 393, 394; 
 as e. Purposive: 390-393. S. Pre- 
 cision : 16, 66, 135, 137. 386. 400, 404. 
 427 (see 'P.'). S. Principle : 63, 347. 
 S. Reasoning : 157. 287. 307-309. S. 
 System : 7, 8, 304, 317-320, 328, 336, 
 424, 426, 430. S. Technology : 16, 66- 
 70. S. Terminology : <S'ce ' T.' S. 
 Testing: 140 (sec 'Verification'). 
 S. Theory: 304, 447 (see ' T.'). S. 
 Thought : 390 (see ' Science '). S. 
 Value of Analogy : 361, 362. S. View 
 of Causation : 383. Sec ' C 
 
 Scripture, E. W. (Prof.) : 253. 
 
 Secondary Quantification : 181. 
 
 Seismometry : 440-443. 
 
 Selection : 314, 406 (ftn.). See ' Natural 
 S.,' ' Selective. . . .' 
 
 Selective Function of Thought : 2, 3. 
 S. Interest : 51, 58, 65, 81, 118, 125, 
 390, 391. See ' I.' 
 
 Self: 208. 
 
 Self-affirmation : 75. See ' Self-evi- 
 dence.' 
 
 Self-consciousness : 28, 368, 458, 459, 
 461, 462. 
 
 Self-consistency : 7, 8, 93, 98. See 
 ' Consistency '). 
 
 Self-contradiction : 7-9, 75, 76, 78, 94, 
 102-104, 122 (ftn.). 126, 128, 150, 188, 
 208, 2S2. 307, 321, 448, 450. See 
 ' Contradiction.' 
 
 Self-definability : 75. 
 
 Self-definition : 75, 76, 80, 81, 84. 
 
 Self-evidence: 75, 76, 82, 84, 85, 99. 
 113, 323 (ftn.). 
 
 Self-identity : 96. 
 
 Self-knowledge : 3, 4, 82. 
 
 Self-limitation : 98, 4.50. 
 
 Self-reference of Thought or Meaning : 
 10. 
 
 Semi-correlative Terms : 35, 36. 
 
 Seneca on Division : 40 (ftn.). 
 
 Sensation : 82-84, 377 (ftn.), 390, 393. 
 
 Sense : 85, 335, 339, 390, 392, 393, 441, 
 450. See ' Common S.,' ' Meaning.' 
 
 Sense-data : 393. 
 
 Sense-experience : 4, 82-84, 87, 377, 
 419. 
 
 Sense-immediacy : 85. See ' Immediate 
 Experience.' 
 
 Sense-individuality: 114. 
 
 Sense-observation ; 299.
 
 INDEX, VERBAL AXD ANALYTIC 
 
 493 
 
 Senso-porception : 453. See ' P.' 
 
 Sense-world : 347. 
 
 Sensible Fact : 94. 
 
 Sensory Support for I\rcaning : 24, 93 
 {see ' Words '). S. Symbolism : 93. 
 
 Sentence: 110, US, \-)4, 102, 103, 1G7, 
 168, 28.5. S. distinguished from Pro- 
 position : 93, 94. 
 
 SeP : 147, 201, 204. 
 
 Sequence : 342. 3.57, 368, 369. 371, 377- 
 379, 382, 384, 455. See ' Cons.' 
 
 Sequence-uniformities : 3G9. 
 
 Series: 61 (ftn.), 69. 
 
 ' Several ' : 168. 
 
 Shaded Diap;rani3 : 150, 151, 1.53. 
 
 Sidgwick, Alfred : viii, 17. 18. 289. 381. 
 
 Sign: 370 (ftn.). See 'Signs.' 'Sym- 
 bol.' ' Verbal Sign.' 
 
 Significance : Sec ' Import,' ' Meaning.' 
 
 Significant General Expression : 80. S. 
 Singular Expression, Name. Term : 
 80, '^Sl. S. Statement: 157. 
 
 Significate : 25, 115. 
 
 Signification : 80 (ftn.), 115. See ' Im- 
 port,' ' Meaning.' 
 
 Signs as Substitutes for Meanings : 14. 
 S. Expressive of Meaning : 14, 23-25. 
 32, 93 {see ' Words '). S. Indicating 
 Objects : 82 {see ' Indication . . .'). 
 Their Definability : 25. 
 
 Sigwart, Christoph (Dr.) : viii, 126, 323 
 (ftn.), 341. 342, 355, 382, 386, 455. 
 456, 458 (ftn.), 459, 460. 
 
 Simple Antecedent : 408. S. Categorical 
 Distributional Syllogism : 216-238, 
 251. S. C. Syllogism: 211-251, 324, 
 325. S. Converse, S. Conversion : 195, 
 198, 237, 249, 259. S. Conversion of 
 Premisses : 325. S. Dilemma : 272, 
 273, 276, 277. S. Enumeration : See 
 ' E.' S. Law : 409. S. Negation : 124 
 {see ' Pure N.'). 
 
 Simplification : 23, 339, 341, 343, 344, 
 419, 437, 441, 443. 
 
 Simultaneity of Cause and Effect : 382. 
 
 Simultaneous F. D.'s : 45. 
 
 Single Agreement : 395-399, 413-417, 
 422. S. Difference: 407, 416, 426. 
 S. Epicheirema: 261. S. Object: 81 
 (see ' Individual '). 
 
 Singleness of F. D. : See ' Unity of F. D.' 
 
 Singular Collective Term : 16'2. S. 
 Concept : 80, 81. S. Form of Divi- 
 dendum : 53. S. Import : 162. S. 
 Meaning, S. Name, S. Term : 74, 79- 
 82, 114. 236 (.see ' Proper N.'s'). S. 
 Proposition: 114, 115. 154. 161-163. 
 168. 173; its Opposition: 181, '200. 
 236, 299 (ftn.). S. Svmbol : 80 (ftu.). 
 
 SiP: 147, 201, 204, 207. 
 
 Situation : 31. 
 
 Smith, Svdnev (Canon) : 283. 
 
 Sociology : 409, 410. 
 
 Socrates : 47, 79, 80, 142, 155. 173, 174, 
 236, 247, 300.302, .340, 341. 
 
 Socratic Method : 316 (ftn.). 
 
 'Some' : 1.32 (ftn.). 152-166, 168, 175- 
 177, 241-243, 280 {see 'Collective 
 Meaning,' ' Conjunctive use . . .,' 
 ' Distributive Moaning,' ' Exclusive 
 use . . .,' 'Particular Judgment'). 
 ' S. A's are either B or C ' : 132. ' S 
 A's are neither B nor C ' : 1.32. ' S.' 
 in relation to ' All' : 155-158. ' S. 
 only ' : 166 (see ' Exclusi>e use . . .'). 
 ' S. S is all P ' : 159 {ste ' Y Pro- 
 position '). ' S. S is not any P ' : 1.59. 
 ' S. S is not some P ' : 159 (see ' w 
 Proposition '). ' S. S is some P ' : 159. 
 ' S. S is-not P ' : 242, 243. ' S. S's 
 are all P's ' : 149, 151 {see ' Y Pro- 
 position '). ' S. S's are P's ' : 147, 149. 
 156, 1.59, 161 (see ' I Proposition '). 
 'S. S's are-not P's: 147. 150, 156, 
 159. 6'ee ' O Proposition.' 
 
 ' Something is ' : 75, 76. 
 
 SoP : 147, 201. 
 
 Sorites: 253-260, 270, 271. 319, .320, 
 325. See ' " Aristolelian " S..' 
 
 ' Goclenian S.,' ' Inverted S.,' ' Stan- 
 dard S.' 
 
 Sound : 374, 388. 
 
 Sound Analogy : 360, 364. S. Defi- 
 nition : 32-36. S. Division : 52. S. 
 Hypothesis : 361. S. Inference : 141, 
 191. 213. 214. See ' Validity.' 
 
 Sound-complex : 21. 81, 115. 
 
 Sounds as expressing Meaning : 14. 
 
 Space: 25, 26, 85, 87, 119 (ftn.), 322, 
 333, 369, 374, 376, 440, 460. S. in 
 relation to Truth : 100, 101. 
 
 Space-intuition : 322. 
 
 Spatial Conditions. S. Relations : 100 
 (see ' Space '). S. Construction : 321, 
 322. S. Form : 89. S. Grouping : 65. 
 66. S. Object : 128. 
 
 Special Rules : See ' R. of " Aristo- 
 telian " Sorites,' ' R. of Goclenian 
 Sorites,' ' R. of the (Syllogistic) 
 Figures.' 
 
 Specialization of a Word's Meaning : 14, 
 15. 
 
 Species (Biological) : See ' Animal.' 
 'Natural Selection,' 'Plant,' S. 
 (Ix)gical) : 19-23, 30, 32, 39-57, 66, 68, 
 72-75, 77, 79, 88, 89, 114, 120, 122 
 (ftn.), 136. 159, 244 (see ' Infima 
 S.'); its Defmition : 30, 35. 
 
 Specific Character : 385. S. Definition- 
 mark : 20, 29. 34 {see ' Differentia '). 
 S. Determination : 131 {see D.). S. 
 Difference : iS'ee ' Differentia." S. 
 Emphasis : 1.34. S. F. D. : Sec ' Unity 
 of F. D.' S. Hj-pothesis: 447. S. 
 Induction : 457. S. Names : 68. S. 
 (or Specified) Possibility : 133, 138.
 
 494 
 
 THE PROBLEM OF LOGIC 
 
 Specification: 90, 121, 12-4. 133, 400 
 (sec ' Differentiation of Meanings.' 
 ' Disjunctive S.'). S. of Cause : 38-i. 
 S. of Interest : 124. 
 
 Specifj-ing Mark : G-1. See ' Differentia.' 
 
 Specimen : G4-G6. 
 
 Spectroscopy : 432. 433. 435, 449. 
 
 Speech: 13, 14. 97, IIG. See 'Lan- 
 guage,' ' Words.' 
 
 Spencer. Herbert : 449. 
 
 Spiritual Being : 120. S. Experience : 
 107, 114 (ftn.), 453. S. Imperative: 
 127. S. Interest : 114 (ftn.). S. Order : 
 33. S. Purpose : 453. S. Structure : 
 84. S. Unity : 6, 25. S. Values : 61 
 (ftn.). S. World : 105. 
 
 Squares of Opposition : 128, 130, 172. 
 
 ' S's alone are P's ' : 166. 
 
 Stages in Answering a Question: 112, 
 113. S. in Causal Inquiry, C. Method, 
 Complete Induction, Inductive Pro- 
 cedure. Scientific Explanation : 326, 
 327, 361. 395, 398, 400, 407, 411. S. 
 in Development of Meaning : 71, 88, 
 89 (see ' D. of M.'). S. (or Steps) in 
 Division : See ' Continued D.' S. in 
 Logic, S. in Thought : 5-7, 9, 24, 25, 
 30. S. (or Steps) in MiU's Deductive 
 Method : 409, 419 ; in M.'s Hypo- 
 thetical Method : 411, 419. S. in 
 Organic Development : 400, 461. 
 
 Standard : 103, 347, 362 (see ' Inductive 
 Criterion '). ' S.' Figure : 257, 325 (see 
 ' Fig. I.'). S. Sorites: 270, 271, 319, 
 320, 325. 
 
 Standards in Dilemma: 275-277, 293, 
 294. 
 
 Star : 38, 69, 70, 234, 329-333, 339, 362, 
 386, 4.30-435, 437, 449. 
 
 State : 31, 379. 
 
 'Statedly': 132 (ftn.), 1.55-158, 262, 
 263. 
 
 Statement: 94, 97-99, 102, 109, 113, 
 117-124, 129-i:}7. 140, 141, 146, 148- 
 158, 160-162. 166-168, 170, 239. 301, 
 310, 321, 370 (ftn.), 386 (see ' Accep- 
 tance of S.,' 'Affirmation,' 'Asser- 
 tion,' '' Circulus . . .,' 'Falsehood,' 
 ' Negation,' ' Proposition,' ' Rejec- 
 tion (of S.),' 'Tautology,' 'Truth in 
 relation to Judgment,' ' State- 
 ments '). S. as bearing on Fact : 94, 
 96, 99, 138, 1.39, 309, .392. 439. S. as 
 Invalid : 8 (see ' Invalidity '). S. as 
 Self-consistent : 7, 8. S. (or Pro- 
 position as Self-contra^lictory : 103, 
 104. 122 (ftn.). S. as Valid : 8 (see 
 ' Validity '). S. of Hvpothc-jis : 421, 
 422, 442 (see 'Formulation of H.'). 
 S. of Limitation : 1.58. 
 
 Statement-import: 7, 94, 95, 99, 123, 
 132 (ftn.), 14.5, 146, 1.56-158, 240, 
 241, 262 (Note). 
 
 Statements as Consistent : See ' Con- 
 sistency.' S. as Contradictory : See 
 ' C. S.' S. as Inconsistent: See 'In- 
 consistency.' S. as Inter consistent: 
 7, 8 (see ' Consistency '). 
 
 Static Continuity : 44. 
 
 Statical Identity : 21, 96, 97. 
 
 Statistical Unit : 43. 
 
 Statistics : 43, 313, 353-355. 
 
 Status of the Dicta : 241-244. 
 
 Steps : See ' Stages.' 
 
 Stock, St. George, M.A. : viii, 47 (ftn.), 
 152 272. 
 
 Stout,' G. F. (Prof.) : vii, 14, 15, 82, 100, 
 102-104, 118, 131, 245, 268 (ftn.), 
 295, 308, 349 (ftn.), 385. 
 
 Strengthened Forms of Syllogism : 226, 
 227, 237, 268. 
 
 Stress : See ' Emphasis.' 
 
 Strict Logical Form : See ' L. F.' 
 
 Strong Converse : 195, 198. S. Educt : 
 189, 192, 195, 200, 204. S. Forms of 
 Syllogism : 226, 247. 
 
 Structural Rules of the Syllogism : 217, 
 218. S. Significance: 146 (see 'Im- 
 port'). 
 
 Structure: 57, 59-64, 67, 74, 75, 363, 
 401, 402. Sec ' Ambiguous S.,' 
 'Grammatical S.,' 'Spiritual S.,' 
 ' Verbal S.' 
 
 Strudwick, Miss Florence : viii. 
 
 Stuart: 390, 391. 
 
 Study of Nature : Sec ' Science.' 
 
 Sturt, Henry : 387. 
 
 Subaltern : 128, 130, 171, 172 (see 
 ' S.ation '). S. Forms of Syllogism : 
 226. S. Genus: 42, 43, 73, 74, 80 
 (see ' Continued Division '). 
 
 Subalternans : 172, 265. 
 
 Subalternate : 128, 172, 265. 
 
 Subalternation : 128, 130, 171, 172, 176, 
 177, 188. 
 
 Sub- class : 39, 59. See ' Continued 
 Division.' 
 
 Subcontrariety : 171, 172, 174-176. 
 
 Sub-differentiation, Subdivision : 52, 
 55, 56. See ' Continued Divi- 
 sion.' 
 
 Subduction : 411. 
 
 Siib-fundamcntum : 45. 
 
 Subject: 129, 208, 400 (see 'Gram- 
 matical S.,' ' Logical S.'). S. and 
 Predicate in the Categorical Pro- 
 position : 115-122 (see ' S. -Predicate 
 Relation'). S. of Discourse: See 
 ' formal S. of D.,' ' Suppositio.' S. 
 Term: 22, 23, 28, 31, 96, 103, 104, 
 106, 111, 112, 114-122, 124-126, 128, 
 1.32, 146-152. 1.54, 1.59-162, 165, 166. 
 168, 170, 320; its Relation to the 
 Suppositio : 118- 120. 
 
 Subject-concept: 116. Sie '.Subject 
 Term.'
 
 IXDEX, VERBAL AND ANALYTIC 
 
 495 
 
 Subjoctivo Aspect of Cleaning : 23-2j 
 {see ' Subject-Object Relation '). S. A. 
 of Truth : 45. S. Association : 17 (5ce 
 'A.'). S. Certainty : 157. S. Inten- 
 sion, S. Meaning : 70, 71. 81. S. 
 Intent : 23, 24, 27. S. Interest : 23- 
 27. S. Possibility : 130. S. Purpose : 
 See ' P.' S. Reference to Reality : 
 309. (see ' formal R. to R '). S. 
 Sj-steniatization : 318. 
 
 Subject-matter : 125, 447. 454. 
 
 Subject-Object Relation : 4-6. 23, 27, 
 51, 77, 78. 82, 94, 97 (ftn.). 
 
 Subject-Predicate Relation : 28, 31, Ofi, 
 112, IIG, 118, 120, 124-126, 159. 239. 
 See ' Predicate Term,' ' Subject T.' 
 
 Subject- thing : 116. 
 
 Sub-Kingdom : 57, 74. 
 
 Sublation : 262, 264. See ' Rejection.' 
 
 Sub-order : 59. 
 
 Subordination of Characters : 58, 59. 
 
 Sub-species : <See ' Sub-differentiation.' 
 
 Substance : 31, 361, 368, 369, 459. S. 
 as Summum Genus : 47 (ftn.). 
 
 Substantial Explanation : 368. 
 
 Substitute Sign : 14. 
 
 Sub-variety : 42. 
 
 Sufficient JReason : 378. 
 
 Suggestion : 28. 71. 81, 105, 113, 157. 
 
 Sum Total {or Totality) of Antecedents 
 or Conditions : 370-381, 383. 
 
 Summum Genus : 42-45, 47, 72-79, 82, 
 88, 95 {see ' Classification,' ' Con- 
 tinued Division '). Its Definability : 
 73-79, 89. 
 
 Sun : 34, 38. 134, 139, 234, 316, 329-333, 
 338, 342, 343, 355, 362, 376, 377, 380, 
 393, 394, 412. 431, 434. 
 
 Superfluity in Definition : 33, 38. 
 
 Superinduction : 350, 453. 
 
 Superordinate Genus : 74, 75, 78. 
 
 Supplementary Causal Methods : 405- 
 407. 
 
 Supposiiio {or Topic or Subject of Dis- 
 course or Universe of D.) : 49, 54, 95, 
 140 (ftn.), 190-192, 328, 450. Its 
 Development: 88, 89, 119, 328. Its 
 relation to the Subject of a Propo- 
 sition : 118-120. 
 
 Supposition : 94, 136, 327, 328, 337, 
 338, 378, 4(X), 406, 455, 456. 459. 
 See ' Hypothesis.' 
 
 Suppositional Use of the Indefinite 
 Term : 49-52. 
 
 Supreme Idea : 452. 
 
 Surmise : 395. -See ' Conjecture,' ' Sup- 
 position.' 
 
 Surreptitious Assumption : 289, 290. 
 S. Conclusion : 291. 
 
 Suspended Hj-pothesis : 361. 
 
 Suspense of Judgment : 99, 123. 
 
 Syllogism: 7, 107. 211-277, 285, 2S6, 
 291. 299-305, 321, 324. 325. 327 
 
 (ftn.). See 'Complex Categorical S,' 
 'Dilemma," Disjunctives.,' 'Epichei- 
 rema,' ' Hvpothetical S.,' ' Mixed 
 H. S.,' 'Pure H. S..' 'Rules of S.,' 
 ' Simple Categorical Distributional 
 S.,' ' Simple Categorical S.,' 'Sorites,' 
 ' Unorthodox Ss.' 
 
 Syllogistic Figure: 215, 222, 224-2.30, 
 232-238. 240-244, 246-250, 255-200. 
 269, 324, 325 {see ' Fig. I.,' etc., 'Rules 
 of the Fs '). S. Form : 221, 230-235, 
 250 {see ' F. of the S.,' ' Fs of S.'). 
 S. Inference: 216 {see 'Syllogism'). 
 S. Mood : 215, 224-226, 228-230, 233, 
 242, 243, 249, 262-264, 208. 
 
 Symbol : 14, 25, 32, 60, 115, 362, 434 
 {see 'Sign'). S. for Unspecified 
 Subject: 163. 
 
 Symbolic Designation of Denial : 98, 
 99. S. Forms of Disjunctive Pro- 
 position : 135. S. Function of the 
 Individual: 115; of Words: 14, 23- 
 25, 32, 93, 115. S. Logic: ix, 251. 
 S. Nomenclature : 69. 70, 201. 
 
 Symbolism : 93. 
 
 Symbolization of A, E. I, O : 147. 201. 
 -See ' SaP,' ' SeP,' ' SiP,' ' SoP.' 
 
 Symptom : 384. 
 
 Synthesis: 30, 31, 61, 122, 125, 343. 
 395, 407. S. in Judgment, S. in 
 Proposition : 93, 111. 122, 123. 125, 
 126. 
 
 Synthetic Copula-mark : 124 (ftn.). S' 
 Function of the Copula : 126. ' S.. 
 Judgment, ' S.' Proposition : 120-123. 
 S. Principle : 140. 
 
 System : 245, 246, 304, 305, 308. 313, 
 317-320, 328-332, 336, 337, 340, 362, 
 369, 372, 375, 386, 387, 408, 424, 426, 
 428, 430, 432, 435, 444, 459, 461 (see 
 ' Copemican S.,' ' Ptolemaic Astro- 
 nomy,' ' Scientific S.'). S. of Causal 
 Connexions : 420. S. of Classification : 
 See ' C S. of Concepts. S. of Mean- 
 ings : -See ' Conceptual Ordering . . .' 
 S. of History : 119. S. of Hypotheses : 
 447. S. of Inferences : 96. S. of Judg- 
 ments : 124. S. of Linnaeus : 63, 64. 
 S. of Nature: 61, 119 (see 'Sys- 
 tematic Conception . . .'). S. of 
 Nomenclature : -S'ce ' N.' S. of 
 Predicablcs : 120 (see ' Pretlicables'). 
 S. of Reality : 308 (see * Systematic 
 Conception . . .'). S. of Spatial Rela- 
 tions : 25, 26, 322. 
 
 Systematic Application of Law : 246. 
 S. (or Organic) Conception of Realitv, 
 etc. : 5. 7, 8. 62, 114. 115, 119. 309, 
 357, 368, 443, 444, 462 (see 'Real 
 Reference to R.'). S. Division : See 
 ' Continued D.' S. Eduction : 200- 
 20S. S. Explanation : 422 (see ' E.'). 
 S. Explication : 95 (see ' Classifica-
 
 49G 
 
 THE PROBLEM OF LOGIC 
 
 tion,' 'Continued Division'). S. 
 Interest: 114. S. Intimacy of Sub- 
 ject and Predicate : 1*20. S. Science : 
 393 (sec ' Science'). 
 Systematization of Conduct: Gl (ftn.). 
 * S. of Experience : 6, 82. S. of Fact : 
 340, 344 (see ' Systematic Concep- 
 tion . . .'). S. of Immediate Ex- 
 periences : 82. S. of Kinds : 67 (see 
 " Classitication '). S. of Knowledge : 
 29, 246. S. of Laws : 342, 369. S. of 
 Science : 317, 31S. S. of the Cate- 
 gories : 31. S. (or Organization) of 
 Thought : 1, 2. 
 
 Table of Opposition : 177. 
 
 Tabulation : 353-355. 
 
 Tautology : 23, 97, 120, 121, 188, 291, 
 304. See ' Circulus . . .' 
 
 Technical Terms : See ' Technology.' 
 
 Technology : 16, 66-70. 
 
 Teloological Definition : 62, 63. T. Ex- 
 planation : 461. T. Investigation : 
 453. T. Law : 104. T. ' I\Iay ' : 128, 
 129. T. Meaning of ' Cause ' : 370. 
 T. Necessity : 128. T. Possibility : 
 128, 129. T. Scheme of Opposition : 
 128. 
 
 Teleology : 96. 97 (ftn.). See ' Toleo- 
 logical . . .' 
 
 Temporal Conditions : lOO. See ' Time.' 
 
 Tendency : 62, 342, 373, 377, 399, 453, 
 454 (ftn.), 457. 
 
 Tense: 101, 163, 164. 
 
 Tense-mark : 163. 
 
 Tentative Analysis : 343. T. Argument : 
 326,327. "f. Conclusion : .362. T. 
 Conjecture, T. Hypothesis : 348, 367. 
 {see 'C' 'H.'). T. Construction: 
 328. T. Explanation : 453. T. 
 Generalization : 350. 
 
 Term : 31, HI, 112, 118. 125, 150, 216, 
 447 (sec ' Abstract Concept,' ' Amlu- 
 guity,' ' Application of Meanings,' 
 * Categorical Assertion,' ' Collective 
 Expression,' ' Concrete Terms,' ' Con- 
 no-denotation,' ' Conno-denotative 
 Ts,' ' Connotation,' ' Definability,' 
 ' Definite Species,' ' Definition,' ' De- 
 notation,' ' Description,' ' Distribu- 
 tion of Terms,' ' Division.' ' Ele- 
 mentary Ts,' ' Extension,' ' Ceneral 
 Concept, ' Indefinite Species,' 'Indica- 
 tion . . ..' '"Infinite T.," ' 'Inten- 
 sion.' ' Negative Subject-t.,' ' Predi- 
 cate T.,' Significant Singular Ex- 
 pression,' ' Singular Collective T.,' 
 ' Singiilar Meaning,' 'Subject T.'). 
 Scientific T. : 70 («ee ' Technology '). 
 T. as Abstract or Deri%'ative : 111, 
 112. T. in Syllogism : 2l7. 218 (see 
 ' Categorical S.. ' Major T..' ' Middle 
 T.,' ' Minor T."}. 
 
 Terminology : 66, 67, 281, 282. 
 
 Terminus : See ' Goal.' 
 
 Terms as Complementary : 48. T. of a 
 Definition : 33, 34, 77. T., their 
 Meaning : 31. Their Opposition : 48. 
 
 Terms-in-relation : 125. Sec ' Subject- 
 Predicate Relation.' 
 
 Test : See ' Criterion,' ' Experimental 
 T.,' ' Verification.' T. of Reasoning ; 
 107. ^ 
 
 Testing : 393-395, 403, 421 (see ' Veri- 
 fication '). T. of Definitions: 32-39. 
 T. of Divisions : 52-56. 
 
 Test-instance : 352, 403. 406, 407, 415, 
 416. 
 
 ' That ' : 75 (ftn.). 82. 356. 357, 369. 
 
 Thatness: 82. 
 
 ' Then ' in Hypothetical Proposition : 
 141. See ' H. Judgment.' 
 
 Theoretical Character of Logic : 13. T. 
 System : 428. 
 
 Theory : 29, 314, 315, 319, 329-.339, 341, 
 378, 386, 391, 394, 409, 410, 418, 420- 
 422, 428-439, 443, 447, 452 (see 
 
 ' Causal T.,' ' Corirascular T ' 
 
 ' Geocentric T.,' ' Heliocentric T.,' 
 ' Hypothesis,' ' Nebular T.,' ' Scien- 
 tific T.,' ' UndulatoryT. . . .'). T. of 
 Categories : 31, 32. T. of Deductive 
 Inference: 385 (see ' D. I.'). T. of 
 Dynamics : 318. T. of Existential 
 Import : 115 (ftn.). T. of Induction : 
 295, 313-327, 377, 379 (see ' I.'). T. 
 of Knowledge : 78. T. of Mechanical 
 Equivalence : 387, 389. T. of Pro- 
 positional Import : 119, 134 (see ' I.'). 
 T. of Recapitulation : 60. T. of Scien- 
 tific Method : 410 (ftn.). T. of the 
 Proposition : 95 (see ' P.'). T. of 
 Vortices : 318, 332, 333, 336. 
 
 Thing : 86-88, 114, 341, 353, 357. 362, 
 392, 449. ' T.' as Summum Genus : 
 47 (ftn.). 
 
 Things classed : »See ' Classing.' Their 
 Definition : 24. Their Division : 40. 
 
 Thinker : 99, 106. 
 
 ' This S is (a) P ' : 161, 162. 
 
 Thomas a Kempis : 284. 
 
 Thought: 112, 113, 139, 208, 390, 391, 
 449, 461 (see ' Scientific T.'). T. and 
 Identity: 97, 98. T. and Intelligi- 
 bility : 78, 79. T. and Self-contra- 
 diction : 78, 79. T. as a Generalizing 
 Process : 339. T. as Consistent : 1, 7- 
 9, 16, 99, 112 (.see ' Consistency '). T. 
 as Continuous : 106 (sec ' Coherence 
 of T.'). T. as Correct, T. as Right : 
 See ' R. Thinking.' T. as Discon- 
 tinuous, T. as Fragmentary : 100. T. 
 as Liconsistent : 99, 106, 139, 140 
 (see 'Inconsistency'). T. aslrreflcc- 
 tive : 28, 29. T. as Obligatory : 107. 
 T. as Refiectivc : 29. T. as True : 1-5,
 
 IXDEX, VERBAL AXD ANALYTIC 
 
 497 
 
 8. 9. T. as Valid : 6-10, 107 (see 
 * Validity '). T. as Vital : 15. T. Con- 
 trolled bv Relevant Fact: 1, 2, 39 
 (see ' R. F.'). T. in Judgment : 93. T. 
 in relation to Areument : 7 ; to In- 
 terest : 95, 96 {see 1.') ; to Language : 
 13-17, 60. 93 ; to Reality : 1-7, 9, 10, 
 23, 24, 31, 99 (see ' formal Reference 
 to R.,' ' Real Reference to R.,' 
 'Systematic Conception of R.') ; to 
 Statement : 7 ; to Universals : 24. 
 Its Abstracting Power : 85. Its De- 
 velopment : 31, 112-114, 138 (see ' D. 
 of Meaning ' ). Its Laws or Principles : 
 <See ' L. of T.'). Its Point of Depar- 
 ture : 83, 84. Its Selective Function : 
 2, 3. Its Unit : 94, 90. T. Regulated 
 by Logic : See under ' L.' T. Syste- 
 matized : 1, 2. T. Unified : 1-3. 
 
 Thought-contradiction : 102, 106. 
 
 Thought-existence : 85. 
 
 Thought-immediacy : 85. 
 
 Thought-process: 113. -See ' Thought.' 
 
 Threefold Scheme of Categorical Pro- 
 positions : 150. 
 
 Three-term Syllogism : 250, 251. See 
 ' Categorical S.' 
 
 Time : 31. 82-85, 87. 119 (ftn.), 127, 129, 
 371, 376, 382, 380, 387, 394, 404. 431- 
 434, 441, 442. 454 (see 'Past,' 
 ' Present '). T. in relation to the Law 
 of Excluded Middle : 100-102. 
 
 Time-element in Prediction : 335. 
 
 Time-experience : 85. 
 
 Time-relations : 102. 
 
 Time-stages in Causal Process : 382. 
 
 Topic : 88, 89, 95, 90, 98, 114, 115, 
 118, 119. See ' Supposilio.' 
 
 Total Antecedent, T. Cause : 372, 373. 
 375, 378-380, 383, 384, 412, 418 (see 
 ' Sum T. . . .'). T. Coincidence, T. 
 Non-c. : 150, 152, 240. T. Eclipse: 
 394. T. Effect : 375, 376, 382, 387, 
 412. T. Exclusion : 153, 240. 242, 
 244. T. Extension-reference : 147, 
 148, 151, 152, 155. 219 (see ' Dis- 
 tributed Term '). T. Impression : 110. 
 T. Inclusion : 152, 153. 240-242, 244. 
 T. Interest, T. Topic: 95, 90. T. 
 Meaning : 98. 
 
 Totality : See ' Sum Total . . .' 
 
 Traditional Logic: 147, 220, 249-251, 
 306 (ftn.), 324. 5ee ' Aristotelian L.' 
 
 Train of Argument. T. of Reasoninor : 
 255. 259, 321. See ' Sorites.' 
 
 Transcendence: 79, 85, 107, 397, 
 462. 
 
 Transference of Meaning : 33. 
 
 Transformation of Energy : 386-389. 
 
 Transit Circle : 413. 
 
 Transition to the Formal Treatment of 
 Logic : 143-140. T. to the Method of 
 Causal Explanation : 389. 
 
 Transitions between Propositional 
 Forms: 112. 113. 
 
 Translation : 17, 32. 
 
 Transposition of Premisses : 237, 248, 
 249. 257, 258. 270, 271. 
 
 Transversion of the Hyix)thetical Pro- 
 position : 138, 139. T. of the H. 
 Syllogism : 268, 209. 
 
 Tree of Concepts, T. of Porphyry : 42, 
 47, 78-80, 82 (see ' Conceptual 
 Ordering . . .'). T. of Self-knowledge : 
 82. 
 
 True . . .: See under ' Truth.' ' T. 
 Analogy ' : 302-364. T. Hypothetical : 
 139 (see ' H. Judgment '). T. I«len- 
 tity : 97. T. Logical Predicate : 6'ee 
 ' L. P.' T. L. Subject : See. ' L. S.' 
 
 Truism: 97, 120, 121, 101. See 
 ' Tautology. ' 
 
 Truth : ix, 304, 321, 348 (ftn.), 351, 361, 
 386, 449, 450, 454, 455, 458, 401 (see 
 ' Necessary T.,' ' Universal T.'). T. 
 as a Problem : 1-4. T. as a Unity : 
 1-3. T. Conventionally Conceived, 
 T. formally Conceived : 5-7, 9, 141. 
 T. Hypothetically Treated : 145, 140. 
 T. Ignored by Formal Logic: 3 (ftn.), 
 7-10, 145, 140, 157, 158, 309, 310, 321. 
 T. in relation to Common Sense : 4, 5 ; 
 to Hypotheticals : 179, 180 ; to Judg- 
 ment, Proposition, Statement : 7, 8, 
 93, 94. 99-104. 122, 123, 132, 137, 140, 
 141, 145, 146, 152, 157, 158, 322-325, 
 406 (see ' Truth-import ') ; to Know- 
 ledge : 3-5 ; to Language, 16 ; to 
 Logic: 1-5, 7-10, 13, 304, 307; to 
 Personality : 3-5 ; to Purpose : 1-3, 5, 
 9 ; to Reality : 1, 2, 4-6, 9 ; to Space : 
 100, 101 ; to Time : 100-102. T. 
 Materially Conceived : 7, 8, 290. T. 
 Objectively Conceived : 4, 5. T. of 
 Assumption : 434. T. of Conclusion : 
 214, 247, 305, 324, 325. T. of Fact : 
 97, 443. T. of Hypothesis, T. of 
 Theory: 318, 328 (ftn.), 336, 337, 
 386, 421, 427, 432, 434, 437, 443. T. 
 of Premisses : 214, 247, 310, 324, 325. 
 T. of the Antecedent, T. of the Conse- 
 quent : 325. T. of the Reason : 97. 
 T. Philosophically Conceived : 3-5, 
 10. T. Provisionally Defined : 1. T. 
 Really Conceived, T. Scientifically 
 C. : 3-6, 9, 141 (see ' Real Refer- 
 ence ...,'' Systematic Concep- 
 tion ...'). T. Subjectively Conceived: 
 4,5. 
 
 Truth-Ideal: 3-5. 
 
 Truth-import : 94, 99, 132 (ftn.). 
 
 Truth-Lifcrcnce : 247, 306-310, 324. 
 
 Truth-Interest : 3-10, 145, 247, 289, 
 300-310, 326, 328. 
 
 Twofold F. D. : See ' Unity of F. D.' 
 
 Tycho-Brahe: 331. 
 
 32
 
 49S 
 
 THE PROBLEM OF LOGIC 
 
 TTOilall. John (Dr.) : 371. 373. 374. 3SS. 
 Tvpe : 57. 01. 62. 64. 65, OS, 130, 341. 
 
 'See ' Definition by T.' 
 Typical Forms of Proposition : See ' A 
 "Proposition.' 'E P..' 'I P..' 'O P.* 
 T. Definition : See ' D. by T.' T. Dis- 
 junctive : 132. 
 
 U Proposition: 153, 159-161, 19S. Its 
 Contradictory : 1S2. 
 
 Ultimate Catesory, U. Concept : 75-79. 
 U. Cause : See ' First Cause.' U. 
 Character of Relevancj- : 120. U. 
 Concept, U. Meaning : 82-85 (.^ee 
 'Fimdamental C). U. Context: 
 llS-120. U. Basis of Thought: 83, 
 84. U. Forms of Being: 31 {sec 
 ' Category '). U. Forms of Extensive 
 Proposition: 147, 148 (see 'A. P.,' 
 ' E P.,' ' I P.,' ' O P.' U. Logical 
 Postulate: 78, 79. U. Major Pre- 
 miss : 457, 458. U. Meaning of a 
 Proposition: 118. U. Subject : 118- 
 120. 
 
 Unambiguous Application of Terms : 
 209. U. Mark : 74. U. Meaning : 14, 
 16, 17, 20, 24, 51, 52, 81, 121. 309, 
 U. Reference: SO, 81. U. Result: 
 402. U. Specification : 116. U. State- 
 ment : 148, 15S. 
 
 Unanalytic Method : 397, 455, 456. See 
 ' Enumeration.' 
 
 Unbelievability : 449, 
 
 Uncertainty : 130, 435. 
 
 Unconditional Antecedent, U. Cause : 
 378, 383. U. Consequence, U. Se- 
 quence : 370, 377-383, 453. U. 
 Validity: 141, 142. 
 
 Undefined Subject-term : 121. 
 
 Under-exhaustive Division : 46. 
 
 Understanding : 6, 448. 
 
 Undetermined . . . : See ' Indetermi- 
 nate . . .' 
 
 Undeveloped Hypothesis: 319, 320, 
 326, 433. U. Meaning, Undifferen- 
 tiates] M. : See ' Development of M.,' 
 ' Differentiation of M.'s,' ' Indetermi- 
 nate Concept,' ' I. Elements ..." 
 
 Undistributed Meaning of ' Some ' : 
 15.5, 160, 175, 170. U. Middle: 218, 
 222, 225, 228. 233, 234, 238, 257, 259, 
 269. U. Term: 148-15.3, 155, 156, 
 l.-^s, 100, 161. 193. 194. 216, 228-230, 
 237. 2:J8. 
 
 Undistributetlness : 155, 1.58. 
 
 Umlulatory Theory of Light : 87, 335, 
 336. 339, 432, 434. 
 
 Unfavourable Event, L^. Instance : 391. 
 
 Unidetermining Definition : 77. 
 
 Unideterminism : 389, 400, 420, 449. 
 See ' Reciprocity . . .' 
 
 Unification : See ' Unity.' 
 
 Uniformities: 342, 3.55, 3.57, 367-309. 
 
 U. of Co-existence : 368. U. of Suc- 
 cession : 377. Their Interrelations : 
 369. 
 
 Uniformity (of Nature) : 307, 455-458. 
 See ' Determinism,' ' Inductive Pos- 
 tulate,' ' Law of Causation.' 
 
 Unifying Agency of Interest : 125. 
 
 Unimaginability : 450. 
 
 Unintelli^ibilit}' of a Contradiction : 
 107. U. of Nature : 139, 140. 
 
 Unit : 43, 94, 97, 111. 353, 387, 389 {see 
 ' Logical U.,' ' Statistical U.'). U. of 
 Speech : 93. U. of Thought : 96. 
 
 Unity: 35 {see 'Identity'). U. of a 
 Conceptual System : 95. U. of a 
 System of Judgments : 124. U. of a 
 Total Meaning : 98. U. of a Work of 
 Art: 118, 125. U. of Defining Pro- 
 cess : 30. U. of Experience : 78. 
 Unity (or Singleness) of F. D. : 43- 
 45, 51-53, 55, 56, 74, 75. U. of Idea 
 with Fact : 6. U. of Interest : 124. 
 U. of Purpose : 2. U. of Syllogistic 
 Process : 223. U. of the Causal 
 Method : 395. U. of the Judgment, 
 U. of the Proposition : 120. 122, 124, 
 125. U. of the Universe : 2, 25, 385. 
 U. of Thought : 1-3, 96, 106 (see ' Co- 
 herence of T.'). U.ofTopic:96, 98. 
 
 Universal: 22, 24, 85, 115, 244, 246, 
 299, 302, 304, 326, 339, 351 (see 
 ' Common Nature '). U. Affirmative 
 (Proposition) : 147, 161, 163, 230 (see 
 'A P.'). U. Conclusion: 224, 226, 
 230, 232, 235. U. Development : 460. 
 U. Extension-reference : 148 {see 
 ' Distributed Term '). U. Judgment, 
 U. Proposition, U. Statement : 147, 
 148 (ftn.), 150, 153-156, 101, 162, 165- 
 107, 299, 300. 302, 303. 357 (see 
 'A P.,' 'E P.,' 'General Maxim'). 
 U. Law : 406, 455. U. Negative (Pro- 
 position) : 147, 149, 168 (see ' E P.'). 
 U. Premiss : 227, 228, 230. 237, 238, 
 243, 248, 300. U. Quantity-mark: 
 154, 158 {see ' " All " as Q.-m.'). U. 
 Truth : 454, 455. 
 
 Universality : 421, 455 (see ' Uni- 
 versal . . .'). 
 
 Universalization : 348, 351, 352. 
 
 Universe : 2, 25, 78, 79, 105, 127, 385- 
 387, 410, 455, 456, 459, 460 {see 
 ' Identity of the U.,' ' Intelligibility 
 of Experience,' ' Unity of the U.'). 
 U. of Discourse : See ' Sitppositio.' 
 
 Unknowable : 77-79, 207, 208. 
 
 Unknown Cause : 412. 
 
 Unlikeness : 57. 
 
 Unmeaning Mark : 81, 82. 
 
 Unorganized Experience : 114. 
 
 Unorthodox Syllogisms : 250, 251. 
 
 Unscientific Method : 454, 455. See 
 ' Enumeration.'
 
 INDEX, VERBAL AND ANALYTIC 
 
 409 
 
 Unsound Analogy : See ' Illegitimate 
 A.' U. Inference : 141 {see ' Inva- 
 lidity '). 
 
 Unspecified . . . : <See ' Indetermi- 
 nate . . .' 
 
 Unstated Possibility : 132 (ftn.). 
 
 Unsystematized Fact : 246. 
 
 Unthinkable: 104, lOG, 107, 115, 173, 
 207. 208. 
 
 Unverified Generalization, U. Hypo- 
 thesis : 99, 326, 350. 
 
 Untruth : 145. See ' Falsehood.' 
 
 Uranus : 337. 412, 451, 452. 
 
 Uses of the Four (Syllogistic) Figures : 
 235-238. 
 
 Vagueness : See ' Indeterminacy.' V. 
 in the Use of Words : 17. 
 
 Valid .. .-.See' Validity.' Valid Forms 
 of Syllogism : 216, 224-229, 242, 243, 
 247, 249, 250. 324, 325. 
 
 Validation : 141. 
 
 Validity: 6-10, 107, 141. 142, 145, 187- 
 189, 193, 208. 209, 213, 223, 244, 250, 
 286, 288, 290, 300. 302-306, 309, 310. 
 321, 326, 327, 456 (see ' Conditional 
 V.,' 'Law of Formal V.,' ' L. of 
 Logical V.,' ' Unconditional V.'). V. 
 distinguished from Consistency : 8, 9. 
 V. in Dilemma : 274. V. in Syllogism : 
 214, 218, 221, 222. 224, 225, 230, 232- 
 234. 236. 242, 243, 247. 248, 251, 
 263, 267, 268, 301. 324. 325. 327 
 (ftn.), (see ' Valid Forms of S.'). V. 
 of Hypotheticals : 140. 141. V. of 
 Propositions : 8, 94. V. of the ' Im- 
 perfect ' Figures : 247. 
 
 Validity-connexion : 142. 
 
 Validity-interest: 6-10, 145, 148, 132, 
 306. 307, 309, 310, 326. 
 
 Validity-Principle: 191. See 'P. of 
 Identity,' ' P. of Non-Contradiction.' 
 
 Value of Enumeration in Inductive 
 Liquiry : 352-355. 
 
 V'ariability : 341, 456. 
 
 Variation : 62. 314, 341, 353, 354 (see 
 'Algol,' 'Concomitant Variations'). 
 V. in F. D. : 44, 45. V. in Meaning of 
 Terms or Words : 8, 14-17, 06, 101, 
 102, 281. 282. V. of Circumstances : 
 394. V. of Conditions : 404. 
 
 Variety : 42, 57. 79. 89. 354, 400. V. of 
 Instances : 396, 397, 399, 407. 
 
 Vegetable Mould : 328, 421, 422. 
 
 Venn, John, Sc.D., F.R.S. : viii, 49 
 (ftn.). 152, 340, 344, 349 (ftn.), 350, 
 370. 371, 380, 381. 
 
 Verbal Contradiction : 100-103. V. 
 Definition : 24, 25. 93. V. Disjunc- 
 tion : 272. V. Division : 37. 52. 
 56. V. Expression of Meaning : 93 
 (see 'Language,' 'Words'). V. Fal- 
 lacies : 281-280. V. Name : 69. V. 
 
 Proposition: 115. V. Sign, V. Sym- 
 bol : 25, 32. 93. 115 (see 'Sign,' 
 'Words'). 'V. Structure: 93. V. 
 Truism : 120 (see 'Tautology'). 
 
 Verifiable Hypothesis : 328. 
 
 Verificandum : 319. See ' Hypothesis.' 
 
 Verification (or Testing) : 114. 140. 141, 
 204, 306, 314, 317-321, 326. .328. 334- 
 339. 343. 347-352. 355-364, 369. 371. 
 394. .397, 400, 401, 406-411, 418, 419, 
 422-428, 431-433. 436-443, 453, 450. 
 
 Verification-method : See ' Method of 
 Verification.' 
 
 Vicious Circle : See ' Circulus . . .' 
 
 Violation of Laws : See ' Inviolability 
 . . .' V. of Rules : 107. 
 
 Vitality : 119 (see ' Deductive V.'). V. 
 of Spiritual Experience : 107. V. of 
 Thought : 15. 
 
 Vivisection : 402. 
 
 Vortex Theory of Descartes : 318, 332, 
 333, 336. V. T. of Matter : 333. 
 
 Ward, James (Dr.) : 386, 387. 
 
 Watts. Isaac, D.D. : 28, 67, 288. 
 
 Wave-theory : See ' Undulatory T. . . .' 
 
 Weak Converse: 194, 198, 206, 229, 
 237. W. Educt : 189. 194. 200, 201. 
 
 Weakened Conclusion : 229. W. Forms 
 of Syllogism : 226. 227. 229. 
 
 Weighing Properties W. Resemblances : 
 358, 360, 361. 
 
 Wclton, J. (Prof.) : viii, 150, 152, 159, 
 165 (ftn.), 193, 261, 268, 286, 336 
 (ftn.), 386 (ftn.), 421. 
 
 ' \\liat ' : 75 (ftn.), 82. 
 
 ^1,^lately, R. (Archbishop) : 250, 291. 
 
 ' Whatever is. is ' : 97. 
 
 WTiewell, W. (Dr.) : 62, 349, 350, 449. 
 
 ' Whole ' and ' Parts ' in Division : 40. 
 W. Extent: See 'Total Extension- 
 reference.' 
 
 ' WTiy ' : 342, 370. 
 
 Will : 140. See ' Free-w.' 
 
 Wish : See ' Optative.' 
 
 Word - complex : 111. 
 
 Words : See ' Ambiguity,' ' Application 
 of Meanings,' ' Definition,' ' Descrip- 
 tion,' ' Division,' ' Meanings ot 
 Terms,' ' Nomenclature,' ' Significa- 
 tion,' ' Term,' ' Terminology," ' Ver- 
 bal Sign.' W. as Expressive Signs : 
 14. 23-25, 32. W. as Fixing Atten- 
 tion : 14 ; as F. Meanings : 13-16, 23, 
 93. W. as Sensory Supports for 
 Meaning : '24, 93. W. as Sound-com- 
 plexes : 24, 25, 81. W. in a Sentence : 
 116. W. in relation to Judgment : 93 ; 
 to Logic : 13, 16, 17 ; to Rhetoric : 
 13; to Thought: 13-17. Their 
 Function : 13-18. Their Right Use: 
 13, 16-18. Their Subjective Inten- 
 sion : 70, 71. Their Variation in
 
 500 
 
 THE PROBLEM OF LOGIC 
 
 Jfeaning : 8, 14-17, 66, 101. 102, 281, 
 
 282. 
 Workinj: Concept. W. Idea: 328-335. 
 
 337. 387, 447-449. 451, 452, 459. Its 
 
 relation to the W. Hypothesis : 328. 
 
 W. Hypothesis : 328, 329, 334, 335, 
 
 337. 'W. Method : 448. 
 World : See ' Real W.' 
 Wright. H. W. : 61 (ftn.). 
 
 r as a Substitute Sign : 14. As Definite 
 or Positive Term : 47-49. As Genus : 
 
 52. As representing an Unspecified 
 Subject : 1G3. As Ultimate Suyyimum 
 Genus : 76. 
 : 201. 
 
 Y Proposition : 159-'6I. 
 Young, Charles Augustus 
 
 339. 
 
 Zcno the Eleatic : 273, 290, 291 
 Zero Extent : 149. 
 
 Zoology: 37, 42, 57, 60-62, 68. 71, 
 73-75. 
 
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