GIFT OF MICHAEL REE&E ff INDUCTIVE LOGIC INDUCTIVE LOGIC By JOHN GKIEB HIBBEN, Ph.D. ASSISTANT PROFESSOR OF LOGIO IN PRINCETON UNIVERSITY WILLIAM BLACKWOOD AND SONS EDINBURGH AND LONDON MDCCCXCVI All Rights reserved 7/* ?L COPYRIGHT, 1896, BY CHARLES SCRIBNER'S SONS NorfoooU ^30 J. S. Cushing & Co. - Berwick & Smith. Norwood Mass. U.S.A. CONTENTS Chap. I. The Nature of Inference ... 1 Psychological and Logical Elements in Inference, page 1 ; Objective and Subjective Necessity, 4 ; Data of Presen- tation, 5 ; System as Ground of Inference, 6 ; The Im- plicit and Explicit, 11 ; Inference mediated through the Universal, 12 ; Conceptual Processes, 13 ; Explana- tion, 14. Chap. II. Induction and Deduction ... 16 Various Opinions concerning their Relative Importance, page 16 ; Regarded as Different Phases of One and the Same Process, 17 ; Their Relation to the Ground of Inference regarded as a System, 17 ; Their Relation to the Universal, 18 ; Difference between Truth and Fact, 19 ; Mutual Dependence of Induction and De- duction, 20. Chap. III. The Essentials of Induction . . 24 Th e Inductive Hazar d, page 24 ; Basal Postulate of Induc- tion, 25 ; Its Epistemelogical Nature, 26 _ ; Induction rSfarded as an Inverse Process, 27 ; Law an d Rule, 30 ; Law in Terms of an Hypothetical Universal,1h ; Induc- tion in the Conduct of Human Affairs, 32 ; The Scien- tific Spirit, 33. Chap. ivr TypUs of Inductive Inference . . 34 The Method of Enumeration, page 35: a. Perfect Induc- tion, 36 ; b. Incomplete Enumeration, 37 ; c. Proba- y/ h ? VI CONTENTS bility, 38 ; The Method of Analogy, 39 ; The Method of Scientific Analysis, or Causal Determination, 40; The C ausal Po stulate u nderlying All the Methods, 43 ; Relation of^Mental itabit to Choice of Method, 47; Generalization, 48. Chap. V. Causation 50 Logical Significance of the Causal Concept, page 50; Its Phenomenal Significance The Conservation of En- ergy, 51 ; Its Philosophical Significance, 53 ; Its Logical Significance, 54 ; Its Epistemelogical Ground, 58 ; Pop- ular and Scientific Idea of Cause, 58 ; Causal Analysis, 60 ; Limitations of K nowle dge, 62. Chap. VI. Causal Analysis and Determination 64 Seauence, page 64 ; Concurrence, 66 ; Co-e xistenc e, 66 ; v ital Growth and Development, 68 ; CollocationT^ ; Different Modes of Transfer of Energy, 7lT"Quantita- tive Determination, 72; Observation and Experiment, 73; Negative Determination, 78 ; ^seuHo^causal Con- nection, 82. * Chap. VII. Mill's Inductive Methods The Method of Agreement 84 The Eive Methods, page 84; The Method of Agreement, 86 ; Symbolic Representation, 87 ; Variation of In- stances, 90 ; The Method of Agreement and Observa- tion, 91 ; Relation to Simple Enumeration, 91 ; Se- quence and Co-existence, 92 ; Defects of this Method, 93 ; Its Chief Function, that of Suggestion, 96 ; Illus- trations, 97. Chap. VIII. The Method of Difference . . 101 Relation to Method of Agreement, page 101 ; Its Charac- teristics, 101 ; Symbolic Representation, 103 ; Similar CONTENTS Vll to Negative Determination, 104 ; Relation to the The- ory of Combinations, 105; Criticisms of this Method, 106; Practical Difficulties, 109; Illustrations, 113; Blind Experiments, 115. Chap. IX. The Joint Method of Agreement and Difference ........ 117 Relation to Method of Difference, page 117 ; Symbolic Rep- resentation, 118; Difficulty of Elimination, 121; Illus- trations, 124 ; Advantage of this Method over the Simple Method of Agreement, 128. Chap. X. The Method of Concomitant Variations 130 Its Characteristics, page 130; Its Symbolic Representation, 131 ; Quantitative Determination, 131 ; Graphic Repre- sentation, 133 ; Psychological Impressions, 133 ; Illus- trations, 134; The Comprehension of the Intensity of Unknown Forces facilitated by this Method, 141 ; Lim- itations of this Method, 142. Chap. XL The Method of Residues . . . 146 A Method of Elimination, page 146 ; Symbolic Representa- tion, 146 ; A Deductive Method, 147 ; The Complexity of the Residual Element, 148 ; Illustrations, 149 ; Re- sidual Error in Experiments, 153 ; The Mental Habit of inspecting All Remainders, 154. Chap. XII. Verification and Prediction . . 156 The Inducto-deductive Method, page 156 ; Verification, 167 ; Prediction, 159 ; Illustrations, 160 ; Bacon's Anticipa- tions of Nature, 163 ; Scientific Thought, 164 ; Indirect Method*of Prediction, 166 ; Exception Phenomena, 170; Generalization, 171 ; Mathematical Method, 172. viii CONTENTS Chap. XIII. Hypothesis 174 Hypothesis, as Preliminary to Experiment, page 174 ; Hy- pothesis, in place of Experiment, 176 ; Illustrations, 177 ; Function of the Imagination in Hypothesis, 184 ; Analysis and Synthesis, 186 ; Eequirements of a Legit- imate Hypothesis, 187 ; Postulate and Hypothesis, 189 ; Fictions, 196 ; Suggestions through Failure of Hypoth- eses, 197 ; Consilience of Inductions, 198 ; Experimen- tum Crucis, 199 ; Whewell and Mill, 201. Chap. XIV. Analogy 204 Analogy as Suggestive of Inductive Inquiry, page 204 ; Analogy in Generalization, 204 ; Formation of Con- cepts and Analogy, 205 ; Natural Kinds, 205 ; Classifi- cation, 207 ; Teleology, 208 ; False Analogies, 220. Chap. XV. Probability Complexity of the Causal-nexus, page 226 ; Relation to Enumerative Induction, 228 ; Calculation of the Proba- bility of a Particular Event, 230 ; Adverbial Probabil- ity, 232 ; Estimate of Aggregates, 234 ; Chance and Coincidence, 243 ; Circumstantial Evidence, 247 ; Rela- tion to the Method of Residues, 251. Chap. XVI. Empirical Laws .... 252 Three Classes of Laws of Varying Degree of Probability, page 252 ; Empirical Law as Expression of Causal Relation in Process of Determination, 253 ; Colloca- tions giving Rise to Empirical Laws, 254 ; Generaliza- tions expressing an Aggregate of Qualities in the Same Individual, 256 ; Probability and Empirical Laws, 258 ; The Method of Agreement, 259 ; T he Empi rical ^Saterre of the CangaT Relation, 260. CONTENTS ix Chap. XVII. Fallacies 262 Of Perception, page 263 : a. Failure to comprehend the Entire Field of Vision, 263 ; b. Failure to concentrate Attention, 265 ; c. Errors due to Apperceptive Projec- tion, 266 ; Of Judgment, 266 : a. False Associations, 267 ; b. Emotional Perturbation, 267 ; c. General Frail- ties of Human Nature, Bacon's Idols, 269; Of Imagi- nation, 271 ; Of the Conceptual Processes, 275 : a. Hasty Generalization, 276 ; b. Interpolation in a Series, 277 ; c. Provincialisms, 278 ; d. False Analogies, 278 ; e. In- correct Classification, 279 ; Psychological Character of these Fallacies, 279. Chap. XVIII. The Inductive Methods as applied to the Various Sciences 281 Nature of Method will vary with Nature of the Phenomena, page 281 ; Complication of the Doctrine of the Conser- vation of Energy, 287 ; The Phenomena of One Science to be interpreted in the Light of the Results of Another Science, 290 ; Growing Tendency to supplement Deduc- tive Method by Inductive, 292. Chap. XIX. Historical Sketch of Induction . 297 Socrates, 297 ; Plato, 297 ; Aristotle, 298 ; Roger Bacon, 300 ; Leonardo da Vinci, 301 ; Telesius, 302 ; Campa- nella, 303 ; Csesalpinus, Copernicus, Gilbert, Kepler, Brahe - , Galileo, 304 ; Francis Bacon, 304 ; Locke, 307 ; Newton, 307 ; Herschel, 308; Whewell, 310 ; Mill, 311. Chap. XX. Logical Exercises .... 313 PREFACE It has been my aim, in the following pages, to present the essential features of inductive logic, in the hope that this work may prove a fitting supplement to the elementary courses in formal or deductive logic. The impression is too often left in the minds of those who have pursued the study of deductive logic exclusively that the for- mal laws of the syllogism constitute the entire body of logical doctrine, and that reasoning con- sists solely in drawing conclusions from given premises. There is danger here lest reasoning become associated with an artificial procedure that seems to find its proper sphere in the solu- tion of verbal quibbles and logical puzzles. In the actual experiences of life, we do not find our premises ready made. They are the result of wide observation and patient investigation and experiment. We challenge premises that are given, and weigh their significance. We meet particular facts before we do the general laws. Xll PREFACE The former must be tested and interpreted, before we can rise to the general laws which underlie them, and which may stand as the major prem- ises of our syllogisms. Thus within the very sphere of deduction itself there naturally opens a wide field for inductive inquiry. Therefore I have emphasized the necessity of a thorough knowledge of the principles of inductive logic in order to comprehend the material as well as the formal elements in inference, and without which no firm grasp of the general process of reasoning is possible. I have also insisted upon regarding induction and deduction as mutually dependent ; not as separate modes of inference, but rather as different phases of one and the same logical procedure. I have endeavored, also, to indicate in some measure at least, the salient characteristics of the modern logic, especially as presented in the works of Lotze, Sigwart, Jevons, Green, Bosan- quet, and Venn. In the illustrations of the various inductive methods I have sought fresh material as far as possible, with the view of representing the actual modes of reasoning and methods of investigation employed by those who have become eminent in their several spheres of research, such as Faraday, Tyndall, Darwin, and PREFACE Xlll Lubbock; and especially the different methods which have led to important discoveries in the various sciences. This applies not only to the illustrations in the text proper, but also to those which I have collected in Chapter XX. under the head of Logical Exercises. It seems to me, moreover, that inasmuch as the principles of inductive investigation are in such accord with the scientific spirit of our age, their importance as a logical discipline cannot be too highly valued. J. G. H. Princeton, N.J., March 2, 1896. INDUCTIVE LOGIC CHAPTER I The Nature of Inference Induction is a particular mode of inference in general ; and therefore before its nature and scope can be adequately defined, it will be necessary to give some account of the theory of inference, and its precise logical signification. Moreover, it is not possible to appreciate the distinction between the processes of induction and deduction, until we have first examined the characteristic features which are common to the two, and which constitute the essential elements of inference itself. The nature of inference may be unfolded in two ways. We may consider what it is in its outward aspect that is, through its phenomenal manifestation in what it effects ; or it may be more strictly defined in terms of its warrant, or ground. From the first point of view we examine inference as regards its psychological significance; that/ is, what is infer- ence considered as a psychical experience, its na- ture and characteristics? But we must consider also the second question, whether there is any B 1 2 INDUCTIVE LOGIC necessity limiting and determining the subjective experience, which presents the character of a law having universal validity. What goes on in the mind during the process of inference ? Also, what is the rationale of such a process? These ques- tions we will examine more closely, in order to show the nature of inference under the two aspects, the one psychological, and the other logical. It is a well-recognized fact in psychology, that in our simplest as well as the more complex per- ceptions, the interpretation of the data of presenta- tion always goes beyond the strict content of the data themselves. We see more than is given in the field of vision immediately before us. The mind supplies here and there the necessary parts that are lacking in the actual elements of presenta- tion, and yet which are necessitated by the known nature of that which is actually given. We form our judgment of distance indirectly, and not through direct presentation. So, also, our idea of a third dimension is acquired by a process, marvellously complex, in which the data both indicate and yet are transcended by the results. Whether the nativ- ist or empiricist holds the true position concern- ing original psychical experience, it still must be conceded according to either theory that the devel- opment of our perceptions corresponds to a law of growth based upon accumulated inferences. Infer- ence has been defined as the indirect reference of a content to reality, and as such, we see the be- ginnings of inference in the most simple of our perceptions. Every perception contains a direct THE NATURE OF INFERENCE 3 reference to reality, but also something which in a greater or less degree is referred indirectly to reality. The fact that our knowledge as given in the complete perception contains more than is actu- ally mediated through the avenues of the senses, is due to the apperceptive processes of conscious- ness. Mind is active in perception, and not a mere passive receptacle. That which is given, the raw material of the senses, is elaborated and extended, as it is combined with the wealth of representative and conceptual material which the mind brings to every new perception. To this extent, at least, the mind possesses a creative function. A certain appearance of sky, combined with peculiar condi- tions of mind and temperature, leads one to assert, with some degree of certitude, that it will rain before morning. The prediction is an inference based upon, and growing out of, the actual data of perception, and yet far outrunning them. We recognize a friend from his step or voice. The mere presentation is only a sound. That it is associated with a person, and not an animal, or a thing, is an inference; that this is the particular person whom we recognize as a friend and can call by name, even before we turn around to confirm the opinion by direct testimony of vision, this is a still further inference. And even when we open our eyes in simple vision itself, we fill up many a gap in our minds, and give depth and distance, and interpret the contrasts of light and shade, and the play of colors, through the process of inference, although we may not be aware of the process itself, 4 INDUCTIVE LOGIC which is automatically operative through long-con- tinued habit. When we thus regard inference as a psychological phenomenon, it may be readily ex- plained by the laws of comparison, association, recognition, generalization, etc. And, as such, in- ference has a subjective force at least, and leads to the habit of prediction and expectation. The will, influenced by the resulting belief, leads to activi- ties consistent with such expectation. Here, however, the question arises, which is urged with such force by Hume, Is there objective validity as well as subjective necessity ? This leads to a consideration of inference, from the sec- ond point of view, above mentioned. We may be constrained to believe certain things concerning the great world lying beyond the sphere of immediate consciousness ; but what warrant have we in so doing, or what assurance that our conclusions are correct ? May we not be deceived, after all, and by some psychological trick be led to regard the phe- nomena of consciousness as quite otherwise than that which obtains in reality? We may have a strong aversion to sitting down at a table where the number of persons will be thirteen. But has the subjective conviction, that one of the thirteen will die in the course of the year, any value when Ave come to refer it to reality, and ask ourselves the nature of the ground upon which the conviction is based ? On the other hand, however, it is quite a differ- ent kind of necessity which constrains us to judge that if a person jumps off of the roof of a house^ THE NATURE OF INFERENCE 5 he must surely fall to the ground below. Some grossly superstitious and ignorant people may be- lieve the former with as obstinate a conviction as the latter, so that a purely psychological criterion of the strength of conviction is not at all adequate or satisfactory. Is there any other criterion ? In what instances does this subjective constraint pro- ceed from the necessities of reality ? or, in other words, in what cases are we able to discover a logi- cally grounded warrant which compels the infer- ence, in distinction from the mere psychological compulsion which is occasioned by the psychical tendencies of association and generalization ? This leads us to consider the logical, in distinction from the psychological, nature of inference. Inas- much as the characteristic feature of inference con- sists in this, that while depending upon certain data of presentation, it nevertheless wholly transcends them, the question naturally suggests itself, whether it is something within the data themselves, or with- out, by virtue of which the mind thus goes beyond them in the process of inference. If it lies wholly without the data, it must be something imposed upon them by the mind, and as such can have only a psychological force and value. For instance, the belief that if thirteen sit down together at a table, one will die in the course of the year, can have only a subjective value and significance. This is true in all cases where the necessity of conviction finds its origin in prejudice or in superstition, or it may be in the force of authority. In all such instances we feel the lack of a satisfactory logical ground. How- 6 INDUCTIVE LOGIC ever, on the other hand, if the data of consciousness contain within themselves that which enables us to transcend them at the same time that we interpret them, there is external validity for our inference that has a logical worth. This seems at the first glance to be a paradox. How can any content enable us to state concerning it more than is con- tained within it ? The answer to the seeming para- dox is that every concept, and every perception as well, have both an explicit and implicit content. We never attain complete vision or perfect apprehension. There are, moreover, many points of view, each giving additional knowledge concerning any phe- nomenon present in consciousness. We see, there- fore, only in part, and yet that which is seen contains certain necessary implications concerning that which is not seen. In the progress of knowledge, subse- quent observations, different points of view, are ever confirming and amplifying our inferences, enabling us to perceive immediately what formerly was only inferred. The process by which the implicit is becoming explicit indicates a necessary relation existing between that which is known mediately and that which is known immediately. Moreover, consciousness has been represented as a stream, or an intricately interwoven web, something extremely complex. Every part is related both proximately and remotely. There is no such thing as an isolated perception ; every perception has its complex rela- tions and connections. So also every concept which is formed by generalization through comparison and abstraction, of our presentations as interpreted by THE NATURE OF INFERENCE 7 us, possesses this characteristic of greater or less complexity. In this manner the world of conscious- ness is constructed ; that is, the world as it is for us. This forms a complex whole made up of parts, which in themselves may be regarded as wholes, and yet which may be still further divided and subdivided. Such an interrelated whole we may style a sys- tem, or, in other words, a complex whole whose parts are congruently arranged. The idea of system finds expression in the "Law of Totality," that our knowledge is capable of arrangement in a self-con- sistent and harmonious system, and which moreover in its content and form faithfully represents objec- tive reality. 1 We find, therefore, that in the focus of consciousness at any one time, whether in the sphere of presentation or in the region of representa- tive or the conceptual processes, whatever is given carries with it always certain implications, and there- fore certain necessary relations. This is specially emphasized in Bosanquet's definition of system : "System is a group of relations, or properties, or things, so held together by a common nature that you can judge from some of them what the others must be.' 7 2 Two facts regarded as independent and considered separately may give no information be- yond their explicit contents ; but when conjoined, they imply more than the sum of their parts. How often two ideas in separate minds yield no result ; but brought together, they give light. Isolation 1 Ueberweg, A System of Logic and History of Logical Doc- trine, pp. 540 f . 2 Bosanquet, The Essentials of Logic, p. 140. 8 INDUCTIVE LOGIC negatives inference. To unfold any data in all their manifold implications is the process of infer- ence. Its warrant lies in the fundamental postulate of knowledge which we are constrained to assume; namely, that our consciousness must be self-con- sistent throughout. Whatever is admitted as true must find a congruent place in the system to which it is possible to refer it. The necessity of fitting it in its proper place gives rise to certain implications which necessitate corresponding relations and attri- butes. And if it could not be put into such a place, we would feel that we should have to surrender the idea of self-consistency in the variously related ele- ments of our consciousness. The very integrity of our mental life necessitates this conviction. Therefore a part being given, we supply in our minds other parts, or the whole to which the given part must necessarily belong. To achieve this, with logical warrant, our knowledge of the part must be adequate to the extent that we know that the ele- ment under consideration cannot be complete in itself, but must be supplemented by its appropri- ately related elements which with it go to make up the complete system. We infer the nature of the flower not yet in bud, by the sprouting leaf. The one necessitates the other by virtue of their com- mon inherence in the same plant system. We know that figs do not come from thorns, nor grapes from thistles. Columbus, noting the seaweed, and birds, and the drift of the sea, inferred a shore beyond, to which he was constrained by the neces- sities of thought to refer them. It is said of Cuvier THE NATURE OF INFERENCE 9 that lie was able to reconstruct part for part the entire frame and organism of an animal whose fossil tooth alone formed the original datum. He knew the system to which it must have belonged and to which it alone could possibly be referred. An interesting quotation from Cuvier himself illus- trates most appropriately this function of inference. He says in his Ossemens Fossiles, " I doubt if any one would have divined, if untaught by observation, that all ruminants have the foot cleft, and that they alone have it. I doubt if any one would have divined that there are frontal horns only in this class ; that those among them which have sharp canines for the most part lack horns. However, since these relations are constant, they must have some sufficient cause ; but since we are ignorant of it we must make good the defect of the theory by means of observation: it enables us to establish empirical laws which become almost as certain as rational laws when they rest on sufficiently repeated observations ; so that now whoso sees merely the print of a cleft foot may conclude that the animal which left this impression ruminated, and this conclusion is as certain as any other in physics or morals. This footprint alone, then, yields to him who observes it the form of the teeth, the form of the jaws, the form of the verte- brae, the form of all the bones of the legs, of the thighs, of the shoulders, and of the pelvis of the animal which has passed by." 1 In the common conduct of every-day life we infer 1 Quoted by Jevons, Principles of Science, 2d ed. p. G83. 10 INDUCTIVE LOGIC beyond the immediate present experience to future happenings and in a similar manner. My train is half an hour late. I know I must miss my connec- tions at the station ahead ; for the train I am hop- ing to catch at that place is scheduled to leave five minutes after the time of arrival of the train I am now on. The time relations here necessitate my missing my connections. This is rendered still more certain if they are rival roads ; on no account will one wait for the other. Moreover, the train I hope to make is made up and leaves the station in question, and so I cannot fall back upon the favoring chance that it also may be detained en route, and so enable me, after all, to reach it in time. Thus, with every additional knowledge of the system which forms the ground of my inference, and the various conditions which affect it, the validity of my infer- ence is thereby increased. Inference regarded as the analysis of a system of interrelated parts is illustrated in the following paragraph of Professor James : " The results of reasoning may be hit upon by accident. Cats have been known to open doors by pulling latches, etc. But no cat, if the latch got out of order, could open the door again, unless some new accident at random fumbling taught her to asso- ciate some new total movement with the total phe- nomenon of the closed door. A reasoning man, however, would open the door by first analyzing the hindrance. He would ascertain what particular feature of the door was wrong. The lever, e.g., does not raise the latch sufficiently from its slot case of insufficient elevation raise door bodily on THE NATURE OF INFERENCE 11 hinges ! Or door sticks at top by friction against lintel press it bodily down! I have a student's lamp of which the flame vibrates most unpleasantly unless the collar which bears the chimney be raised about a sixteenth of an inch. I learned the remedy after much torment, by accident, and now always keep the collar up with a small wedge. But my procedure is a mere association of two totals, dis- eased object and remedy.- One learned in pneumat- ics could have named the cause of the disease and thence inferred the remedy immediately." 1 Inference, therefore, may be regarded as a deep penetrating insight. The explicit is that which lies upon the surface, which the mind immediately grasps, for it lies directly in the focus of conscious- ness. Whereas the implicit is beneath the surface, and is revealed only through a searching analysis. This difference may be exhibited through the dis- tinction between the actual and the potential. A child regards gunpowder merely as a pile of coarse- grained sand. The man sees what the child sees, but also the existing possibilities under certain con- ditions of explosive force. He apprehends the potential as well as the actual ; and his inference as to the possible results is based upon his superior insight. It is therefore the well-furnished mind which sees things as most widely related, and dis- cerns the potential as well as the actual manifesta- tion, which will prove the most fertile in accurate inference, in prophetic suggestion, and in inventive resource. i James, Psychology, Vol. II. pp. 339, 340. 12 INDUCTIVE LOGIC The whole world of reality, as well as that of knowledge, may be considered as one system, em- bracing within the unity of its totality all the vari- ous systems with their complicated parts. From this point of view everything bears relations to everything else in the universe. The original sig- nification of the term universe is thus emphasized. This thought, no doubt, Tennyson had in mind in the following verse : Flower in the crannied wall, I pluck you out of the crannies, I hold you here, root and all, in my hand, Little flower but if I could understand What you are, root and all, and all in all, I should know what God and man is. We can, in this connection, best exhibit the pre- cise nature and function of the universal in infer- ence. The possibility of unfolding the properties or relations of anything in all its implications depends upon our knowledge of the universal con- cept to which the properties or relations in ques- tion are naturally referred. While a singular proposition is the statement of the mere occur- rence of a phenomenon, the universal always implies a knowledge of the conditions and rela- tions of the phenomenon. 1 Insight is only pos- sible where there is a wealth of universal concepts. We see an animal which we observe to be cloven- footed. We infer that it also chews its cud. We do not observe this. The assertion does not arise i See Green, Philosophical Works, Vol. II. pp. 284, 285. THE NATURE OF INFERENCE 13 directly from observed reality, but indirectly through the generic concept that has grasped to- gether the two attributes of chewing the cud and cloven feet as always and necessarily coexisting in one and the same animal Inference, in this sense, may be regarded as the indirect reference of knowledge to reality, and this is always medi- ated through the universal. The universal has this characteristic feature, that it preserves an identity in the midst of manifold differences. The same thought may be expressed by saying that the universal manifests a unity in the midst of diver- sity. However widely different, in many respects, the animals may appear that chew the cud, as the cow, deer, sheep, etc., there is always the constant characteristic that they are cloven-footed. Such a point of identity furnishes the constant factor which determines the nature and the validity of the inference. Were it not for this conceptual power of the mind, this ability to grasp phenom- ena in their universal essence, and consider them as interrelated and connected, we could never pass beyond individual and particular experiences which would form a series of wholly disconnected events. Knowledge could not then form a self-consistent system, or inference possess any higher worth than a haphazard guess. As Green says, "A 'mere fact/ a fact apart from relations which are not sen- sible, would be no fact, would have no nature, would not admit of anything being known or said about it." 1 1 Green, Philosophical Works, Vol. IL p. 301. 14 INDUCTIVE LOGIC Moreover, inference is not merely employed to extend the field of consciousness in unfolding sup- plementary elements lying beyond the sphere of direct cognition; the elements may all be given immediately, and inference employed to discover their connection and interrelations, by virtue of what bond they belong in one or the same sys- tem. Inference here functions as explanation. A man is found dead ; there are many wounds upon his person, evidences of a struggle in an out-of-the- way place upon a lonely road. Such a combina- tion of facts calls for an explanation which shall be consistent with them. The facts must all be cor- related in a system whose related facts and the unity of the whole will completely satisfy the mind. The mind is satisfied only when all hang together in what seems the only possible self- consistent co-ordinated system. The facts being given, they must be read backward to their origin. The other aspect of inference is the reading of facts forwards, or unfolding them in their neces- sary consequences. Inference is the reply to the natural questions of the mind, whence and whither ? And the process is essentially the same, whether its peculiar mode consists in the evolu- tion or the involution of that which is given in consciousness. Moreover, the mere psychological inference, the subjective extension of the data of consciousness without any objective ground or warrant, should ever be corrected, or even at times wholly set aside by means of the truly logical inference. Where THE NATURE OF INFERENCE 15 the psychological experience, in transcending simple presentation, proceeds upon strictly logical grounds, and has objective validity as well as subjective necessity, we possess a warrant of the highest pos- sible worth. CHAPTER II Induction" and Deduction There have been divergent tendencies in the history of logic, to make either deduction or in- duction alone the whole of logical procedure in the process of inference. The fact that the Aristotelian logic, which is essentially deductive, has been for centuries exclusively associated with logic as a whole, has left the impression upon many minds that it is the beginning and end of the logical en- cyclopedia. On the other hand, J. S. Mill and his followers have attempted to analyze the syllogism to prove its essentially inductive character; and they have maintained that all reasoning is induc- tive. This is the position in the main of Bacon, Locke, and Bain. Locke, for instance, insists that the syllogism is of less value than external and internal experience, induction, and common sense. 1 - So also, in a similar vein, Schleiermacher says: "The syllogistic procedure is of no value for the real construction of judgments, for the substituted judgments can only be higher and lower ; nothing is expressed in the conclusion but the relation of two terms to each other, which have a common 1 Essay on Human Understanding, Book IV. p. 7. 16 INDUCTION AND DEDUCTION 17 member, and are not without, but within, each other. Advance in thinking, a new cognition, can- not originate by the syllogism ; it is merely the re- flection upon the way in which we have attained, or could attain, to a judgment, the conclusion; no new insight is ever reached." 1 The two opposed views thus indicated do not necessitate conflicting or mutually exclusive processes. It is better to regard them, not as radically different types of inference, but rather as different phases of one and the same inferential process. We have seen that inference consists in interpreting the implications of the system to which the given in consciousness belongs. In the light of this definition we can best indicate the relative functions of induction and de- duction in the process of inference. When the system can be considered as a whole, and is appre- hended in its entirety, then it may become the ground upon which the inference is based, resulting in unfolding the necessary nature or relations of any of the parts considered in themselves, or in reference to the system as a whole. The procedure in such a case is from the nature of the whole system, to the nature of the several parts, and their existent relations, and this is deductive in its es- sential features. On the other hand, when we know the various parts, and proceed from them as data to construct the system which their known nature and rela- tions necessitate, it is induction, or procedure from elementary parts to the whole thus necessitated. 1 See Ueberweg, System of Logic, etc., p. 345. c 18 INDUCTIVE LOGIC From a knowledge of the planetary system, we can infer the necessary positions of sun, moon, and earth at any required time, as, for instance, in the calculation of an eclipse. This is deduction. But when we begin with investigating the several move- ments of the different planets, and from them infer the necessary nature of the system of which they are parts, we have the process of induction. Such processes we see must be complementary, and mu- tually dependent. As Lavater says, " He only sees well who sees the whole in the parts, and the parts in the whole." Moreover, the distinction between deduction and induction may be shown through their respective relations to the universal, which we have seen is the ground of inference. The question whose answer leads to the deductive process in reasoning, is, What does the universal necessitate ? In induction, the question which starts the investigation is, Into what system may I construct the given material properties or relations, so as to reach a universal concept that will be consistent with itself and with the whole of knowledge which forms the world of consciousness ? In this there is an analytical dis- crimination of the essential and accidental elements, and the gathering together of the former into the complex whole which is the universal. Induction, therefore, is inference viewed from the side of the differences ; deduction is inference viewed from that of the universal. For instance, we may investigate the characteristic features of a diamond, and find that a certain specific gravity, 3.53 as compared with INDUCTION AND DEDUCTION 19 water, is a constant and determining attribute, and as such must be incorporated as an essential element of the general concept diamond. We can then form the universal judgment. Whatever stones possess this specific gravity are diamonds. Their differences, regarding size, brilliancy, etc., may all be set aside as accidental, but the one constant determining feature indicates a oneness in which they all agree. And so with the other essential attributes. After possessing such knowledge gained inductively, we may use it practically in a deductive manner; and it is so used in discriminating between true and imitation stones, as described in the following process : " Diamonds, rubies, and sapphires are now tested by floating to prove their genuineness. The liquid used has five times the density of water, and is composed of double nitrate of silver and thallium. The tests are rapidly made, as allstones of the same nature have the same specific gravity, while none of the bogus ones have the same weight as those they are made to imitate." Another view of the relation of induction to de- duction may be gained by calling attention to the difference of significance between the terms, a truth and a fact. A fact carries with it only the special and individual character of the particular occur- rence in which it is manifested. A truth, however, is always universal in its very nature, admitting of universal application, and capable of illustration in an indefinite number of different facts which embody its essence. In deduction we have given some truth of universal nature that leads to indi- 20 INDUCTIVE LOGIC vidual facts that may be subsumed under it. In induction, we interpret a fact or a number of facts in the light of their universal implication, on the ground that there can be no such thing as an isolated fact, but every fact must have some relation to a universal to which it must be referred. While considering the distinctions between in- duction and deduction, we must not overlook their y mutual dependence. We cannot proceed in de- duction irrespective of induction, because the uni- versal upon which the deductive process is based arises in the majority of cases from a previous induction. It is true that the universal term may be in a proposition that is known a priori, as the axioms of geometry and certain space and time postulates ; but a very small proportion of major premises can be said to have such an origin, and their resulting conclusions have very slight ma- terial significance. Deduction that reaches other than purely abstract and formal conclusions must rest upon induction for the material to form its premises. We find this even in the technical con- struction of the syllogism, where, for instance, the question of the distribution of the terms is raised. We may insist that a certain middle term is dis- tributed as it is the subject of an universal affirma- tive proposition ; but then the further question naturally suggests itself, How do we know that the proposition in question is really a universal ? Its material significance alone tells us that we may write it as an A or i" proposition, as the case may be. The matter is a function of the form, INDUCTION AND DEDUCTION 21 and the form a function of the matter. They can- not be separated in fact, unless we conceive reason- ing as a purely formal process of determining a conclusion, irrespective of the truth or falsity of the premises. If we regard the premises as given, and we accept them with unquestioning credence, the deduction is purely formal ; so also if the various terms are expressed by letters A, B, C, etc., and devoid of any material signifi- cance. Any process of reasoning based upon a slavish acceptance of premises can only reach artificial and even false results. In the actual ex- periences of life our premises are not made for us. They must be constructed by us through our interpretation of reality. Disregard of this has brought formal logic into much disrepute, and it has often degenerated into the barren discussion of logical puzzles and quibbles. Grant a person any premises he may choose to assume, irrespective of an inductive test of their validity, he can prove black white, and white black. On the other hand, induction is dependent upon deduction; for we cannot reason from particular instances to a universal proposition, unless we as- sume as basis of the whole inductive process some postulate which has real universal significance. Otherwise, we reach only a high degree of prob- ability, but not necessity; a rude generalization, but not universality. When we assert some such general statement as this, that arsenic always acts as a poison, we have the universal character of the proposition upon an underlying postulate that is 22 INDUCTIVE LOGIC understood even though it is not expressed, such as the uniformity of nature, that under identical con- ditions we always look for identical effects. This will be discussed later more in detail ; it is re- ferred to at this point merely to illustrate the de- ductive basis of induction. Bradley insists that there can be no such thing as induction, because it always rests upon an implied universal which gives to the process as a whole a deductive character. 1 His criticism has the force only of proving that induction cannot be independent of deduction. This dependence does not, however, necessarily vitiate the integrity of induction as a mode of the inferential process. Lotze has placed special em- phasis upon this dependence of induction upon deduction. He says : " It is the custom in our day to collect into one body the numerous opera- tions which assist us in ascending from particulars to generals, or to call this inductive logic, and to set it against the deductive or demonstrative logic along with much disparagement of the latter. Such disparagement rests on a mistake. The inductive methods, it is certain, are the most effectual helps to the attainment of new truth, but it is no less certain that they rest entirely on the results of deductive logic." 2 Moreover, in induction the results obtained and formulated in general propositions may be extended, and often modified by a deduction which is based 1 Bradley, Principles of Logic, p. 332. 2 Lotze, Logic, p. 288. See also Bosanquet, Logic, Vol. II. p. 119. INDUCTION AND DEDUCTION 23 upon them as major premises ; for the deduction thus proceeding from them reveals new instances which conform or perhaps modify the simple induc- tive results themselves. What is popularly called a hasty generalization, if made a major premise of a syllogism, will often lead us astray through the deductions drawn from it. As soon as we are aware of this, we return to question the validity of the generalization, whose weakness is not appreciated until thus tested and revealed. Thus deduction serves to extend and correct the results of induc- tion, and at the same time it itself is dependent upon the results of inductive generalization for the material to form its premises. We come to see, therefore, how intimately associated these two proc- esses are in actual reasoning. For convenience of illustrating their individual characteristics, they may be considered as separate, and each investigated as an independent mode of inference. But they are in reality mutually related and dependent, and are always found manifesting their functions together. In any course of reasoning concerning the conduct of our every-day affairs, or in scientific investigation, anywhere, indeed, outside of the artificial examples of logical text-books, we reason both inductively and deductively in one complex process. CHAPTER III The Essentials of Induction We now proceed to a more precise determination of the nature of induction. Its point of view in all reasoning regards concrete instances. They are the data, and from them general propositions are to result. The procedure is from given facts to laws which are the ground and explanation of these facts. We are here, however, at once struck with the evident break in the course of our reasoning. Pro- cedure from the particular to the universal cannot be a continuous process. There is a gap somewhere. The conclusion contains more than the premises. In deduction, we are proceeding from the greater to the less, and we experience no violation of our logical sense ; but at once we appreciate the diffi- culty which attends the reverse process, from the less to the greater. Here we soon reach a point where we pass beyond the sphere of our experience to the generalization which necessarily embraces far more than our experience. This is the so-called inductive leap ; or it is sometimes referred to as the inductive hazard. But is this a leap in the dark a wild guess concerning all that lies beyond the sensuous sphere of our immediate experience ? This 24 THE ESSENTIALS OF INDUCTION 25 would be the case, were we compelled to use the mere data of experience as sole ground for our inferences. John Stuart Mil l insists that nothing whatever is given in consciousness but particular sensations, and these are but subjective states of feeling, and with no assurance of any definite cor- respondence with the external world. With such purely empirical data it is impossible to proceed to truths of universal validity. It is necessary to post- ulate some universal truth which the mind through strictly a priori considerations is constrained to formulate, and which will serve to bridge the gulf between the particular and the universal. This postulate has been variously expressed by different authors, yet with substantially the same significance in all. In the older logic, it is put under the convenient formula of the uniformity of nature; that is, that beyond the sphere of experi- ence, phenomena will behave in the same manner, under like conditions, as in the sphere of immediate observation and experiment. In the modern logic this is somewhat differently expressed. The phrase "uniformity of nature," being somewhat indefinite and implying a point of view purely objective, is not used. Modern writers have omitted it largely from their terminology. Lotze says : " The logical idea upon which induction rests is by no means merely probable, but certain and irrefragable. It consists in the conviction, based upon the principle of identity, that every determinate phenomenon M can depend upon only "one determinate condition, and accordingly that, where under apparently dif- 26 INDUCTIVE LOGIC ferent circumstances or in different subjects P, S, T, IT, the same M occurs, there must inevitably be in them some common element % which is the true identical condition of M, or the true subject of MP 1 We have a somewhat similar description of the basis of the inductive process given by Sigwart: "The logical justification of the inductive process rests upon the fact that it is an inevitable postulate of our effort after knowledge that the given is necessary, and can be known as proceeding from its grounds according to universal laws." 2 Bosanquet considers as the basis of inductive inference that which he calls the postulate of knowledge, that " the universe is a rational system, taking rational to mean not only of such a nature that it can be known by intelligence, but further of such a nature that it can be known and handled by our intelligence." 3 I have, quoted these passages from Lotze, Bosan- quet, and Sigwart, that we may appreciate the mod- ern tendency to derive the inductive postulate from an epistemological source ; namely, that our knowl- edge must be consistent throughout with itself, part to part, and parts to whole, and that the world for us is the world as constructed by our knowledge. Whatever is given in consciousness must belong therefore in the one place where it appropriately and necessarily belongs. Here also there must be a place for everything, and everything in its place. There must be a uniformity of consciousness ; that 1 Lotze, Logic, p. 102. 2 Sigwart, Logic (Eng. translation), Vol. II. p. 289. 3 Bosanquet, The Essentials of Logic, p. 166. THE ESSENTIALS OF INDUCTION 27 is, the primary postulate and the uniformity of nature is secondary to this, and implied in it. This postulate may also be expressed as follows : What is once true, is always true. Here true is used in the sense of the universal significance of a fact. Whenever a concrete instance is present in con- sciousness, its existence must be considered as necessitated by some antecedent which can satis- factorily account for it, and which can at the same time be appropriately adjusted to the whole of our knowledge in interpreting it. Bosanquet says that " ideally speaking every concrete real totality can be analyzed into a complex of necessary relations." * These necessary relations of course have a universal significance, and therefore in every concrete in- stance, if we can rightly interpret it, we may dis- cern the universal element which is contained in it, and gives it a place and meaning in the world as cognized by us. There is a sense in which induction may be re- garded as the inverse process of deduction. In deduction the problem is concerned with the ques- tion, What does the universal necessitate ? In induction, the instance is given, and the problem is, What universal can be discovered which could give rise to the instance in question ? This view of induction is especially associated with the name of Jevons, whose inductive system is described as the inverse of deduction. He calls it the decipher- ing of the hidden meaning of natural phenomena. 2 1 Bosanquet, Logic, Vol. II. p. 82. 2 Jevons, Principles of Science, p. 124. 28 INDUCTIVE LOGIC The name commonly used to designate this view of induction is that of "reduction/' originally sug- gested by Duhamel. 1 This process was known to the old logicians, who called it " Method " to denote the process of hunting for middle terms by the aid of which a given conclusion could be proved. 2 Like all inverse processes, it is by itself an indeterminate one. Given All A is B, and All B is C, we infer by the direct process of deduction that All A is a But in the indirect or inverse process we have given all A is C, and the problem, to find a middle term which necessitates such a conclusion, is an indeterminate one. There may be a number of middle terms. This is analogous to the method of integral calculus ; while differentiation leads to a definite result, the inverse process of integration leads to an indeterminate result. So also we mul- tiply two numbers, producing one determinate re- sult ; but inversely, when we have given a certain number, and ask what factors multiplied together could produce this number, we may reach several different solutions. The answer is indeterminate. Professor Jevons, in his scheme of inductive infer- ence, falls back upon probability to indicate which of several possibilities is the most likely one in the i Duhamel, Methodes, Vol. I. p. 24. 2 Venn, Empirical Logic, p. 361. i9 ^ Y- OF THK UNIVERSI r THE ESSENTIALS OF INDUCTION given case. 1 But before the inverse operation can re- sult in determinate results, the given terms such as A and must be subjected to some analysis in order that their material signification may give sugges- tion as to the nature of the middle term. For in- stance, a man is found dead, washed ashore by the tide ; the natural supposition would be that he met his death by drowning. And yet it might possibly happen that the man died through injuries inflicted by blows, or by poison, or heart failure. The at- tendant circumstances and bodily indications must suggest the most probable cause to account for the given effect. Venn criticises Jevons' view of induc- tion, making it the inverse process of deduction, on the ground that it is purely a formal process, and therefore can lead only to indeterminate results. 2 / It is always possible, however, to make some analysis of the material significance of the data, as has been above indicated, which relieves the purely formal processes from the indefiniteness of the re- sults. Bosanquet criticises Jevons' theory of in- ductive inference, in that the hypothesis proposed to account for the given in reality can at best be only highly probable. 3 However, Venn, Lotze, Bo- sanquet, Sigwart, all allow a place to the inverse function of all inductive reasoning; their conten- tion, however, is this, that it does not furnish an adequate account of the whole matter. 4 1 Jevons, Principles of Science, p. 219. 2 Empirical Logic, p. 359. 3 Bosanquet, Logic, Vol. II. p. 175. 4 Venn, 361 ; Bosanquet, Vol. II. p. 175 ; Sigwart, Vol. II. p. 203, 289. Lotze, Outlines of Logic, p. 93. 30 INDUCTIVE LOGIC It is interesting to note that Whewell's theory of induction corresponds in the main to this idea of reduction, or inverse process. He finds in induc- tion a twofold operation of the mind, consisting in the colligation of facts and the explication of con- ceptions. By the colligation of facts he refers to that insight which is able to see the connections and relations which necessarily exist between the different phenomena present in consciousness ; and by explication of conceptions he refers to the ap- propriate fitting in of these related facts to some conception of the mind which most readily ac- counts for them. 1 Such a process is merely the reading of given facts backward to their origin, or substantially an inverse process, where the pro- cedure is from the given concrete to the explanation of the same in terms of the universal to which it can be most appropriately referred. So also Mill's account of procedure by hypothesis, as we shall see later on, presents characteristics similar to this process of reduction. The end of induction is to discover a law having objective validity and universal application. There is a distinction which must be noticed and clearly kept in mind ; namely, the distinction between a law and a rule. Induction seeks a law, and not a rule. A law expresses the essential and universal rela- tions subsisting between given phenomena, elimi- nating entirely all accidental and local coloring. A law has objective validity, and preserves a constant nature. There can be only one law in reference 1 Whewell, Philosophy of the Inductive Sciences, pp. 172, 202. THE ESSENTIALS OF INDUCTION 31 to one and the same connection of facts. A rule, however, is subjective, dealing with the individual's attitude to phenomena, rather than an explanation of the essential features of the phenomena them- selves. It often is determined in the concrete by that which is external, local, and accidental. There may be many rules, varying with many minds and many climes. Fundamental and universal laws of political economy become maxims and rules in dif- ferent communities. The laws of morality, univer- sal and immutable, become rules of conduct in in- dividual experience admitting of wide difference of opinion and diversity of application. 1 In the proc- esses of induction, therefore, the law is the desidera- tum, and not the rule. Law, however, is used rather loosely in our ordi- nary terminology. Law as used in jurisprudence has a meaning quite different from law as used in physical science. And so, also, the laws of biology, the laws of political economy, the laws of ethics, are referred to with different shades of meaning in each sphere. However ambiguous may be the sig- nificance of " law " in ordinary thought and usage, nevertheless in induction it has a constant and a simple significance, which if carefully adhered to will avoid confusion, and obscurity as well, in our inferential processes and results. Law in induction is always in the form of an hypothetical universal : If A is, B is. It does not assert what has happened, but what 1 Lotze, Logic, p. 335. 32 INDUCTIVE LOGIC should happen under certain conditions. Given the antecedent A, a certain determinate consequent B is always necessitated. The relation is constant and invariable, and therefore has a universal signifi- cance. Induction holds a peculiar and important place in our every-day life, because it has to do with the analytical treatment of instances as they appear in experience. The large part of our conscious think- ing has to do with the concrete, the raw material of experience ; this, induction alone can handle. Leonardo da Vinci's maxim was " to begin with ex- perience and by means of it to direct the reason." 1 Thus the superstructure of knowledge is raised day by day. The given is continually being interpreted and referred to its appropriate place, as the stones of the quarry are hewn and fitted in their proper position in the building for which they have been designed. There are certain individual experiences which it is impossible to determine through our syllogistic forms. They cannot be judged deduc- tively. There is no general category under which they can be subsumed. They may be formally illogical if thus expressed, and yet admit of direct investigation and experiment in an inductive man- ner, for the purpose of disclosing the law under- lying them and as yet unknown. It often happens that through indifference or indolence, we are content to refer many phenomena to long-established and convenient categories, which, if investigated independently, we would find could 1 Ueberweg, Logic, p. 42. THE ESSENTIALS OF INDUCTION 33 not possibly be so treated. The convenient pigeon hole, because near at hand, receives much that does not properly belong there. It is the office of induc- tion to investigate anew the old material, and then to reclassify in accordance with the revised gen- eralizations which such investigations may neces- sitate. The procedure by induction is in keeping with the scientific spirit of the day, to interpret the phenomena of nature as given, and not to antici- pate nature through preconceptions, and wrest fact in order to fit theory. It comes to the sources in nature with empty vessels to draw and carry away that which nature alone can give. CHAPTER IV Types of Inductive Inference The process of induction, as we have seen, is a procedure from given instances to the discovery of the law which underlies them, and which is the ground of the connection of the various attributes and relations that unite in the one concrete whole. Viewed from the standpoint of the direction of the process, we have found that it is always towards some general expression of individual experiences, and in this respect it is the inverse of deduction, which proceeds from the general to the particular which is embraced in it. There is, however, another and important point of view that should not be over- looked. We have to consider the mode of the proc- ess as well as its direction ; not merely the result to be attained, but also the peculiar manner of realizing the same must be considered. Difference in method here gives rise to various kinds of induc- tive inference. The end proposed in all is to gen- eralize our experiences as they occur in the concrete and particular. When I find a given phenomenon, A, given in consciousness, and characterized by several distinctive features among which I note specially the mark jB, the question at once most 34 TYPES OF INDUCTIVE INFERENCE 35 naturally suggests itself Is there a reasonable expectation that I shall always find B as an insep- arable accompaniment of A, so that I can assert confidently that whenever A is found, B also will be found ? There are three ways of satisfying ourselves as to the existence of any constant rather than coincidental connection between antecedent and consequent, as A and B. These give rise to three different methods of inductive research, and they are as follows : I. The Method of Enumeration. II. The Method of Comparison, or Analogy. III. The Method of Scientific Analysis, or Search after Causal Connection. Failure to distinguish between the three methods has given rise to confusion in the definition of and corresponding reference to inductive inference; some authors use induction in one, and some in another of these senses. It is necessary to dis- criminate carefully, and to maintain a strict con- sistency in the usage of the terms as defined. I. . The MetJwd of Enumeration. We observe the various instances in which certain attributes, as A and B, are conjoined in our experience. We count them in the sense of noting to what extent they accumulate, without noticing any exception to what seems at least an invariable connection. We do not necessarily count by precise enumeration reach- ing a numerically definite result. We notice merely to what extent the observed instances of like nature accumulate; that is, whether a few, a considerable number, or a very large number. The 36 INDUCTIVE LOGIC mere number of instances produces a certain psycho- logical impression, whatever may be their logical force. This is brought about through the laws of association, and creates an expectation of a con- tinuous repetition of the experience in question. This arises from a natural tendency of the mind to generalize. We observe that crows are black ; and the increasing number of confirming instances goes far to establish a connection between the crow and its color which seems to have universal validity. The enumeration of instances may lead us to any one of three results : 1. We may meet with no exception whatsoever, until the scope of observation completely embraces the sum of all possible instances. This is complete enumeration, and when enumeration reaches this limit, it passes over into deductive reasoning, by virtue of the logical canon that whatever is true of the parts is true of the whole distributively ; that is, provided the summation of the parts has been an exhaustive one. We assert that all the sheep of a given flock are white ; for we have observed each separately and no one has been missed in the count. So, also, the judgment that all planets move around the sun, resulting from an enumeration of the planets one by one. It is possible also to have a perfect in- duction with an infinite enumeration of parts. This is possible in two cases, as pointed out by Beneke. 1 First, when the parts are connected together contin- uously in space, so that a survey of all is possible in a finite, and often a very short time. This occurs 1 Quoted by Ueberweg, Logic, p. 482. TYPES OF INDUCTIVE INFERENCE 37 in geometrical demonstration when the inference, based upon the simple figure it refers to, is extended to all figures falling under the like definition. And second, when the parts are not continuously con- nected, if it can be proved syllogistically that what is true of a definite ?ith part, must also be true for the (n + l)th part. Perfect induction also embraces arithmetical method and computation. Here the whole, which is the sum of the facts in each case, is a totality or universal whose differences, which are all sepa- rate and distinguishable, are yet homogeneous and equal. 1 There is no qualitative differentiation of parts, only a quantitative one. The total is the sum of the units, and each unit is like every other one. If we have one hundred units making a totality, the one that may be the twenty-seventh is precisely like the sixty-seventh. It is a case' where each one counts for one, and no one for more than one in an absolutely literal sense. It has been urged against perfect induction that it affords no new information, and, therefore, its results are not valuable. However, the summation of particulars in abbreviated forms is always an advantage. It is a labor-saving process to the mind. It enables the mind to retain a large number of facts by throwing them into one and the same cate- gory; and it facilitates arithmetical processes by convenient comprehending of units within a totality. 2. The second result that is possible, is that, in counting instances, our enumeration should prove 1 Bosanquet, Logic, Vol. II. p. 54. 38 INDUCTIVE LOGIC incomplete. From the necessities of the case, we are often not able to observe the entire sphere of possible occurrences and cover the whole ground. It may be that beyond the sphere of our expe- rience, the constant connection between certain phenomena may be disturbed by the appearance of some variable factor of which we have been wholly ignorant. It is the possibilities beyond the sphere of observation which render uncertain the results of our count. We are sure as far as we have observed ; but we have not gone far enough perhaps. Such results, formulated in general prop- ositions, are termed empirical laws ; that is, gen- eralizations from an experience necessarily limited. 3. We have still a third case ; where in our enumeration of positive instances we meet with exceptions to a greater or less extent. Here we cannot even sum up the actual experience in terms of a generalization. There are outstanding excep- tions which will invalidate it. We must, therefore, fall back upon the theory of probability and the calculation of chances, presuming that, in general, we will meet with the same proportion of excep- tions to positive instances in the future, that we have already observed in the past. So we make, in our minds at least, comparative tables of posi- tive cases over against exceptions, and reach a summary of the result in the form of a ratio, whose numerator will be the number of positive cases observed, and the denominator the total num- ber of instances including positive instances and the corresponding exceptions. We observe that TYPES OF INDUCTIVE INFERENCE 39 some cryptogamous plants possess a purely cel- lular structure ; others, however, do not, being partially vascular. The probability that a new cryptogam will be cellular can be estimated only on the ground of the comparative number of known cryptogams which are cellular, as over against the total number of cryptogams, both cel- lular and vascular, previously observed. 1 II. The Method of Analogy. Here, also, we start with the experience that A is characterized by the mark B. But there is additional knowledge of which we may avail ourselves in the generaliza- tion of some past experience already effected, such as the following : that A very closely resembles C, in that the two have many properties or attributes in common. The inference by analogy is that C also, as well as A, will have the mark B. It may be that we cannot examine C in a number of various instances to see in how many the mark B occurs. Our only resource is the inference which is based upon the known resemblances, or analogies. This kind of inference, for example, was employed by Sir Isaac Newton in a very interesting manner. He had ob- served that certain "fat, sulphureous, unctious bodies," such as camphor, oils, spirit of turpentine, amber, etc., have refractive powers two or three times greater than might be anticipated from their densities. He noticed also the unusually high re- fractive index of diamond, and from this resem- blance, based upon similarity in reference to one attribute only, he inferred that diamond also would 1 Jevons, Principles of Science, pp. 146, 147. ^cftE LIBffi^ 40 INDUCTIVE LOGIC prove to be combustible. His prediction in this regard was verified by the Florentine Academicians in 1694. 1 Brewster made a striking comment upon Newton's inference, to the effect that if Newton had drawn a like analogy in reference to greenock- ite and octahedrite as he did concerning diamond, inasmuch as they, too, have a very high refractive index, he would have been wholly incorrect. This is an indication of the fact that argument by anal- ogy is not conclusive. Bosanquet has very strikingly expressed the es- sence of the analogical method in saying that " in Analogy we weigh the instances rather than count them." 2 The distinction between analogy and enumeration of instances lies in this, that in the former we count similar attributes in the contents of two instances, and balance them against the dis- similar or unknown. In induction by enumera- tion we count similar instances, considering them in their totality without examination and compari- son of their respective attributes. III. The Method of Scientific Analysis. The instance in question, A, which is characterized by the mark B, is subjected to a vigorous analytical examination, to show that A and B are related through a causal connection. This analysis is effected either through a minute observation or by means of exact experiment. The end to be attained by such analysis is to separate a complex phenomenon into its several elements, by which 1 Jevons, Principles of Science, p. 527. ' 2 Bosanquet, The Essentials of Logic, p. 155. TYPES OF INDUCTIVE INFERENCE 41 i process a causal connection may be revealed, whose very existence is disguised by the complexity of the phenomenon. For instance, the phenomenon of death following the taking of arsenic is an event so complex as to evade a precise determina- tion of the causal relation. When analyzed into simpler elements, it is found that the immediate effect of arsenic upon the bodily tissues is to harden them so as to prevent their normal function- ing. This is the causal ground of the death due to arsenic. Moreover, this analytic process, which may be appropriately called a material one, is sup- plemented by a formal process of negation, that is, an instance in which the suspected causal element is absent in the complex phenomenon under in- vestigation, and the related effect, before observed, now no longer appears. This formal process acts as a check, and as a verification as well, of the material analysis of the phenomenon. For ex- ample, an antidote, as sesquioxide of iron, being administered, no death from arsenic occurs ; and it is also observed that no hardening of the tissues has resulted, therefore the former result, hardening of tissues producing death, has been thus corrobo- rated negatively by the fact that where no harden- ing of tissues has resulted, death does not follow. We see at once the advantage of such a method over that of counting all instances where taking of arsenic has caused death. The latter is a phenom- enally adjudged result ; the former penetrates with analytic insight to the ground of the phenome- non itself. Thus one instance, if its parts and their 42 INDUCTIVE LOGIC manifold relations are adequately comprehended, may suffice for a universal conclusion based upon it. It is true, however, as remarked by Bosanquet, that " number of observations does, as a rule, assist analy- sis and contribute to eliminating error. Scientific analysis as such, however, does not deal with in- stances, but only with contents." 1 In cases where the phenomenon does not reveal its component elements under observation, and it is impossible to subject it to experiment, the most likely cause of the effect in question is tentatively judged to be the real cause, until it can be verified in reality. This is procedure by hypothesis, and is always resorted to as preliminary to a subsequent experiment which is its test, or else in lieu of such an experiment when it is by the nature of the case precluded. It is a form of ideal analysis. The ex- periment is constructed mentally. The phenome- non is separated into what we would reasonably imagine its simpler elements would be. We are constrained to believe that if the hypothetical ante- cedent existed, it would be adequate to produce the effect. Although rising in the sphere of the imagi- nation, it is that with which the mind is, for the time at least, satisfied as an explanation of the facts which demand some cause to account for them. Regard- ing induction as a process of reduction, hypothe- sis is the assumed universal, or middle term, which will necessitate the phenomenon under investiga- tion as its logical conclusion. We will now proceed to a further examination 1 Bosanquet, Logic, Vol. II. p. 118. TYPES OF INDUCTIVE INFERENCE 43 of these methods, considered both singly and to- gether. 1. They all proceed upon the supposition that what is given in consciousness has some necessary ground for its being. In enumerative induction, there is some causal connection presupposed, yet in a very general and indefinite manner, and accom- panied by no analysis of the various concepts either by a systematic observation or experiment. It is a vague sense of uniformity which, when observed for many times, we feel will continue indefinitely. That which has happened often and not contradicted car- ries with it a certain convincing power by dint of bare repetition, especially to persons of narrow experi- ence, and unaccustomed to discriminating observa- tion. Ueberweg has made the following comment in reference to the so-called imperfect induction. " The conclusion is made universal with more or less prob- ability, and the blank which remains over in the given relations of spheres is legitimately filled up partly on the universal presupposition of a causal- nexus in the objects of knowledge, partly on the particular presupposition that in the case presented such a causal-nexus exists as connects the subject and predicate of the conclusion. The degree of prob- ability of the inductive inference depends in each case on the admissibility of this last presupposition, and the various inductive operations, the extension of the sphere of observation, the simplification of the observed conditions by successive exhaustion of the unessential, etc., all tend to secure its admissibility." 1 1 Ueberweg, Logic, pp. 483 f . 44 INDUCTIVE LOGIC Analogy likewise proceeds upon the assumption of an underlying cause among the observed phenomena, and.this is more definitely in the foreground through- out the process than in that of induction by enumer- ation. Analogy is based upon the postulate that sim- ilar phenomena have similar causes ; the greater the agreement of the various attributes of the different phenomena compared, the greater will be the result- ant probability that causes capable of producing them as effects will be similar. The similarity of the lightning flash to the electric spark suggested to Benjamin Franklin the possibility that they were due to a like origin, and by experiment his analogical reasoning was actually confirmed, as is well known. Upon the theory that the world as it exists for us in knowledge forms a system to some place in which every phenomenon given in experience must be appropriately and necessarily referred, it follows, therefore, that a simple experience, devoid of any complexity of parts, may fit into several possible places in our world of consciousness, and remain so far forth indeterminate. However, a complex phe- nomenon, with many parts intricately connected, will fit into one unique place only in the system to which it must be referred. It is like a key that will fit into only one lock. The presumption, there- fore, is that any other phenomenon which resem- bles the first through much of its entire content, part for part, attribute for attribute, will also re- semble it further as regards other attributes not yet examined, so as it will likewise fit into the peculiar place in the system of knowledge to which the TYPES OF INDUCTIVE INFERENCE 45 first has been found to belong. There is always a strong probability that agreement in spheres of great complexity is not a mere coincidence, but the result of a causal relation. One characteristic of a system, which we have found to be the ground of inference generally, is the co-ordination of like things under one concept. Analogy, therefore, is based upon the view of causal connections within the system which comprises the world as given in consciousness. In the third method, the causal relation is more prominent still, and the search for it characterizes the procedure employed. That, which in the other methods may exist merely as a vague impression, is here formulated and made the direct and sole object of research. 2. The three methods in the order here presented show an increasing prominence given to the causal connection in the phenomena of experience. And therefore they possess a relatively increasing scien- tific value. As the first has only indirect reference to the causal connection of its facts, it is the least trustworthy and has no claim as a scientific method. It breaks down as soon as an exception is noted ; and is even weakened by the fact that it is con- stantly menaced by the possibility at least of the appearance of an exception. " How do we know," says Green, " that the instances, with the examina- tion of which we are always dispensing on the strength of the rule (that is, our generalization), might not be just what would invalidate it, if they were examined ? " x We may arrive at the conclu- i Green, Phil. Works, Vol. II. p. 282. 46 INDUCTIVE LOGIC sion, based upon our observation and consequent record, that all sheep are white, and yet black sheep do occur, even in every flock, as the proverb has it. According to Aristotle, the proposition that all swans are white, was a perfectly general one, and yet in recent times black swans have been discovered in Australia. Bacon's criticism upon this method has become classic : " Indue tio quae procedit per enumerationem simplicem, res puerilis est et precario concludit et periculo exponitur ab instantia contradictoria et plerumque secundum pauciora quam par est et exiis tantummodo quae presto sunt pronunciat." 1 The validity of this method of procedure depends largely upon the probability of our meeting and noticing exceptions were they to occur. As Lotze puts it: "A man who never observes a place of public resort but once in every seven days, and that on a Sunday afternoon, has no right to suppose, because it is crowded then, that it is as crowded on a week-day." 2 He is here in no position to note the exceptions even should they occur. Analogy, unless confirmed by experiment, or upon the ground of resemblance established by a verifiable hypothesis, has no claim to be considered as a scien- tific method. There may be false analogies depend- ing upon surface resemblances. A child might conclude that oil would put out fire because it so closely resembles water, which he knows can ex- tinguish the flames. The difference between essen- tial and accidental agreement between phenomena 1 Novum Organon, i. 105. 2 Lotze, Logic, p. 343. TYPES OF INDUCTIVE INFERENCE 47 can be revealed only when the underlying cause is ascertained. The third method alone has scientific worth. True induction must be a continued search to dis- cover a causal relation. 3. The two first processes fulfil their functions largely as tentative and suggestive methods. In enumeration of instances, we are often led to note resemblances which become the basis of analogy. And analogy suggests, in turn, hypothesis which is capable of verification through subsequent experi- ment. The question may be put, "Which of the three processes is induction proper ? " The fact is that it may involve all three, but it is not complete until it reaches the third, the experimental method. Analogy is especially fertile in suggestion. Scien- tific minds most carefully trained and versed in scientific methods of research are often most keen in noting resemblances, and detecting analogies which become the basis of their experiments. New- ton possessed that rare insight which, in spite of the manifest dissimilarity of the two phenomena, could yet discern an essential likeness between the fall of an apple and the gravitating force of the moon towards the earth. 4. It is also to be observed that the choice of method will depend largely upon mental habit. Some minds naturally or by special training and surroundings are given to experiment. They have a testing facility and inventive capacity. Others naturally are susceptible in an unusual degree to 48 INDUCTIVE LOGIC contrasts and resemblances. Others again are ac- customed to accurate observation that is ever push- ing beyond and seeking to extend its sphere. Thus we have a natural division of these methods accord- ing to psychical proclivities. The choice of method is often conditioned by the force of circumstances. Experiment is not alway possible. Are all crows black ? There is no connection between the general organism of the crow and its color that has thus far been revealed through analysis or experiment. The only recourse is to number instances over the widest possible field. We say, moreover, that Mars may be inhabited ; for it has an atmosphere similar to the earth and therefore capable of sustaining life. Analogy is the only guide in such a case, and it is impossible to verify it either by observation or experiment. 5. All the methods tend to one end, that of ef- fecting a generalization of experience. The gen- eralization may be either a numerically general one, or one expressed in terms of a generic concept. (1) The former consists in the extension of several instances to their repetition under like conditions. (2) The second consists in the extension of several instances to all cognate species under the same genus. Examples of these two kinds of generalization are as follows : The general proposition that all sulphur is combustible is of the former kind; all instances are substantially of the same nature, and do not differ as distinguishable species under the same genus, but rather a repetition of like phenom- TYPES OF INDUCTIVE INFERENCE 49 ena. The general concept in the above proposition is of the nature of an infima species. On the other hand, the proposition that all mammals are verte- brates, has the subject-term in form of a generic concept. Many species, differing widely among them- selves, may be embraced under it. 1 i Sigwart, Logic, Vol. II. pp. 310, 311. CHAPTER V Causation We have seen that induction as a truly scientific method consists in the analytical determination of the relations of cause to effect in any complex phe- nomenon, accompanied by a generalization of the result obtained. The final outcome of such a proc- ess is an universal concept which embodies a law, expressed in terms of a constant connection between antecedent and consequent. As Green has said, " The essence of induction consists in the discovery of the causes of phenomena." 1 A causal view of the universe gives rise to logical concepts, whereas a mythological view of the universe, as in ancient times, resulted in mere empirical concepts, which gave no assurance either of stability or invari- ability. It will be necessary, therefore, to de- termine more precisely the logical significance of the causal idea, which seems to underlie all induc- tive inference. This is no easy task. According to Clifford, cause has sixty-four meanings in Plato, and forty-eight in Aristotle. 2 i Green, Phil. Works, Vol. II. p. 284. 2 Clifford, Lectures and Essays, Vol. I. p. 149. 50 CAUSATION 51 The causal idea has sometimes found expression in the phrase, the uniformity of nature, or it is often referred to as the doctrine of universal causation. These two phrases. are often used interchangeably; this gives rise to confusion of thought, for their meanings are quite distinct. Uniformity of nature, strictly interpreted, means that like antecedents, under precisely the same conditions, will be fol- lowed by like effects ; this idea expresses one phase of causation, viz. its invariability. The doctrine of universal causation, however, expresses the impos- sibility of phenomena rising spontaneously, without an antecedent, or antecedents, sufficient rationally to account for them. The two ideas lie at the root of the causal idea. As Tennyson has put it : For nothing is that errs from Law. Some confusion has also arisen from the failure to discriminate precisely between the philosophical and the purely logical questions relative to the general subject of causation. Causation may be viewed from three different points of view : 1. What it is phenomenally, that is, as regards its physical aspects. 2. What it is essentially, as regards its real nature. This is a metaphysical question. 3. What it is in respect to its characteristic attribute of invariability. This is a purely logical question. (1) As to the first, what is causation phenome- nally? What is its purely physical significance? Investigations in this line have led to the doctrine 52 INDUCTIVE LOGIC of the conservation of energy. This is substan- tially the assertion that, in every event, no new energy is called forth which did not exist before, potentially at least, nor can any energy be ulti- mately lost; nothing new is created, there is only a change or transfer from one state or condition to another. Moreover, the sum total of energy in the universe is a constant quantity; it can neither be added to, nor subtracted from. There is an excel- lent illustration of this theory in the admirable chapter on " Conservation of Energy " by Professor Tait. I give it somewhat in full : " I allow an electric current to pass through a galvanic battery and there is for the moment a certain quantity of zinc consumed, or, as we may put it, a certain quan- tity of potential energy in the battery has been converted into the kinetic energy of a current of electricity. That current of electricity passes round some yards of copper wire, coiled round a bar of iron or a number of fine iron wires which are stand- ing vertically inside this apparatus. The moment the current passes, these iron wires are converted into magnets, but, in consequence of the conserva- tion of energy, while this is going on they weaken the current. The current of electricity becomes weaker in the act of making the magnet, but the moment the magnet springs into existence, it again is weakened, because, from the necessities of its position, its mere coming into existence necessitates the passage of a new current of electricity in an- other coil of wire which surrounds this externally, and finally this last current we can use to produce CAUSATION heat, or light, or sound." 1 In this cycle of changes we see how closely connected even disparate phenom- ena are, and how the appearance of energy in any one definite state is dependent upon its previous existence in some other state. The doctrine of conservation of energy, we shall see later on, may be suggestive as to the nature of the analytical treatment of cause and effect. (2) The philosophical question as to the inner nature of causation met with one answer generally until the time of Hume; namely, that the idea of cause signified that the antecedent was efficient in producing the corresponding consequent, implying the transfer of power sufficient to bring about the effect. Hume, however, contended that in the greatest possible extent of our knowledge, all that we certainly know is this, that one event follows another. We have no ground for an assertion concerning the manner in which the sequence is effected, nor assume any real tie between them. Hume insisted that phenomena were conjoined, but never connected. 2 His opponents, as Kant and others, deny him, however, his fundamental posi- tion, that the origin of the causal concept comes from experience alone. They urged that it has an a priori origin, a concept simple and unanalyzable, given through intuitive insight; developed in the sphere of experience, but not dependent upon expe- rience for its warrant. It is an interesting fact that the idea of the conservation of energy devel- 1 Tait, Recent Advances in Physical Science, pp. 76, 77. 2 Hume, Essay on Idea of Necessary Causation. 54 INDUCTIVE LOGIC oped subsequent to Hume's time. It seems to give evidence which Hume insisted was not and could not be forthcoming, namely, concerning the idea of the antecedent as an efficient power. Through the modern doctrine, the impression of a transfer of real power is produced, though its mode and man- ner still remain a mystery. (3) The logical aspect concerns not the phenome- nal manifestation of cause and effect, nor their inner nature, but rather the element of invariability in causation. Two questions here suggest themselves : First, Is invariability a fact, a constant ele- ment in causation ? Second, How do we account for its existence ? The first only has truly logical significance. The invariability of causation, that like antecedents under precisely the same condi- tions produce like effects, alone makes induction possible. Mill says that it is the belief in the uniformity of nature which stands as the ultimate major premise in every process of induction. Hume accepted it, and based inferences upon it, and never challenged it as a working basis as regards the affairs of every-day life. He acknowl- edged the element of invariability, and only denied the bond of connection. This element has peculiar logical significance : without it, it would be impos- sible to extend our knowledge beyond the seen and the heard, indeed that which is seen and heard would then have no meaning, and no basis for their interpretation and appreciation. Being assumed, however, in our logical postulate, we have a basis for induction, a constant to be sought for, and to CAUSATION 55 be depended upon, in explanation of the past and in prediction of the future. When we come to the second question, which is essentially a genetic one, how the belief in the uni- formity of nature arose, we find two classes which answer respectively that the belief arose a priori, and on the other hand, from experience simply. The former is the opinion especially associated with the Scottish School of philosophy. Hume holds that it proceeds from a psychological law of custom or habit, an unbroken line of mental associations ^ inducing a belief within, concerning the uniformity of nature without. Mill has also a like empirical basis for a belief in the uniformity of nature ; he holds that having observed uniformity in many experiences, in fact never contradicted, we general- ize so as to cover a sphere beyond our experience. Moreover, we possess the consensus of testimony, coextensive with the history of humanity, of the indefinitely wide extent of the sphere of causation, and the accompanying characteristic of uniformity. His position is fortified by the fact that in the process of incomplete induction, its probability is strengthened where there has been exceptionally abundant scope for observation, so that there is the overwhelming conviction that if there had been a time or place where the law would prove untrue, it wOuld have been noticed. Instead of universal causation, Mill and his followers make a more cautious statement ; causation as coextensive with the sum total of human experience. This is abundantly adequate to embrace all possible cir- 56 INDUCTIVE LOGIC cumstances of practical inference. The immensely high degree of probability en^en^ers a subjective certitude which in every-day conduct of affairs, and even in the more exact requirements of scientific investigation, is never questioned. Preyer has given an interesting account of the extremely early appearance of the appreciation of the causal relation in the case of his child, " who, at the three hundred nineteenth day of its life, struck several times with a spoon upon a plate. It hap- pened accidentally, while he was doing this, that he touched the plate with the hand that was free ; the sound was dulled, and the child noticed . the differ- ence. He now took the spoon in the other hand, struck with it on the plate and dulled the sound again, and so on. In the evening the experiment was renewed with a like result. Evidently the function of causality had emerged in some strength, for it prompted the experiment. The cause of the dulling of the sound by the hand was it in the hand, or in the plate ? The other hand had the same dulling effect, so the cause was not lodged with the one hand. Pretty nearly in this fashion the child must have interpreted his sound-impres- sion and this at a time when he did not know a single word of his later language." l The theoretical soundness of Mill's speculations, however, has a flaw, although the practical results may not be thereby invalidated. The inductive proc- ess, which is supposed to be a truly scientific method, and superior to induction by simple enumeration 1 Preyer, The Senses and the Will, pp. 87, 88. CAUSATION 57 must, according to Mill, at the last analysis, rest upon a principle which is itself based upon an in- complete induction. A very fair and searching criti- cism of Mill is that of Venn's in his Empirical Logic} Whately insists that the whole question concerning the nature of our belief in uniformity is irrelevant, as it is a purely psychological and not a logical one. Mansel holds a mediating position in insist- ing that the idea of universal causation is intuitive, while that of uniformity is necessarily empirical. Sigwart has very trenchantly criticised Mill in that "taking away with one hand what he gives with the other, he shows in the uncertainty of his views the helplessness of pure empiricism, the im- possibility of erecting an edifice of universal propo- sitions on the sand-heap of shifting and isolated facts, or, more accurately, sensations ; the en- deavor to extract any necessity from a mere sum of facts must be fruitless. The only true point in the whole treatment is one in which Mill as a logi- cian gets the better of Mill as an empiricist 5 namely, that every inductive inference contains a universal principle; that if it is to be an inference and not merely an association of only subjective validity, the transition from the empirically universal judg- ment all known A's are B to the unconditionally universal all that is A is B, can only be made by means of a universal major premise, and that only upon condition of this being true are we justified in inferring from the particular known A 9 & to the still unknown ^4's." 2 1 Venn, Empirical Logic, p. 130. 2 Sigwart, Logic, Vol. II. p. 303. 58 INDUCTIVE LOGIC The whole tendency of the modern logic is to base the causal postulate upon a ground which is epistemological ; namely, inasmuch as our knowl- edge is one and self-consistent throughout all its separate elements, there must be a corresponding invariability in the phenomena themselves, as there is in the system of knowledge which results from the interpretation of these phenomena. This is the general view of Sigwart, Bosanquet, Lotze, and Green. 1 This view may be considered also as an expres- sion of the Law of Sufficient Eeason ; namely, that there is an inherent characteristic of intelligence which demands that every element of conscious- ness must be referred to some other element for its explanation, and that it is only when the logical connection of ideas corresponds to a real causal connection, that the mind discovers a reason for its several experiences which is satisfying. It has been said by Ueberweg, as given expression to this view : " The external invariable connection among sense phenomena is, with logical correctness, ex- plained by an inner conformability to law, accord- ing to the analogy of the causal connection perceived in ourselves between volition and its actual accom- plishment." 2 There is a distinction that is of importance to note between the popular and the scientific idea of 1 Sigwart, Logic, Vol. II. pp. 119, 120; Bosauquet, Logic, Vol. II. pp. 220, 221 ; Lotze, Logic, p. 68 ; Green, Phil. Works, Vol. II. p. 286. 2 Ueberweg, Logic, pp. 281, 282. CAUSATION 59 cause. The former is the outcome of the supposi- tion that whatever immediately precedes the effect has evidently produced it, and that this is sufficient wholly to account for it. Such an idea of causes leads, at the best, but to a loose and superficial determination of the relation between any ante- cedent and its consequent, and there is the danger, moreover, of a hasty inference which results in the fallacy of post hoc ergo propter hoc. In order to attain a true view of causation, we must especially attend to the extreme complexity of the causal con- nection. There is no such thing as a simple cause followed by a simple effect. The cause is always a combination of several elements, circumstances, and conditions; the effect is always manifold. This characteristic has been admirably presented in Mill's chapter on the " Plurality of Causes and the Intermixture of Effects. It is well known that the variation in the height of a barometer is due partly to the variation of the atmospheric pressure, and partly to the variation of the expansion of the mer- curial column due to heat. In exact determination, some experiment or calculation must precede, before there can be a discrimination between the elements of the joint effect. And so also, a number of cir- cumstances may combine to restore an invalid to health, no one of which alone being capable of effecting his recovery. The cause of any phenomenon has been defined by Mill, as also by Brown and Herschel, as the sum total of all its antecedents. This statement has been criticised, inasmuch as the sum total of all 60 INDUCTIVE LOGIC antecedents is indeterminate, and that there is no end to the possible ramifications in all directions which an exhaustive analysis of any complex cause will yield. However, the problem is one of reduc- tion to simplest possible terms within the range of our powers of observation and experiment. There is much in the sum total of all the antecedents of any given effect which is irrelevant. It is the peculiar function of logical analysis to discriminate between the relevant and irrelevant. The temper- ature of the laboratory will not affect, one way or the other, experiments with falling bodies ; but will essentially influence certain chemical experiments, and must enter as one of the determining factors in the sum total of antecedents. It may be that cer- tain elements of a complex whole may seem to us ultimate and unanalyzable, and yet be themselves systems of more or less complexity. There is always a limit to analysis, both experimental and mental. The analysis is to extend to the ultimate parts as far as possible. It is not an exact process, but a process which tends to exactness to the ex- tent which the scope of finite intelligence will per- mit. The reason is not at fault so much as the natural limitations of observation and experimental analysis. The end of our research in causal analy- sis is to discover an invariable relation that can be expressed in the form of an hypothetical universal, If A, then B. In order to effect this, the complex A must be separated into its parts, a, b, c, etc., and the effec- tive, and necessary, and indispensable element pro- CAUSATION 61 ducing B must be determined. Suppose it proves to be a, it may be possible to subject this to further analysis, and reduced to simpler elements, such as x, y, z, etc., and x found to be the significant ele- ment of the real cause. Each analysis determines a narrower and still narrower sphere within which the cause lies. A man is shot. We say the bullet killed him ; then the driving force behind the bul- let; then the explosive power of the gunpowder; this in turn was occasioned by the combined chemi- cal and mechanical energy of its ingredients, where- by a solid is transformed into a gaseous substance many times its original bulk. Sooner or later we must reach the end of our an- alysis, and the investigation be necessarily checked. No explanation is ultimate ; we only transfer our point of view from a less to a more familiar sphere of interpretation. We do not feel the need of ex- plaining the very familiar ; though the most famil- iar is hardest satisfactorily to explain, because there is nothing simpler in whose terms we may para- phrase it. We feel this in giving a definition of terms whose meaning we best know, and which we most frequently use. Mr. Barrett, a former assist- ant at the Royal Institution, said of Faraday : " I well remember one day when Mr. Faraday was by my side, I happened to be steadying, by means of a magnet, the motion of a magnetic needle under a glass shade. Mr. Faraday suddenly looked most impressively and earnestly, as he said : i How won- derful and mysterious is that power you have there ! The more I think over it, the less I seem to know/ y 62 INDUCTIVE LOGIC And yet, he who said this knew more of it than any living man." 1 Although our knowledge is limited as in all cases of causation, however simple, nevertheless, as far as it goes, the several elements are related logically, that is, necessarily and universally. We may only know in part, but still we know, and the world, as interpreted for us in knowledge, is a world of invariable sequences. The process of inductive analysis, therefore, consists in reducing a complex antecedent to its ultimate parts, in order to reveal the element or elements in it which have caused the given effect. It sometimes happens that differ- ent elements in an antecedent may be regarded severally as the cause, according to the psychologi- cal point of view as regards the interests of the investigator. It is not always that a scientific determination of the cause is required ; it may be that all that is desired is a knowledge of that part of the antecedent which is most closely and prom- inently connected with the event in question. An inquiry may be started in reference to the cause of an epidemic in a community. One may discover the cause in the carelessness of sanitary engineers ; another may say the cause lies in the poor construc- tion of the sewerage ; another says that the cause of the epidemic is a certain kind of bacilli. Each one is looking at the chain of events related as cause and effect ; but they all look at different links of the same chain. One element, therefore, of a complex antecedent may be brought into more or 1 Gladstone, Michael Faraday, p. 180. CAUSATION 63 less prominence as the efficient element of the cause, according as the point of view is shifted. If, in the search for the cause of phenomena, the sum total of antecedents were always given exhaus- tively, the explanation might become so loaded down with details as to burden the mind and con- fuse, rather than clear, the understanding. CHAPTEK VI The Method of Causal Analysis and Deter- mination It will be well to consider the various problems which will confront us in seeking to analyze a com- plex antecedent for the purpose of discovering its cause. 1. There are problems where cause and effect appear in evident sequence. There is an antece- dent which is followed by a consequent. If A happens, then B will happen. Instances of this kind most readily yield themselves to the process of analysis, because a change in any given phe- nomenon is occasioned by the efficiency of the ante- cedent which is observed in connection with the change itself. It is easier to note active than passive relations, the dynamic rather than the static. The attention is attracted and held by change. The bird flying across our path is ob- served, and the one perched upon the tree near at hand, however conspicuous may be its position, is passed by without any notice taken of it. It is easier to connect the moisture of the grass with falling rain, than when the same is occasioned by the dew. In one case, the causal relation is ex- 64 CAUSAL ANALYSIS AND DETERMINATION 65 hibited in operation ; in the other, the connection is veiled. We find the grass wet ; what preceded it we are not able to see. There are several in- stances of sequence among observed phenomena which must be carefully discriminated in order to avoid confusion of thought. They are as follows : (1) When we have A followed by B, and A ceases wholly while B endures for an appreciable time afterwards, or it may be permanently. A billiard ball strikes another, the second goes on by virtue of the newly acquired energy transferred by impact from the first, which, however, stops altogether. I throw a ball which lodges on the top of a building; the effect produced lasts per- manently, for the ball has gained a gravity poten- tial due to the energy imparted to it by the initial throwing. The old formula, therefore, does not always hold : " Cessante causa cessat effectus." (2) Cases where A ceases, and thereupon B immediately ceases also. If we cut off the supply of gas which feeds a flame, the flame at once dis- appears. There are cases, however, when an ap- preciable time must elapse in order that the transferred energy in the effect may be dissi- pated. When we shut our eyes the stimulus caus- ing the perception is cut off, and the perception at once is at an end ; however, there are cases where the stimulus being very strong, after-images are induced which remain for some time in the dark field after the eyes are closed. (3) Cases where the antecedent is wholly in- adequate to produce the effect, but whose function 66 INDUCTIVE LOGIC is merely to liberate potential energy already stored, and waiting an occasion for its active manifestation. A slight blow upon a piece of dynamite causes an explosion wholly dispropor- tionate to the striking force employed. As is well known, heat is often an exciting cause of chemical action. In such cases the real cause is more or less concealed, while that which is appar- ent upon the surface is not a cause so much as an occasion of the phenomenon in question. I touch the pendulum and a clock starts and so continues for many hours ; the swinging pendulum, however, is only the occasion of liberating the potential energy of the wound-up spring, and thence the power which runs the clock, pendulum, wheels, hands and all. 2. We have also instances not so much of se- quence as of concurrence. The planets revolve around the central sun ; here the cause is constant, attended by constant effect. The machine never runs down ; nor has to be wound up, so that it can be seen that the cause antedates the effect. 3. Cases of Coexistence. These are more diffi- cult to analyze, for the phenomena do not here appear as antecedent and consequent in the midst of changing conditions and circumstances. We have coexistence of two kinds. (1) Coexisting attributes in one and the same organism. They are always found together. They form one generic concept and are called by one name. Cows have horns, cloven feet, are rumi- nant, etc. Dogs have their distinct and constant characteristics. The orange has its correlation of CAUSAL ANALYSIS AND DETERMINATION 67 color, taste, smell. And so we have the so-called " natural kinds," i.e. organisms presenting an unique and characteristic appearance, differenti- ated thereby from all others. There are also certain correlations of growth which present a constant relation between certain attributes, as the fact, however we may explain it, that cats with blue eyes are invariably deaf. There are, moreover, illustrations of the same in an inorganic sphere, as the law which connects the atomic weight of substances and their specific heat by an inverse proportion; or that other law which obtains between the specific gravity of substances in the gaseous state, and their atomic weights, they being either equal or the one a multiple of the other. In many cases, the bare fact of co- existence must be accepted without being able to explain the causal ground of it. The several ele- ments present a constant association, and that is all that can be said about it. In other cases, however, a cause may be found as regards, for instance, the correlation of warm-blooded animals always possessing lungs. The connection between respiration and the generation of heat is found to depend upon chemical action as its causal basis. (2) A relation of statics rather than dynamics, as, for instance, a pillar supporting a roof or arch, is said to be the cause in the sense of the sustaining cause of the superstructure. So also the cohesive force which holds together the particles of a stone. In such cases the energy inherent in the cause is of the nature of a stress and strain. 68 INDUCTIVE LOGIC 4. Under this head are embraced the phenom- ena of vital growth or development. These are the most difficult of all the causal problems to deter- mine ; for it is required to discover the inner neces- sity of essence, and how the succeeding stages of development unfold through the play of the central forces inherent in the very nature and being of the organism itself. Mill is content with classifying organisms as different natural kinds, and he is not concerned with the reason why there should be such and such kinds, nor does he attempt to discover any law concerning these natural correlations and the mode of their growth. In inductive analysis, our concepts must not merely grasp what the natural kinds are, but also what has determined them to be what they are. Darwin puts special emphasis upon the environment as affecting changes in organisms and producing differentiating modifications among species. This, however, must be considered not as sole factor, but one which is combined with inner needs and necessities. Moreover, Darwin has drawn attention to the fact that individual differences need scientific explanation as well as the common attri- butes, as, for instance, why some sheep are black, and why some pigeons are fan-tailed and others are not. In all such considerations we must not lose sight of the fact that there are two determining factors, the inner necessity of development ; and the external necessity of causality, as organisms are acted upon by their environment. 1 5. Cases of collocation where no one element of 1 Sigwart, Logic, Vol. II. pp. 322, 330, 331, CAUSAL ANALYSIS AND DETERMINATION 69 the cause is efficient, but all together they combine to produce the effect. In searching for the cause, we must not only find a certain amount of energy- capable of producing the effect, but we must also discover what peculiar arrangement of the elements concerned must exist before the energy in question can become operative. Chalmers says that " the existing collocations of the material world are as important as the laws which the objects obey, that many overlook this distinction and forget that mere laws without collocations would have afforded no security against a turbid and disorderly chaos. " ' We would naturally say that the sole cause of water boiling at 212 is the enveloping heat ; it has, how- ever, been observed that on top of Mont Blanc, water boils at 180 instead of 212. This indicates that, in addition to the fire, the atmospheric press- ure is an element in the cause, very easily over- looked. Charcoal and diamond are of the same substance ; a difference only in the arrangement of the molecules results in such radically different combinations. There are, in the main, three special kinds of collocations, as follows : (1) Cases of modifying circumstance. A strong wind blows down a tree ; this would not have oc- curred had not the tree been hollow. The hollow- ness of the tree is here a co-operative circumstance that is combined with the efficient cause, the force of the wind. An instance where arrangement of the elements concerned rather than their efficient en- ergies is productive of the effect, is that of capil- 1 Quoted by Jevons, Principles of Science, p. 740. 70 INDUCTIVE LOGIC larity, the rising of liquid in a tube of exceedingly- small bore. Here form is more essential to the effect than the expenditure of any visible energy. (2) Cases in which certain negative conditions prevent the realization of the effect. The plants and shrubs die in a long drouth, because it did not rain. A train collides with another, because the red signal was not exposed as it should have been. A match will ignite gunpowder generally, but it fails to do so should the powder prove to be wet. (3) There are also cases of counteracting causes, where the effect of cause A is not realized, as cause B neutralizes the force of cause A) as when an anchored boat will not respond to the pull of the oar. Sometimes the cause is not wholly counter- acted, or it may be the counteracting cause more than holds the positive cause in check, and is itself operative. The rise of a balloon in the air is due to the fact that the force of gravity is more than overbalanced by the expansive force of the gas within the balloon; one force pulling downwards, the other bearing up, and the latter prevailing. Mechanical forces acting in combination admit of a resolution of their joint effect according to the theory of the parallelogram of forces. Chemi- cal and vital forces cannot be treated in such a way at all. From the character of the elementary forces in mechanics, one can calculate their com- bination. In chemistry, however, when the ele- ments are given, the resulting compound cannot be thus determined. So, also, in vital and mental phenomena, the necessarily complex nature of the CAUSAL ANALYSIS AND DETERMINATION 71 elements involved prevents not only prediction of resulting combinations, but even adequate explana- tion of that which may be immediately given in consciousness. 6. It is necessary, in the investigation of causal relations, to understand the different modes of the transfer of energy, which are as follows : (1) Molar or mechanical, as in the case of a billiard-ball transferring its energy to another through impact. (2) Molecular, as heat, chemical and electrical and magnetic forces, light, etc. One passes into another, as chemical force producing electric, elec- tric producing magnetic, or producing heat and light. (3) Cases where mechanical force becomes mo- lecular, as friction inducing heat; or cases where molecular becomes mechanical, as heat transferred into the driving power of an engine, or electricity applied as a motor. A precise determination of equivalents can be made between molar and molec- ular energy; as, for example, it has been found that it takes the same amount of energy to raise 772 pounds a distance of one foot that it does to raise the temperature of one pound of water 1 F. ; or the heat requisite to boil a gallon of freezing water would lift 1,389,600 pounds through a dis- tance of one foot. As a consequence of the doctrine of the transfer of energy, a causal law can be so stated as to ex- press the fact that variations in the antecedents will call for corresponding variations in the effect, 72 INDUCTIVE LOGIC as, for instance, such a law as the following : " Re- sistance in a wire of constant section and material is directly proportional to the length and inversely proportional to the area of the cross-section." 1 The neglect of quantitative determination of the proportionate variations of the antecedent and conse- quent was a glaring defect in the inductive systems \both of Mill and of Bacon. Through the representation of the various stages of such variation, it is also possible to establish the upper and lower limits beyond which the cause does not produce the corresponding effect ; as in Weber's law concerning the relation of stimulus to sensa- tion, that stimulus must increase geometrically in order that the sensations increase arithmetically. There is an upper and lower limit beyond which this proportion does not hold. The doctrine of conservation of energy creates, the impression of continuous change in causation, in which the effect unfolds out of the cause. We do not think of phenomena under this aspect as discrete events. More than ever, in the light of modern science, does the old saying obtain, "Natura non facit saltum." We no longer look for catas- trophic results in nature but regard causation as a continuous transfer of potential energy into kinetic or actual energy. We come now to the consideration of the method by which the causal analysis is mediated. This is effected through observation and experiment. Observation is something more than mere looking at 1 Jenkin, Electricity and Magnetism, p. 83. CAUSAL ANALYSIS AND DETERMINATION 73 phenomena; it means concentration of attention for the purpose of research ; it means discriminating insight, an appreciation of likeness and difference ; it means a penetration beneath surface appear- ances, and an apprehension of the essential features of the objects of perception. Experiment consists in modifying the elements which form the complex antecedent in order to observe the resultant effect upon the corresponding consequent. Forces may be added or subtracted; their intensity may be varied, increased, or decreased; the circumstances or conditions may be altered. Herschel speaks of observation and experiment as passive and active observation. When we interfere to change the course of nature, or to bring natural forces within the range of our observation, we are experimenting. Observation is preliminary to experiment, and sug- gests the lines along which experiment should pro- ceed. An observation that sees the parts in the whole and the whole in the parts, is in itself an analysis of a phenomenon, in course of which proc- ess causal relations must be disclosed. The scien- tific spirit demands absolute veracity in observation. One ought not to be blind to facts even though they tend to contradict preconceived theories. Bacon has observed that " men mark when they hit, never mark when they miss." We must strive against a natural tendency to see things as we would have them, and not as they strictly are. We must also carefully distinguish between ob- served facts, and inferences which we instinctively draw from these facts. Observation is preliminary 74 INDUCTIVE LOGIC to an inductive inference, therefore it must not it> self involve an inference, or we should be arguing in a circle. An interesting illustration of the dif- ference between observation and inference based upon it, is narrated in the life of Faraday: "An artist was once maintaining that in natural appear- ances and in pictures, up and down, and high and low, were fixed indubitable realities ; but Faraday told him that they were merely conventional accep- tations, based on standards often arbitrary. The disputant could not be convinced that ideas which he had hitherto never doubted, had such shifting foundations. ' Well/' said Faraday, ' hold a walking- stick between your chin and great toe ; look along it and say which is the upper end.' The experiment was tried, and the artist found his idea of perspec- tive at complete variance with his sense of reality ; either end of the stick might be called upper, pictorially it was one, physically it was the other." ' This indicates how readily our inferences and observations blend, and how difficult it is to separate them in consciousness. De Morgan has pointed out that there are four ways of one event seeming to follow another, or to be connected with it, with- out really being so : (1) Instead of A causing B, our perception of A may cause B. A man dies on a certain day which he has always regarded as his last through his own fears concerning it. (2) The event A. may make our perception of B follow, which would otherwise happen without 1 Gladstone, Michael Faraday, pp. 165, 166. CAUSAL ANALYSIS AND DETERMINATION being perceived. It was thought that more comets appeared in hot than cold summers; no account, however, was taken of the fact that hot summers would be comparatively cloudless, and afford better opportunities for the discovery of comets. (3) Our perception of A may make our percep- tion of B follow. This is illustrated by the fallacy of the moon's influence in the dissipation of clouds. When the sky is densely clouded, the moon would not be visible at all ; it would be necessary for us to see the full moon in order that our attention should be strongly drawn to the fact, and this would happen most often on those nights when the sky is cloudless. (4) B is really the antecedent event, but our perception of A, which is a consequence of B, may be necessary to bring about our perception of B. Upward and downward currents are continually cir- culating in the lowest stratum of the atmosphere; but there is no evidence of this, until we perceive cumulous clouds, which are the consequence of such currents. 1 There are certain natural limitations to obser- vation, as things too minute to be seen, too swift to be carefully examined; there are sounds which some ears can detect, while others cannot, and shades that some eyes cannot discriminate. There are effects proceeding from certain causes that are so slight that we fail to observe them, and yet erro- neously infer that they do not exist. Professor Tyndall has given a striking illustration of the dif- 1 Quoted by Jevons, Principles of Science, pp. 409-411. 76 INDUCTIVE LOGIC ference of auditory power in two individuals; lie says : " In crossing the Wengern Alp in company with a friend, the grass at each side of the path swarmed with insects which to me rent the air with their shrill chirruping. My friend heard nothing of this, the insect music lying quite beyond his limit of audition." 1 Much has been done by in- ventive skill to increase our powers of observation, and at the same time to render them more accurate, as the telescope, microscope, the vernier for precise measurement of minute differences of magnitude, the chronograph for time measurements, self-registering thermometers, the thermopile, galvanometers, etc. One of the chief problems of scientific method is to overcome natural limitations of observation through mechanical devices. . Observations on a large scale and over a consid- erable period of time must sometimes be taken in order to disclose tendencies as seen only in the average or the mean of the observed results. Thus meteorological, vital statistics, and others of a like kind must extend over .a large area, and embrace a large number of instances in order to reach results of any value. It is known that Tycho Brahe made an immense number of most exact records of the positions of the heavenly bodies with the aid of the best of astronomical instruments, and these records afterwards became the foundation of Kepler's laws and of modern astronomy. 2 The faculty for accurate observation can be in- i Tyndall, On Sound, pp. 73, 74. 2 Gore, The Art of Scientific Discovery, p. 316. CAUSAL ANALYSIS AND DETERMINATION 77 creased by acquiring the habit of examining care- fully everything within the field of vision. We fail to see many things because we fall into the easy way of passing them by without noting their presence or appreciating their significance. It was said of Charles Darwin by his son that " he wished to learn as much as possible from every experiment, so that he did not confine himself to observing the single point to which the experiment was directed, and his power of seeing a number of other things was wonderful." * The open-eyed vision is the prime requisite for scientific investigation. The limitations of observation naturally lead to experiment, whose special function is to so modify phenomena as to bring a suspected causal element more prominently into notice. This can be done by intensifying the force in question, or by neutralizing all other elements in combination with it, so that the sole effect of this force in actual operation can be observed. When the cause is not a simple ele- ment, but a combination, then the problem is to vary the conditions so that but one possible com- bination, then another, can be operative alone, and note the corresponding effect. Given a certain number of elements, the number of possible combi- nations is mathematically determinate, and can be tried seriatim until all possibilities are exhausted. Venn has given a long and interesting illustration of this in his Empirical Logic. 2 All combinations need not be tried, however ; for many will be seen 1 Life and Letters of Charles Darwin, Vol. I. p. 122. 2 pp. 402 ff . 78 INDUCTIVE LOGIC to be either impossible or irrelevant. The aim is to obtain an antecedent which shall consist either of a simple element, or a combination such that with its presence the effect in question is present also, but with its disappearance the effect is wanting. It is not sufficient to note merely the presence of an antecedent connected with a corresponding consequent ; scientific determination consists, in ad- dition, in proving the absence of the suspected cause in cases where the given effect is not present. This is called determination by negation. A proposition which is held affirmatively has only a vague sig- nificance; it must be determined within definite limits assigned to it by virtue of what it is not. Defining means to set limits to a term ; these limits grow out of the nature of the thing itself. The negative judgment marks a transition always from that which is indefinite and incoherent to that which is definite and coherent. For instance, we have a vague notion of chemical affinity that ele- ments combine to form compounds. That is the nucleus of our knowledge ; it grows in definiteness through a continuous process of limitation by nega- tion. We find that not all elements combine with each other, that they do not combine except in cer- tain proportions, and that even those which do in certain definite proportions will not combine in the presence of others having greater affinity, as, for instance, in the presence of oxygen, and so on. Every negative proposition established renders the original one more accurate. This may be illustrated also in the concrete, when CAUSAL ANALYSIS AND DETERMINATION 79 in dissection one is tracing a nerve ; it is followed throughout its course by a series of negative judg- ments though they be unexpressed: This is not a nerve, but an artery ; this is not a nerve, but a vein ; this is not a nerve, but a filament, or shred of muscle, etc. So we rise through negative discrim- ination to a clear apprehension of an object under investigation. The original proposition must be readjusted with every new negative determination. It sometimes happens that the original proposition is completely negatived by the negative determin- ation, sometimes again it is confirmed. A proposition that has not been worked over through such a process has no real logical worth or scientific value. Therefore in the analysis of phenomena when the suspected cause and effect are combined in a proposition, it can at first be held only tentatively. It must be confirmed negatively, or else readjusted to conform to the negative re- quirements. Suppose we have given that A is followed by B as far as we have been able to ob- serve. We may proceed by experiment to multiply instances of A's connection with B, but still the causal relation is not absolutely proved. We must go on to show that in all cases of not-^1 there is not- B, or in all cases of not-I? there is not- A. Negative experiment produces the contrapositive, or the con- verse contrapositive of the proposition under inves- tigation, which deductively necessitates the validity of the original proposition. This is substantially Mill's method of difference, that if an instance in which the phenomenon under 80 INDUCTIVE LOGIC investigation occurs, and an instance in which it does not occur, have every circumstance save one in common, and that one occurring only in the former ; the circumstance in which alone the two instances differ, is the effect or cause or a necessary part of the cause of the phenomenon. This method will be described later ; it is the main inductive method, the others being largely modifications of it. A negative instance which is established concerning relations of not-^1 and not-B, is absolutely conclu- sive, inasmuch as not-^4 is the contradictory of A, and not-B is the contradictory of B. They are mutually exclusive. No other possibility can be forthcoming, and the experimental analysis is ex- haustive. Professor Tyndall gives the following account of an experiment to determine the cause of resonance. " I hold a vibrating tuning-fork over a glass jar eighteen inches deep ; but you fail to hear the sound of the fork. Preserving the fork in its position, I pour water with the least possible noise into the jar. The column of air underneath the fork becomes shorter as the water rises. The sound augments in intensity, and when the water reaches a certain level, it bursts forth with extraordinary power. I continue to pour in water, the sound sinks, and becomes finally as inaudible as at first." 1 Prom this it is inferred that a certain column of water of definite height is necessary to the produc- tion of the sound, for above and below the limits no sound is heard. This experiment also indicates that which is most important in causal determina- i Tyndall, On Sound, p. 172. CAUSAL ANALYSIS AND DETERMINATION 81 tion, a variation in canse accompanied by a vari- ation in effect, as also a maximum and minimum as regards the intensity of the sound. Experiment proceeds upon the supposition of the measurable- ness of phenomena, and seeks numerically expres- sible results in this regard. For instance, by different experiments, Tyndall proved that the length of the column of air which resounds to the fork in a maximum degree of intensity is equal to one-fourth of the length of the wave produced by the fork. 1 The negative determination of a suspected con- nection of cause and effect must be precise in order to establish the causal relation with that degree of accuracy which is demanded in a truly logical and scientific method. Upon this point, Bosanquet has a very suggestive passage : " The essence of signifi- cant negation consists in correcting and confirming our judgment of the nature of a positive phenome- non by showing that just when its condition ceases, just then something else begins. The i Just-not' is the important point, and this is only given by a positive negation within a definite system. You want to explain or define the case in which A be- comes B. You want observation of not-i?, so that you are lost in chaos. What you must do is to find the point within A where A 1 which is B, passes into A 2 which is C, and that will give you the just- not-B which is the valuable negative instance." 2 For example, in Professor Tyndall's experiment, the i Tyndall, On Sound, p. 174. 2 Bosanquet, The Essentials of Logic, p. 134. 82 INDUCTIVE LOGIC significant negative instance was this, when the water in the tube reached just that height when for the first time during the experiment no sound was audible. The discriminating observation that can mark and measure the precise point of transition from sound to no sound, has determined accurately the conditions necessary to produce the sound, and precisely define their limitations. In all observation and experiment, the following possibilities should be kept before the mind in order to avoid a hasty conclusion in reference to a seeming causal connection. We may think that we have dis- covered the relation that if there is A, then there must be B, and the one therefore the cause of the other, but it may happen that 1. Both A and B are effects of another cause and are thereby related co-ordinately in reference to it. 2. A may be merely a liberating circumstance, or an invariable accompaniment of B. 3. A may not be the cause of B, but only an element of a complex collocation which is the cause of B. 4. Each separate instance of B may so differ as to respond to the action of A in a manner different from the others. 5. A may be related to B in a system of such a nature, that the system in continuously developing new effects causes B, as the introduction of medicine into an organism whose forces are themselves effect- ing a healing process. 6. It is often very difficult to tell whether A is the cause of B, or B the cause of A, as in districts CAUSAL ANALYSIS AND DETERMINATION 83 where drunkenness and poverty are prevalent, or cases of moral and intellectual feebleness. Which is the cause ? and which the effect ? In many cases such as these, the forces react upon each other, the effect tending to increase the intensity of the cause. 7. The connection of A and B may be one of mere coincidence, and not of a causal nature what- soever. Newton was much impressed with the apparent connection between the seven intervals of the octave, and the fact that the colors of the spec- trum divide into a like series of seven intervals. And yet there is no causal connection that can be proved to exist between the two. The more we dwell upon these various possibilities, the more are we impressed with the extreme com- plexity in which the relation of cause and effect is involved. The investigator must bring to his re- search the spirit of patience and perseverance, as well as a clear vision and discriminating insight. Sir John Lubbock, in his observations upon the habits of ants, says that at one time he watched an ant from six in the morning until a quarter to ten at night, as she worked without intermis- sion during all that time. 1 It is to such patient investigators that Nature reveals her secrets. i Sir Jobn Lubbock, Scientific Lectures, p. 73. CHAPTER VII Mill's Inductive Methods The Method of Agreement There are various methods of causal research which have received the name of inductive methods and have been especially associated with the con- tribution of John Stuart Mill to the history of logic. There are five of these methods or inferen- tial processes as given by Mill, and forming the in- tegral part of his system of induction. They are as follows : 1. The Method of Agreement. 2. The Method of Difference. 3. The Joint Method of Agreement and Inf- erence. 4. The Method of Concomitant Variations. 5. The Method of Residues. The method of agreement consists in inferring the existence of a causal relation, when in a num- ber of varying instances it is observed that the supposed cause is always accompanied by the phe- nomenon in question, as corresponding effect. The method of difference is the comparing of an in- stance where the supposed cause is present, accom- panied by the corresponding effect, with an instance 84 THE INDUCTIVE METHODS 85 having precisely the same setting, but where the supposed cause is withdrawn, the effect also disap- pearing ; the inference of a causal relation is then permissible. The joint method of agreement and difference is the comparing of instances where the supposed cause is present, with similar instances where it is absent; if the corresponding effect is present in the former, and absent in the latter group of instances, a causal relation may be inferred. This differs from the method of difference, that in the latter the same instance, now with, and again without the presence of the suspected cause, is the subject of observation; in the joint method it is a number of instances with, compared with a number of similar instances without, the presence of the sup- posed cause. The method of concomitant variations consists in so modifying any given phenomenon that the supposed cause will vary in intensity ; then a corresponding variation in the accompanying effect is evidence of a causal relation. The method of residues consists in the analysis of a given complex phenomenon, in which all elements save one of the antecedent are known to be related in a causal manner to all elements save one of the conse- quent ; then the residual element of the one may be regarded as the cause of the residual element of the other. We will now examine these methods more in detail. The brief outline above is intended merely to give a general idea of the methods, that it may lead to a better understanding of the more exact statement of their nature and characteristics. 86 INDUCTIVE LOGIC The Method of Agreement. The more precise statement of this method is given in the first canon of Mill, which is substantially as follows : If 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 probable cause (or effect) of the given phenomenon, or sustains some causal relation to it. The above is based upon the causal axiom that the constant elements which emerge in any given series of similar phenomena are to be considered as connected in some manner with the cause of the phenomena ; but that the variable elements are not connected with the phenomena in any causal man- ner whatsoever. The method of agreement is illustrated in the in- vestigation of the very common phenomenon of the transformation of substances from the solid to the liquid state. What is the one circumstance which is always present when we consider the melting of such widely different substances as butter, ice, lead, iron, etc. ? In all instances, to whatsoever extent they may be multiplied, of the change from solid to liquid states, heat has been observed to be present, and is thereby indicated as the likely cause of the phenomenon in question. The method may be represented through the use of symbols which, ac- cording to Mill, are the capital letters to denote antecedents, and the smaller letters to denote cor- responding consequents. Let the following be a number of different instances with the antecedents THE METHOD OF AGREEMENT 87 and consequents arranged in order, and represented as above indicated : ABC abc ADE ade AMN amn etc. etc. By inspection of such a table of instances thus analyzed, and symbolically represented, it will be readily seen that A is the only element common to all the antecedents, while a is the only one common to all the consequents. The inference, therefore, is that A is the cause of a. It has been objected to this system of representation that it artificially ar- ranges the elements of antecedent and consequent, as though there were a number of distinct cause- elements, each connected with a correspondingly distinct effect-element, and it produces the impres- sion that it is quite an easy matter to see how these causal pairs are thus separately related. 1 As nat- ure presents her phenomena to us, however, there is such complexity throughout, that the analysis cannot readily distribute part to part in appropri- ate causal relations. To avoid such an error in notation, I have adopted the following symbols, which will be used hereafter to describe the various methods. Let us take C as the letter to represent the supposed causal element, and S, the entire set- ting of accompanying circumstances; let e denote the corresponding effect, and s the sum total of the attendant consequences. The causal relation will 1 Venn, Empirical Logic, p. 411. 88 INDUCTIVE LOGIC be then indicated, according to the method of agree- ment, as follows : S +C s +e S' + O s' +e S"+C s" + e Here the setting changes throughout, as indicated by S, S' f S", etc., but C remains constant in the antecedents ; also the corresponding setting in the consequents changes, as indicated by s, s', s", etc., but e remains constant throughout. Such a nota- tion does not attempt to represent just which parts of S cause corresponding parts of s, nor by what elements precisely S differs from S' and S", etc. It does represent, however, the difference between the variable and constant elements of the table of instances which are arranged for comparison, and this is the key to disclose the causal relation. As an example of this method, let us take the physical law that different bodies tend at the same time to absorb and to emit the same waves of light. It is known that every substance in burning gives its own lines in the spectrum analysis, sodium, for instance, producing a very bright line in the yellow portion of the spectrum in a definite locality (Line D, of Fraunhofer). If now, instead of burning sodium, we interpose the vapor of sodium in the path of the ray which should give a continuous spectrum, the phenomenon is completely reversed; at the exact point where there was a bright line in the spectrum, a dark line now appears. Thus the vapor of sodium, acting as a screen, absorbs the THE METHOD OF AGREEMENT 89 rays which it emits when it acts as the luminous source. A similar effect is observed in the case of vapors of iodine, of strontium, of iron, etc. ; and is a phenomenon, therefore, admitting of generaliza- tion by induction. 1 This is according to the method of agreement; and we may make the following representation : Vapor of sodium acting as a screen = S + " iodine " " " = S' + C iron " " = S" + C " strontium " " " = JS'" + etc. etc. The corresponding consequents are : Reversing bright sodium line to dark = S +e " " iodine " " = S' + e iron " " = S" + e strontium" = S" ! + e etc. etc. Therefore we have : JS +0 8 +e S' +C a' +e S" + O s" +e S'" + C s'" + e etc. etc. In this the constant O of the antecedents is the vapor of any substance acting as a screen ; the con- stant e is the reversal in each case of the bright 1 Saigey, The Unity of Natural Phenomena, pp. 94, 95. 90 INDUCTIVE LOGIC line of the substance in the spectrum to the corre- sponding dark line of the same. From this it is inferred that the vapor of any substance acting as a screen absorbs exactly those rays which it emits when it acts as the luminous source. It is of great importance that the instances se- lected for observation or experiment be as varied as possible, so that widely differing phenomena may be gathered together. Then if running through them all there is one common element observed among the antecedents, and one common element among the consequents, the greater the variation among the instances the more pronounced will be the significance of the constant elements. In the illustration given the substances which are so differ- ent as iron, strontium, sodium, iodine, etc., preclude the possibility of the resultant phenomenon de- scribed being due to the peculiar properties of any one metal, or group of metals. So many, and so dif- ferent in kind, are taken as to eliminate the peculiar- ities attached to any one in particular. In this re- spect, the method is one of elimination. By varying the instances, the non-essential is eliminated, and the essential, which remains as the element common to all, is thereby emphasized, and differentiated from all attendaot circumstances. This method also is one of discrimination, of discerning the constant element under the various changing forms which it can assume, and as such it is similar to the logical process of the formation of a concept. The concept is the grasping of * the universal element which is present through the THE METHOD OF AGREEMENT 91 particular and concrete manifestations of the same. Through them all there is the like common element which is the basis of the concept itself. So out of many particular instances the mind grasps the elements which are common to all, and considers them as related in a constant and therefore causal manner, which has in itself the character of a uni- versal concept and so admits of being formulated in the form of a law universal, which is the end of all induction. This method, moreover, is peculiarly adapted to observation, the collating of a number of instances, rather than to experiment. Instances cannot al- ways be manufactured, and so it may be beyond the power of experiment to reproduce them. They can, however, always be the objects of research, and as such fall naturally into the field of obser- vation. The question may properly be asked at this point, How does this method differ from that of induction by simple enumeration ? The latter we have seen is never satisfactory because the enumeration can- not be complete, and may be contradicted by an enlarged experience. This method, however, is superior in that it provides for more than simple enumeration of instances in which the phenomenon in question has occurred; there must be a corre- sponding analysis of the instances, accompanied by a discriminating insight to distinguish the essential from the unessential. Number of instances in- creases the probability that the variable elements have been eliminated, and enables the mind to con- 92 INDUCTIVE LOGIC centrate upon the constant elements that remain and are thereby disclosed. This method primarily admits of application to instances where a sequence is observable ; that is, where antecedent can be distinguished from conse- quent by an appreciable time element. It is, how- ever, possible to apply this method to the investi- gation of coexistences, where it may show that either the coexisting elements are related as cause and effect, or that in some causal manner they are the correlated effect of some cause sufficient to account for them both. Many instances may be adduced of the prevalence of poverty and crime associated together. This may indicate a causal relation be- tween them, and yet a sequence cannot be observed of sufficient definiteness to indicate which is the cause, and which the effect. The problem is thus left indeterminate, with the suggestion of some other cause which may possibly account for them both. All that the method of agreement can at- tain, is by collecting a number of instances of di- verse nature to indicate that in some way at least poverty and crime are connected by causal ties. The constant coexistence of attributes in one indi- vidual admits of a similar treatment and similar results. The fact of the high coloring of male but- terflies in a large number of instances, in reference to a variety of species, indicates a constant relation be- tween the fact of its being a male and the possession of brilliant coloring. This inseparable association indicates a causal relation, which, however, cannot be more precisely determined by this method. The THE METHOD OB" AGREEMENT 93 full explanation of the phenomenon requires some supplementary hypothesis depending upon condi- tions not disclosed by this method, an hypothesis such that the high coloring has the special function of attracting the female butterfly and has been intensified and developed by natural selection. The method of agreement is open to criticism at several points, and yet it must be at the beginning understood that this method does not rank as a final method. We shall soon see that it serves rather as suggestive of and leading to experiments according to the method of difference, to corroborate or disprove the results which, the method of agree- ment may have attained. The chief criticisms that have been made of this method may be summed up as follows : 1. The cause indicated by the method of agree- ment is not thereby proved to be the sole cause of the phenomenon in question. We may gather to- gether a number of varied instances where an ex- tensive failure of crops in the summer has caused hard times during the winter following. And yet there may be, and as a fact there are, many other causes which engender periods of industrial depres- sion. We may say, therefore, that this method is capable of establishing,. tentatively at least, an uni- versal proposition of the form, All x is y ; it does not, however, attempt to give any indication one way or the other, regarding the validity of the converse, All y is x. Knowing the limitations of a method, does not by any means destroy its legitimacy as a method ; it rather increases its efficiency within its 94 INDUCTIVE LOGIC proper sphere, by the more exact knowledge as to the precise extent of that sphere itself. 2. It is urged that while it is possible to recog- nize in most, if not in all cases, the common element in the several effects of similar phenomena, it is not so easy a matter to differentiate the common ele- ment in the corresponding antecedents by the sim- ple method of agreement alone. For instance, in Bacon's illustration of the investigation of the cause of heat, he cites such disparate phenomena as the sun's rays, friction, combustion, etc. The element of heat is readily discernible through them all; but what is the common element which operates as cause in each case? There is the difficulty. Sig- wart illustrates this in the case of the phenomenon of death. The effect can be easily detected as sim- ilar throughout, but in all the antecedents the only property common to them all is life, and, therefore, we are led into the fallacy of attributing to life the cause of death. 1 We must therefore acknowl- edge that some phenomena may occur in such a variety and such a number of manifestations as to disguise the nature of the cause under the mask of a generality too indefinite to be recognized. In all such instances, the method of agreement must operate upon suggestions received from some other source, as to the nature of the common element in the antecedents. Or, some minor circumstances attending the effect may indicate more precisely the nature of the cause, as, for instance, the peculiar symptoms associated with death by drowning. 1 Sigwart, Logic, Vol. II. p. 341. THE METHOD OF AGREEMENT 95 3. The common element in the antecedents may prove to be an unessential accompaniment of all the instances examined. Its presence, therefore, may have nothing whatsoever to do with the observed effects. A number of different medicines, for ex- ample, may produce a certain effect alike in all instances. The only common element that can be detected in the various medicines examined, may be the alcohol which is used as the common vehicle of the different drugs, and yet its effect may be entirely inert as regards the medicinal qualities in question. The common element really efficient may be overlooked, and another common element which is easily discernible may nevertheless remain wholly inoperative. This difficulty may be overcome by a more thorough analysis of the phenomena observed, which may be attained by a judicious variation of the instances, so as to reveal, in turn, the precise effect of the various simple elements which together constitute the complex whole of the phenomenon in question. The defects of the method in this respect are, in a word, the defects of induction by simple enumeration. 4. The cause may be present in all the antece- dents, and, notwithstanding the corresponding ef- fect, not appear, and this, not because the two are not related in a causal mariner, but because the cause is neutralized by the associated elements which appear in combination with it in the various antecedents. For instance, diphtheria germs are the cause of diphtheria, and have been found accompa- nying this disease in all cases which have been ob- 96 INDUCTIVE LOGIC served. And yet their presence is often noted when the disease itself does not develop. The tendency existing is counteracted by the condition of the organism at the time, so that the dread bacilli are inoperative and therefore harmless. As we have seen before, the presence of the effect necessitates the presence of the corresponding cause ; but by no means is it always true, that the presence of the cause necessitates the effect. The cause always produces the tendency at least, which, however, may be neutralized. 5. This method is often applied in a very care- less way to the observations of persons who do not possess the power of accurate discrimination, and therefore observed coincidences are hastily assumed to be particular instances of an universal law. Such procedure leads to superstition and prejudice. It not only warps the judgment, owing to its illogi- cal nature, but it also affects indirectly the man's moral view, as it implies a weakness in character as well as in mind. This criticism refers, however, to the abuse rather than the legitimate use of this method under such restrictions as have been already indicated. The chief function of this method is that of sug- gestion. It indicates often only the possibility of the existence of a causal relation ; in other cases it leads to an inference of high probability. In all cases, however, it marks but the preliminary steps of an investigation which should be followed up by painstaking experiment. As it is the method of observation chiefly, it is natural that it should pre- THE METHOD OF AGREEMENT 97 cede experiment ; for it is only by reflection upon our observations that we discover the nature and relations of phenomena, which serve as data for subsequent experiment. I have selected several illustrations to indicate the various fields of research in which this method of agreement has led to satisfactory results. The first refers to the relation between the occurrence of financial crises and the prevalence of over-production. Guyot, in his Principles of Social Economy, gives the following instances.: An enormous consumption of capital in the United States in the seventies for the construction of rail- roads, was followed by unusual commercial depres- sion. Then the like outlay in India for railway construction by means of loans and taxes which absorbed the whole circulating capital of the Indian population, was followed by a devastating famine and general commercial paralysis. Again in Ger- many there was an enormous consumption of capi- tal in forts and armaments and general military equipment, bringing on the crisis of 1876-1879. England at the same time was unduly supplying circulating capital to the United States, Egypt, and her colonies, and a financial crisis was the result. Through all these varying instances and others of a like nature which might be added, the constant relation of over-consumption in the ante- cedents to the industrial depression evident in the effect, indicates the one to be the cause of the other, either in whole or in part. Again, it is narrated in Brewster's Treatise on 98 INDUCTIVE LOGIC Optics that lie accidentally took an impression from a piece of mother-of-pearl in a cement of resin and beeswax, and, finding the colors repeated upon the surface of the wax, he proceeded to take other impressions in balsam, fusible metal, lead, gum arabic, isinglass, etc., and always found the irides- cent colors the same. His inference was that the form of the surface is the real cause of such color effects. 1 The common element which appears in all the antecedents is evidently the same form impressed upon each, which was originally received from the mother-of-pearl. The cause is, moreover, independent of the nature of the substance in each case which received the impression upon its sur- face, because such a variety of substances was chosen as to eliminate the individual nature of each as an influencing factor in the result. In this experiment we see the advantage of varying the instances as far as possible for this very purpose of eliminating all irrelevant elements. Similar experi- ments have proved like results in reference to the colors exhibited by thin plates and films. Here the rings and lines of color have been found to be nearly the same whatever may be the nature of the substance. A slight variation in color is due to the refractive index of the intervening substance. With the exception of this, the nature of the sub- stance is not operative in producing the color effect, but the form alone. The celebrated scientist, Pasteur, in the year 1868 was carrying on his investigations as to the 1 Quoted by Jevons, Principles of Science, p. 419. THE METHOD OF AGREEMENT 99 cause of the blight then devastating the silkworms of France. One of his experiments consisted in selecting thirty perfectly healthy worms from moths that were entirely free from the corpuscles, which latter are the germs of disease, or at that time sus- pected to be the germs of disease. Then, rubbing a small corpusculous worm in water, he smeared the mixture over the mulberry leaves. Assuring himself that the leaves had been eaten, he watched the consequences day by day. One after the other the worms languished; all showed evidences of being the prey of the corpusculous matter, and finally, within one month's time, all died. Pas- teur's inference naturally was that the corpuscles had produced the death. Of course his results were not founded upon this experiment alone, but other experiments, carried on in many different ways, served to corroborate the causal relation which the experiment just described had suggested as at least highly probable. In medicine also the method of agreement is often used with effect. Certain drugs are adminis- tered in a number of cases and the results noted. An uniform effect consequent upon the administra- tion of a given drug indicates a causal connection capable of generalization. Not only are subjects in disease, but also in health, selected, and the effects upon both the normal and morbid natures compared. Thus a variation in instances is secured. If a num- ber of different drugs produce like effects, the ques- tion at once suggests itself, What is the property common to them all ? The method of agreement 100 INDUCTIVE LOGIC often gives some indication of this, when the elim- ination of the inert properties can be accomplished through a sufficient variation of instances. The difficulty lies, however, in this very thing, to so vary the instances as to disclose the efficient ele- ment present in them all. Various medicines pre- sent a complex nature of such a character that it is extremely difficult to ascribe the precise effects which the several component parts individually exercise. The method of agreement is also used, perhaps unconsciously, in the conduct of the every-day affairs of life. Whenever different phenomena in our experience present certain characteristics of a constant nature, we are at once led to suspect a causal connection, and to start upon a more search- ing investigation of the same. Too often, however, the supplementary investigation is omitted, and the . mind rests content with a few surface resemblances that lead to a hasty generalization, without being more precisely and adequately determined. CHAPTER VIII The Method of Difference The method of agreement, as we have seen, pre- sents a causal relation as a suggestion, admitting of a high degree of probability it may be, but requir- ing to be tested by some more scientific method. This is accomplished by the method of difference. Here a phenomenon is observed, in which the supposed cause-element and effect-element appear ; then while all other circumstances and conditions remain unaltered, the supposed cause-element is withdrawn, or its force adequately eliminated ; the immediate disappearance of the supposed effect- element consequent upon this, indicates a causal relation existing between the two. Or the experi- ment may be made in a different manner, but to the same end ; that is, a phenomenon may be char- acterized by the absence of both cause-element and effect-element; then, if the introduction of the cause-element does not disturb the phenomenon in question, except immediately to produce the effect- element, the inference may be drawn that the one is the veritable cause of the other. Canon of the Method of Difference. If an in- stance in which the phenomenon under investigation 101 102 INDUCTIVE LOGIC occurs, and an instance in which it does not occur, have every circumstance, save one, in common, that one occurring only in the former, the circumstance in which alone the two instances differ is the effect, or it may be the cause, or a necessary part of the cause, of the phenomenon. This method has manifold illustration in our every-day inferences. A person is asleep in the room with us, and we hear the loud noise of a slam- ming door, and observe the person at once awakening with a start and exclamation. We have no hesitancy in ascribing the awakening to the noise immediately preceding it. We observe again some one receiving a letter or telegram, and immediately upon opening, it the face grows white with anxiety and fear, the hands tremble, and there are shown general symp- toms of perturbation. The message received, we say, has caused the mental shock and physical accompaniments. Or, taking a simple experiment in quite another sphere, it was observed by Boyle, in 1670, that an extract of litmus was immediately turned red by the introduction of an acid. This subsequently became a test for the presence of acids, the infer- ence being that an acid has this capacity of chang- ing the litmus to a red color from its original blue. Professor Tyndall describes an experiment to prove that waves of ether issuing from a strong source, such as the sun or electric light, are compe- tent to shake asunder the atoms of -gaseous mole- cules, such as those of the sulphur and oxygen which constitute the molecule of sulphurous acid. THE METHOD OF DIFFERENCE 103 He enclosed the substance in a vessel, placing it in a dark room, and sending through it a powerful beam of light. At first nothing was seen; the vessel containing the gas seemed as empty as a vacuum. Soon, along the track of the beam, a beautiful sky-blue color was observed, due to the liberated particles of sulphur. For a time the blue grew more intense; it then became whitish; and from a whitish-blue it passed to a more or less per- fect white. Continuing the action, the tube became filled with a dense cloud of sulphur particles which, by the application of proper means, could be ren- dered visible. 1 In this series of continuous changes, we find the one antecedent, giving initiative causal impulse, to be the beam of light. It Avas the one element introduced which started the several changes leading to the appearance of the sulphur visibly manifested. The one, therefore, is to be regarded as the cause of the other. It is possible to represent this method by means of symbols in a manner similar to that of the method of agreement. Let C be the supposed cause and e the effect corresponding, while S and s denote the setting of antecedent and consequent respectively. We have, therefore, the following : S + C s + e Then, withdrawing O, we have the absence of e. S a 'he inference then is that C is the cause of e. 1 Tyndall, Use and Limit of the Imagination in Science, p. 33. 104 INDUCTIVE LOGIC Iii the method of agreement, a number of in- stances were taken agreeing only in the posses- sion of two circumstances, the cause and effect elements common to them all. In this method, only two instances are taken, and they must be precisely alike, with the one exception, the pres- ence of two circumstances in one, that is, the cause and the effect elements, and the absence of the same in the other. In the method of agree- ment, we compare the various phenomena, to note wherein they agree; in the method of difference, we compare the two phenomena, to note wherein they differ. The logical axiom underlying the two methods is substantially one and the same, differ- ing only in its special adaptation in each case. The method rests on the assumption, which must be accepted as a fundamental postulate, that what- ever can be eliminated from the various instances is not connected with the phenomenon under in- vestigation in any causal manner ; and the method of difference is based on the postulate that what- ever cannot be eliminated is connected with the phenomenon by a causal law. The method of difference is evidently the method by negation, which has already been indicated as the truly scientific process in induction. It is also pre-eminently the method of experiment rather than observation; for the withdrawal or introduc- tion of forces can only be accomplished at will when we bring the phenomena under experimental control. At times, Nature herself may perform the experiment for us, and we stand simply as THE METHOD OF DIFFERENCE 105 observers to note the results. This is especially "the case in the catastrophic phenomena, such as volcanic eruption, earthquakes, etc. Generally- speaking, however, the method of difference is the process of man's manipulation to secure purposed results in which a causal relation is disclosed. A question naturally suggests itself, What is there to determine the precise mode of experiment ? We may have given a concrete whole of extreme com- plexity. In our experiment, which element shall we proceed to eliminate, in order to note the re- sult? An answer may be given us through sug- gestions received from the results of the method of agreement which has been already applied to the problem. If it is not possible to avail one's self of this contribution from another sphere of in- vestigation, then the complex whole must be broken up, as far as possible, into its simplest component parts, and one after another these parts, singly, then in pairs, and all other possible combinations, caused to be withdrawn, or their force neutralized, and the results in each case noted, as to whether the effect under investigation disappears. The exhaus- tion of all possible combinations must yield some definite result. Suppose, for instance, there is a complex antecedent involving four separable ele- ments, as A, B, C, D. Withdraw severally A, B, C, and D, noting results; then withdraw, in turn, AB, AC, AD, BC, BD, CD, that is, the pos- sible combinations of four elements taken two at a time ; then withdraw ABC, then BCD, ABD, and ACD, that is, combinations of four elements 06 INDUCTIVE LOGIC taken three at a time. By such a process there will be disclosed whether one element alone or whether a combination of two or more have pro- duced the effect under investigation ; also whether more than one element or combination of elements may have caused the effect. 1 The practical diffi- culty in separating the elements of a complex whole, and withdrawing the several combinations from the whole, renders this process in many cases impossible. The cause, however, is generally sus- pected. It may be suggested by the method of agreement, by analogy, or by that insight which at once declares certain combinations to be impossible and others irrelevant. There is generally a mental experiment in which the judgment rejects unlikely combinations, thus narrowing the field of investiga- tion as preliminary to the experiments proper. The method of difference is open to various criti- cisms ; the most important are the following : 1. In applying this method, we may be so easily misled, in supposing our two instances are precisely alike, with the one exception of the presence or absence of the supposed cause, and yet in reality the instances may differ radically, and yet we may be unable to detect this. A patient may have medicine administered to him, and begin at once rapidly to recover, and yet the very taking of the medicine in itself may have made such a mental impression, inducing confidence and hope, that the real cause of the recovery may be due wholly to 1 This process has been illustrated and criticised at length in a striking manner by Venn, Empirical Logic, pp. 401 ff. THE METHOD OF DIFFERENCE 107 this mental reaction. Persons taking pills com- posed of inert substances have often given evidence of bodily effects wholly impossible to trace to the medicine itself. And yet this criticism is one of caution rather than of censure ; for the defects are but difficulties which extreme care and insight may overcome. 2. It has been objected that this method may point out the cause in the concrete instance before the experimenter, but that this furnishes no basis whatsoever for a wider generalization that the effect in question is always produced by this cause. Sig- wart has illustrated this objection by the instances in which typhus fever has been traced to the drink- ing of impure water. 1 The causal relation may be fully established in the cases investigated, but the universal proposition does not follow that wherever typhus fever appears, impure water has been drunk. This objection applies especially to cases of extreme complexity, where proximate causes alone can be discovered, and their ultimate nature, which may appear in various forms, is not revealed; for in- stance, the impure water is not in itself the ultimate cause of the typhus fever. It contains the poison germs, the real cause; they may be introduced into the system in some other way. Care, therefore, should be taken to reveal the cause in and by itself, and not the cause of the cause. The objection, therefore, may be in a measure overcome. To effect a generalization, moreover, of logical valid- ity, it is necessary to supplement the method of 1 Sigwart, Logic, Vol. II. p. 420. 108 INDUCTIVE LOGIC difference by hypothesis and subsequent verifica- tions, which will be described later on. 3. This method may lead to error in cases where the supposed causal element is regarded as the cause in its entirety, when it is in reality but a part of the cause. If one should plant seed in a garden and water only one half of the plot, and it should follow that only the watered part brought forth the leaf and flower, then an inference according to the method of difference might be drawn that the water caused the sprouting of the young plants. And yet it must be regarded simply as contributory to the real cause. Such a difficulty may be obvi- ated by a careful discrimination in the analysis of the phenomenon investigated. 4. Sometimes a liberating cause may be revealed by a strict interpretation of the method of differ- ence, when the real cause is more obscure, and may be overlooked. A stone may strike a can of dyna- mite and the explosion which occurs may be traced to the impact of the stone. It is the one element of difference introduced in the sphere of the ob- served phenomena, with the consequent result. The force existing as a potential is naturally ob- scure, and apt to elude observation. Therefore, whenever a cause disclosed by the method of dif- ference seems to be out of all proportion to the effect, it at once suggests the probability that a potential force not discerned by our powers of ob- servation has been the real cause, and the former a conditioning cause merely. Another illustration of this is the experiment of Priestley which led to THE METHOD OF DIFFERENCE 109 his discovery of oxygen in 1774. He placed some oxide of mercury npon the top of quicksilver in an inverted glass tube filled with that metal and standing in mercury; he then heated the oxide by means of a glass lens and the sun's rays, and ob- tained a gas, which he called " nitrous air," after- wards designated as oxygen. The heat in this case was the sole element of difference between the two instances, one in which there was no gas, and the second after application of the heat, when the gas was present. Here the heat must be regarded as the liberating and not in any sense the producing cause. Again, as Lotze says, "the fact that with the destruction of a single part of the brain a defi- nite psychical function ceases, is no proof that just this single part was the organ which alone pro- duced that function." * In addition to the difficulties attending this method which have been enumerated and which have to do with the logical theory of the method, there are also difficulties of a practical nature which arise in the actual application of this method in ex- perimental inquiry. They are as follows : 1. Care must be taken that, in the two phe- nomena compared, with and without the supposed cause, there shall not be an interval of time elaps- ing, in which period some other cause might be introduced unknown to the investigator, and yet capable of producing the result, or else of neutraliz- ing some force that is present and itself capable of producing the result. For instance, if a chemical i Lotze, Logic, p. 322. 110 INDUCTIVE LOGIC compound be left for an appreciable time, we may notice certain changes and be able to assert posi- tively that no new element has been introduced, and yet the action of the air may in itself have been sufficient to work these changes. When the two phenomena to be compared can be presented for inspection simultaneously, this difficulty is obvi- ated. This is illustrated in an experiment devised to exhibit the presence of light effects in the spectrum beyond the violet rays; that is, beyond the place where the spectrum seems to end. A sheet of paper is taken, the lower part of which is moistened with a solution of sulphate of quinine, while the upper part remains dry. Let the image of the solar ray fall upon this sheet ; the spectrum preserves at the top of the sheet in the dry portion of the paper its ordinary appearance, while in the moistened portion a brilliant phosphorescence ap- pears beyond the region of the violet rays. Here the dry and wet portions are simultaneously pre- sented, and there is but one point of difference be- tween the two. The inference, therefore, is readily drawn that the solution of sulphate of quinine is a substance sensitive to the ultra-violet portion of the sun's rays, the phosphorescence being the effect of these rays upon the solution. 2. Extreme care must be taken that, in the with- drawing of any element in the course of the experi- ment, no other element is inadvertently introduced, and that, in adding any element, no existing element or combination of elements is destroyed, or their effect neutralized. Mr. Venn has admirably illus- THE METHOD OF DIFFERENCE 111 trated this difficulty, and I give the following quotation in full from him : " We suppose that when we have put a weight into one pan of a pair of scales we have done nothing more than this, or can at any rate by due caution succeed in doing nothing more. But if we exact the utmost rigidity of conditions, we easily see that we have done a great deal more. Our bodies are heavy, and there- fore the mere approach to the machine has altered the magnitude and direction of the resultant attrac- tion upon the scales. Our bodies are presumably warmer than the surrounding air ; accordingly, we warm and therefore lighten the air in which the scales hang, and if the two scales and their con- tents are not of the same volume, we at once alter their weight as measured in the air. Our breath produces disturbing currents of air. Our approach affects the surface of the non-rigid floor or ground on which the scales stand, and produces another source of disturbance, and so on through the whole range of the physical forces." 1 In the Keport of the British Association, 1881, an account is given of Professor GK H. Darwin's exper- iments to measure the lunar disturbance of gravity at the Cavendish Laboratory by means of an ex- tremely delicate pendulum. It was found that approaching the pendulum in order to observe its reading, the surface level of the stone floor on which the instrument stood was deflected by the weight of the observer. As he stood to take the reading, the shifting of his weight from one leg to 1 Venn, Empirical Logic, p. 416.. 112 INDUCTIVE LOGIC the other was perceptible ; so it became necessary to observe the reading by a telescope from a dis- tance, or to adopt some similar plan. 1 Faraday was able at will to produce or remove a magnetic force, through the revealed properties of the electromagnet. Many of his experiments would have been impossible if it had been neces- sary to remove a cumbersome magnet and reinstate it again and again in his experiments. The electro- magnet, however, could produce or destroy the presence of magnetic force without any incidental perturbations. Thus Faraday was enabled to prove the rotation of circularly polarized light by the fact that certain light ceased to be visible when the electric current of the magnet was cut off, and instantly reappeared when the current was re-established. Faraday says of the experiment himself: "These phenomena could be reversed at pleasure, and at any instant of time, and upon any occasion, showing a perfect dependence of cause and effect." 2 3. In some cases it is impossible to remove an element which is supposed to be the cause of an effect under investigation. Its removal might cause the destruction or the impairing of the whole phe- nomenon. The force, therefore, that cannot be eliminated must be neutralized by an equal and opposing force. For instance, the force of gravity cannot be eliminated; it must, therefore, be coun- terbalanced by some device of the investigator. 1 Quoted by Venn in Empirical Logic, p. 419. 2 Experimental Researches in Electricity, Vol. III. p. 4. THE METHOD OF DIFFERENCE 113 In chemistry the removal of an element from a compound may be impossible without destroying utterly the compound itself; in such a case, also, a neutralizing agent must be introduced. Darwin wished to prove that the odor of flowers is attrac- tive to insects irrespective of the attraction of color. He therefore covered certain flowers with a muslin net, which still attracted the insects to them. 1 The following illustrations may serve further to exhibit the various features of the method of difference : Mr. Robert Mallet gives the following interesting account of his visit to Faraday : " It must be now eighteen years ago when I paid him a visit, and brought some slips of flexible and tough Muntz's yellow metal, to show him the instantaneous change to complete brittleness with rigidity pro- duced by dipping into pernitrate of mercury so- lution. He got the solution and I showed him the facts ; he obviously did not doubt what he saw me do before and close to him ; but a sort of experimental instinct seemed to require he should try it himself. So he took one of the slips, bent it forward and backward, dipped it, and broke it up into short bits between his own fingers. He had not before spoken. Then he said, 'Yes, it is pliable, and it does become in- stantly brittle.' " 2 Here the experiment with and without the significant antecedent and consequent 1 Darwin, Cross and Self Fertilization, p. 374. 2 Gladstone, Michael Faraday, p. 175. 114 INDUCTIVE LOGIC result indicates the causal relation, especially as the instantaneous effect precludes the possibility of the operation of any other cause. Another experiment of Faraday's is that of his investigation of the behavior of Lycopodium pow- der on a vibrating plate. It had been observed that the minute particles of the powder collected together at the points of greatest motion, whereas sand and all heavy particles collected at the nodes, where the motion was least. It occurred to Fara- day to try the experiment in the exhausted re- ceiver of an air pump, and it was then found that the light powder behaved exactly like heavy pow- der. The inference was that the presence of air was the condition of importance, because it was thrown into eddies by the motion of the plate, and carried the Lycopodium powder to the points of greatest agitation. Sand was too heavy to be carried by the air. 1 Sir John Lubbock gives an account of experi- ments performed upon insects to prove that the sense of smell is in some way connected with their antennae. One experiment was performed by Forel, who removed the wings from some blue- bottle flies and placed them near a decaying mole. They immediately walked to it, and began licking it and laying eggs. He then took them away, and removed the antennae, all other circumstances re- maining the same as before, after which, even when placed close to the mole, they did not ap- pear to perceive it. Another experiment similar 1 Jevons, Principles of Science, p. 419. THE METHOD OF DIFFERENCE 115 to this was tried by Plateau, who put some food of which cockroaches are fond on a table and surrounded it with a low circular wall of card- board. He then put some cockroaches on the table ; they evidently scented the food, and made straight for it. He then removed their antennae, after which, as long as they could not see the food, they failed to find it, even though they wandered about quite close to it. 1 Another experiment is that of Graber to prove the sense of hearing in insects. He placed some water-boatmen (Gorixa) in a deep jar full of water, at the bottom of which was a layer of mud. He dropped a stone on the mud, but the beetles, which were reposing quietly on some weeds, took no notice. He then put a piece of glass on the mud, and dropped a stone on to it, thus making a noise, though the disturbance of the water was the same as when the stone was dropped on the mud. The water-boatmen, however, then at once took flight. 2 An illustration of the method of difference occurs in the so-called blind experiments, which are often made in chemistry especially. As Professor Jevons has described such an experiment : " Suppose, for instance, a chemist places a certain suspected sub- stance in Marsh's test apparatus and finds that it gives a small deposit of metallic arsenic, he cannot be sure that the arsenic really proceeds from the suspected substance; the impurity of the zinc or 1 Lubbock, On the Senses, Instincts, and Intelligence of Ani- mals, p. 45. 2 Lubbock, On Senses, etc., p. 75. 116 INDUCTIVE LOGIC sulphuric acid may have been the cause of its ap- pearance. It is therefore the practice of chemists to make what they call blind experiments, that is, to try whether arsenic appears in the absence of the suspected substance. The same precaution ought to be taken in all important analytical operations. Indeed it is not merely a precaution, it is an essen- tial part of any experiment. If the blind trial be not made, the chemist merely assumes that he knows what would happen." 1 1 Jevons, Principles of Science, p. 433. CHAPTER IX The Joint Method of Agreement and Dif- ference It has already been shown that the method of difference is sometimes not available, inasmuch as it may be neither possible nor practicable to remove from the phenomenon to be investigated the sus- pected causal element without destroying the phe- nomenon itself. Sometimes, too, it is impossible even to neutralize the effect of the causal element if it is allowed to remain as an integral part of the phenomenon. This is especially the case in all vital phenomena, and also in many chemical phe- nomena. Therefore another method is resorted to, which is known as the joint method of agreement and difference. Inasmuch as the suspected causal element cannot be removed, we must select another phenomenon as much like the former as possible, which is, however, characterized by the absence of the causal element. By the simple method of dif- ference, two instances only need be compared, the one with and the other without the causal element, but agreeing precisely in every other particular. In the joint method, the instances with and without 117 118 INDUCTIVE LOGIC the causal element, differ it may be in several par- ticulars. A number of varying instances must therefore be selected so as to eliminate the possi- bility of any of these differing characteristics being the cause in question. Therefore two sets of in- stances are collected, and compared. The one set comprises all the positive instances having the pres- ence of the supposed causal element, and the second set consists of the negative instances having the supposed causal element absent altogether. The validity of the method depends upon the similarity of the two sets of instances. As the similarity increases, the method approximates to the simple method of difference. The Canon of the Joint Method. If several instances in which the phenomenon occurs have only one circumstance in common, while several instances in which it does not occur have nothing in common save the absence of that circum- stance; the circumstance in which alone the two sets of instances differ, is the effect, or cause, or a necessary part of the cause, of the phenomenon. The symbolical representation of this method may be exhibited as follows, using a similar nota- tion to that employed in the previous methods : I. Table of positive instances. 8 + C 8 +e S' +C s' +e S" +C s"+e S'"+Q s'" + e etc. etc. METHOD OF AGREEMENT AND DIFFERENCE 119 II. Table of negative instances. *i s , Su ' *h Bui s m etc. etc. In the two sets of instances, the following con- ditions mnst be observed in order to render the method valid : 1. 8 + G, S' + C, S" + C, &'" + C, etc., must be so varied that they reveal but one constant element, common to them all, as C. It may be that S will resemble IS' in more marks than the one, namely C, and this may be true of any two or more instances ; however, taken all together, they must possess but the one common element, G. 2. In the same way S t may resemble S ti in more marks than merely the absence of C and so for any two or more instances in the series S p S lP S Ui , etc. However, the one characteristic common to them all must be the absence of C. 3. If in the instances chosen an element is com- mon to all in addition to C, or in the second set its absence, then additional instances must be added to the tables both positive and negative in order to secure this all-important condition of elimination through suitable variation. 4. Moreover, the two series, positive and negative, must have their settings similar. S ti S n , JS UP etc., must resemble S', S", S'", etc. ; otherwise the nega- tive instances would not be significant. They must 120 INDUCTIVE LOGIC be chosen from the same sphere as the positive, that they may be similar. It is possible to multiply neg- ative instances ad infinitum, which, however, would furnish no ground for any inference, because they would be wholly irrelevant to the problem under investigation. 5. If S, is so similar to /S' as to be identical with it, and also s t pass over into s'; then we have the method of difference in its pure form : S'+C s' + e S' s' Here the setting, instead of being similar in the two cases, is the same in each. The following is an experiment of Sir John Lub- bock's concerning the sense of smell in insects, which I have chosen as illustrating this method of induc- tive research. He took a large ant and tethered her on a board by a thread. When she was quite still, he brought a tuning-fork into close proximity to her antennae, but she was not disturbed in the least. He then approached the feather of a pen very quietly, so as almost to touch first one and then the other of the antennae, which, however, did not move. He then dipped the pen in the essence of musk and did the same ; the antenna was slowly retracted and drawn quite back. He then repeated the same with the other antenna, and with like result. Care was taken throughout not to touch the antennae. Lubbock then repeated the experi- ment with a number of different ants, and using various substances. The results in all cases were METHOD OF AGREEMENT AND DIFFERENCE 121 the same, and the inference was naturally drawn that the antennae possessed the sense of smell. In these experiments, various substances were taken having nothing in common save the odor of musk that had been placed upon them. In some cases it is not possible to discover posi- tive instances in which the only common element is the suspected cause. In such cases the method is not conclusive in its results, although it may attain a high degree of probability, if all the com- mon elements save the suspected cause-element are known to be irrelevant, or can in any other way be proved to have no influence whatsoever upon the result. For instance, an illustration is often given of this method, which fails in the manner just de- scribed. A man is attempting to discover whether a particular article of food disagrees with him. He notices several occasions, a large number if you please, when he has eaten this particular kind of food, and has soon after experienced distress. These are the positive instances. This peculiar distress has never been experienced when he has abstained from the food in question. The inference is that this food has caused the distress. In the various instances, however, the sole element in common is not merely the taking or not taking the food. The person's whole bodily organism is com- mon to all the instances. Within it, unforeseen complications independent of this article of food might have caused the trouble. In such cases, number of instances must be resorted to in order to render the possibility of a coincidence impossible. 122 INDUCTIVE LOGIC So also in such cases as the treatment of any given disease in a hospital. An experiment may be tried in the treatment, say, of typhoid fever. One ward may be subjected to a particular kind of treatment, and another ward not subjected to that treatment. If recovery is hastened in the one and retarded in the other case, an inference may be drawn as to efficacy of this treatment. In these instances again, while they are all different patients, still the nursing, surroundings, etc., are common to them all. It must be shown that these are present both in the negative and positive instances, and equally capable of accomplishing the effect if they had been real causes. They may therefore be eliminated in comparing the two sets of instances, because common both to the negative and positive cases. In this also resort must be had to the num- ber of instances in order to eliminate chance coin- cidences. The presence of common elements in excess of the common causal element may be rep- resented according to the symbolical notation of the joint method, by the introduction of another symbol x. Let x stand for that which is common to all instances in addition to the common element C. We then have : I. Set of positive instances. 8 -f C + x s +e S' + C + x s' + e S" + C + x s' +e S> + C + x s'" + e etc. etc. METHOD OF AGREEMENT AND DIFFERENCE 123 II. Set of negative instances. S, + x s t S +x ........ . s u ^tu + x s m etc. etc. We observe x in all instances both positive and negative. Being present when the effect occurs and when it does not, indifferently, we can at once infer that x is not the whole cause of e. However, it may have united with C in the first set of instances to produce the effect e, so that C without x, or some part or parts of x, could not alone produce the effect e. In all such cases the exact force of x must be estimated in some other way. If x is extremely complex, or subject to change from forces emanating from within itself, as in the case of organic phenom- ena, then it becomes extremely difficult to determine x ; and consequently the method of agreement and difference does not yield as exact results. As long as the force of x remains unknown, it becomes the source of possible disturbance, which may wholly vitiate the results attained. Mr. Darwin, in his ^experiments upon cross and self fertilization in the vegetable kingdom, placed a net about one hundred flower heads, thus protect- ing them from the bees and from any chance of fertilization by means of the pollen conveyed to them by the bees. He at the same time placed one hundred other flower heads of the same variety of plant where they would be exposed to the bees, and, as he observed, were repeatedly visited by them. 124 INDUCTIVE LOGIC Here we have the two sets of instances, in one the flowers accessible to the bees, and in the other, not accessible. He obtained the following result. The protected flowers failed to yield a single seed. The others produced 68 grains weight of seed, which he estimated as numbering 2720 seeds. Cross-fertiliza- tion as the cause in this case is thus proved. The common element in all these instances, however, is not merely the presence in one case and the absence in the other of the bees ; there is also the element of the common plant structure running through all of the two hundred instances. This element is, how- ever, of such an unvarying nature in all the instances, and the number observed so many as to eliminate the possibility of any given plant structure possess- ing unobserved peculiarities sufficient to produce the result in question. It may therefore be considered as an inert element as regards the effects noticed in the one and absent in the other set of instances. Sir John Lubbock, in his researches concerning the different functions of the two kinds of eyes in insects, illustrates the joint method in its general features. The two kinds of eyes are the large com- pound eyes, situated one on each side of the head, and the ocelli, or small eyes, of which there are generally three, arranged in a triangle between the other two. He wished to determine the precise function of the small eyes, the ocelli ; and he has gathered together the following facts. Plateau has shown that caterpillars, which possess ocelli, but no compound eyes, are very short-sighted, not seeing above one to two centimetres. He has also proved METHOD OF AGREEMENT AND DIFFERENCE 125 by experiments that spiders, which have ocelli but no compound eyes, are very short-sighted ; they were easily deceived by artificial flies of most inartistic construction, and even hunting spiders could not see beyond ten centimetres (four inches). Lubbock experimented on this point with a female spider, which, after laying her eggs, had rolled them into a ball, and had enveloped the whole with a silken bag which she carried about with her. Having captured the female and having taken the bag of eggs from her, he placed it on a table about two inches in front of her. She evidently did not see it. He then pushed it gradually towards her, but she took no notice till it nearly touched her, when she eagerly seized it. He then took it away a second time, and put it in the middle of the table, which was two feet four inches by one foot four, and had nothing else on it. The spider wandered about for an hour and fifty minutes before she found it, apparently by accident. He then took it away again and put it down as before, when she wandered about for an hour without finding it. Like experiments were tried with other spiders and with the same results. Plateau also experimented with scorpions which had ocelli and no compound eyes. They appeared scarcely to see beyond their own pincers. More- over, the ocelli are especially developed in insects, such as ants, bees, and wasps, which live partly in the open light and partly in the dark recesses of nests. Again, the night-flying moths all possess ocelli. On the other hand, however, they are en- tirely absent in all butterflies, with, according 126 INDUCTIVE LOGIC to Scudder, but one exception, namely, the genus Pamphila. Forel, moreover, varnished the com- pound eyes of various insects which had ocelli as well. The latter, however, he allowed to remain in their natural state. Inasmuch as their habits of flight required powers of vision in these insects extending to a considerable distance, it happened that when placed on the ground they made no at- tempt to rise; while, if thrown into the air, they flew first in one direction and then in another, striking against any object that came in their way, and being apparently quite unable to guide them- selves. They flew repeatedly against a wall, falling to the ground, and unable to alight against it, as they did so cleverly when they had their compound eyes to guide them. All these instances, taken together in their positive and negative aspects, led Sir John Lubbock to infer that the ocelli were use- ful in dark places and for near vision, while the compound eyes were for the light and more distant vision. 1 Another illustration of this method may be found in Darwin's account of the extreme tameness of the birds in the Galapagos and Falkland islands. I quote some extracts from his narrative, in which it will be seen that Darwin's inferences follow from his comparison of the positive and negative instances before him. He says : " This tameness of dispo- sition is common to all the terrestrial species of these islands in the Galapagos Archipelago ; namely, 1 Lubbock, On the Se?ises, Instinct, and Intelligence of Ani- mals, pp. 175 ff. METHOD OF AGREEMENT AND DIFFERENCE 127 to the mocking-thrushes, the finches, wrens, tyrant flycatchers, the dove, and carrion-buzzard. All of them often approached sufficiently near to be killed with a switch, and sometimes, as I myself tried, with a cap or hat. A gun is here almost super- fluous ; for, with the muzzle, I pushed a hawk off the branch of a tree. In Charles Island, which had been colonized about six years, I saw a boy sitting by a well with a switch in his hand, with which he killed the doves and finches as they came to drink. He had already procured a little heap of them for his dinner ; and he said that he had constantly been in the habit of waiting by this well for the same purpose. The Falkland Islands offer instances of birds with a similar disposition. The snipe, upland and lowland goose, thrush bunting, and even some true hawks, are more or less tame. The black- necked swan is here wild, and it was impossible to kill it. It, however, is a bird of passage, which probably brought with it the wisdom learned in foreign countries. From these several facts, we may, I think, con- clude that the wildness of birds with regard to man, is a particular instinct directed against him and not dependent on any general degree of caution arising from other sources of danger; secondly, that it is not acquired by individual birds in a short time, even when much persecuted, but that in the course of successive generations it becomes hereditary. With domesticated animals we are accustomed to see new mental habits or instincts acquired and rendered hereditary, but with animals 128 INDUCTIVE LOGIC in. a state of nature it must always be most difficult to discover instances of acquired hereditary knowl- edge. In regard to the wildness of birds towards man, there is no way of accounting for it except as an inherited habit : comparatively few young birds, in any one year, have been injured by man in England, yet almost all, even nestlings, are afraid of him ; many individuals, on the other hand, both at Galapagos and at the Falklands, have been pur- sued and injured by him, but yet have not learned a salutary dread of him." * I have given this quotation somewhat at length in order to show the method of a great investigator in the realm of nature; and that it may be seen how naturally he falls into the method of compar- ing positive and negative sets of instances relative to the object of research. The animal and vege- table kingdoms are especially adapted to the appli- cation of this joint method, and therefore it is in biology that it is most frequently employed and where it has yielded the most fertile results. The advantage of the joint method over the sim- ple method of agreement is that it largely elimi- nates the possibility of there being any other cause of the given phenomenon than the one disclosed by the operation of this method. The method of agreement, as we have seen, often fails of a definite result owing to the plurality of causes. The joint method tends to indicate the one and only cause, and when the instances are rigorously selected ac- cording to the conditions of the canon, there is a 1 Darwin, Voyage of a Naturalist, Vol. II. pp. 172 f. METHOD OF AGREEMENT AND DIFFERENCE 129 high, degree of probability that the sole cause is discovered. Mr. Mill at this point claims too much for the method in insisting that it gives a certainty regarding the sole cause, when the requirements are perfectly realized. It is impossible to realize the requirements perfectly. In selecting the nega- tive instances, we are never sure that we have compassed the entire sphere of significant negative instances. We may, however, attain results highly probable in this regard, though they may not reach an absolute certainty. Such a statement is more moderate in its expression, and practically it assures as satisfactory results. CHAPTER X The Method of Concomitant Variations The method of concomitant variations is a process of determining a causal relation when, as an element in an antecedent varies in intensity, greater or less, there is observed a corresponding or concomitant variation in the consequent. Canon of the Method of Concomitant Variations. Whatever phenomenon varies in any manner, when- ever another phenomenon varies in some particular, is either a cause or an effect of that phenomenon, or is connected with it through some fact of causation. The latter clause of this canon provides for that circumstance in which the varying elements may both be concomitant effects of a common cause. When we are assured of the absence of any possi- ble common cause to which we can assign the two phenomena as effects, then they must be related between themselves as cause and effect. A simple illustration of this method is the rise of the mer- cury in the thermometer owing to the increase of heat; its fall, whenever there is decrease of heat. One varies as the other concomitantly, and we infer a causal relation that we at once proceed to gen- eralize without hesitation. 130 r ERSlTY METHOD OF CONCOMITANT VARIATIONS 13] The symbolical representation of this method is as follows : S+C s + e S +0 dC s + ede etc. etc. Then C is the canse of e. I have used dC, and de to denote the increments or decrements of the cause and effect respectively. This method is used generally when the method of difference is impossible, owing to the fact that the supposed causal element cannot be made to vanish wholly. In all such cases a variation of the ele- ment is resorted to, and the corresponding result observed. Heat is relative and not absolute, as also the height of mercury in the tube. The relation is determined, therefore, by variations, greater and less. This method is also used to supplement the results of other methods by which a causal relation has been determined, but not in exact quantitative terms. It may be known that a certain phenome- non C is always the cause of a certain effect e, and the method of concomitant variations will then be of use in determining precisely how much of a vari- ation in C will cause a specified variation in e. A law finds scientific expression only when stated in terms of exact quantitative relation between varia- tions in antecedent and consequent. We may ex- press the law of universal attraction in a vague way that bodies always attract each other and the greater attraction when the bodies are nearer together, and the larger they are. But this statement needs to 132 INDUCTIVE LOGIC be recast in terms exhibiting the precise quantita- tive variation. Bodies attract each other directly as the product of their masses, and inversely as the square of their distance. It is evident that the special function of this method of concomitant vari- ations consists in just this quantitative determina- tion. In one respect, therefore, it may be regarded as a substitute for the method of difference, and in another way as a supplement to the method of difference in leading to quantitatively determinate results. The quantitative variation between antecedent and consequent may be either direct or inverse vari- ation. The former is when one increases as the other increases, or when one decreases as the other decreases. The inverse is when one decreases as the other increases, or vice versa. This may be expressed symbolically S+CdC . . . . s+eTde We have, for instance, Boyle's law as regards the variation of volume of gases according to the press- ure ; that is, when we double the pressure, we halve the volume. This may be proved experimentally. The method also was used by Kicardo to prove his law that the rate of profits varies in inverse ratio to the rate of wages. We have also the tendency ob- served in respect to increase of crimes, when there is decrease of opportunities for labor. The expression of a law in terms of the quantita- tive relation between antecedent and consequent may be facilitated by a graphic representation of the METHOD OF CONCOMITANT VARIATIONS 133 same, through corresponding abscissas and ordinates. The varying antecedents, for instance, may be laid off on the axis of X, and each several consequent represented by the corresponding ordinates. The resulting curve thus obtained will represent the law of their mutual relation. If the equation of the curve can be determined, it will represent the math- ematically exact expression of the law in question. If this is not possible, it may prove at least sug- gestive of the law which otherwise might have remained concealed. This graphical method is especially useful in dealing with physical phenom- ena. " If the abscissae represent intervals of time, and the ordinates corresponding height of the ba- rometer, we may construct curves which show at a glance the dependence of barometric pressure upon the time of day. Such curves may be accurately drawn by photographic processes on a sheet of sen- sitive paper placed behind the mercurial column, and made to move past it with a uniform horizontal velocity by clockwork. A similar process is applied to the temperature and electricity of the atmosphere, and to the components of terrestrial magnetism." J This method, moreover, has the advantage of the psychological impression which it makes. The mind is more susceptible to the perception of varia- tion in forces where the change is apparent to the senses, than to the perception of a constant force, whose constant character thereby conceals its nat- ure and function from the senses. Synchronous 1 Thomson and Tait, Elements of Natural Philosophy, Vol. I. p. 119. 134 INDUCTIVE LOGIC changes attract the attention, and admit of ready comparison, as we follow ont the variations from point to point. We may ring a bell in a vacuum, and detect no sound whatsoever, and then allow the air to enter gradually. We notice that as the air enters more and more freely, the sound grows louder and louder. The relation of cause and effect is thus demonstrated to the senses in the most vivid manner possible. The variations are exhibited side by side, and thus, presented together in their con- comitant relation, produce the deeper impression. This method is of special advantage in all experi- ments where the intensity of the forces can be varied at will and the consequent effects exhibited in some palpable manner. The determination of the heat rays in the solar spectrum is accomplished by this method. The spectrum may be received upon a plate pierced with a narrow slit, through which the rays can act upon a thermo-electric pile, which will indicate by deflections of a needle the varying intensity of the heat in the several rays of the spectrum. Now, move the slit through the whole extent of the spectrum, beginning with the violet portion. While in the violet, the indigo, the blue, and even the green, the needle of the ther- moscopic apparatus will be deflected but slightly, it will indicate an amount of heat increasing as the slit crosses the yellow, next the orange, then the red ; and then beyond the red, and entering the dark portion of the spectrum, we find the greatest deflection of all. The maximum of heat is there- fore in a region beyond the observation of the METHOD OF CONCOMITANT VARIATIONS 135 senses when unaided by experimental device ; and yet revealed conclusively by this method. 1 Professor Tyndall performed a very interesting experiment to prove that the cloud of darkness sur- rounding flames of great heat was due to the fact that the heat consumed the floating motes in the air which serve to scatter the light which is visible only when thus diffused. The phenomenon which he endeavored to explain was somewhat as follows : Beneath a beam of electric light, a red-hot poker was placed, and from it black wreaths as of smoke were seen to ascend. A large hydrogen flame being employed, it produced whirling masses of darkness far more copiously than the poker. Of this Pro- fessor Tyndall remarked : " Smoke was out of the question ; what then was the blackness ? It was simply that of stellar space ; that is to say, blackness resulting from the absence from the track of the beam of all matter competent to scatter its light. When the flame was placed below the beam, the float- ing matter was destroyed in situ; and the air freed from this matter rose into the beam, jostled aside the illuminated particles, and substituted for their light the darkness due to its own perfect transpar- ency. Nothing could more forcibly illustrate the invisibility of the agent which renders all things visible. The beam crossed, unseen, the black chasm formed by the transparent air, while at both sides of the gap the thick-strewn particles shone out like a luminous solid under the powerful illumination. " 2 1 Saigey, The Unity of Natural Phenomena, p. 61. 2 Tyndall, Fragments of Science, p. 280. 136 INDUCTIVE LOGIC Such being the phenomenon and Professor Tyndall's explanation, it will be seen that he proceeded accord- ing to the method of concomitant variations in the following experiment of many which he performed to substantiate this theory : A platinum tube, with its plug of platinum gauze, was connected with an experimental tube, through which a powerful beam could be sent from an electric lamp placed at its end. The platinum tube was heated till it glowed feebly but distinctly in the dark. The experimental tube was then exhausted, and filled with air that had passed through the red-hot tube. A considerable amount of floating matter which had escaped com- bustion was revealed by the electric beam. Then the tube was raised to a brighter redness and the air permitted to pass slowly through it. Though diminished in quantity, a certain amount of floating matter passed into the exhausted ex- perimental tube. The platinum tube was rendered still hotter; a barely perceptible trace of the floating matter now passed through it. The experiment was repeated, with the difference that the air was sent more slowly through the red-hot tube. The floating matter was totally destroyed. The platinum tube was now lowered until it bordered upon a visible red heat. The air, sent through it still more slowly than in the last ex- periment, carried with it a cloud of floating mat- ter. Professor TyndalPs commentary upon this experiment is as follows: "If, then, the sus- METHOD OF CONCOMITANT VARIATIONS 137 pended matter is destroyed by a bright red heat, much more is it destroyed by a flame, whose tem- perature is vastly higher than any employed in this experiment. So that the blackness intro- duced into a luminous beam where a flame is placed beneath it is due, as stated, to the destruc- tion of the suspended matter." l Professor Tyndall also supplemented this experi- ment by one which was according to the joint method of agreement and difference. He prepared oxygen so as to exclude all floating particles, and found that when blown into the beam, darkness was produced ; also that hydrogen, nitrogen, car- bonic acid, and coal-gas, when prepared in a similar way, each produce darkness when poured or blown into the beam. These instances, combined with various positive instances of illumination of mote- strewn currents of air, illustrate the method of agreement and difference. An additional experiment, according to the method of difference, was as follows : Professor Tyndall placed an ordinary glass shade in the air with its mouth downward. This permitted the track of the beam to be seen crossing it. Letting coal-gas, or hydrogen, enter the shade by a tube reaching to its top, the gas gradually filled the shade from the top downward. As soon as it occupied the space crossed by the beam, the luminous track was in- stantly abolished. Lifting the shade so as to bring the common boundary of gas and air above the beam, the track flashed forth. After the shade was 1 Tyndall, Fragments of Science, pp. 283, 284. 138 INDUCTIVE LOGIC full, he inverted it; thereupon the gas passed up- ward like a black smoke among the illuminated particles. 1 The method of concomitant variations is not only capable of illustration by laboratory methods and devices ; it finds abundant illustration as well in the realm of nature, where observation alone becomes the instrument of investigation and where experiment is impossible or impracticable. Lyell, in his Principles of Geology, gives a very interest- ing account of the alternate elevation and subsi- dence of the temple of Jupiter Serapis, at Pozzuoli, on the Bay of Naples. 2 It is situated in proximity to several volcanoes, Vesuvius, however, being at some distance. It has been observed that there is a certain connection between each era of upheaval, and a local development of volcanic heat; and on the other hand, between each era of depression, and the local quiescent condition of volcanic phenomena. Before the Christian era, when Ischia was in a state of eruption, and A vermis and other points in the Phlegrsean fields were celebrated for their volcanic character, it was observed that at that time the ground on which the temple stood was several feet above water. Vesuvius was then regarded as a spent volcano. After the Christian era, Vesuvius became active and then scarcely a single eruption occurred in Ischia or around the Bay of Baiae. It was ob- served that at that time the temple was sinking. Vesuvius then became quiet for five centuries pre- 1 Tyndall, Fragments of Science, pp. 284, 285. 2 Chapter XXX. METHOD OF CONCOMITANT VAKIATIONS 139 ceding the eruption of 1631, and during that period the Solf atara was in eruption in 1198, Ischia in 1302, and Monte Nuovo was formed in 1538. Then the foundations of the temple were observed to be ris- ing again. Vesuvius became active after that, and has continued so ever since, and during this time the temple has been subsiding. The inference is that as the subterranean heat increases, and lava forming without obtaining an easy vent like that afforded by Vesuvius, the surface is elevated, but when the rocks below are cooling and contracting, the pent-up fire being withdrawn in the eruption of the great Vesuvius, then there is a corresponding subsidence. The observation of concomitant variations is furthermore illustrated in Darwin's researches con- cerning the formation of coral reefs, as regards the question which some naturalists have raised as to which part of the coral reef is most favorable to the growth of coral. 1 He adduces the following facts, most of which are the direct result of his observa- tions : " The great mounds of living Porites and of Millepora round Keeling atoll occur exclusively on the extreme verge of the reef, which is washed by a constant succession of breakers. At the Mar- shall Islands the larger kinds of coral which form rocks measuring several fathoms in thickness pre- fer the most violent surf. The outer margin of the Maldiva atolls consists of living corals, and here the surf is so tremendous that even large ships have been thrown, by a single heave of the sea, 1 Darwin, Coral Reef s, pp. 87 f. 140 INDUCTIVE LOGIC high and dry on the reef, all on board thus escap- ing with their lives. In the Ked Sea the strongest corals live on the onter reefs and appear to love the snrf. From these facts it is certain that the strong- est and most massive corals nourish where most exposed. The less perfect state of the reef of most atolls on the leeward and less exposed side, com- pared with its state to the windward, "and the anal- ogous case of the greater number of breaches on the rear sides of those atolls in the Maldiva Archipelago, which afford some protection to each other, are obvi- ously explained by this circumstance." There seems to be here a combination of the method of agree- ment with that of concomitant variations. And such a combination may be employed to advantage in cases where the phenomena under investigation show forces under varying degrees of intensity; the causal relation is more apparent, and the pos- sibility of fortuitous coincidence is largely elimi- nated if a number of instances can be collected in which the forces manifest themselves in vary- ing degrees. Accumulation of instances, showing concomitant variations in the forces observed, cor- responds to the actual variations which in an experi- ment are effected by the investigator himself. In such observed instances, however, we cannot always have before us the variations expressed continuously. There are evident gaps that must be interpolated mentally. In the experiment, however, of whatever nature, the degrees' of intensity can be exhibited continuously, one degree merging into another through inappreciable increments. There is thus METHOD OF CONCOMITANT VARIATIONS 141 a gradation which has no gaps to be filled, and the psychological impression is thereby heightened. By the method of concomitant variations it is possible also to represent to the mind the magni- tude of an unknown force, or unobservable force by comparison with the intensity of a known force, which lies within the sphere of observation. Tor instance, Mr. Darwin gives an interesting account in his narrative of the finding near the shores of the Plata a group of vitrified silicious tubes which had been formed by lightning entering loose sand. The internal surface of such tubes is completely vit- rified, glossy, and smooth, and the tubes themselves are generally compressed, and have deep longitu- dinal furrows so as closely to resemble a shrivelled vegetable stalk, or the bark of an elm or cork tree. Their circumference is about two inches, but in some fragments which are cylindrical and without any furrows, it is as much as four inches. Judging from the uncompressed fragments, the measure or bore of the lightning proved to be about one inch and a quarter. In contrast with the force of light- ning as thus revealed in its effects, Mr. Darwin cites some experiments performed in Paris by an artifi- cial force of great magnitude indeed and yet with results that seem insignificantly small in compari- son. He says : " At Paris, M. Hatchette and M. Beudant succeeded in making tubes in most respects similar to these fulgurites by passing very strong shocks of galvanism through finely powdered glass : they failed, however, both with powdered felspar and quartz. One tube, formed with pounded glass, 142 INDUCTIVE LOGIC was very near an inch long, namely, .982, and had an internal diameter of .019 of an inch. When we hear that the strongest battery in Paris was used, and that its power on a substance of such easy fusi- bility as glass was to form tubes so diminutive, we must feel greatly astonished at the force of a shock of lightning, which, striking the sand in several places, has formed cylinders in one instance at least thirty feet long, and having an internal bore, where not compressed, of full an inch and a half ; and this in a material so extraordinarily refractory as quartz ! " 1 The method of concomitant variations may be used in regard to phenomena whose nature is such as seemingly to indicate a constant law of variation, and yet inferences based thereupon lead to false results. It is, therefore, well to note some of these instances by way of general precaution in applying this method. 1. It does not necessarily follow that having observed two forces varying in a constant ratio through several concomitant modifications, the same ratio will be preserved indefinitely through all subsequent changes. Water contracts as it is cooling. Suppose we begin to note this continued contracting of water from 100 F. to 90 ; we natu- rally expect to find it continuing through 90 to 80. And as we observe, we find our expectations con- firmed. And so on through to 40, we find that water continues to contract. It is, therefore, most natural for us to expect to find water contracting 1 Darwin, Voyage of a Naturalist, Vol. I. pp. 76 f . METHOD OF CONCOMITANT VARIATIONS 143 at 39. But just at this point in the series, there is a break in the continuity of variation ; at 39 water begins to expand and so continues until it passes into the solid form at the freezing-point. The same also is illustrated in Weber's law, already mentioned, which expresses the quantitative relation between the stimulus and the corresponding sensation. The law is that the force of the stimulus must increase geometrically, in order that the intensity of the sensation should increase arithmetically. This law, however, breaks down towards the upper or lower limits, with a stimulus of slight degree of intensity and with one of extreme intensity. We find also an increase of temperature as we proceed towards the centre of the earth of about one degree to every fifty-three feet of descent. This by no means warrants us in inferring that this ratio continues constant to the very centre itself. In certain phenomena, moreover, there are natural limits, as in sound, for example, where the pitch rises as the number of vibrations increases. At a certain point, varying according to different individuals, increase of vibrations gives no resulting sound whatsoever ; and so there is a lower limit, vibrations may decrease to a point beyond which no sound is heard. An illustration of this fallacy, though not strictly of the method of concomitant variations, is given by Jevons. He takes the following series of prime numbers: 41, 43, 47, 53, 61, 71, 83, 97, 113, 131, etc. It will be seen that they all agree in being values of the general expression x 2 + x -}- 41, 144 INDUCTIVE LOGIC where we put for x the successive values of 0, 1, 2, 3, 4, etc. For instance, let x = in x 2 -f- x + 41, we get 41 ; let a; = 1 in the same, we get 43 ; when x = 2, we get 47 ; and so on. It seems as though we could keep this up indefinitely, produc- ing an increasing series, always of prime numbers. It is found, however, that if we take x = 40, in the formula x 2 -f x + 41, we shall have 40x40 + 40 -f- 41, which equals 1681, and this number is the square of 41 and therefore not a prime number. At this point the law breaks down. 1 In the sphere of political economy also we might be led into an easy yet false inference. Suppose a certain farm yield 500 bushels of corn with a given amount of expenditure and labor. We might think that if we doubled the expenditure and labor, we will also be able to double the results, and obtain a yield of 1000 bushels as over against the 500 of the previous year. Here, however, what is known as the law of decreasing returns obtains ; to double the product it may be necessary to triple or quadruple the labor and expense. " Thus in the production of any plot of land there is a point of equilibrium, which marks an impassable limit, not of course a limit which could not be passed if it were wished, but one that no one wishes to pass, because there is nothing to be gained by so doing." 2 To know that such false inferences are at least possible in the application of this method of con- comitant variations to the unknown regions beyond 1 Jevons, Principles of Science, p. 230. 2 Gide, Political Economy, p. 325. METHOD OF CONCOMITANT VARIATIONS 145 our experience, may serve at least to keep us on guard under similar circumstances. 2. There are certain phenomena, moreover, in which an increased intensity of the force in ques- tion may give rise to incidental effects which tend to neutralize the chief effect to be attained. For instance, an overdose of arsenic causes violent contractions of the stomach so that its contents are immediately ejected, and thus the system is relieved of the noxious substance. 3. Two elements in a given phenomenon may vary together constantly and yet they may not be related at all as cause and effect, but appear as coin- cidental effects of one and the same cause. It has been observed that the occurrence of the Aurora Borealis has been accompanied by pronounced mag- netic disturbances. It, however, cannot be inferred that the former has been the cause of the latter ; they are probably the varied effects of some widely operating magnetic force. The precaution above mentioned has already been referred to as provided for in the canon of this method which states that the observed concomitant variation may indicate not always a direct causal element between the two varying elements, but that they are at least connected with the phe- nomenon under investigation through some fact of causation. CHAPTER XI The Method of Residues The method of residues consists in the analysis of a given phenomenon based upon previous induc- tions, through which it has been determined that certain elements in the antecedent, have caused certain elements in the consequent; the inference is then drawn, that the remaining elements of the antecedent are necessarily the cause of the remain- der of the consequent. It is a method of elimina- tion of the known relations so as to simplify the complex character of the phenomenon and disclose the relations that are unknown in the light of a causal connection which we are constrained to be- lieve must obtain. Hie Canon of the Method of Residues. Subduct from any phenomenon such part as is known by previous inductions to be the effect of certain ante- cedents, and the residue of the phenomenon is the effect of the remaining antecedents. The symbolical representation is as follows : Given S+C s+e If it is known that there exists the causal relation S *, 146 THE METHOD OF RESIDUES 147 we may then infer that C is the cause of e. In this C may be simple or complex ; if it is simple, the causal relation established is expressed in its simplest terms and is therefore a determinate result. If, however, the residue C is complex, it must be reduced by experimental analyis to its simplest elements, and their relation to the elements into which e can be analyzed further determined. The most striking illustration of this method, and one of the most brilliant achievements of science as well, is the discovery of the planet Neptune by Adams and Le Verrier, working on the problem in- dependently and reaching the same result. These astronomers had observed certain perturbations in the planet Uranus. It did not keep in its proper orbit as determined by their mathematical calcula- tions based upon the presence of the known stellar bodies. It behaved as though beyond its orbit was an outer planet, whose presence alone could account for the observed perturbations. Adams and Le Verrier then proceeded to calculate the exact posi- tion of such a disturbing body as determined by the nature and magnitude of the perturbations of Uranus. The telescope was then pointed to the exact point in the heavens, as thus indicated, and the planet Neptune was revealed to the eye accord- ing to the determination of far-reaching prophecy, which confidently asserted that it must be there. The method of residues is really a deductive method based upon the law of sufficient reason ; so many elements on the one hand producing so many elements on the other ; if, then, a part of the former 148 INDUCTIVE LOGIC is to be checked off as cause of a part of the latter, then the remainder on one hand must be the cause- of the remainder on the other, t This is pure de- duction. For we ask, Why are we constrained to account for the remainder on one side by the re- mainder on the other ? The only possible answer is that it must be accounted for within the system to which it is referred ; and but one part therein is left which can possibly account for it, because all the others are specifically determined in the known effects which they have produced. This method, however, has a proper place among the inductive methods, inasmuch as it is based on previous induc- tions, and leads to investigations that can be prose- cuted only by the various inductive processes of experiment. When the residue of the antecedent is a simple element, and no other possible causal element can lie concealed from our observation, then the infer- ence is simple and conclusive. A difficulty, how- ever, may present itself, owing to the fact that the residual element is apt to be complex and leave the phenomenon still indeterminate, or there may be a lurking element unnoticed by us which is the real cause in question. The function of this method is, therefore, largely suggestive. It says the effect is not wholly accounted for by the known causal ele- ments ; there is a residue unaccounted for, and its cause is to be sought in the residue of the antece- dent, and if it is thought that the whole of the antecedent is comprehended, the question is started, May there not be unobserved circumstances of the THE METHOD OF RESIDUES 149 antecedent that further experiment will be calcu- lated to reveal? In many cases, therefore, this method must be supplemented by some other ex- perimental method in order to secure more precise determination, generally the method of difference. It often happens in investigations in chemistry, astronomy, and physics, that the actual phenomena vary in greater or less degree from their expected behavior according to established theory. This must lead either to a reconstruction of theory, or to a search for some unobserved force sufficient to account for the discrepancy. Herschel was the first to point out the significance of such discrep- ancies in scientific research, and he called them residual phenomena. An illustration of such a situation and the solu- tion of the problem thus presented is that of Sir Humphry Davy's experiments upon the decomposi- tion of water by galvanism. "He found that besides the two components of water, oxygen and hydrogen, an acid and alkali were developed at the two opposite poles of the machine. As the theory of the analysis of water did not give reason to ex- pect these products, they were a residual phenome- non, the canse of which was still to be found. The insight of Davy conjectured that there might be some hidden cause of this portion of the effect ; the glass containing the water might suffer partial decomposition, or some foreign matter might be mingled with the water, and the acid and alkali be disengaged from it, so that the water would have no share in their production. Assuming this, he 150 INDUCTIVE LOGIC proceeded to try whether the total removal of the cause would destroy the effect produced. By the sub- stitution of gold vessels for the glass, without any change in the effect, he at once determined that the glass was not the cause. Employing distilled water, he found a marked diminution of the quantity of acid and alkali evolved ; yet there was enough to show that the cause, whatever it was, was still in opera- tion. The impurity of the water, then, was not the sole, but a concurrent cause. He now conceived that the perspiration from the hands touching the instruments might affect the case, as it would con- tain common salt, and an acid and alkali would result from its decomposition under the agency of electricity. By carefully avoiding such contact, he reduced the quantity of the products still further, until no more than slight traces of them were per- ceptible. What remained of the effect might be traceable to impurities of the atmosphere decom- posed by contact with the electrical apparatus. An experiment determined this ; the machine was placed under an exhausted receiver, and when thus secured from atmospheric influence, it no longer evolved the acid and alkali." 1 By means of the suggestions incident upon this method, Bunsen, in 1860, discovered two new alka- line metals, caesium and rubidium. He was ex- amining alkalies produced by the evaporation of mineral water from Durkheim. The flame of these salts was examined by the spectroscope. Bunsen discovered several bright lines which he had never 1 Gore, The Art of Scientific Discovery, pp. 432, 433. THE METHOD OF RESIDUES 151 noticed before, and which he knew could not be produced by potash or soda, whose corresponding lines were in close proximity. He then subjected the mixture to a searching analysis and succeeded in obtaining two new alkaline substances. When he had separated them, he then tested them by the method of difference, by which he found that they were capable of producing the lines at first noticed ; but when withdrawn, the lines instantaneously dis- appeared. Thomson and Tait, in their Elements of Natural Philosophy, have the following reference and illustration of this method. "When, in an ex- periment, all known causes being allowed for, there remain unexplained effects (excessively slight it may be), these must be carefully inves- tigated, and every conceivable variation of ar- rangement of apparatus, etc., tried ; until, if possible, we manage so to exaggerate the residual phenomenon as to be able to detect its cause. It is here, perhaps, that in the present state of science we may most reasonably look for exten- sions of our knowledge ; , at all events, we are warranted by the recent history of natural phi- losophy in so doing. Thus, to take only a very few instances, and to say nothing of the discovery of electricity and magnetism by the ancients, the peculiar smell observed in a room in which an electrical machine is kept in action was long ago observed, but called the ' smell of electricity/ and thus left unexplained. The sagacity of Schon- bein led to the discovery that this is due to the 152 INDUCTIVE LOGIC formation of ozone, a most extraordinary body, of enormous chemical energies; whose nature is still uncertain, though the attention of chemists has for years been directed to it." 1 Another illustration of this method is seen in the comparison of the observed and calculated positions of Encke's comet. It was found that the comet returned a little sooner than it should have done, the period regularly decreasing from 1212.79 days, between 1786 and 1789, to 1210.44 between 1855 and 1858. The inference has been that there is a resisting medium, as the ether, filling the space through which the comet passes. What the resisting medium is, and its nature, is of course a matter of conjecture as far as re- vealed by this method alone. The method merely indicates some resisting medium to account for the observed discrepancy. 2 Herschel has observed that all great astronom- ical discoveries have been disclosed in the form of residual differences. The practice was intro- duced by Halley, when astronomer royal, of comparing systematically the positions of the heavenly bodies as actually observed with what might have been expected theoretically. His re- ductions of the lunar observations gave a series of residual errors, extending from 1722 to 1739. These were carefully tabulated, and formed the basis for certain modifications of the lunar theory. 3 1 Thomson and Tait, Elements of Natural Philosophy, Vol. I. pp. 113 f . 2 Jevons, Principles of Science, p. 570. s Ibid. p. 572. THE METHOD OF RESIDUES 153 A discrepancy was observed by Newton between the theoretical and actual velocity of sound; the former being 968 feet per second, and the latter 1142. Newton's experiments and calculation were both inaccurate; nevertheless, a real discrepancy has been proved to exist, the theoretical being 916 and the real velocity 1090 feet per second. In 1816 La Place showed this difference to be due to the heat evolved by the sudden compres- sion of the air during the propagation of the sound wave, the heat having the effect of in- creasing the elasticity of the air, and therefore appreciably accelerating the sound impulse. It sometimes happens that in repeating an ex- periment, we are confronted with evidently different results. Then, we may be sure, the experiment has been carelessly or inaccurately performed ; or else there are some disturbing causes not observed by us. On the other hand, however, if there is no likelihood of coincidence on repeated trials, yet, nevertheless, a marked agreement is noticed in the results of various trials, the mind should be at once alert to discover the hidden cause of such agreement, -and possibly may be led to new truths of great importance. The following illustration is given by Thomson and Tait : " With a very good achromatic telescope a star appears to have a sensi- ble disc. But, as it is observed that the discs of all stars appear to be of equal angular diameter, we of course suspect some common error. Limiting the aperture of the object-glass increases the appear- ance in question, which, on full investigation, is 154 INDUCTIVE LOGIC found to have nothing to do with discs at all. It is, in fact, a phenomenon due to diffraction of light." x It was said of Darwin that in his researches the residual phenomena were always the special objects of his attention. His son, Francis Darwin, says of him: " There was one quality of mind which seemed to be of special and extreme advantage in leading him to make discoveries. It was the power of never letting exceptions pass unnoticed. Every- body notices a fact as an exception when it is strik- ing or frequent, but he had a special instinct for arresting an exception. A point apparently slight and unconnected with his present work is passed over by many a man almost unconsciously, with some half-considered explanation, which is in fact no explanation. It was just these things that he seized upon to make a start from. In a certain sense there is nothing special in this procedure, many discoveries being made by means of it.- I only mention it, because, as I watched him at his work, the value of this power to an experimenter was so strongly impressed upon me." 2 This is striking testimony as to the practical worth of this method as an instrument of research. This method has also been applied to the more practical usage of examining the refuse of manu- factured and other products in order to discover 1 Thomson and Tait, Elements of Natural Philosophy, Vol. I. p. 114. 2 F. Darwin, Life and Letters of Charles Darwin, Vol. I. p. 125. THE METHOD OF RESIDUES 155 some concealed utility. The analysis of coal-tar refuse has led to the discovery of many valuable substances that have proved of use in the arts, and in medicine as well. Glauber, the eminent chemist, and a discoverer of several chemical compounds, said he made it a rule to examine what every other chemist threw away. vgpS=- LIB** /T9^ OF THK CHAPTER XII Verification and Prediction Tlie Inducto-deductive Method. We have seen that the inductive methods are efficient in revealing the cause of a given phenomenon under investiga- tion; and yet they do not warrant us in general- izing the special instance so as to formulate a universal law. There is always the possibility that while the special case which we experiment upon may give us indications of an existing causal relation, still a wider experience might disprove, or else modify materially our conclusions. The well- recognized fact of the plurality of causes and the intermixture of effect further embarrasses us in the attempt to rise to a law having universal sig- nificance and validity. The results of the induc- tive methods, therefore, need to be supplemented by some corroborative observations or experiments that will conclusively verify the results as obtained. This supplementary method is one which combines deduction with induction. Mr. Mill calls it the Deductive Method. It is, however, more ade- quately designated by the name, the Inducto-de- ductive Method. It consists of three stages : 1. Obtaining, by the inductive methods already 156 VERIFICATION AND PREDICTION 157 described, the evidence of some existing causal connection, tentatively expressed in the form of a universal law. 2. Kegarding this universal law as the basis for subsequent deductions, by which we gain a knowl- edge of the nature of unknown phenomena, as necessitated by the conditions of this law. 3. Verifying the results thus obtained by their correspondence with the phenomena as actually observed. Where this correspondence is wanting, then either the law was not correctly expressed, or there must have been some error in our deduction based upon it. When we are assured that the lat- ter is not the case, then a discrepancy between the theoretically deduced result and the actual facts as observed, always discredits our original induction. This method of verification serves as a check upon hasty generalization, on the one hand; and on the other, it serves to extend our knowledge into un- known regions, and is valuable as a means of scien- tific prediction. In the development of scientific knowledge, it has been a potent factor in enlarging the bounds of knowledge beyond the sphere of im- mediate observation. This combined process of reasoning is the one commonly employed by us all. Induction and de- duction are not separate processes, but, as before remarked, they are complementary factors in the one actual process of reasoning. We are con- tinually using our inductions as a deductive basis, inferring how things should be before they are really seen; and, when seen, at once instinctively 158 INDUCTIVE LOGIC comparing prior inference with present fact, we are either confirmed in our reasoning process, or compelled to discard our previous inference as false or inadequate as the case may be. Our world, the world of knowledge, is built up of the seen, and the unseen as well, because necessitated by inferences growing out of the seen which we are constrained to make; the unseen which we thus are continually building into the seen and regard- ing it as though the known, we are, however, from time to time compelled to alter, and here and there tear down what we have too rashly builded up, as the structure is put to the test of verifying fact. This method of verification was used to decide between inferences drawn by Newton and Huy- ghens respectively, regarding the nature of light. Newton's observations led him to infer that light consisted of particles of matter shot out from the sun. Huyghens insisted that light consisted in the propagation of some kind of disturbance in the man- ner of a wave-motion. Newton's theory being taken as established, it would necessitate that light on entering a denser body of water, being refracted more nearly in a direction perpendicular to the surface, should, accordingly, move faster in the denser body than in the rarer one outside. On the other hand, according to Huyghens' theory, the opposite effect should take place, light being re- fracted towards the vertical at the horizontal sur- face of a dense body such as water, its velocity in the dense body should be less than its velocity in VERIFICATION AND PREDICTION 159 the rare body. The experiments separately made by Fizeau and Foucault, both gave the result that in water light moves slower than in air, and therefore the theory of Huyghens, which was in accord with snch a fact, was verified, and the theory of Newton, which was radically out of harmony with such a fact, was discredited. 1 We cannot theorize concerning nature to any con- siderable extent without resorting to nature again to correct aberrations of reason, and the false fancies of the imagination. Theory, if correctly formulated, will always lead to a representation of facts as they are ; just as facts as they are, if rightly interpreted, will always lead to correct theory. The following are illustrations of the value of this method in predicting results before unknown. " Halley had the glory of having first detected a periodic comet in the case of that which has since borne his name. In 1705, Halley explained how the parabolic orbit of a planet may be determined from three observations; and joining example to precept, himself calculated the positions and orbits of twenty-four comets. He found, as the reward of his industry, that the comets of 1607 and 1531 had the same orbit as that of 1682. And here the intervals are nearly the same, namely, about seventy- five years. Are these three comets then identical ? In looking back into the history of such appear- ances, he found comets recorded in 1456, in 1380, and 1305; the intervals are still the same, sev- 1 Tait, Recent Advances in Physical Science, pp. 65, 66. 160 INDUCTIVE LOGIC enty-five or seventy-six years. It was impossible now to doubt that they were the periods of a revolv- ing body, its orbit a long ellipse, not a parabola. If this were so, the comet must reappear in 1758 or 1759. Halley began his laborious calculations and predicted that the comet would reach its perihelion April 13, 1759, but claimed the license of a month for the inevitable inaccuracies of a calculation in which, in addition to all other sources of error, was made in haste, that it might appear as a prediction. The comet justified his calculations and his caution together ; for it arrived at its perihelion on March 13, 1759." ! Another illustration of a like nature is the pre- diction of Faraday, based upon Wheatstone's ex- perimental proof that the conduction of electricity required time; namely, "that if the conducting wires were connected with the coatings of a large Leyden jar, the rapidity of conduction would be necessarily lessened. This prediction was made in 1838 and was not verified until, sixteen years later, a submarine cable was laid beneath the English Channel. A considerable retardation of the electric spark was then detected by Siemens and Latimer Clark. Faraday at once pointed out that the wire surrounded by water resembles a Leyden jar on a large scale : so that each message sent through the cable verified his remark of 1838." 2 In Pasteur's experiments with silkworms already referred to, he made a prediction in 1866, when, 1 Whewell, History of Inductive Science, 3d ed. Vol. II. p. 182. 2 Jevons, Principles of Science, p. 543. VERIFICATION AND PREDICTION 161 having inspected fourteen parcels of eggs intended for incubation, and having examined the moths which produced these eggs, he wrote out the pre- diction of what would occur in 1867, and placed the prophecy as a sealed letter in the hands of the mayor of St. Hippolyte. In 1867, the cultivators communicated to the mayor their results. The letter of Pasteur was then opened and read, and it was found that in twelve out of fourteen cases there was absolute conformity between his predic- tion and the observed facts. Many of the groups had perished totally ; the others had perished almost totally ; and such was Pasteur's prediction. In two out of the fourteen cases, instead of the prophesied destruction, half an average crop was obtained. 1 Another interesting illustration concerns Dar- win's speculations regarding the formation of coral reefs and atolls. Before Darwin wrote on the subject, it was generally believed that the coral atolls were formed by the coral polypes growing upon submerged volcanic craters. Darwin insisted that as the polypes cannot live below a depth of 100 feet, and are killed by exposure to sunshine and air, and could not therefore have grown upward from the vast depths to which the coral masses extend, each atoll must have begun as a f ringing- reef about an island almost touching the shore, with only a narrow and shallow channel of water be- tween ; and then became a barrier reef, that is, one with a wider and deeper channel of water separat- 1 Tyndall, Fragments of Science, pp. 291, 292. M 162 INDUCTIVE LOGIC ing from the shore, owing to the slow but progres- sive subsidence of the island round which the polypes first began to build. Then with the further and complete subsidence of the island beneath the water, there remained a ring of coral with a central lagoon forming the so-called atoll. Darwin says in his Autobiography that the main features of his theory were conceived while on the voyage, and that even previous to seeing a true coral reef. 1 He says : " No other work of mine was begun in so deductive a spirit as this, for the whole theory was thought out on the west coast of South America, before I had seen a true coral reef. I had only to verify and extend my views by a careful examina- tion of living reefs. But it should be observed that I had during the two previous years been in- cessantly attending to the effects on the shores of South America of the intermittent elevation of the land, together with denudation and deposition of sedi- ment. This necessarily led me to reflect much on the effects of subsidence, and it was easy to replace in imagination the continued deposition of sediment by the upward growth of corals. To do this was to form my theory of the formation of barrier reefs and atolls." It will thus be seen that Darwin's deduction was based upon previous inductions in other spheres, the result of his own observation; he also tells us in the same connection, that he had, in the prepa- ration of his work on Coral Reefs, spent twenty months of hard labor, reading every work on the 1 Life and Letters of Charles Darwin, 1887, Vol. I. p. 58. VERIFICATION AND PREDICTION 163 islands of the Pacific and consulting many charts. He thus made the widely extended observations of other men tributary to his inferences concerning coral-reef formations. Dr. Williams says of Dar- win's insight in this particular: "He saw more clearly than his precursors had done the validity of the dictum of Johannes Mliller in this, and indeed all his works, that the most important truths in natural science are to be discovered, neither by the mere analysis of philosophical ideas, nor by simple experience, but by reflective experience, which distinguishes the essential from the acciden- tal in the phenomena observed, and thus finds principles from which many experiences can be derived." * This is a very satisfactory and strik- ing account of what may be styled the combined inducto-deductive temper of mind, and especially as embodied in so eminent a student of nature as Darwin. Bacon insists that anticipations of nature are sources of innumerable errors, and that the truly scientific method consists in an interpretation of nature as it is revealed to the perception through direct observation and experiment. It is, however, largely through these " anticipations " that progress in science is attained. There may be anticipations which are considered final, and all attempts at veri- fication regarded as unnecessary and even as im- pertinent. Results deductively attained are then asserted with dogmatic insistence, as though pos- 1 Darwin, Coral Reefs. Prefatory note by Dr. J. W. Williams, p. ix. 164 INDUCTIVE LOGIC sessing the convincing power of facts themselves; and all appeal to controverting or exceptional cases are set aside, without even so much as a respectful hearing. Such anticipations of nature rightfully fall under the scornful reprehension of a Bacon. But there are other anticipations which serve as a spur to a more penetrating observation, and more painstaking experiment, in order to square theory to facts. Such anticipations are the glory of science ! Suppose such anticipations are disproved by subsequent experiment or observation ; they have served a high purpose in suggesting investigation along lines which otherwise would have remained unthought of. Anticipations alone are barren ; an- ticipations leading to verification are productive of valuable results. To this the history of scientific thought bears abundant testimony. Professor Clif- ford has made the power of prediction one of the essential characteristics of scientific thought. He says, in his essay on the Aims and Instruments of Scientific Thought, that " the difference between sci- entific and merely technical thought is just this : Both of them make use of experience to direct hu- man action; but while technical thought or skill enables a man to deal with the same circumstances that he has met with before, scientific thought en- ables him to deal with different circumstances that he has never met with before." 1 He cites two illus- trations, which are admirable examples of scientific prediction. The first relates to the suggestion of Meeming Jenkin, regarding structural bracing. It i Clifford, Lectures and Essays, Vol. I. p. 128. VERIFICATION AND PREDICTION 165 had been known before that in an arch every part is compressed or pushed by other parts ; and every part of a chain is in a state of tension, that is, pulled by the other parts. In many cases these forms are united in the common girder, which con- sists of two main pieces, of which the upper acts as an arch, and is compressed, while the lower one acts as a chain and is pulled. " Now," says Profes- sor Clifford, " suppose that any good, practical engi- neer makes a bridge or a roof upon some approved pattern which has been made before. He designs the size and shape of it to suit the opening which has to be spanned ; selects his material according to the locality ; assigns the strength which must be given to the several parts of the structure, accord- ing to the load which it will have to bear. There is a great deal of thought in the making of this design, whose success is predicted by the applica- tion of previous experience ; it requires technical skill of a very high order, but it is not scientific thought. On the other hand, Mr. Fleeming Jenkin designs a roof consisting of two arches braced to- gether, instead of an arch and a chain braced together; and, although this form is quite different from any known structure, yet before it is built he assigns with accuracy the amount of material that must be put into every part of the structure in order to bear the required load, and this prediction may be trusted with perfect security. What is the natural comment on this ? Why, that Mr. Fleeming Jenkin is a scientific engineer." 1 1 Clifford, Lectures and Essays, Vol. I. pp. 127, 128. 166 INDUCTIVE LOGIC The second illustration which Professor Clifford gives is as follows : " You know that if you make a dot on a piece of paper, and then hold a piece of Iceland spar over it, you will see not one dot, but two. A mineralogist, by measuring the angles of a crystal, can tell you whether or no it possesses this property without looking through it. He requires no scientific thought to do that. But Sir William Eowan Hamilton, the late astronomer royal of Ire- land, knowing these facts, and also the explanation of them, which Fresnel had given, thought about the subject, and he predicted that by looking through certain crystals in a particular direction we should see not two dots, but a continuous circle. Mr. Lloyd made the experiment and saw the circle, a result which had never been even suspected. This has always been considered one of the most signal instances of scientific thought in the domain of physics. It is most distinctly an application of experience gained under certain circumstances to entirely different circumstances." 1 There is also an indirect method of prediction, varying somewhat from the one already described and yet similar to it. It is called prediction by inversion of cause and effect. There are many cases, in which cause and effect are related in a reciprocal manner, so that not only will the cause produce the effect, but the effect, operating as a cause, will bring about the original cause as an effect, it may be in a modified form but clearly recog- nizable as such. Professor Tyndall said of Faraday 1 Clifford, Lectures and Essays, Vol. I. pp. 128, 129. VERIFICATION AND PREDICTION 167 that " the strong tendency of his mind to look upon the reciprocal actions of natural forces gave birth to his greatest discoveries." 1 For instance, Oersted had proved that an electric current will produce magnetism, and Faraday, taking this as a suggestion, inferred that magnetism might produce an electric current; in the year 1831 he devised a suitable experiment of introducing a bar-magnet into a coil of insulated copper wire, and then withdrawing the magnet whilst the two ends of the wire were con- nected with a distant galvanometer, which indicated the presence of the electric current. Thus, his in- ference received substantial verification. 2 It has, moreover, been found that when a given cause produces a certain effect, then if the effect be produced in some other manner, the process will tend to produce the original cause, but inverted as regards its direction or nature. For instance, it is known that heat will expand gases ; now, if a gas be re- lieved of the pressure of the vessel enclosing it, it will expand by virtue of its own elastic power, produc- ing, however, cold in the surrounding atmosphere. So also heat will cause a bar of iron to expand. Dr. Joule proved that if iron were expanded by mechanical force, it would be accompanied by cold. Inasmuch as india-rubber is related to heat in an opposite manner to that of iron, being contracted by heat instead of expanded, we would, according to the law above expressed, naturally expect that a mechanical expansion of india-rubber would give 1 Tyndall, Fragments of Science, p. 338. 2 Gore, The Art of Scientific Discovery, p. 594. 168 INDUCTIVE LOGIC heat, and a contraction produce cold. An experi- ment may be tried by suddenly stretching a rubber band while the middle part is in the mouth ; when stretched, it grows warm ; when relaxed, it seems cold. 1 Again, as heat will melt many substances, if we can reduce the same substance from the solid to the liquid state, we would expect, as a result, the negative of heat, namely, cold. This occurs in all freezing mixtures, as the affinity of salt for water causes it to melt ice, thus producing cold in the surrounding atmosphere, sufficient to freeze cream or other similar substance, inasmuch as, passing from solid to liquid, water absorbs heat from all substances near it ; this absorption producing arti- ficial cold surrounding it. The reciprocal action of heat and cold is illustrated in an interesting ex- periment described by Tait. 2 He took a bar of ice, supported horizontally at either end, and over the middle of the bar he put a fine wire, and put equal weights to the two ends of the wire. The wire gradually, by the action of the weights, cut through the bar of ice, and yet it was observed that the path of the wire was instantly replaced by the freezing again of the melted portion produced by the press- ure, and when the wire had wholly traversed the entire thickness of the bar, the bar itself was intact, and even stronger along the line of the cutting than before. The explanation of this experiment is that inasmuch as heat melts ice, then when ice is melted 1 Jevons, Principles of Science, p. 545. 2 Tait, Recent Advances in Physical Science, pp. 99, 100. VERIFICATION AND PREDICTION 169 by pressure, as in this case of the weighted wire, cold, the negative of heat, is induced ; thus, as the wire was forced by the weights into the ice, the press- ure upon the ice melted it, making it colder, so that the water produced, passing around the chilled wire, and being thus relieved of pressure, froze again. Faraday predicted certain magnetic phenomena by this method, which are specially interesting as illustrations of this kind of prediction. It seems that Arago had observed in 1824 that the number of oscillations which a magnetized needle makes in a given time, under the influence of the earth's magnetism, is very much lessened by the proximity of certain metallic masses, and es- pecially of copper. Employing the latter substance in an experiment upon a magnetized needle, he suc- ceeded in reducing the number of its vibrations in a given time from three hundred to four. Taking the experiment as a basis for his inference, Fara- day predicted that since the presence of a metal at rest stops the oscillations of a magnetic needle, the neighborhood of a magnet at rest ought to stop the motion of a rotating mass of metal. He there- fore proceeded to put his inference to the test of actual experiment, by suspending a cube of copper to a twisted thread which was placed between the poles of a powerful electromagnet. When the thread was left to itself, it began to spin round with great velocity, but stopped the moment a powerful current passed through the electromag- net. 1 Again, as heat applied to the junction of i Ganot, Physics, pp. 797, 798. 170 INDUCTIVE LOGIC two metallic bars, as antimony and bismuth, pro- duced an electric current, it was inferred that if an electric current was made to pass through such a junction, it would produce cold, and such proved to be the case. 1 In the general process of verification, it often happens that seeming exceptions occur which are direct contradictions of the law we are attempting to prove. And it is in dealing with such cases that one's power of discrimination is most fully taxed. It is necessary to make a most careful distinction between seeming and real exceptions. Professor Jevons has given a very exhaustive clas- sification of the different kinds of exceptional phenomena, which it is well to have in mind, in order to know in any investigation the various pos- sible complications that may rise. 2 The excep- tional phenomena, as given by Jevons, are : 1. Imaginary, or false exceptions ; that is, facts, objects, or events which are not really what they are supposed to be. 2. Apparent but congruent exceptions, which, though apparently in conflict with a law of nature, are really in agreement with it. 3. Singular exceptions, which really agree with a law of nature, but exhibit remarkable and unique results of it. 4. Divergent exceptions, which really proceed from the ordinary action of known processes of 1 Jevons, Principles of Science, p. 547. 2 See Jevons, Chapter XXIX., in his Principles of Science, on " Exceptional Phenomena." VERIFICATION AND PREDICTION 171 nature, but which are excessive in amount or monstrous in character. 5. Accidental exceptions, arising from the inter- ference of some entirely distinct but known law of nature. 6. Novel and unexplained exceptions, which lead to the discovery of a new series of laws and phe- nomena, modifying or disguising the effects of pre- viously known laws without being inconsistent with them. 7. Limiting exceptions, showing the falsity of a supposed law to some cases to which it had been extended, but not affecting its truth in other cases. 8. Contradictory, or real, exceptions, which lead us to the conclusion that a supposed hypothesis or theory is in opposition to the phenomena of nature, and must therefore be abandoned. It will be seen that among so many possibilities of interpretation an exception does not necessarily prove the rule, as the old adage would have it; nor does the exception, on the other hand, neces- sarily disprove the rule or law. It must be in each case strictly and adequately interpreted, which re- quires a penetrating sagacity and a thorough knowl- edge of the phenomena under investigation. In the process of verification, the question nat- urally suggests itself: How many verifying in- stances are sufficient to determine the universal validity of a given law? This question will be recognized as an old difficulty, now presented in another form; but in reality it is the perplexing problem of determining the logical ground of in- 172 INDUCTIVE LOGIC duction. What is our warrant for proceeding from known and verified instances to unknown phenom- ena, of the same kind it is true, but as yet beyond the pale of our experience ? The warrant for our generalization does not lie wholly in the number of verifying instances. In addition to the effect which mere number produces in confirming our belief, there is the confidence which we feel in the con- stancy of the order of nature, and which we are constrained to assume as a fundamental postulate. 1 Therefore, we say that the verifying facts must be of such a number, and of such a nature as well, that they give evidence of a uniformity which transcends all supposition of mere coincidence, and compels us to attribute it to the uniformity of nature itself, in which we find a warrant for our generalization. As Professor Clifford has remarked : " The aim of scientific thought is to apply past ex- periences to new circumstances. The instrument is an observed uniformity in the course of events. By the use of this instrument it gives us informa- tion transcending our experience, it enables us to infer things that we have not seen from things that we have seen ; and the evidence for the truth of that information depends on our supposing that the uniformity holds good beyond our experience." * In extending knowledge and predicting results beyond the sphere of experience, modern scientific investigation is largely indebted to the principles and methods of mathematics. Mathematical laws, 1 See Sigwart, Logic, Vol. II. p. 348. 2 Clifford, Lectures and Essays, Vol. I. pp. 131, 132. VERIFICATION AND PREDICTION 173 applied to the data given in sense-perception, give indications of the necessary relations that must exist in the observed phenomena, and all that they involve. Thus, that which is given directly in con- sciousness is supplemented by that which is given indirectly as mathematically necessitated. The mathematico-experimental method in physics has led to very rich and important results which have proved practically its efficiency as a scientific method. CHAPTER XIII Hypothesis The inductive process cannot proceed to any- great extent or attain satisfactory results without the aid of some hypothesis. An hypothesis is a supposition regarding the cause of a phenomenon, which we make either as preliminary to an experi- ment which will prove or disprove the supposition, or in lieu of an experiment or systematic observa- tion when such are impossible, owing to the peculiar conditions of the phenomenon itself. We see, there-, fore, that the framing of hypotheses has a double function. First, considered as preliminary to ex- periment. We found that in eases where two, three, or more elements enter into a complex antecedent, it is impossible often, and always impracticable to test the various possible combinations separately in order to note their different results. The combinations in complex phenomena are indefinitely great, and the isolation of certain elements in order to. estimate tl^e exact result of the combined force of the other combinations is extremely difficult and often im- possible. Therefore the mind discards some com- bin ations as irrelevant, others as impossible, and selects one perhaps as the most likely cause of the 174 HYPOTHESIS 175 given effect. This selective function of the mind, therefore, indicatesTthe line of experiment in a de- terminate manner and does not leave the phenom- ena to indeterminate and haphazard investigation. Consider, for instance, so eminent an experimenter as Charles Darwin, so fertile in all kinds of experi- mental resources ; yet it is said of him that every experiment was the result of a tentative theory, thought out in advance of all actual test by a saga- cious insight into the necessary conditions of the interrelated phenomena before him. His son, Fran- cis Darwin, says of him in his Reminiscences: "He often said that no one could be a good observer unless he was an active theorizer. 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 mag- nified into importance. In this way, it naturally happened that many untenable theories occurred to him; but fortunately his richness of imagination was equalled by his power of judging and condemn- ing the thoughts that occurred to him. He was just to his theories, and did not condemn them un- heard ; and so it happened that he was willing to test what would seem to most people not at all worth testing. These rather wild trials he called i fool's experiments/ and enjoyed extremely. As an example, I may mention that, finding the cotyledons of Biophytum to be highly sensitive to vibrations of the table, he fancied that they might perceive the vibrations of sound, and therefore made me play 176 INDUCTIVE LOGIC my bassoon close to a plant. The love of experiment was very strong in him, and I can remember the way he would say, ' I shan't be easy till I have tried it/ as if an outside force were driving him." ! Hypothesis and experiment were in the hand of Darwin like a two-edged sword, which he employed with rare skill and effect. An hypothesis is to be regarded not only as the precursor of experiment, but it also functions as a method of explanation when actual verification is impossible. We see this constantly in our every-day life. We are com- pelled again and again to account for situations which occur that are impossible for us to reproduce in the form of an experiment, that we are able to observe but once. Some explanation is required to satisfy mental demands which are imperative in this regard. The explanation which seems most in keeping with the sum of facts in our possession, is the hypothesis which we frame ; so also in explain- ing the conduct of others by conjecture as to the most reasonable motives that will satisfactorily account for the same ; such hypotheses we are con- stantly compelled to assume. We are not always able to perceive the relations existing between facts as they come into the sphere of our experi- ence, and yet we are constrained to think of them as related; but in order to systematize them, we must supply mentally the lacunae which appear in the phenomena as perceived. This supposition that is necessary to construct facts into system is an hypothesis. 1 Life and Letters of Charles Darwin, Vol. I. p. 126. HYPOTHESIS 177 An illustration of an hypothesis suggesting syste- matic observation and experiment is found in the history of the discovery of vaccination by Jenner. It seems that while a mere youth, pursuing his studies at Sodbury, he chanced to hear a casual remark made by a country girl who came to his master's shop for advice. The small-pox was men- tioned, when the girl said, " I cannot take that dis- ease, for I have had cow-pox." l This observation, expressing the common superstition of the simple country folk, appealed to Jenner's mind as an in- choate hypothesis. Seizing upon it as a suggestion of possible value, he proceeded to make diligent inquiries and careful observations, which finally led him to the discovery of vaccination. An illustration of hypothesis as explanation of phenomena beyond the range of experiment is found in the hypothesis as to the source of the sun's energy. An enumeration of the different hypothe- ses advanced upon this subject is given by Tait in his Recent Advances in Physical Science. 2 "The old notion that the sun is a huge fire, or something of that kind, is one which will only occur to one thinking of the matter for the first time ; but with our modern chemical knowledge, we are enabled to say that, massive as the sun is, if its materials had consisted of the very best materials for giving out heat, that enormous mass of some 400,000 miles in radius could have supplied us with only about 5000 years of the present radiation. A mass of 1 Gore, The Art of Scientific Discovery, p. 495. 2 pp. 151 ff. N 178 INDUCTIVE LOGIC coal of that size would have produced very much less than that amount of heat. Nor would the most energetic chemicals known to us, combined in proportion for giving the greatest amount of heat by actual chemical combination, supply the sun's present waste for even 5000 years. There- fore as we all know that geological facts, if there were no others, point to at least as high a radiation from the sun as the present, for at all events a few hundreds of thousands of years back, and per- haps also indicate even a higher rate of radiation from the sun in old time than at present, it is quite obvious that the heat of the sun cannot possibly be supplied by any chemical process of which we have the slightest conception. "Now, if we can find, on the other hand, any physical explanation of this consistent with any present knowledge, we are bound to take it and use it as far as we can, rather than say : This question is totally unanswerable unless there be chemical agencies at work in the sun of a far more power- ful order than anything we meet with on the earth's surface. If we can find a thoroughly in- telligible source of heat, which, though depending upon a different physical cause from the usual one, combustion, is amply sufficient to have sup- plied the sun with such an amount of heat as to enable it to radiate for perhaps the last hundred millions of years at the same rate as it is now radiating, then I say we are bound to try that hypothesis first, and argue upon it until we find it inconsistent with something known. And if we HYPOTHESIS 179 do not find it inconsistent with anything that is known, while we find it completely capable of explaining our difficulty, then it is not only philo- sophic to say that it is most probably the origin of the sun's energy, but we feel ourselves con- strained to admit it. Newton long ago told us this obligation in his Rules of Philosophizing. Now it is known that if we were to take a mass of the most perfect combustibles which we know, and let it fall upon the sun merely from the earth's distance, then the work done upon it by the sun's attraction during its fall would give it so large an amount of kinetic energy when it reached the sun's surface as to produce an impact which would represent six thousand times the amount of energy which could be produced by its mere burn- ing. "It appears, then, that our natural and only trustworthy mode of explaining the sun's heat at present, in time past, and for time to come must be something closely analogous to, but not identical with, what was called the nebular hypothesis of Laplace, the hypothesis of the falling together (from rudely scattered distribution in space) of the matter which now forms the various suns and planets. We find by calculation in which there is no possibility of large error, that this hypothesis is thoroughly competent to explain one hundred millions of years' solar radiation at the present rate, perhaps more ; and it is capable of showing us how it is that the sun, for thousands of years together, can part with energy at the enormous 180 INDUCTIVE LOGIC rate at which it does still part with it, and yet not apparently cool by perhaps any measurable quantity. "In confirmation of this, not only is the hy- pothesis itself capable of explaining the amounts of energy which are in question, but also recent in- vestigations, aided by the spectroscope, have shown us that there are gigantic nebular systems at great distances from our solar system, in the process of physical degradation in that very way, by the falling together of scattered masses, and with numerous consequent developments of heat by impacts. What are called temporary stars form another splendid and still more striking instance of it, as where a star suddenly appears, of the first magnitude, or even brighter than the first, out- shining all the planets for a month or two at a time, and then, after a little time, becomes invisi- ble in the most powerful telescope. Things of that kind are constantly occurring on a larger or smaller scale and they can all be easily explained on this supposition of the impact of gravitating masses." Such a hypothesis, it will be seen, embraces all the facts observed in one self-consistent system. The other hypotheses are inadequate to account satisfactorily for the phenomena. The validity of this hypothesis lies in its being both adequate and congruent as well ; experiment or corroborative observation being out of the question, we are, as Tait says, " constrained to admit it." Mr. Darwin gives an enumeration and criticism HYPOTHESIS 181 of the different hypotheses which have been sug- gested to explain the extinction of the gigantic animals known to have existed upon the earth. His account will give an indication of the natural propensity of the mind to frame hypotheses con- cerning phenomena which lie outside the sphere both of observation and experiment. Mr. Darwin says: "It is impossible to reflect on the changed state of the American Continent without the deep- est astonishment. Formerly it must have swarmed with great monsters ; now we find mere pigmies compared with the antecedent allied races. The greater number, if not all, of these extinct quad- rupeds, lived at a late period, and were the con- temporaries of most of the existing sea-shells. What then has exterminated so many species and whole genera ? The mind at first is irresistibly hurried into the belief of some great catastrophe ; but thus to destroy animals, both large and small, in Southern Patagonia, in Brazil, on the Cordillera of Peru, in North America up to Behring's Straits, we must shake the entire framework of the globe. "An examination, moreover, of the geology of La Plata and Patagonia leads to the belief that all the features of the land result from slow and gradual changes. It appears from the character of the fossils in Europe, Asia, Australia, and in North and South America, that those conditions which favor the life of the larger quadrupeds were lately coextensive with the world. What those conditions were, no one has yet even con- jectured. It could hardly have been a change of 182 INDUCTIVE LOGIC temperature, which at about the same time de- stroyed the inhabitants of tropical, temperate, and arctic latitudes on both sides of the globe. In North America we positively know from Mr. Lyell that the large quadrupeds lived subsequently to that period when boulders were brought into lati- tudes at which icebergs now never arrive; from conclusive but indirect reasons we may feel sure that in the southern hemisphere the Macrauchenia also lived long subsequently to the ice-transporting boulder-period. Did man, after his first inroad into South America, destroy, as has been suggested, the unwieldy Megatherium and the other Eden- tata ? We must look at least to some other cause for the destruction of the little tucutuco at Bahia Blanca, and of the many fossil mice and other small quadrupeds in Brazil. No one will imagine that a drought, even far severer than those which cause such losses in the provinces of La Plata, could destroy every individual of every species from Southern Patagonia to Behring's Straits. What shall we say of the extinction of the horse ? Did those plains fail of pasture which have since been overrun by thousands and hundreds of thou- sands of the descendants of the stock introduced by the Spaniards ? Have the subsequently intro- duced species consumed the food of the great antecedent races ? Can we believe that the Capy- bara has taken the food of the Toxodon, the Guanaco of the Macrauchenia, the existing small Edentata of their numerous gigantic prototypes ? Certainly no fact in the long history of the world HYPOTHESIS 183 is so startling as the wide and repeated exter- minations of its inhabitants." 1 Mr. Darwin's own hypothesis concerning this phenomenon is rather indefinite, but nevertheless as definite as the ex- treme complexity of the facts will allow. He says that there are certain causes operating in nature, their exact character remaining unknown, such that the too rapid increase of every species, even the most favored, is steadily checked, pro- ducing in some cases rarity and in others ex- tinction, if these causes operate with unusual efficacy. His hypothesis marks a tendency whose nature, nevertheless, remains concealed. In all these widely differing hypotheses we see a certain mental constraint to offer some explanation, even though it be but a disguised confession of ignorance, as in Mr. Darwin's hypothesis. An illustration of an hypothesis to explain ob- served phenomena that cannot be further tested is that given in the following instance cited by Pro- fessor Tyndall : " At Erith, in 1864, there occurred a tremendous explosion of a powder magazine. The village of Erith was some miles distant from the magazine, but in nearly all cases the windows were shattered ; and it was noticeable that the windows turned away from the origin of the explosion suf- fered almost as much as those which faced it. Lead sashes were employed in Erith church ; and these, being in some degree flexible, enabled the windows to yield to pressure without much fracture of glass. Every window in the church, front and back, was 1 Darwin, Voyage of a Naturalist, Vol. I. p. 223. 184 INDUCTIVE LOGIC bent inwards. In fact, as the sound-wave reached the church, it separated right and left, and, for a moment, the edifice was clasped by a girdle of in- tensely compressed air, which forced all its win- dows inwards. After compression, the air in the church no doubt dilated, and tended to restore the windows to their first condition. The bending in of the windows, however, produced but a small condensation of the whole mass of air within the church; the force of the recoil was, therefore, feeble in comparison with the force of impact, and insufficient to undo what the latter had ac- complished." 1 Here also is a set of conditions that must be satisfied by a correct hypothesis. The phenomenon was not capable of repetition by any experiment. Professor Tyndall, therefore, pict- ures to his mind what must have happened beyond that which was observed, in order to account for the result which actually happened. He fills up the unseen from what he knows of the nature of sound-waves, and thus constructs one self-consistent system which includes both the seen and the un- seen, the known and the unknown, the observed and the inferred. It will be noticed in this and other illustrations of hypothesis, how large a part is played by the imagination. It is the imagination which fills out the vacant spaces in the picture of perception. With some, the function of imagination is asso- ciated with fancy rather than fact. It must, in this connection, however, be clearly emphasized i Tyndall, On Sound, p. 23. HYPOTHESIS 185 that the imagination which constructs hypotheses must be throughout in touch with fact. It must represent to the mind, not what fancy suggests, but what the known facts necessitate. The unseen is constructed out of the determining conditions of the seen. It is this deductive function of the imagination that gives to it a strictly logical sig- nificance. For instance, Professor TyndalPs reason- ing concerning the Erith church was somewhat as follows : The windows are all bent inward, there- fore the pressure must have operated on all sides from without, inward; such pressure could only occur upon the supposition that the sound-waves, separating right and left, wholly encompassed the church, etc. In each case, that which he pictured to his mind as happening, was regarded by him as actually necessitated by the facts as observed. Professor Tyndall has most admirably discussed the " Scientific Use of the Imagination ; " and his lecture under that title every student, both of logic or of science, should read. I quote one passage from it, which has special bearing upon what has just been said : " We are gifted with the power of Imagination, combining what the Germans call Anschauungsgabe and Einbildungskraft, and by this power we can lighten the darkness which sur- rounds the world of the senses. There are tories in science who regard imagination as a faculty to be feared and avoided rather than employed. They had observed its action in weak vessels and were unduly impressed by its disasters. But they might with equal justice point to exploded boilers as an 186 INDUCTIVE LOGIC argument against the use of steam. Bounded and conditioned by co-operant Reason, imagination be- comes the mightiest instrument of the scientific dis- coverer. Newton's passage from a falling apple to a falling moon was, at the outset, a leap of the imagination. When William Thomson tries to place the ultimate particles of matter between his compass points, and to apply to them a scale of millimetres, he is powerfully aided by this faculty. And in much that has been recently said about protoplasm and life, we have the outgoings of the imagination guided and controlled by the known analogies of science. In fact, without this power our knowledge of nature would be a mere tabula- tion of coexistences and sequences. We should still believe in the succession of day and night, of summer and winter; but the soul of Force would be dislodged from our universe ; causal relations would disappear, and with them that science which is now binding the parts of nature to an organic whole." * In all the illustrations which have been given, and, in fact, in all examples of the framing of hypotheses, it will be seen that the mental functions specially in operation are those of analysis and synthesis, a separation of the elements as far as possible into their simplest forms of expression, and the building them together into some one sys- tem whose unity lies in the assumed hypothesis. Mr. Venn has especially emphasized this aspect of 1 Tyndall, Use and Limit of the Imagination in Science, p. 16. HYPOTHESIS 187 hypothesis, and his chapter on this subject will well repay a careful reading. 1 Every supposition, however, is not necessarily an hypothesis in the logical or scientific significance of that term. It will be necessary, therefore, to mention some of the requirements which a logical .hypothesis should satisfy. \ 1. An hypothesis should be plausible ; that is, it should be no fanciful, or merely conjectural, expla- nation of the phenomena in question. The sup- positions of the interference of spirits, or in a mythological age of the gods, to account for per- plexing situations or obscure happenings, have no rank as hypotheses ; so, also, Fate is often referred to as a convenient confession of ignorance in lieu of a satisfactory explanation. Spinoza has remarked upon this as follows : " They who have desired to find scope for the display of their ingenuity in assigning causes, have had recourse to a new style of argument to help them in their conclusions, namely, by reduction, not to the impossible or ab- surd, but to ignorance or the unknown, a procedure which shows very plainly that there was no other course open to them." The difference between a scientific hypothesis and a popular explanation concerning the same phenomena may be found in Darwin's account of "a strange belief which is general amongst the inhabitants of the Maldiva atolls, namely, that- corals have roots, and therefore that if merely broken down to the surface, they grow up again ; 1 Venn, Empirical Logic, Chapter XVI. 188 INDUCTIVE LOGIC but if rooted out, they are permanently destroyed. By this means the inhabitants keep their harbors clear ; and thus the French governor of St. Mary's in Madagascar ' cleared out and made a beautiful little port at that place.'" 1 Their explanation, however, is purely fanciful, having no basis in fact. In contrast, Darwin's hypothesis to explain the facts in the case is of a logically scientific nature, and is as follows : Inasmuch as loose sediment is injurious to the living polypifers, and as it is prob- able that sand would accumulate in the hollows formed by tearing out the corals, but not on the broken and projecting stumps, therefore in the former case the fresh growth of coral might be thus prevented by the deposited sediment. 2. The second requirement is that the hypothesis must be capable of proof or disproof. This does not demand a test by experiment necessarily ; for that, as we have seen, may be impossible. It does, however, require that some facts should be forth- coming that will either confirm the hypothesis or disprove it. There are cases, however, as Lotze suggests, whose very nature precludes the possi- bility of proving or disproving the hypothesis framed to account for them. For instance, the very common and simple hypothesis of regarding the stars, which are apparently but small points of light, as bodies of vast size, only very remote from us, is in itself incapable of being either refuted or confirmed by subsequent discovery. Lotze says: "We must abide content if our hypotheses are 1 Darwin, Coral Reefs, p. 89. HYPOTHESIS 189 thinkable and useful, if they are capable of ex- plaining all interconnected appearances, even such as were still unknown when we constructed them, if, that is to say, they are indirectly confirmed by the agreement of all that can be deduced from them in thought with the actual progress of experience. But if we would be so fortunate as to find an hypothesis which will not lack this subsequent confirmation, we must not simply assume anything that can be barely conceived as real ; we must only assume that which, besides being thinkable, con- forms, so to speak, to the universal customs of real- ity, or to the special local customs which prevail in that department of phenomena to which the object we are investigating belongs." ' It is to be specially observed that while the re- quirement of proof of an hypothesis may be waived in the sphere of phenomena where proof is mani- festly impossible, still, where proof is available, an hypothesis must never be so framed as to render the required test either impossible or impracticable. 3. The hypothesis must be adequate. It must cover all the facts in the case. An outstanding fact which it cannot explain is sufficient to contro- vert such an hypothesis. A knowledge of the dis- tinction between postulate and hypothesis, and of the relation which nominally exists between the two, will help us to appreciate more clearly the force of this requirement of adequacy. As defined by Lotze, a postulate "expresses the conditions which must be set up, or the ground of explanation which 1 Lotze, Logic j p. 353. 190 INDUCTIVE LOGIC must be given by some reality, force, or event, before we can think the phenomenon in the form in which it is presented to us ; it thus requires or postulates the presence of something that can account for the given effect. An hypothesis is a conjecture which seeks to fill up the postulate thus abstractly stated by specifying the concrete causes, forces, or pro- cesses out of which the phenomenon really arose in this particular case, while in other cases maybe the same postulate is to be satisfied by utterly dif- ferent though equivalent combinations of forces or active elements." 1 According to this distinction as applied to the problem of the source of the sun's energy, the postulate would be the sum of conditions which require explanation; namely, the tremendous radiation of heat extending through thousands and thousands of years. The postulate therefore requires a force adequate to supply for so long a period so great an amount of energy. We found that ordinary combustion of the most highly combustible materials would not, as an hypothesis, satisfy the conditions which obtain in the postulate ; nor would the liberation of chemical energy stand as an hypothesis adequate to satisfy the postulate the hypothesis of impact of masses upon the sun's surface from immense distances presents a force sufficient to meet the requirements of the postulate. Moreover, we see in this illustration how the hy- pothesis is a particular and concrete expression of the conditions expressed in general and in abstract terms in the postulate. The essential characteristic 1 Lotze, Logic, pp. 349, 350. HYPOTHESIS 191 therefore of the hypothesis is that it shall perfectly satisfy all the conditions expressed in the postulate. The hypothesis that nature abhorred a vacuum, in order to account for the rise of water in a tube or pump, was seen to break down utterly when it was found that the water did not rise beyond some thirty-three feet. The demand of the postulate in the case was a force of precisely such magnitude that it would balance a column of water thirty-three feet in height. This force, precisely satisfying the conditions of the postulate, is found in the hypoth- esis that the atmospheric pressure is such a magni- tude as to exert a pressure equivalent to a column of water some thirty-three feet in height. The strength of the hypothesis lies in its exact and appropriate fitting into the facts of the problem. Another illustration of the fitting of hypothesis to postulate, and one where the conditions of the postulate are extremely complex, I have chosen from Mr. Wallace's work, On Natural Selection: "There is a Madagascar orchis the Angrcecum sesquipedale with an immensely long and deep nectary. How did such an extraordinary organ come to be developed ? Mr. Darwin's explanation is this. The pollen of this flower can only be removed by the base of the proboscis of some very large moths, when trying to get at the nectar at the bottom of the vessel. The moths with the longest probosces would do this most effectually; they would be rewarded for their long tongues by get- ting the most nectar ; whilst, on the other hand, the flowers with the deepest nectaries would be the best 192 INDUCTIVE LOGIC fertilized by the largest moths preferring them. Consequently the deepest-nectaried orchids and the longest-tongued moths would each confer on the other an advantage in the battle of life. This would tend to their respective perpetuation, and to the constant lengthening of nectaries and pro- bosces. In the Angrcecum sesquipedale it is neces- sary that the proboscis should be forced into a particular part of the flower, and this would only be done by a large moth burying its proboscis to the very base, and straining to drain the nectar from the bottom of the long tube, in which it occu- pies a depth of one or two inches only. Now let us start from the time when the nectary was only half its present length, or about six inches, and was chiefly fertilized by a species of moth which ap- peared at the time of the plant's flowering, and whose proboscis was of the same length. Among the millions of flowers of the Angrsecum produced every year, some would always be shorter than the average, some longer. The former, owing to the structure of the flower, would not get fertilized, because the moths could get all the nectar without forcing their trunks down to the very base. By this process alone the average length of the nec- tary would annually increase, because the short- nectaried flowers being sterile, and the long ones having abundant offspring, exactly the same effect would be produced as if a gardener destroyed the short ones, and sowed the seed of the long ones only ; and this we know by experience would pro- duce a regular increase of length, since it is this HYPOTHESIS 193 very process which has increased the size and changed the form of our cultivated fruits and flowers. But this would lead in time to such an increased length of the nectary that many of the moths could only just reach to the surface of the nectar, and only the few with exceptionally long trunks be able to suck up a considerable portion. This would cause many moths to neglect these flow- ers, because they could not get a satisfying supply of nectax, and if these were the only moths in the country the flowers would undoubtedly suffer, arid the further growth of the nectary be checked by exactly the same process which had led to its increase. " But there are an immense variety of moths, of various lengths of proboscis, and as the nectary became longer, other and larger species would be- come the fertilizers, and would carry on the process till the largest moths became the sole agents. Now, if not before, the moth would also be affected ; for those with the longest probosces would get the most food, would be the strongest and most vigorous, would visit and fertilize the greatest number of flowers, and would leave the largest number of descendants. The flowers most completely fertil- ized by these moths being those which had the longest nectaries, there would in each generation be, on the average, an increase in the length of the nectaries, and also an average increase in the length of the probosces of the moths ; and this would be a necessary result from the fact that nature ever fluct- uates about a mean, or that in every generation o I 194 INDUCTIVE LOGIC there would be flowers with longer and shorter nectaries, and moths with longer and shorter pro- bosces than the average. I may here mention that some of the large Sphinx moths of the tropics have probosces nearly as long as the nectary of Angrce- cum sesquipedale. I have carefully measured the proboscis of a specimen of Macrosila cluentius from South America, in the collection of the British Museum, and find it to be nine inches and a quarter long. One from tropical Africa (Macrosila mor- ganii) is seven inches and a half. A species having a proboscis two or three inches longer could reach the nectar in the longest flowers of Angraicum ses- quipedale, whose nectaries vary in length from ten to fourteen inches. That such a moth exists in Madagascar may be safely predicted ; x and natural- ists who visit that island should search for it with as much confidence as astronomers searched for the planet Neptune, and I venture to predict they will be equally successful." 2 I have given this quotation at length in order to indicate not only the fitting of hypothesis to the facts observed, but also the large and important part performed by the imagination in reproducing along parallel lines the natural history of the orchid and moth. The hypothesis reaches back over an indefinitely long past, by virtue of the necessities 1 It is interesting to note that since Mr. Wallace wrote the above, Kirby, in his European Moths and Butterflies, makes mention of one of the Sphingidae with a proboscis twelve inches long! 2 Wallace, On Natural Selection, pp. 271-275. HYPOTHESIS 195 observed in the present, and in accordance with well-established analogies and approved inductions. The function of the imagination especially promi- nent is that of its deductive insight, which is able to picture to the mind the inevitable results of this and that condition as furnished by the postulate, and then to fit such necessitated results into one self- consistent system, with nothing left unexplained, incongruous, or contradictory. Another illustration of an hypothesis covering a large number of complex facts is that of the ferti- lization of certain flowers by means of the wind. As given by Sir John Lubbock, Ave have the follow- ing facts and the corresponding explanation of them : " Wind-fertilized flowers, as a rule, have no color, emit no scent and produce no honey, and are regular in form. Color, scent, and honey are the three characteristics by which insects are attracted to flowers. Again, as a rule wind-fertilized flowers produce much more pollen than those which are fertilized by insects. This is necessary, because it is obvious that the chances against any given pollen grain reaching the stigma are much greater in the one case than in the other. Every one has observed the showers of yellow pollen produced by the Scotch fir. Again, it is an advantage to wind- fertilized plants to flower early in the spring before the leaves are out, because the latter would catch much of the pollen, and thus interfere with its access to the stigma. Again, in these plants the pollen is less adherent, so that it can be easily blown away by the wind, which would be a disad- 196 INDUCTIVE LOGIC vantage in most plants which are fertilized by insects. Again, snch flowers generally have the stigma more or less branched, or hairy, which evi- dently must tend to increase their chances of catching the pollen." 1 There is here a structural adaptation of these plants to the circumstances designed to explain them, so that the consequent self-consistent system thus formed carries with it the weight of conviction. There are some explanations which do not per- fectly correspond to reality, and yet, when their nature is known, they may be profitably used, not to represent reality, but to assist the mind by an approximate representation to better appreciate the facts as they really are related one to another. These so-called " fictions " are useful, especially in mathematics. We suppose, for instance, inscribed and circumscribed polygons of a circle, with ever- increasing number of sides, gradually approaching and becoming coincident finally with the curve itself. This latter we know to be impossible, and yet we may treat that which happens only approxi- mately as though really happening, merely as an aid to the imagination ; and a fiction, if always so understood, may thus prove helpful in the repre- sentation of reality more clearly to our minds. 4. The hypothesis, moreover, should involve no contradiction. This is clearly a requirement that is deductive rather than inductive, depending upon the fundamental principle of contradiction lying at the basis of the deductive system of logic. 1 Lubbock, Scientific Lectures, pp. 9, 10. HYPOTHESIS 197 5. The hypothesis should be as simple as possible. No involved explanation that mystifies rather than clears the difficulties presented can rank as a true hypothesis. Simplex veri sigillum. This require- ment, of course, cannot in all cases be strictly com- plied with ; for the phenomena to be explained may present such a degree of complexity that a simple hypothesis would be altogether out of the question. For instance, the hypothesis of a substance filling the universe, and pervading all particles of matter, however solid and closely knit together, a substance itself more solid than steel, and more elastic as well, such a supposition seems not only too involved, but also even to belie the ordinary judgments of common sense. And yet this undulatory hypothesis is more and more confirmed by every advance of science in the knowledge of the phenomena of light and heat. It sometimes happens that the very failure of an hypothesis forms a substantial contribution to the progress of thought, leading to the readjustment of a received theory, or stimulating research in order to discover the true in place of the false hypothesis. As Mr. Tait says: "We all know that if there had not been a pursuit after the philosopher's stone, chemistry could not yet have been anything like the gigantic science it now is. In the same way we can say, that modern physics could not yet have covered the ground it now occupies had it not been for this experimental seeking for the so-called perpetual motion, and the consequent establishment of a defi- nite and scientifically useful negative." l The cir- 1 Tait, Recent Advances in Physical Science, p. 69. 198 INDUCTIVE LOGIC cular theory of the orbits of the planets, while incorrect, yet made the transition easier from the hypothesis of circular motion to that of motion in an elliptical orbit, which is the true theory. It often happens that an hypothesis may not be wholly wrong but may need correction, and this is often provided for, not by a total rejection of the hypothesis in question, but by supplementing it by so-called subsidiary hypotheses. As to the tests of a correct hypothesis in addition to the fulfilment of the requirements already men- tioned, Dr. Whewell has especially emphasized the importance of what he has styled a " Consilience of Inductions." An hypothesis receives a confirmatory strengthening of its validity, when it enables us to explain and determine cases not only of the same kind as the phenomena out of which the hypothe- sis itself has developed, but cases which arise in a sphere entirely different from that which gave material originally for the formation of the hy- pothesis. An hypothesis that can thus be carried into new territory as an effective instrument of research, is thereby doubly accredited. As Dr. Whewell remarks: "Accordingly the cases in which inductions from classes of parts altogether different have thus jumped together, belong only to the best established theories which the history of science contains. And as I shall have occasion to refer to this peculiar feature in their evidence, I will take the liberty of describing it by a particular phrase ; and will term it the Consilience of Induc- tions. It is exemplified principally in some of the HYPOTHESIS 109 greatest discoveries. Thus it was found by Newton that the doctrine of the attraction of the sun vary- ing according to the inverse square of the distance, which explained Kepler's Third Law, of the pro- portionality of the cubes of the distances to the squares of the periodic times of the planets, ex- plained also his First and Second Laivs, of the elliptical motion of each planet; although no connection of these laws had been visible before. Again, it appeared that the force of universal gravi- tation, which had been inferred from the perturba- tions of the moon and planets by the sun and by each other, also accounted for the fact, apparently altogether dissimilar and remote, of the precession of the equinoxes. Here was a most striking and surprising coincidence which gave to the theory a stamp of truth beyond the power of ingenuity to counterfeit." * When two rival hypotheses can be submitted to the test of an experiment which negatives one and confirms the other, such a testing is called an ex- perimentum crucis. The name was first given by Bacon, and has met with universal acceptance in scientific phraseology. A crucial test, as decisive between the emission and the undulatory theory of light, is given in an experiment first tried by Father Grimaldi, a Bolognese monk, in 1665. If a shutter be pierced with a very small hole, and the luminous cone which passes through the orifice be examined, the cone will be found to be much iWhewell, Novum Organon Renovatum, Bk. II. Ch. V. Art. 110. 200 INDUCTIVE LOGIC less acute than would be expected, considering only the rectilinear transmission of the rays, as according to the emission theory. If there be interposed in the path of the luminous ray a second shutter, pierced with a hole also, it will be noticed that the rays of the second cone are even more divergent than those of the first. If the image of the orifice be received upon a screen, a white circle is seen surrounded by a dark ring, next a white ring, even more brilliant than the central portion, then a second dark ring, and finally another very faint white ring. If in the shutter with which the experiment is made, two very small holes are pierced at a dis- tance from each other of one or two millimetres, and the two images received upon a screen in such a manner that they overlap each other, it is found that in the cuticular segment formed by the over- lapping of the images, the circles are more obscure than in the part where they are separated. Thus by adding light to light darkness is produced. 1 These phenomena are now known to be consistent only with the undulatory theory, and directly in contradiction to the emission hypothesis. M. Komanes performed several experiments upon bees which had the force of crucial tests of two opposed hypotheses, one, that bees possess a general sense of direction, irrespective of any special knowl- edge of their particular surroundings; the other, that they are guided in their flight by a knowledge of the localities which they have been wont to fre- quent. M. Eomanes took a score of bees in a box out 1 Saigey, The Unity of Natural Phenomena, p. 66. f9^ OF THK y I UNIVERSITY HYPOTHESIS ^^|c to sea, where there could be no landmarks to guide the insects home. None of them returned home. Then he liberated a second lot of bees on the sea- shore, and, none of these returning, he liberated another lot on the lawn between the shore and the house. None of these returned, although the dis- tance from the lawn to the hive was not more than two hundred yards. Lastly, he liberated bees in different parts of the flower-garden on either side of the house, and these at once returned to the hive; and with repetition of the experiment, a similar result, even arriving at the hive before he himself had time to run from the place where he had liberated them to the hive. As the garden was a large one, many of them had to fly a greater distance, in order to reach the hive, than those liberated on the front lawn. Their uniform suc- cess, therefore, in finding their way home so im- mediately was no doubt due to their special knowledge of the flower-garden, and not to any general sense of direction. 1 The hypothesis that leads to verification by ex- periment represents true scientific procedure, and that which has actually been the most effective instrument of research in all the various spheres of human investigation. The old controversy between Mill and Whewell admits of a ready adjustment in this regard. Whewell emphasized discovery as the heart of the system of induction, leading to the framing of hypotheses whose chief test was not 1 Lubbock, On the Senses, Instincts, and Intelligence of Animals, pp. 269, 270. 202 INDUCTIVE LOGIC experimental so much as the capability of account- ing for the given phenomena. Mill, on the other hand, insisted that logic was essentially proof, and not discovery. He, accordingly, emphasized the experimental testing by means of his several meth- ods, as being the all-important part of the in- ductive method. He had little concern for the origin of the suggestions as to the most likely causal elements in the midst of a complex phe- nomenon. The primary function of logic, according to him, is merely to prove or disprove. The ideas of Whewell and Mill are not necessarily contradic- tory; they can be regarded as mutually supple- mentary, which gives us a true account of the ideal logical method, where hypothesis suggests the line of experiment, and experiment in turn confirms hypothesis. In such a method, as can be seen in the illustration given, there is a blending of deduc- tive and inductive reasoning, which is the general characteristic of all actual processes of thought. As Sigwart has so admirably put it: "Without quickness of combination, by which we can call up a number of possible analogies and apply them to the unexplained case; without a happy power of divination which is guided by unanalyzable associa- tions to discover that analogy which embraces most aspects of the event; finally, without imagination to construct connections for which the only ground may be a hidden similarity, our thoughts, if com- pelled to proceed strictly according to method, would frequently be condemned, by the impossibility of discovering in this way a sufficiently grounded con- HYPOTHESIS 203 nection, to complete stagnation. Bnt the fact is in no way contrary to the nature of induction ; it is a necessary consequence of it. We cannot even begin the process of inference without making general assumptions ; and the general proposition which we get by summing up a number of instances is really a hypothesis, to which, it is true, we are led clearly and certainly in this case. But between these most general presuppositions, upon which all induction is grounded, and the simplest cases to which they can be applied, there is a wide region within which the hypotheses which are always necessary for in- duction can only be formed tentatively, in order to give some definite direction to investigation, to serve in our analysis of phenomena into their elements as a means of breaking up complete phenomena on cer- tain lines, and to invent the experiments which will make it possible to confirm or refute an opinion." 1 1 Sigwart, Logic, Vol. II. p. 423. CHAPTEE XIV Analogy It often happens that the cause of a phenomenon is disclosed by the fact that the cause of a similar phenomenon is known, and the inference then fol- lows that the similar phenomena have similar causes. Such a process of inference is determina- tion by analogy. Analogy, considered in its rela- tion to the inductive processes, occupies a twofold position. In the first place, when a complex phe- nomenon is given, as preliminary to the formation of any hypothesis, as to the probable cause which will, in turn, lead to experimental determination by one of the inductive methods, the mind instinc- tively examines, with sweeping glance, every detail of the phenomenon for the purpose of discovering some familiar features that may prove suggestive of known relations and functions occurring in other spheres. Analogical suggestion, therefore, initiates every inductive inquiry. In the second place, in every inductive general- ization there is an extension of the known into unknown regions, by virtue of the principle of analogy expressed in what we may style its limit- ing case. For instance, when we have examined 204 ANALOGY 205 a number of A's and find them always character- ized by the mark B, and then by generalization rise to the proposition, All A's are B, we do so by reason of postulating an analogy between all the individual A's of so strictly an accurate nature, that it amounts to essential identity. I have therefore called this the limiting case of analogy ; and this resemblance of particulars is the ground of all uni- versal whereby they manifest an identity in the midst of differences. We are therefore justified in affirming that all inductive generalizations present an aspect of analogical inference. Analogy, considered as a mental process, is grounded in the law of similarity. This tendency of noting resemblance makes possible the extension of knowledge. The formation of our concepts is, in the main, an analogical procedure; just as the generalization of an universal depends upon our discrimination of the elements which are similar from those which are different. While analogy thus functions in all the logical processes of thought, it is used in a more restricted sense to indicate that mode of inference especially which proceeds from a number of observed characteristics that are similar, to others which are thereby judged to be similar also. This method is very potent as an instrument of discovery. In 1845, Faraday dis- covered the magnetic rotary polarization of light ; by analogical reasoning, Waitmann in the following year inferred that a similar result would be attained with a beam of heat, which was afterwards experi- mentally verified. The so-called " natural kinds" 206 INDUCTIVE LOGIC furnish manifold illustrations of conclusive analo- gies. They possess numerous properties, some of them known and others unknown. Through large groups of them are found similar characteristics side by side with manifest differences, and yet the similarities are so striking that often, when new properties are discovered in certain members of the group, there seems to be ground for inferring their existence in other members of the group also. Certain properties known to exist in potassium and sodium were inferred to be present in rubidium and caesium ; the carbonates of sodium and potassium are not decomposed by a red heat, and it was in- ferred that the same would prove true of the car- bonates of rubidium and caesium ; and such proved to be the case. Some of the statements which are true of chlorine are found to be true of bromine and iodine. Mr. Gore, having found the molecu- lar change in antimony electro-deposited from its chloride, he inferred and discovered the same in that deposited from its bromide and iodide. Sir Humphry Davy, having discovered that potassium might be isolated by means of electrolysis, imme- diately inferred and proceeded to prove by experi- ment that it would be possible also to isolate sodium and other substances of analogous properties. 1 The principle of analogy lies at the basis of all classification, the separating and grouping together in appropriate divisions individuals which possess certain salient attributes in common. Professor Jevons' definition of classification em- 1 Gore, The Art of Scientific Discover, p. 522. ANALOGY 207 bodies at the same time a full statement of its exact logical significance as an instrument of re- search, and therefore I give it in full : " By the classification of any series of objects, is meant the actual or ideal arrangement together of those which are alike and the separation of those which are unlike, the purpose of this arrangement being, primarily, to disclose the correlations or laws of union of properties and circumstances, and secon- darily, to facilitate the operations of the mind in clearly conceiving and retaining in the memory the characters of the object in question." x In describ- ing the purpose of classification, the latter clause is more a psychological desideratum than logical ; the former specification contains its logical pur- pose ; namely, to disclose the correlations or laws of union of properties and circumstances. This may be illustrated in the grouping together of potassium, sodium, caesium, rubidium, and lithium, and calling them the alkaline metals. This was done by virtue of the common characteristics in the midst of their individual peculiarities ; namely, they all combine very energetically with oxygen to decompose water at all temperatures, and form strongly basic oxides, which are highly soluble in water, yielding powerful caustic and alkaline hy- drates from which water cannot be expelled by heat; their carbonates are also soluble in water, and each metal forms only one chloride. The manifest advantage of classifying these metals together lies in its suggestive capacity, as we have 1 Jevons, Principles of Science, p. 677. 208 INDUCTIVE LOGIC already noted in illustrations above given. So many observed similarities suggest inferences by analogy ; when, for instance, a new property is discovered in any one or two of the metals of this class, the idea immediately suggests itself that the same property may possibly extend over all the metals of the same class. Not only is such an idea suggested, but along with it there exists an ante- cedent probability respecting its solution in accord- ance with the suggestion which analogy starts. An excellent illustration of the practical results attained through a scientific use of classification is found in Mr. Lockyer's researches on the sun. 1 As a guide as to what elements to look for in the sun's photosphere, he prepared a classification of elements according as they had or had not been traced in the sun, together with a detailed statement of the chemical nature of each element. He was then able to observe that the elements found in the sun were, for the most part, those forming stable com- pounds with oxygen. He then inferred that the other elements which were known to form stable compounds with oxygen would, in all probability, be found present in the sun. Starting upon this suggested track, he succeeded in discovering five such metals. Analogical inference carries special weight when it is based upon the principle of teleology; that is, when any observed phenomena seem to possess structural contrivances adapted to ends, in some degree, at least, similar to human contrivances 1 Quoted by Jevons in Principles of Science, p. 676. ANALOGY 209 designed to produce certain proposed ends. When this similarity is apparent, it suggests the possi- bility that an observed contrivance in nature may subserve ends beyond the possibility of observa- tion, and which, therefore, may be inferred really to exist. We have seen that the ground of all inference lies in the representation of any given phenomena of consciousness as cohering in one system, which comprehends the several parts in a common unity of such a nature that, knowing some of the parts and their relations, we infer the character and function of other parts not known, and yet which that already known necessitates. And among the many kinds of relation that may obtain between part and part, or part and whole, the teleological is a very common one, and, more- over, by its nature necessitates certain consequences that lie beyond the sphere of observation, and yet, nevertheless, may very properly be supplied by in- ference. In other words, the causal connections in a system are not merely those of an efficient or a formal cause; they may, with a like force and suggestiveness, be considered in the light of a final cause; that is, the presence of means adapted to certain ends, or of organs adapted to certain neces- sary functions, or of contrivances of a mechanical nature as though designed for a specific purpose. Janet has specially emphasized the importance and prevalence of this kind of inference, and, as an illustration of the cogency of inference based upon finality, he urges that the certitude which the belief in the intelligence of our fellow-men gives us is p 210 INDUCTIVE LOGIC based upon analogical reasoning of this type j and that, moreover, this belief, resting npon such a basis, is one of the strongest beliefs which we possess. He says: "Now, if we ask ourselves why we suppose that other men think, we shall see that it is in virtue of the principle of final causes. In effect, what is it that experience shows in the actions of other men, but a certain number of phenomena co-ordinated in a certain manner, and bound not only together, but also to a future phe- nomenon more or less remote ? Thus when we see a man prepare his food by means of fire, we know that this assemblage of phenomena is connected with the act of taking food ; when we see a painter drawing lines on a canvas, we know that these apparently arbitrary acts are connected with the execution of a picture; when we see a deaf mute making signs which we do not understand, we be- lieve that these gestures are connected with a final effect, which is to be understood by him to whom he makes them ; in fine, when men speak, we see that the articulations of which a phrase is com- posed are co-ordinated to each other so as to pro- duce a certain final effect, which is to awaken in us a certain thought and sentiment. Now we cannot see such co-ordinations, whether actual or future, without supposing a certain cause for them; and as we know by internal experience that with our- selves such co-ordinations only take place under the condition that the final effect is previously represented in our consciousness, we suppose the same thing in the case of other men; in a word, ANALOGY 211 we suppose for them the consciousness of an end, a consciousness reflecting more or less, according as the circumstances more or less resemble those that accompany in ourselves the reflecting con- sciousness. Thus when we affirm the intelligence of other men, we affirm a truth of indisputable cer- titude ; and yet we only affirm it on the ground of analogy, and of analogy guided by the principle of final causes." l In this illustration of Janet's we have the idea of a system of co-ordinated parts especially promi- nent; and for a satisfactory account of the rela- tions obtaining in such a system, it will be seen how indispensable it is to postulate the theory of final cause. This mode of inference finds a striking illustration in the famous discovery of Harvey, con- cerning the circulation of the blood. In the early part of the seventeenth century, while Harvey was his pupil, the celebrated anatomist, Fabricius Aqua- pendente of Padua, observed that many veins con- tain valves which lie open as long as the blood is flowing towards the heart. Harvey, learning of this fact, saw in it the suggestion of an adaptation of means to an end; namely, a contrivance so fash- ioned by nature as to permit the blood to flow always in one direction only, and to prevent its flow in an opposite direction. Observation of other portions of the circulatory mechanism led to a con- firmation of the idea, and to the discovery of the circulation of the blood. 2 1 Janet, Final Causes, pp. 113, 114. 2 Gore, Art of Scientific Discovery, p. 571. 212 INDUCTIVE LOGIC Again, many flint substances have been discov- ered, as though curiously wrought, with sharp edges and a place as though designed for a handle, with which to wield the stone as a weapon or a tool ; it has been inferred from these general char- acteristics that the stones were so constructed by human effort, and used by human beings for the purposes for which they evidently seem to be adapted. This inference is based upon an analogy between the peculiar shapes of such stones, and known shapes designed and used by man. This form of analogy has proved specially sug gestive in researches regarding plant and animal life. Sir John Lubbock gives the following description of the common white dead-nettle, with the explanation of its functions that is evidently a teleological in- ference: "The flower consists of a narrow tube, somewhat expanded at the upper end, where the lower lobe of the corolla forms a platform, on each side of which is a small projecting lobe. The upper portion of the corolla is an arched hood, under which lie four anthers in pairs, while between them and projecting somewhat downwards is the pointed pistil. At the lower end, the tube contains honey, and above the honey is a row of hairs almost clos- ing the tube. Now, why has the flower this pecul- iar form ? What regulates the length of the tube? What is the use of this arch ? What lessons do these lobes teach us ? What advantage is the honey to the flower ? Of what use is the fringe of hairs ? Why does the stigma project beyond the anthers ? Why is the corolla white, while the rest ANALOGY 213 of the plant is green ? Similar questions may of course be asked with reference to other flowers. At the close of the last century, Conrad Sprengel published a valuable work, in which he pointed out that the forms and colors, the scent, honey, and general structure of flowers, have reference to the visits of insects, which are of importance in trans- ferring the pollen from the stamens to the pistil. Mr. Darwin developed this theory and proved ex- perimentally that the special service which insects perform to flowers, consists not only in transferring the pollen from the stamens to the pistil, but in transferring it from the stamens of one flower to the pistil of another." * The line of subsequent ob- servation and experiment was thus originally sug- gested by the structural appearance of these flowers which seemed formed for some specific end. The questions, once started, To what end ? To what purpose ? For what use ? led to the theory of Sprengel and the corroborative experiments of Darwin. This is further illustrated in some very inter- esting flower structures, also described by Sir John Lubbock, which indicate peculiar contrivances for the destruction of insects. The peculiarity of formation first suggested some such end as this, which has since been proved by careful observation to be the case. "The first observation on insect- eating flowers was made about the year 1868 by Ellis. He observed that in Dionsea, a North American plant, the leaves have a joint in the 1 Lubbock, Scientific Lectures, pp. 1, 2. 214 INDUCTIVE LOGIC middle, and thus close over, kill, and actually di- gest any insect which may alight on them. An- other case is that of Utricularia, an aquatic species which bears a number of utricles or sacs, which have been supposed to act as floats. Branches, however, which bear no bladders float just as well as the others, and there seems no doubt that their real use is to capture small aquatic animals, which they do in considerable numbers. The bladders, in fact, are on the principle of an eel-trap, having an entrance closed with a flap, which permits an easy entrance, but effectually prevents the unfortunate victim from getting out again. In the genus, Sar- racenia, some of the leaves are in the form of a pitcher. They secrete a fluid, and are lined inter- nally with hairs pointing downwards. Up the out- side of the pitcher there is a line of honey glands which lure the insects to their destruction. Flies and other insects which fall into this pitcher cannot get out again and are actually digested by the plant." 1 In the example where the idea of an eel-trap sug- gested the possible function of the similar struct- ure in the plant, Utricularia, we find one of the most striking illustrations of this mode of ana- logical inference. It was an easy and natural transition from similarity of structure to similarity of function. To give an idea of the great number of teleological phenomena in the vegetable and animal world, and the wealth of possible sugges- tion stored away in these various structures, and disclosed by a sagacious analysis, I quote a remark 1 Lubbock, Scientific Lectures, pp. 4, 5. ANALOGY 215 of Sir John Lubbock's in commenting upon the variation of color and markings of caterpillars : " I should produce an impression very different from that which I wish to convey, were I to lead you to suppose that all these varieties have been explained or are understood. Far from it ; they still offer a large field for study; nevertheless, I venture to think the evidence now brought forward, however imperfectly, is at least sufficient to justify the con- clusion that there is not a hair or a line, not a spot or a color, for which there is not a reason, which has not a purpose or a meaning in the economy of nature." 1 An illustration given by Darwin shows this mode of inference applied to the sphere of animal life also. He says: "The great size of the bones of the megatherioid animals was a complete puzzle to naturalists until Professor Owen lately solved the problem with remarkable ingenuity. The teeth indicate, by their simple structure, that these mega- therioid animals lived on vegetable food, and prob- ably on the leaves and small twigs of trees ; their ponderous forms and great, strong, curved claws seem so little adapted for locomotion that some emi- nent naturalists have actually believed that, like the sloths, to which they are intimately related, they subsisted by climbing back downwards on trees, and feeding on the leaves. It was a bold, not to say preposterous, idea, to conceive even ante- diluvian trees with branches strong enough to bear animals as large as elephants. Professor Owen, 1 Lubbock, Scientific Lectures, pp. 66, 67. 216 INDUCTIVE LOGIC with far more probability, believes that, instead of climbing on the trees, they pulled the branches down to them, and tore up the smaller ones by the roots, and so fed on the leaves. The colossal breadth and weight of their hinder quarters, which can hardly be imagined without having been seen, become, on this view, of obvious service, instead of being an encumbrance: their apparent clumsiness disappears. With their great tails and their huge heels firmly fixed like a tripod on the ground, they could freely exert the full force of their most powerful arms and great claws. Strongly rooted, indeed, must have been that tree which could have resisted such force ! The Mylodon, moreover, was furnished with a long extensile tongue like that of the giraffe, which, by one of those beautiful provisions of nature, thus reaches, with the aid of its long neck, its leafy food." 1 Throughout we observe analogical inference based upon these teleological marks, and furnishing a basis for a satisfactory hypothesis. We see what a wide field thus opens in the region of biology alone for the discovery of resem- blances leading to the appreciation of the fuller teleological significance of plant and animal life. In the illustrations given, both of the teleologi- cal and other forms of analogy, we notice that its chief logical function is that of suggestion of some hypothesis which may or may not be afterwards confirmed by subsequent experiment. Some of the most important discoveries of science have arisen from analogical suggestions. Sir John Herschel 1 Darwin, Voyage of a Naturalist, pp. 106, 107. ANALOGY 217 was led by observed analogies to predict certain phenomena afterwards verified experimentally by Faraday. Herschel had noticed that a screw-like form, known as helicoidal dissymmetry, was ob- served in three cases, namely, in electrical helices, plagihedral quartz crystals (that is, crystals having an oblique spiral arrangement of planes), and the rotation of the plane of polarization of light. As Herschel himself said : " I reasoned thus : Here are three phenomena agreeing in a very strange peculiarity. Probably this peculiarity is a connect- ing link, physically speaking, among them. Now, in the case of the crystals and the light, this prob- ability has been turned into certainty by my own experiments. Therefore, induction led me to con- clude that a similar connection exists, and must turn up, somehow or other, between the electric current and polarized light, and that the plane of polarization would be deflected by magneto-elec- tricity." Herschel thus anticipated Faraday's ex- perimental discovery of the influence of magnetic strain upon polarized light. 1 Another important discovery the germ-theory of epidemic disease was first suggested by an analogy. In the theory, as expressed by Kircher, and favored by Linnaeus, and afterwards supported by Sir Henry Holland t its special strength, accord- ing to Professor Tyndall, " consisted in the perfect parallelism of the phenomena of contagious disease with those of life. As a planted acorn gives birth to an oak competent to produce a whole crop of 1 Jevons, Principles of Science, p. 630. 218 INDUCTIVE LOGIC acorns, each gifted with the power of reproducing the parent tree, and as thus from a single seedling a whole forest may spring, so, it is contended, these epidemic diseases literally plant their seeds, grow and shake abroad new germs, which, meeting in the human body their proper food and temperature, finally take possession of whole populations." 1 The theory of evolution was first suggested to Mr. Darwin by the analogous phenomena observed in artificial selection and breeding. The transition to natural selection was easily made, especially as, on reading Malthus, On Population, he conceived the idea of a struggle for existence as the inevitable result of the rapid increase of organic beings. This idea necessitated the natural selection, which he needed to account for results similar to the artifi- cial selection, and thus his theory grew out of an analogy as its beginning. Moreover, in the devel- opment of the theory in its manifold details, other analogies proved also suggestive. For instance, there is the supposed analogy between the growth of a species and the growth of an individual. It supposes, for example, as Professor Clifford has put it, "that the race of crabs has gone through much the same sort of changes as every crab goes through now, in the course of its formation in the egg, changes represented by its pristine shape utterly unlike what it afterwards attains, and by its gradual metamorphosis and formation of shell and claws." 2 1 Tyndall, Fragments of Science, p. 287. 2 Clifford, Lectures and Essays, p. 86. ANALOGY 219 The germ-theory of putrefaction, first suggested by Schwann, received confirmation through certain resemblances noted by Professor Lister between fermentation and putrefaction. In his Introduc- tory Lecture before the University of Edinburgh, Professor Lister called attention to the fact that fermentation and putrefaction present a very striking parallel. In each a stable compound sugar in one case, albumen in the other under- goes extraordinary chemical changes under the influence of an excessively minute quantity of a substance which, regarded chemically, would be considered inert. It was pointed out, also, by Pro- fessor Lister, in this connection, that, as was well known, one of the chief peculiarities of living or- ganisms, is that they possess extraordinary powers of effecting chemical changes in materials in their vicinity out of all proportion to their energy as mere chemical compounds. Such being the facts in the case, and, moreover, the fermentation of sugar being generally allowed to be occasioned by the presence of living organisms, Professor Lister's inference was that putrefaction was due to an analogous agency. 1 A discovery in quite a different sphere, that of mathematics, leading to the branch of analytical geometry, was first suggested to Descartes through observing the resemblances existing between geom- etry and algebra. In a similar manner, Boole was led by the resemblances noted between algebra and logic, to give expression to the same in a sys- 1 Tyndall, Fragments of Science, pp. 300-302. 220 INDUCTIVE LOGIC tern which he called the laws of thought, and which has become the basis of a general or symbolic logic. While there are thus unquestionable evidences of the value of analogy as a form of inference, there are also cases of false analogy unfortunately so nu- merous as to discredit the process wholly in some quarters. It will be well, therefore, to indicate some of the requirements of true analogy : 1. In the first place the resemblance must be a preponderating one; that is, the phenomena com- pared must show a more striking agreement than difference. Some writers have balanced agreement against difference upon a purely numerical basis of comparison, forming what may be called an analogi- cal ratio, with points of similarity forming the numer- ator, and the points both of similarity and difference, plus the unknown, that is, the total number, form- ing the denominator. Such a representation of the force of an analogy is given by Mill, Bain, and others. I think, however, that this representation is apt to be misleading in producing the impression that the mere number of points of agreement, irre- spective of their significance, is the chief feature of analogy. Whereas it is the weight of the agreeing attributes, and not the number, that counts. As has been before said, in analogy we weigh instances, and do not count them. The analogical ratio ex- pressed numerically, as above, is really equivalent to the ratio of probability which will be described in the following chapter. I have therefore changed the usual wording of this requirement, so that it reads, the resemblances must be more striking than ANALOGY 221 the differences. This provides for cases when per- haps a few points of resemblance will be of such a nature as to outweigh many points of difference in the total estimate. This requirement also excludes all fanciful anal- ogies and all resemblances resting upon a figurative rather than a real basis. For instance, the advo- cates of annual Parliaments in the time of the Commonwealth, urged their case on the analogical ground that a body politic is similar to a living body and that serpents annually cast their skin, which, being no doubt for a beneficial purpose, might well be imitated. 2. In noting the points of resemblance between two phenomena, all circumstances which are known to be effects of one cause must therefore be re- garded not as many, but as one. For instance, two chemical oxides may be compared ; the effects com- mon to each may be due to the presence of the oxygen which each contains and therefore must not be regarded in the light of independent marks of similarity. 3. If we infer by analogy that a substance pos- sesses a certain property which we know is incom- patible with some one or other known properties of the substance, the analogy is at once discredited. We may infer that the moon is inhabited, by virtue of the many points of resemblance between the moon and the earth. However, the fact that the moon has no atmosphere necessary to sustain life, at once makes such an argument based upon an- alogy wholly out of the question. 222 INDUCTIVE LOGIC 4. There are certain special requirements refer- ring to that particular form of analogy which is based upon teleological considerations. They are as follows : a. This principle must never be used as an argu- ment against an observed fact, or an established law of nature. While this precaution is not neces- sary at the present time, in scientific circles at least, still there was a time when its counsel was sorely needed. When in astronomy it was proved that there were suns gravitating around other suns, without our solar system, this was objected to upon the following ground, as given by one Nicholas Fuss, a celebrated astronomer, at the end of the eighteenth century : " What is the good of some luminous bodies revolving round others ? The sun is the only source whence the planets derive light and heat. Were their entire systems of suns con- trolled by other suns, their neighborhood and their motions would be objectless, their rays useless. The suns have no need to borrow from strange bodies what they themselves have received as their own. If the secondary stars are luminous bodies, what is the end of their motives ? " There is, moreover, another abuse of the principle of final causes, which has also historic interest rather than any present pertinence ; namely, oppos- ing certain false teleological ideas to established discoveries or inventions, with a mistaken zeal, in defence of a Divine Providence. For instance, at the time of Jenner's great discovery, an English physician, Dr. Eowley, said of small-pox : " It is a ANALOGY 223 malady imposed by the decree of heaven, and vac- cination is an audacious and sacrilegious violation of our holy religion. The designs of these vaccina- tors appear to defy heaven itself, and the very will of God." The introduction of winnowing machines into Scotland met with bitter opposition on the ground that the winds were the work of God, and that the wind thus artificially raised was a veritable "devil's wind," as they were wont to call it. Sir Walter Scott, in Old Mortality, has the old Mause say to her mistress : " Your ladyship and the stew- ard are wishing Cuddie to use a new machine to winnow the corn. This machine opposes the de- signs of Providence, by furnishing wind for your special use, and by human means, in place of asking it by prayer, and waiting with patience till Prov- idence itself sends it." b. Final causes should never be employed to explain phenomena which do not exist. As M. Florens has said : " We must proceed not from final causes to facts, but from facts to final causes ; that is, we should not superimpose final causes upon phenomena. We must see them in phenomena themselves, and we must not arbitrarily project a teleological idea, purely subjective, upon an objec- tive ground. Thus in ancient times, Hippocrates is said to ' have admired the skill with which the auricles of the heart have been made to blow the air into the heart? " c. We must distinguish accidental from essential marks of finality, and not be led into fanciful or far-fetched analogies. Voltaire has expressed such 224 INDUCTIVE LOGIC a defect when in satire he made that famous re- mark, " Noses are made in order to bear spectacles." Bernardin de Saint-Pierre says: "Dogs are usu- ally of two opposite colors, the one light, the other dark, in order that whenever they may be in the house, they may be distinguished from the furni- ture, with the color of which they might be con- founded. . . . Wherever fleas are they jump on white colors. This instinct has been given them, that we may the more easily catch them." And again the same writer says : " The melon has been divided into sections by nature, for family eating." I All such grotesque inferences will give an idea of how readily the imagination will run riot if allowed to remain uncurbed by the reason. 5. Analogy should never be regarded as having more weight than that of extremely high probabil- ity, even in cases seemingly most conclusive. Its true function is suggestive, leading to hypothesis and experiment, and it needs this supplementary proof. It was an inference based on analogy, for instance, which suggested the probability that the Binomial Law, having proved to be valid as regards the second, third, and fourth powers, might also be extended to the fifth, and so on to the other powers indefinitely. This suggestion offered no real basis, however, upon which the Binomial Theorem could rest ; it needed mathematical demonstration to con- firm and generalize its expression in the special cases already experimentally tested, so as to cover 1 The illustrations upon the abuse of final causes I have taken from Janet's admirable chapter, Chapter VIII. of Appendix. ANALOGY 225 all possible exponents, both positive and negative, fractional and integral. So also the discovery of the circulation of the blood was first suggested to Harvey, as has been said, by analogical considerations upon observed teleological phenomena. Harvey, however, was not content with this suggestion merely. He was led to experiment upon the veins and arteries ; he tied an artery and vein, and carefully observed the me- chanical effects upon the two sides of the tied parts. Experiments of this nature, with close observation and study, were kept up most diligently, and with rare perseverance, for nineteen years, before he had traced the entire course of the blood through all parts of the human body, and, in a manner wholly satisfactory to himself, verified the first statement of this theory. CHAPTEE XV Probability There are certain phenomena of such a nature that their antecedents, being extremely complex, cannot be adequately comprehended by observation, however searching it may be ; nor can they be sub- jected to any analysis that will disclose the causal elements to which the effect in question is due. Moreover, with seemingly the same antecedents, the event sometimes happens, and sometimes does not ; and even with antecedents associated with an event as cause and effect respectively, nevertheless the event does not occur as we should naturally expect, while with antecedents associated with the contradiction of the event as cause and effect re- spectively, we find the occurrence of the event quite contrary to what we should naturally expect. The evidence of a constant connection between antece- dent and consequent, that we have found in so many cases which we have examined, is here wholly lack- ing. Regularity has been replaced by irregularity respecting such phenomena. For instance, I throw dice repeatedly ; the antecedent shaking of the box, and tossing the dice upon the table, is about the same each time, at least the difference can- 226 PROBABILITY 227 not -be determined, and yet the results vary with each successive throw. The causal determination in each case is so complex as to be beyond com- putation ; the initial position of the dice, the force of their ejection from the box, the height of the box above the table when they leave it, the ine- qualities of the table itself, a variation between the physical and geometrical centres of gravity of the dice, etc., all these make the antecedent so complex that a slight variation in any one of these conditions will affect the result. We find, therefore, double sixes at one time, a three and four at another, and so on indefinitely. Or, again, it sometimes happens that with perfect sanitary conditions a contagious disease will appear, that has always been regarded, and that correctly, as due to imperfect sanitation ; or, an entire disregard of sanitary requirements and of all the laws of health may yet give rise to no disease of special moment. Certain conditions of temperature, at- mospheric pressure, velocity and direction of the wind, may one day bring storm and rain, and as far as observation can detect, similar conditions may again bring fair weather. So, also, the rise and fall in stock and money markets is extremely susceptible to the varying conditions of indefi- nitely complex forces wholly beyond all powers of determination or of prediction. Such phe- nomena present a problem which the methods of inductive inquiry cannot deal with. Observation is not far-reaching enough to provide the data for the solution of the problem, and, even if it were, 228 INDUCTIVE LOGIC our methods of computation and determination are not sufficiently adequate to solve problems of so many terms and of so complex a nature. The experimental methods are designed to test causes suggested by analogy, or a mental analysis ; but in such phenomena as these, the problem is not simply to find a causal connection. The causal connection may be established beyond all reasona- ble doubt, and yet the cause obtains in the midst of so complex a setting that the problem is really this, to determine whether a cause, whose exact nature may be known or unknown, will prove operative or inoperative. The cause may be al- ways present and even its exact nature may be known, and yet the complex circumstances at- tending it may be of such a character that one alone, or two or more combining, may neutralize the operation of the cause, and on the other hand a slight variation of the combined circumstances may promote and even accelerate the operation of the cause in question. The problem then is to determine how often the event happens, and how often it fails of happening, the complex and in- determinate antecedent being present in all the instances examined. When we begin to count instances, we are re- minded that we must be in the near neighborhood of the sphere of enumerative induction. Enumera- tive induction, it will be remembered, treats in- stances by noting the number of observed coincident happenings of the antecedent and consequent under investigation, no attempt being made to analyze PROBABILITY 229 their respective contents, or to determine a causal connection more definitely by means of any one or more of the inductive methods of research and veri- fication. The result of such an investigation may be formulated in a proposition of the form, Every A is B. This, strictly interpreted, has the force of, Every A that has been observed is B. The enu- meration of the kind of instances which we are dis- cussing in this chapter, however, differs from this in that the observation leads to a twofold result, a set of instances in which it is observed that the A's are B's also, another set, however, in which the A'* are not B's. These instances are of such a nature that the observed A is an ante- cedent so extremely complex that the element within it, which is a cause capable of producing B, may either be absent without producing an appreciable change in the general nature of A, or, being present, may be neutralized by some other element of A itself. The result gives a basis for a probable inference only ; and the nature of that in- ference will depend upon the preponderance of the observed happening, or of the failure of the event under investigation. The probability attached to such an inference, however, is different from the probability which characterizes the nature of enumerative induction. In the latter, when the observation has been widely extended and no exceptions noted, it is usual to say the result expressed in the proposition, Every A is B, has the force of a high degree of probability. In the instances, however, whose investigation 230 INDUCTIVE LOGIC shows the result that some A'a are B's, and some not, and yet where the former, for instance, far outnumber the latter cases, then it may be inferred that the ^.'s which in future we may meet with will probably be jB's ; and the degree of probability expressed in such a proposition is commensurate with the preponderance of the number of observed affirmative instances over the negative. Here the probability refers to the validity of an inference concerning certain particular instances, be they many or be they few, which lie beyond the sphere of our present knowledge; in enumerative induc- tion, the probability is attached to the universality of the proposition affirmed as a result of observa- tion that has not so far detected an exception. In the former case, the question of the universality of the result is conclusively answered, and that in the negative ; there can be no universal proposition possible, as some instances give A and B together, others give A with the absence of B; and the question of probability that here arises, therefore, refers to individual cases not yet examined, as to whether they severally will more likely correspond to the set of affirmative, or to that of the negative instances already noted. The comparison of the number of happenings with that of the failures of an event affords a basis for three kinds of inference, all of them in the sphere of probability. 1. We find in such a comparison a basis for the calculation of the probability of a particular event happening, in case there is a repetition of the cir- PROBABILITY 231 curastances which, in former cases, have sometimes produced the event, and sometimes have failed to produce it. If, according to former observation, the event has happened, let us say, seven times, and failed three, the probability, expressed nu- merically, of its happening again is T 7 . The rule is, to express the probability of an event, take as numerator the number of times which the event has been observed to occur, and as denominator the total number observed, both of happening and failure ; the fraction thus expressed will represent the probability of the event happening. The coun- ter-probability may be represented by the number of observed failures of the event divided by the total number of cases observed. The counter-prob- ability, plus the probability, evidently is equal to unity. If, therefore, the probability is unity, the counter-probability will equal zero; that is, the probability in that case has merged into certainty. Zero, therefore, represents absolute impossibility. All fractions between the limits zero and one rep- resent varying degrees of probability from impos- sibility at one extreme to certainty at the other. Not only may there be this inductive basis for the calculation of probability, arising from actually observed instances ; there may be also a deductive calculation of probability based upon the known structure or nature of the phenomena themselves in advance of any observation as to their actual behavior. For instance, we say the probability of a penny turning up heads is . Knowing the form of the penny and that there are but two 232 INDUCTIVE LOGIC possibilities, heads or tails, and , there being no reason why one should more likely turn up than the other, we say there is one chance favorable to heads as over against the two chances which rep- resent the total number of possibilities under the existing circumstances. With a die, in the form of a perfect cube, we say there is one chance of its turning up the face marked 1, as over against the six chances represented by the six faces, the total number ; here the probability is i. Thus the basis for the calculation of probability may be a theoretical as well as an empirical one. In the estimate of the probability of an event in the actual conduct of affairs, we seldom express that probability numerically. I would say that we express a degree of probability adverbially rather than numerically ; that is, we say an event is quite probable, or it is very probable, or it is extremely probable. The fact is that, as regards most phenom- ena, we do not keep an exact or even approximate memorandum of the number of happenings compared with that of the failures. We rather classify our observations in terms of more or less. For instance, certain circumstances we observe produce about as many failures as happenings of an event; other circumstances produce far more happenings than failures ; others far less, and so on. Consequently we receive certain psychological impressions of varying degrees of intensity according to the pre- ponderance of happening over failure, or vice versa ; this impression becomes the basis for estimating the probability in question, and the degree of that PROBABILITY 233 probability is commensurate with the intensity of the original psychological impression arising from concepts of more or of less. In such a sphere, how- ever, as that devoted to the interests of betting, gambling, pool-selling, book-making, etc., probabili- ties are estimated according to observations and theoretical considerations, whose conditions are ex- pressed numerically ; and the amount risked in each case is strictly estimated according to the exact ratio of probability to counter-probability under the existing circumstances. The estimation of probability in terms of a greater or less degree is, however, more usual, and applicable to the conduct of human life generally. It has special force and utility as a mode of infer- ence, when the observed instances so far outnumber the exceptions as to create an impression of such a high degree of probability as to approximate practical if not theoretical certainty. For instance, it has been noted over a wide field of observation, that a second attack of scarlet fever is extremely rare. Exceptions have occurred and, therefore, by enumerative induction it is impossible to general- ize the universal proposition that a second attack will never occur. It is, however, possible to assert with somewhat positive assurance that it is highly probable that a person will be exempt from a second attack. Or, you hear that a person, whose name is un- known to you, has met with an accident in the city of New York, resulting fatally. You are not alarmed, and perhaps the possibility does not even 234 INDUCTIVE LOGIC suggest itself to you, that the unknown person may prove to be a member of your own family, or a friend who at the time is known to be in New York. The probability against such a suggestion is so large as to preclude even the thought of it. Suppose, however, the accident occurred at one of the suburban stations. Your knowledge that your friend rides on one of the suburban trains each day to and from town, may be the ground of some anxiety, because in this case the range of possibili- ties is materially narrowed. Suppose, moreover, that the station where the accident occurred is at the village where your friend resides, your anxiety receives an additional increment; and, again, sup- pose it is at the hour at which your friend ordi- narily reaches this station, there is then increased apprehension on his account. Thus, as further knowledge limits the number of total possible cases, the denominator of the probability fraction is contin- ually decreasing, and therefore the probability itself continually increases, until it has developed from a fraction of insignificant proportions to one which is suggestive of great anxiety and suspense. 2. The comparison of failure and happening of events based upon observation, or theoretical con- siderations of structure and nature, leads also to inferences concerning large numbers of instances considered together. If a memorandum is kept of the number of times an event has happened, and the number of times it has failed, and the total number of instances examined be sufficiently great, then the resulting ratio of favorable instances to PROBABILITY 235 the total number will be found approximately repeated, if a second set of an equal number of instances be likewise examined. There is a law of tendency whereby nature seems to repeat herself even when the attendant circumstances of an event are most complex, and beyond all powers of accu- rate determination. As the result of observations extending over thousands and thousands of in- stances, it is affirmed that about one-fourth of the children born in the world die before the age of six years, and about one-half before the age of sixteen. Take a group of ten children, the ratios would per- haps be deviated from very materially ; in a group of a hundred the deviation is apt to be less ; in a group of a thousand, still less ; and in a group of one hundred thousand, the ratios as above given would be substantially realized. The approxima- tion would be so near that the error would be insig- nificant as compared with total number of cases. The following law, therefore, expresses this ten- dency, that while in a small number of instances there is irregularity in the observed ratio between the number of times a given event has happened and its failures, still in a large number of instances this ratio tends towards a constant limit. This is clearly seen in the pitching of a penny j 10 throws might very possibly result in 7 heads and 3 tails ; in 100 throws, however, the ratio expressing the result as to heads and tails observed will be much nearer i than in the former case ; while if 1000 or 10,000 throws be observed, the result will approxi- mate the ratio i The comparison of observed cases >cP I IRd 236 INDUCTIVE LOGIC with the number given by the calculation of the probabilities in question has been made by Quetelet, also by Jevons. Their results are most significant and interesting. Quetelet made 4096 drawings from an urn containing 20 black balls and 20 white. Theoretically, he should have drawn as many white as black balls, 2048 each; the actual drawings re- sulted in 2066 white balls and 2030 black. Jevons made 20,480 throws of a penny; the theoretical result should have been 10,240 heads; the actual result was 10,353 heads. The tendency towards a constant ratio in aggre- gates containing a considerable number of instances is strikingly illustrated in the record of baptisms taken from an old parish register in England. The number of male baptisms registered to every 1000 female ran as follows for the respective years from 1821 to 1830 : 1048, 1047, 1047, 1041, 1049, 1046, 1047, 1043, 1043, 1034. We see with what surprising accuracy the constant ratio was repeated substan- tially, year after year. This tendency to approximate a constant ratio is seen even in such indeterminate events as railroad accidents. Here the causes pro- ducing the accidents are so numerous, so diverse, so complex and extending over so large an area, as, for example, the whole of the United States, that we should think that the results would exhibit so many variations from any definite ratio as abso- lutely to elude all attempts at accurate determina- tion. The following figures, however, given by the Interstate Commerce Commission, indicate results wonderfully corresponding for year after year: PROBABILITY 237 El * 3 a B '5 s oo a N H iN M CO O CM CO kO OS CO M O 00 CO CO io co~ cT co co*" Q o a N CO a n cc n * * CM CO CO O rH CO O lO CO Nb N CO K o 9 i M g B o 8 E s '5 s IN iO CO CO >0 CO CO O CO iO CO CO iH CM t^ r-l "<*l co tJh >* rti iO y& *3 8 a t^ i-H CO CD t- OS ^* OS t >- -i c3 e8 T3 BO CD CD E a) 5 02 2 CO a O 9 CD CD bC rO P3 09 3 -+l l CM (f ft O o f-i O CO a M bj d CM O CO C5 CM t)H wa . . .OQ0-*'*'OO CO T i i i O >Q *0 ** 3 rl CO WO a o OCHr)('*J o ' " ' Tt* CM 05 C5 kO o CM 0"*H w ... CO "^ fcc 1 t* in C5 O CO 00 CO * Jo * i i t- "* co CM ae CO t-h CM -tf a a H X H O GO 3 d eg ** <* ** '-" *0 CO t^ . . . t l CO t- ii * o 2 ... T* T-I CO d CO CO "*j C5 5 CO OS r^ 'CO^NNOH o iH iH "^ *M o "o3 ... CO T* CI -d . CM i <* CO OS O CO 01 CO o* CO CO t- OS t^ Hcoa CO c-i M ... CM CO 'd -N^NOMOJ G5 i ' l- N lij H O ^ 4> M-*n^O^COhO CM ^ CO CM t-I t-i HO t~ M CM 0) g bC | he fl B- K S'O M M ft $ p 3 o -2 5P 6 Q p ill.. .11 ... ft! I ssi-it o ft g 1 be a g| ^2 H O c3 > o a> +3 -m -u +3 o H OPhOOPO^^O ^ A _ c3 * *- CO O Ci T-I OS O CO ON r~- CO o * 00 00 00 t^ CO CO T-I HH Tfl <* CO 05 o oo O CO TJH CO CM CO CO 05 o o i 00 Tf CO CO *tf CO 05 00 00 CO 00 t^ ^ CO I CM CO 00 CO HHOS O ^ CO 55 co CM i-l CO 00 O O CO O O CO 05 rH oo" 00 00 T-I Ttl^ O CO OS COO CO 05 oo co OS OS r-l 1 .05 * iO CO CO GO HTjiH -^ O O OO T-I t-i r-i CO CM 00 t -*n co CO o o T-I