44 ELEMENTARY LESSONS IN LOGIC THE MACMILLAN COMPANY NEW YORK BOSTON CHICAGO SAN FRANCISCO MACMILLAN & CO., LIMITED LONDON BOMBAY CALCUTTA MELBOURNE THE MACMILLAN CO. OF CANADA, LTE TORONTO ELEMENTARY LESSONS IN LOGIC: DEDUCTIVE AND INDUCTIVE, WITH COPIOUS QUESTIONS AND EXAMPLES AMD A VOCABULARY OF LOGICAL TERMS. W. STANLEY JEVONS, M.A. PROFESSOR OF LOGIC IN OWENS COLLEGE, MANCHKSTBm NE W EDITION. THE MACMILLAN COMPANY 1918 All rights reserved PREFACE. IN preparing these Lessons I have attempted to show that Logic, even in its traditional form, can be made a highly useful subject of study, and a powerful means of mental exercise. With this view I have avoided the use of superfluous technical terms, and have abstained from entering into questions of a purely speculative or metaphysical character. For the puerile illustrations too often found in works on Logic I have generally substituted examples drawn from the distinct objects and ideas treated in the natural and experimental sciences; and in this and other respects have aimed at rendering these Lessons a suitable companion to a series of science school books. 2033685 PREFACE. Logic is not only an exact science, but is the t simple and elementary of all sciences ; it ought therefore undoubtedly to find some place in every course of education. The relations of propositions and the forms of argument present as precise a sub- ject of instruction and as vigorous an exercise of thought, as the properties of geometrical figures, or the rules of Algebra. Yet every school-boy is made to learn mathematical problems which he will never employ in after life, and is left in total ignorance of those simple principles and forms of reasoning which will enter into the thoughts of every hour. Logic should no longer be considered an elegant and learn- ed accomplishment; it should take its place as an indispensable study for every well-informed person. These Lessons I trust will introduce to the science many who have not leisure or inclination to read more elaborate treatises, and many who would not be at- tracted by the numerous but somewhat dry and brief compendiums published in past years, It is desirable that Lessons in Logic should be made the basis of many exercises, and for this pur- pose I have supplied abundance of questions and examples at the end of the book, some of which an selected from the examination papers of the Oxford, PREFACE. Y* London, and Edinburgh Universities. In my owt classes I have constantly found that the working and solution of logical questions, the examination of argu- ments and the detection of fallacies, is a not lew practicable and useful exercise of mind than is the performance of calculations, and the solution of pro- blems in a mathematical class. Except in a few places, where special notice is given, I hare abstained from putting forward any views not commonly accepted by teachers of logic; and I have throughout devoted more attention to describing clearly and simply the doctrines in which logicians generally agree, than discussing the points in which there is a difference of opinion. The recent logical discoveries of Sir W. Hamilton, Archbishop Thomson, Prof, de Morgan, and especially the late ProC Boole, cannot yet be fully adopted in an ele- mentary work, but I have attempted to give a clear notion of the results to which they inevitably lead. In the latter Lessons which treat of Induction I have generally followed Sir John Herschel, Dr Whewell and Mr J. S. Mill, as the recognised authorities on the subject These Lessons in fact may be regarded as an easy introduction to some of the most important parts of Mr Mill's treatise on Logic viii PREFACE. At the end of almost every Lesson will be found references to the works in which the student will most profitably continue ms reading of the subject treated, so that this little volume may serve as a guide tc 1 lore extended course of study. TABLE OF CONTENTS. L DEFINITION and Sphere of the Sdeoot ......... I IL The Three Parts of Logical Doctria* ............ 9 TERMS. III. Terms, and their various Kinds .... 16 IV. Of the Ambiguity of Termi..... ^ 17 V. Of the twofold meaning of term* in Extension and Intension .... *,.. .................... ........ *7 VL The Growth of Language .. .^.. 44 VIL Leibnitz on Knowledge _.........._..._... ff PROPOSITIONS. V1IL Kinds of Proposition* .............. 60 IX. The Opposition of Proposition* 71 X. Conversion of Propositions, and Immediate In- ference . 8* XL Logical Analysis of Sentences - 88 X1L The Predicates, Division, and Definitiom 9! XIII. Pascal and Descartes on Method Iff TABLE OF CONTENTS. SYLLOGISM. XIV. XV. XVL XVIL XVIIL XIX, XX. XXI. The Laws of Thought The Rules of the Syllogism The Moods and Figures of the Syllogism. Reduction of the Imperfect Figures Irregular and Compound Syllogism* .~ v .. Of Conditional Arguments.. ............. FALLACIES. Logical Fallacies... Material Fallacies ... 15* ... i6o 169 RECENT LOGICAL VIEWS. XXII. The Quantification of the Predicate ............ iJ XXIII. Book's System of Logic __...... If METHOD. XXIV. Of Method, Analysis, and Synthesis ___ O! INDUCTION. XXV. XXVI. Perfect Induction and the Inductive Syllogism sic Geometrical and Mathematical Induction, Ana- logy, and Example ........ .................... j8 XXVIL Observation and Experiment .......... , ......... 118 tXVUL Methods of Induction .......... ..................... 130 XXIX. Methods of Quantitative Induction __________ 4 7 TABLE OF CONTENTS. XXX. Empirical and Deductive Methods *5g XXXI, Explanation, Tendency, Hypothesis, Theory and Fact . *6* SUBSIDIARIES OF INDUCTION. XXXII. Classification, and Abstraction 376 Jf 'XIII. Requisites of a Philosophical Language 087 Questions and Exercises 496 Examples of Terms ........ 197^99 Examples of Propositions - 303 Examples of Arguments ~ $i, 315 ...........................0. ...%..........-. 331 INTRODUCTION. LESSON I. DEFINITION AND SPHERE OF THE SCIENCE LOGIC may be most briefly defined as the Science ol Reasoning. It is more commonly defined, however, as the Science of the Laws of Thought, and some logicians think it desirable to specify still more accurately that it is the Science of the Formal, or of the Necessary Laws of Thought. Before these definitions can be of any real use to us we must come to a clear understanding as to the meaning of the expressions; and ft will probably appear that there is no great difference between them. By a Law of Thought we mean a certain uniformity or agreement which exists and must exist in the modes in which all persons think and reason, so long as they do n&t make what we call mistakes, or fall into self-contradiction and fallacy. The laws of thought are natural laws with which we have no power to interfere, and which are of course not to be in any way confused with the artificial laws of a country, which are invented by men and can be altered by them. Every science is occupied in detecting and describing the natural laws which are inflexibly observed DEFINITION AND SPHERE [LESS by the objects treated in the Science. 'Ine science pi astronomy investigates the uniform or similar way in which the heavenly bodies, and in fact all material sub- stances, tend to fall towards each other as a stone falls towards the earth, or to move round each other under the influence of this tendency. The universal law of gravitation is thus the natural law or uniformity treatec in physical astronomy. In chemistry the law of equivalent proportions de- cribes the well ascertained fact that each chemical substance enters into combination with every other che- mical substance only in certain definite proportions ; as when exactly eight parts by weight of oxygen unite with one part of hydrogen to form water, or sixteen parts of oxygen and six parts of carbon unite to form carbonic acid in the ordinary burning of a flame or fire. When- ever we can detect uniformities or similarities we so far create science and amve at natural laws. But there may be, and are, many things so fickle, complicated, and uncertain, that we can never be sure we have detected laws that they will uniformly obey ; in such cases no science, in the proper sense of the word, is possible. There is no such thing, for instance, as a real science of human character, because the human mind is too variable and complicated a subject of investigation. There are no two persons so much alike that you may be sure of one acting in all circumstances as the other would; it thus becomes impossible to arrange persons in classes so that all who are in the same class shall act uniformly in the same manner in any given circumstances. But there is a science of human reason or thought apart from the many other acts of mind which belong to human character, because there are modes in which all persons do uniformly think and reason, and must think **H reason. Thus if two things are identical with a third Ll OF THE SCIENCE. 3 common thing they are identical with each othei. Thit is a law of thought of a very simple and obvious charac- ter, and we may observe concerning it, 1. That all people think in accordance with it, and agree that they do so as soon as they understand iti meaning. 2. That they think in accordance with it whatever may be the subject about which they are thinking. Thus if the things considered arc London, The Metropolis, The most populous city in Great Britain, since "the Metropolis is identical with London," and "London is identical with the most populous city in Great Britain," it follows necessarily in all minds that " the metropolis is identical with the most populous city in Great Britain." Again, if we compare the three following things Iron, The most useful metal, The cheapest metal, and it be allowed that " The most useful metal is Iron," and " Iron is the cheapest metal,* it follows necessarily in all minds that "the most useful metal is the cheapest" We here have two examples of the general truth that things identical with the same thing are identical with each other ; and this we may say is a general or necessary form of thought and reasoning. Compare, again, the following three things, The earth, Planets, Bodies revolving in elliptic orbits. We cannot say, as before, that "the earth is identical -fch the planets;" it is identical only with one of the 12 DEFINITION AND SPHERE [LES& planets, and we therefore say that " it is a planet" Sim* Arly we may say that " the planets are bodies revolving in elliptic orbits," but only a part of the whole number so revolving. Nevertheless it follows that if the earth is among the planets, and the planets among bodies re- volving in elliptic orbits, then the earth is among the latter. A very elementary knowledge of chemistry enables s to argue similarly concerning the following ; Iron, Metals, Elementary substances. Iron is one of the metals, and metals are elements ot simple undecomposable substances, in the sense of being among them or a part of them, but not as composing the whole. It follows necessarily that " Iron is one of the elementary substances." We have had then two exam- ples of a fixed and necessary form of thought which is necessary and true whatever the things may be to which it is applied The form of argument may be expressed in several different ways, and we shall have to consider k minutely in the lessons on the syllogism ; we may express it, for instance, by saying that "part of a part is part of the whole." Iron is part of the class of metals, which is part of the class of elements: hence iron is part of the class of elements. If I now introduce another definition of Logic and say that it is "the science of the necessary forms of thought," the reader will I hope clearl) apprehend the meaning of the expression " necessary forms of thought." A fora is something which may remain uniform and unaltered, while the matter thrown into that torm may be raried. Medals struck from the same dies have exactly the same form, but they may be of various matter, as j.] OF THE SCIENCE. S bronze, copper, gold or silver. A building of exactly the same form might be constructed either of stone or bricks ; furniture of exactly similar shape may be made of oak, mahogany, walnut wood, etc. Just as we thus familiarly recognize the difference of form and substance in common tangible things, so we may observe in Logic, that the form of an argument is one thing, quite distinct from the various subjects or matter which may be treated in that form. We may almost exhibit to the eye the form of reasoning to which belong our two latter arguments, af follows: (X) ...... is ..... .(Z) If within the three pairs of brackets, marked respect- ively X, Y and Z we place three names, such that the one in place of X may be said to come under that in Y t and that in Y under that in Z, then it necessarily follows that the first (X) comes under the last (Z). Logic, then, is the science occupied in ascertaining and describing all the general forms of thought which we must employ so long as we reason validly. These forms are very numerous, although the principles on which they are constructed are few and simple. It will hence appear that logic is the most general of all the sciences. Its aid must be more often required than the aid of any other science, because all the particular sciences treat portions only of existing things, and create very different and often unconnected branches of knowledge. But logic treats of those principles and forms of thought which must be employed in every branch of knowledge. It treats of the very origin and foundations of knowledge itself; and though it is true that the logical method em- ployed in one science may differ somewhat from that era- 6 DEFINITION AND SPHERE ployed in another science, yet whatever the particulai form may be, it must be logical, and must conform to thi laws of thought. There is in short something in which all sciences must be similar; to which they must con- form so long as they maintain what is true and self* consistent; and the work of logic is to explain this common basis of all science. One name which has been given to Logic, namely the Science of Sciences, very aptly describes the all extensive power of logical principles. The cultivators of special branches of knowledge appear to have been fully aware of the allegiance they owe to the highest of the sciences. for they have usually given names implying this allegi- ance. The very name of logic occurs as part of nearly all the names recently adopted for the sciences, which are often vulgarly called the "ologies," but are really the "logics," the "o" being only a connecting vowel or part of the previous word. Thus geology is logic applied to explain the formation of the earth's crust ; biology is logic applied to the phenomena of life ; psychology is logic applied to the nature of the mind ; and the same is the case with physiology, entomology, zoology, teratology, morphology, anthropology, theology, ecclesiology, thalat- tology, and the rest*. Each science is thus distinctly confessed to be a special logic. The name of logic itself is derived from the common Greek word \oyos, which usually means word, or the sign and outward manifesta- tion of any inward thought. But the same word was also used to denote the inward thought or reasoning of which words are the expression, and it is thus probably that latei Greek writers on reasoning were led to call their science Except Philology, which is differently formed, and meaai the love or stud* of words ; the name of this ritnCT. if formed opoa the same plan, would be logology. L) OP THE SCIENCE. oj, or logical science ; also rtynf Aoyuof, ot logical art. The adjective XoytKij, being used alone, sooa came to be the name of the science, just as Mathematic, Rhetoric, and other names ending in "ic" were ori- ginally adjectives but have been converted into substan- tives. Much discussion of a somewhat trifling character has arisen upon the question whether Logic should be con- sidered a science only, an art only, or both at the same time. Sir W. Hamilton has even taken the trouble to classify almost all the writers on logic according as they held one opinion or the other. But it seems substan- tially correct and sufficient to say, that logic is a science in so far as it merely investigates the necessary princi- ples and forms of thought, and thus teaches us to under- stand in what correct thinking consists; but that it be- comes an art when it is occupied in framing rules to assist persons in detecting false reasoning. A science ttachM ua to know and an art to do, and all the more perfect sciences lead to the creation of corresponding useful arts. As- tronomy is the foundation of the art of navigation on the ocean, as well as of the arrangement of the calendar and chronology. Physiology is the basis of the art of medi- cine, and chemistry is the basis of many useful arts. Logic has similarly been considered as the basis of an art of correct reasoning or investigation which should teach the true method to be observed in all sciences. The cele- brated British logician Duns Scotus, who lived in the I3th century, and called logic the Science of Sciences, called it also the Art of Arts, expressing fully its preeminence. Others have thus defined it" Logic is the art of direct- ing the reason aright in acquiring the knowledge o( things, for the instruction both of ourselves and others," Dr Isaac Watts, adopting this view of logic, called hit well-known work " the Art of Thinking." 8 DEFINITION AND SPHERE L LEa It may be fairly said however that Logic has mort the form of a science than an art for this reason all persons necessarily acquire the faculty and habit of rea- soning long before they even know the name of logic. This they do by the natural exertion of the powers of mind, or by constant but unconscious imitation of others. They thus observe correctly but unconsciously the prin- ciples of the science in all very simple cases ; but the con- tradictory opinions and absurd fallacies which are put forth by uneducated persons shew that this unaided ex- ercise of mind is not to be trusted when the subject of discussion presents any difficulty or complexity. The study of logic then cannot be useless. It not only explains the principles on which every one has often reasoned correctly before, but points out the dangers which exist of erroneous argument. The reasoner thus becomes consciously a correct reasoner and learns con- sciously to avoid the snares of fallacy. To say that men can reason well without logical science is about as true as to say that they can live healthily without medi- cine. So they can as long as they are healthy ; and so can reasoners do without the science of reasoning as long as they do reason correctly ; but how many are there that can do so ? As well might a man claim to be immortal in his body as infallible in his mind. And if it be requisite to say a few words in defence of Logic as an art, because circumstances in the past his- tory of the science have given rise to misapprehension, can it be necessary to say anything in its praise as a science ? Whatever there is that is great in science or in art or in literature, it is the work of intellect In bodily form man is kindred with the brutes, and in his perish- able part he is but matter. It is the possession of con- scious intellect, the power of reasoning by general notions that raises him above all else upon the earth ; and wh IL] OF THE SCIENCE. \ can say that the nature and procedure of this intellect if not almost the highest and most interesting subject of study in which we can engage? In vain would an? one deny the truth of the favourite aphorism of Sir W Hamilton IN THE WORLD THERE IS NOTHING GREAT BUT 1CAM IN MAN THERE IS NOTHING GREAT BUT MIND. LESSON II. THE THREE PARTS OF LOGICAL DOCTRINE. IT has been explained in the previous lesson that Logic is the Science of Reasoning, or the Science of those Ne- cessary Laws of Thought which must be observed if we are to argue consistently with ourselves and avoid self- contradiction. Argument or reasoning therefore is the strictly proper subject before us. But the most conve- nient and usual mode of studying logic is to consider first the component parts of which any argument must be made up. Just as an architect must be acquainted with the materials of a building, or a mechanic with the ma* terials of a machine, before he can pretend to be ac- quainted with its construction, so the materials and in- struments with which we must operate in reasoning arc suitably described before we proceed to the actual forms of argument If we examine a simple argument such as that giret in the last lesson, thus Iron is a metal, Every metal is an element, Therefore Iron is an element. to THE THREE PARTS OF [LESS Jcts? The simplest and truest answer is to say that it treats in a certain sense of all three. Inasmuch as no reasoning process can be ex- plained or communicated to another person without words, we are practically limited to such reasoning as is reduced to the form of language. Hence we shall always be concerned with words, but only so far as they are the instruments for recording and referring to our thoughts. The grammarian also treats of language, but he treats it as language merely, and his science terminates with the description and explanation of the forms, varieties, and ILl LOGICAL DOCTRINE. t> relations of words. Logic also treats of language, but only as the necessary index to the action of mind. Again, so long as we think correctly we must think ol things as they are; the state of mind within us must correspond with the state of things without us whenevet in opportunity arises for comparing them. It is im possible and inconceivable that iron should prove not to be an elementary substance, if it be a metal, and every metal be an element. We cannot suppose, and there is no reason to suppose, that by the constitution of the mind we are obliged to think of things differently from the manner in which they are. If then we may assume that things really agree or differ according as by correct logical thought we are induced to believe they will, it does not seem that the views of the logicians named are irreconcileable. We treat of things so far as they are the objects of thought, and we treat of language so far as it is the embodiment of thought. If the reader will bear this explanation in mind, he will be saved from some per- plexity when he proceeds to read different works on logic, and finds them to vary exceedingly in the mode of treat- ment, or at least of expression. If, when reduced to language, there be three parts of logic, terms, propositions, and syllogisms, there must be as many different kinds of thought or operations of mind These are usually called 1. Simple apprehension, 2. Judgment 3. Reasoning or discourse. The first of these, Simple Apprehension, is the act ol Blind by which we merely become aware of something, or have a notion, idea, or impression of it brought into the mind. The adjective simple means apart from othei things, and apprehension the taking hold by the mind. Thus the name or term Iron instantaneously makes tbf I* THE THREE PARTS OF [LESI mind think of a strong and very useful metal, but doe* not tell us anything about it, or compare it with any thing zlse. The words sun, Jupiter, Sirius, St PauPs Cathe- dral, are also terms which call up into the mind certain veil-known objects, which dwell in our recollection even when they are not present to our senses. In fact, the use of a term, such as those given as examples, is merely as a substitute for the exhibition of the actual things named. Judgment is a different action of mind, and consists in comparing together two notions or ideas of objects de- rived from simple apprehension, so as to ascertain whe- ther they agree or differ. It is evident, therefore, that we cannot judge or compare unless we are conscious of two things or have the notions of two things in the mind at the same time. Thus if I compare Jupiter and Sirius I first simply apprehend each of them ; but bringing them into comparison I observe that they agree in being small, bright, shining bodies, which rise and set and move round the heavens with apparently equal speed. By minute examination, however, I notice that Sirius gives a twinkling or intermittent light, whereas Jupiter shines steadily. More prolonged observation shews that Ju- piter and Sirius do not really move with equal and regular speed, but that the former changes its position upon the heavens from night to' night in no very simple manner. If the comparison be extended to others of the heavenly bodies which are apprehended or seen at the same time, I shall find that there are a multitude of stars which agree with Sirius in giving a twinkling light and in remaining perfectly fixed hi relative position to each other, whereas two or three other bodies may be seen which resemble Jupiter in giving a steady light, and aljq in changing then- place from night to night among the fixed stars. I have now by the action of judgment formed in my mind the general notion of faced start, by a] LOGICAL DOCTRINE. 13 bringing together mentally a number of objects which ' agree ; while from several other objects I have formed the general notion ot planets. Comparing the two genera] notions together, I find that they do not possess the same qualities or appearances, which I state in the proposition, ! * Planets are not fixed stars." I have introduced the expression "General Notion" as if the reader were fully acquainted with it. But though pnilosophers have for more than two thousand years con- stantly used the expressions, general notion, idea, con- ception, concept, &c., they have never succeeded in agreeing exactly as to the meaning of the terms. One class of philosophers called Nominalists say that it is all a matter of names, and that when we join together Jupiter, Mars, Saturn, Venus, &c., and call them planets, the common name is the bond between them in our minds. Others, called Realists, have asserted that besides these particular planets there really is something which com- bines the properties common to them all without any of the differences of size, colour, or motion which distin- guish them. Every one allows in the present day how- ever that nothing can physically exist corresponding to a general notion, because it must exist here or there, of this size or of that size, and therefore it would be one particu- lar planet, and not any planet whatever. The Nominal- ists, too, seem equally wrong, because language, to be of any use, must denote something, and must correspond, as we have seen, to acts of mind. If then proper names raise up in our minds the images of particular things, like the sun, Jupiter, &c., general names should raise up general notions. The true opinion seems to be that of the philoso- phers called Conceptualists, who say that the general no don is the knowledge in the mind of the common pro- perties or resemblances of the things embraced undet 14 THE THREE PARTS OF [l the notion. Thus the notion planet really means the consciousness in anybody's mind that there are certain heavenly bodies which agree in giving a steady light and in moving about the heavens differently from thi axed stars. It should be added, however, that there art many, including Sir W. Hamilton, who would be counted as Nominalists and who yet hold that with the general name is associated a consciousness of the resemblance existing between the things denoted by it. Between this form of the doctrine and conceptualism it is not easy to draw a precise distinction, and the subject is of too de- batable a character to be pursued in this work. It will appear in the course of these lessons that the whole of logic and the whole of any science consists in so arranging the individual things we meet in general no- tions or classes, and in giving them appropriate general names or terms, that our knowledge of them may be made as simple and general as possible. Every general notion that is properly formed admits of the statement of general laws or truths ; thus of the planets we may affirm that they move in elliptic orbits round the sun from west to east ; that they shine with the reflected light of the sun ; and so on. Of the fixed stars we may affirm that they shine with their own proper light; that they are incomparably more distant than the planets ; and so on The whole of reasoning will be found to arise from this faculty of judgment, which enables us to discover and affirm that a large number of objects have similar pro pertief, so that whatever is known of some may be in lerred and asserted of others. It is in the application of such knowledge that w< onploy the third act of mind called discourse or reason dig, by which from certain judgments we are enabled, without any new reference to the real objects, to form * new judgment If we know that iron comes under th n.J LOGICAL DOCTRINE. 1} general notion of metal, and that this notion comes under the still wider notion of element, then without furthei examination of iron we know that it is a simple unde composable substance called by chemists an element. Oi if from one source of information we learn that Neptun* is a planet, and from another that planets move in ellip tic orbits, we can join these two portions of knowledge together in the mind, so as to elicit the truth that Nep- tune moves in an elliptic orbit. Reasoning or Discourse, then, may be defined as .he progress of the mind from one or more given propositions to a proposition different from those given. Those pro- positions from which we argue are called Premises, and that which is drawn from them is called the Conclusion, The latter is said to follow, to be concluded, inferred or col- lected from them ; and the premises are so called because they are put forward or at the beginning (Latin pra, be- fore, and mitto, I send or put). The essence of the pro- cess consists in gathering the truth that is contained in the premises when joined together, and carrying it with us into the conclusion, where it is embodied in a new proposition or assertion. We extract out of the pre- mises all the information which is useful for the purpose in view and this is the whole which reasoning accom- plishes. I have now pointed out the three parts of logical doc- trine, Terms, Propositions, and Reasoning or Syllogism, into which the subject is conveniently divided. To the consideration of these parts we shall proceed But it may be mentioned that a fourth part has often been added called Method, which is concerned with the ar- rangement of the parts of any composition. It is sometimes said that what proposition is to term, and what syllogism is to proposition, such is method to syllogism, and that a fourth division is necessary to com* 16 TERMS, AND THEIR [l plcte the doctrine of Logic. It is at any rate ccrtaii however that this fourth part is much inferior in import- ance and distinctness to the preceding three ; and all thai will be s&id of it is to be found in Lesson xxrv, LESSON IIL TERMS, AND THEIR VARIOUS KINDS. IT has been explained in the preceding lesson that ever) assertion or statement expresses the agreement or dif- ference of two things, or of two general notions. In putting the assertion or statement into words, we must accordingly have words suitable for drawing the attention of the mind to the things which are compared, as well as words indicating the result of the comparison, that is to say, the fact whether they agree or differ. The words by which we point out the things or classes of things in question are called Terms, and the words denoting the comparison are said to form the Copula. Hence a com- plete assertion or statement consists of two terms and a copula, and when thus expressed it forms a Proposition. Thus in the proposition " Dictionaries are useful books," the two terms are dictionaries and useful books; the co- pula is the verb are, and expresses a certain agreement ol the class dictionaries with the class of useful books con- sisting in the fact that the class of dictionaries forms part of the class of useful books. In this case each term con- sists of only one or two words, but any number of words nay be required to describe the notions or classes com- III.] VARIOUS KINDS. 17 pared together. In the proposition "the angles at the base of an isosceles triangle are equal to each other, 7 * the first term requires nine words for its expression, and the second term, four words (equal to each other); and tLere is no limit to the number of words which may be em- ployed in the formation of a term. A term is so called because it forms one end (Latin, tt*-minus) of a proposition, and strictly speaking it is a term only so long as it stands in the proposition. But we commonly speak of a term or a name meaning any noun, substantive or adjective, or any combination of words denoting an object of thought, whether that be, as we shall shortly see, an individual thing, a group of things, a quality of things, or a group of qualities. It would be impossible to define a name or term better than has been done by Hobbes : "A name is a word taken at pleasure to serve for a mark, which may raise in our mind a thought like to some thought which we had before, and which, being pronounced to others, may be to them a sign of what thought the speaker had before in his mind." Though every term or name consists of words it is not every word which can form a name by itself. We cannot properly say " Not is agreeable" or " Probably is not true f nothing can be asserted of a preposition, an adverb, and certain other parts of speech, except indeed that they are prepositions, adverbs, &c. No part of speech except a noun substantive, or a group of words used as a noun substantive, can form the subject or first term of a proposition, and nothing but a noun substan- tive, an adjective, the equivalent of an adjective, or a yerb, can form the second term or predicate of a propo- sition It may indeed be questioned whether an adjec live can ever form a term alone ; thus in " Dictionaries are useful," it may be said that the substantive things of books is understood in the predicate, the complete sen* 9 it TERMS, AND THEIR [LESS fence being " Dictionaries are useful books f but as this is a disputed point we will assume that words are divided into two kinds in the following manner : Words which stand, or appear to stand alone as com- plete terms, namely the substantive and adjective, and certain parts of a verb, are called categorematlc words from the Greek word rar^yopcw, to assert or predicate. Those parts of speech, on the other hand, such a* prepositions, adverbs, conjunctions, &c, which can only form parts of names or terms are called syncategorematio words, because they must be used with other words in order to compose terms (Greek 4. Whether it is concrete or abstract 5. Whether it is positive, or negative, or privative. 6. Whether it is relative or absolute. It will be fully pointed out in the next lesson *K* most terms have more than one meaning; and as the one meaning may be general and the other singular, the one concrete and the other abstract, and so on, it is absolute- ly necessary that the reader should first of all choos* one precise meaning of the term which he is examining. And in answering the questions proposed it is desirable he should specify the way in which he regards it Taking the word sovereign, we may first select the meaning in which it is equivalent to monarch ; this is a general term in so far as it is the name of any one of many monarch) living or dead, but it is singular as regards the inhabit- ants of any one country. It is clearly categorematic, concrete, and positive, and obviously relative to the sub- jects of the monarch. Read Mr Mill's chapter on Namts, System of Logic Book I. chap. 2. LESSON IV. OF THE AMBIGUITY OF TERMS. 'HERE is no part of Logic which is more really useful than that which treats of the ambiguity of terms, that is of the uncertainty and variety of meanings belonging to irords. Nothing indeed can be of more importance to the attainment of correct habits of thinking and reason- ing than a thorough acquaintance with the great imper- fections of language. Comparatively few terms have one 98 OF THE AMBIGUITY [LESS single clear meaning and one meaning only, and when ever two or more meanings are unconsciously confuseo together, we inevitably commit a logical fallacy. If, foi instance, a person should argue that " punishment is ar evil," and according to the principles of morality "nc evil is to be allowed even with the purpose of doing good," we might not at the first moment see how to avoid the conclusion that " no punishments should be allowed/ because they cause evil. A little reflection will show that the word evil is here used in two totally different senses in the first case it means physical evil or pain ; in the second moral evil, and because moral evil is never to be committed, it does not follow that physical evils are never to be inflicted, for they are often the very means of pre- venting moral evil./ Another very plausible fallacy which has often been put forth in various forms is as follows : " A thoroughly benevolent man cannot possibly refuse to relieve the poo*", and since a person who cannot possibly act otherwise than he does can claim no merit for his actions, it follows that a thoroughly benevolent man can claim no merit for his actions." According to this kind of argument a man would have less merit in proportion as he was more virtuous, so as to feel greater and greater difficulty in acting wrongly. That the conclusion is fallacious every one must feel certain, but the cause of the fallacy can only be detected by observing that the words cannot possibly have a double meaning, in the first case referring to the influence of moral motives or good character, and ra the second to circumstances entirely beyond a person's control ; as, for instance, the compulsion of the laws, the want of money, the absence of personal liberty. The more a person studies the subtle variations in the mean- ing of common words, the more he will be convinced oi the dangerous natuie of the tools he has to use in alJ V.j OF TERMS. * communications and arguments. Hence I must ash much attention to the contents of this Lesson. Terms are said to be tmlvocal when they can suggesi to the mind no more than one single definite meaning They are called equivocal or ambiguous when they \\.\\ two or more different meanings. It will be observe, however, that a term is not equivocal because it can !< applied to many objects when it is applied in the same sense or meaning to those different objects. Thus cathe- dral is the name of St Paul's, the York Minster, and the principal churches of Salisbury, Wells, Lincoln and a number of other cities, but it is not ambiguous, because all these are only various instances of the same meaning ; they are all objects of the same description or kind. The word cathedral is probably univocal or of one logical meaning only. The word church, on the other hand, is equivocal, because it sometimes means the building in which religious worship is performed, sometimes the body of persons who belong to one sect or persuasion, and assemble in churches. Sometimes also the church means the body of the clergy as distinguished from the laity; hence there is a clear difference in the sense or meaning with which the word is used at different times. Instances of univocal terms are to be found chiefly ID technical and scientific language. Steam-engine, gas- ometer, railway train, permanent way, and multitudes of such technical names denoting distinct common objects, are sufficiently univocal. In common life the names penny, mantelpiece, teacup, bread and butter, have a suf- ficiently definite and single meaning. So also in chemistry, oxygen, hydrogen, sulphate of copper, alumina, lithia, ind thousands of other terms, are very precise, the words themselves having often been invented in very recent years, and the meanings exactly fixed and maintained invariable. Every science has or ought to have a series jo OF THE AMBIGUITY [] of terms equally precise and certain in meaning. (Set Lesson xxxm.) The names of individual objects, build- ings, events, or persons, again, are usually quite certain uid clear, as in J ulius Caesar, William the Conqueror, the irst Napoleon, Saint Peter's, Westminster Abbey, the ireat Exhibition of 1851, and so on. But however numerous may be the univocal terms vhich can be adduced, still the equivocal terms are asto- 11 shingly common. They include most of the nouns and idjectives which are in habitual use in the ordinary intercourse of life. They are called ambiguous from the Latin verb ambigo, to wander, hesitate, or be in doubt; or . gain hontonymousy from the Greek o^os, like, and OVO/AO, name. Whenever a person uses equivocal words in such a way as to confuse the different meanings and fall into error, he may be said to commit the fallacy of Equivoca- tion in the logical meaning of the name (see Lesson XX.) ; but in common life a person is not said to equivocate unless he uses words consciously and deceitfully in a manner calculated to produce a confusion of the true and apparent meanings. I will now describe the various kinds and causes ol ambiguity of words, following to some extent the inter- esting chapters on the subject in Dr Watts' Logic. In the first place we may distinguish three classes of equi- rocal words, according as they are 1. Equivocal in sound only. 2. Equivocal in spelling only. 3. Equivocal both in sound and spelling. Fhe first two classes are comparatively speaking of very ilight importance, and do not often give rise to serious error. They produce what we should call trivial mis- takes. Thus we may confuse, when spoken only, the words right, wright and rite (ceremony) ; also the wordi rein, rain and reign, might and mite, &c. Owing partly IV.] OF TERMS. * to defects of pronunciation mistakes are not unknowi between the four words air, hair, hare and heir. Words equivocal in spelling but not in sound are such as tear (a drop), and tear pronounced tare, meaning a rent in cloth ; or lead, the metal, and lead, as in follow- ing the lead of another person. As little more than mo- mentary misapprehension, however, can arise from such resemblance of words, we shall pass at once to the class of words equivocal both in sound and spelling. These I shall separate into three groups according as the equivo- cation arises i. From the accidental confusion of different words. x From the transfer of meaning by the association of ideas. 3. From the logical transfer of meaning to analogous objects. i. Under the first class we place a certain number of curious but hardly important cases in which ambi- guity has arisen from the confusion of entirely different words, derived from different languages or from differ- ent roots of the same language, but which have in the course of time assumed the same sound and spell- ing. Thus the word mean denotes either that which is medium or mediocre, from the French moyen and the Latin medius, connected with the Anglo -Saxon mid, or middle; or it denotes what is low-minded and base, being then derived from the Anglo-Saxon Gemane, which means " that belonging to the mcene or many," whatever in short is vulgar. The verb to mean can hardly be confused with the adjective mean, but it comes from a third distinct root, probably connected with the Sanscrit verb, to think. As other instances of this casual ambiguity, I may mention rent, a money payment, from the French renU ^ to return), or a tear, the result of the action o/ p OF THE AMBiGUITY [iXSk reiuling, this word being of Anglo-Saxon origin and on of the numerous class beginning in r or wr, which imitate more or less perfectly the sound of the action which the) denote. Pound, from the Latin pondus, a weight, is con- fused with pound, in the sense of a village pinfold foi cattle, derived from the Saxon pyndan y to pen up. />//, a mountain, is a perfectly distinct word from fell, a skin or hide ; and pulse, a throb or beating, and pulse, peas, beans, or potage, though both derived from the Greek or Latin, are probably quite unconnected words. It is curious that gin, in the meaning of trap or machine, is a contracted form of engine, and when denoting the spirit- uous liquor is a corruption of Geneva, the place where the spirit was first made. Certain important cases of confusion have been de- tected in grammar, as between the numeral one, derived from an Aryan root, through the Latin unus, and the in- determinate pronoun, one (as in "one ought to do one's duty"), which is really a corrupt form of the French word homme or man. The Germans to the present day use man in this sense, as in man sagt, i.e. one says. 2. By far the largest part of equivocal words have become so by a transfer of the manning from the thing originally denoted by the word to some other thing habitually connected with it so as to become closely as- sociated in thought Thus, in Parliamentary language, the House means either the chamber in which the mem- bers meet, or it means the body of members who happen to be assembled in it at any time. Similarly, the word church originally denoted the building (/tvpioicoi/, the Lord's House) in which any religious worshippers assem- ble, but it has thence derived a variety of meanings ; it may mean a particular body of worshippers accustomed to assemble in any one place, in which sense it is used in 4cts xiv. 23 ; or it means any body of persons holding iv.] OF TERMS. 31 the same opinions &nd connected in one organization, ai in the Anglican, or Greek, or Roman Catholic Church j it is also sometimes used so as to include the laity as well as the clergy ; but more generally perhaps the clergy and religious authorities of any sect or country are so strongly issociated with the act of worship as to be often called the church par excellence. It is quite evident moreover that the word entirely differs in meaning according as it is used by a member of the Anglican, Greth, Roman Catholic, Scotch Presbyterian, or any other existing church. The word foot has suffered several curious but very evident transfers of meaning. Originally it denoted the foot of a man or an animal, and is probably connected in a remote manner with the Latin pes, pedis, and the Greek ro*;y, iro86s ', but since the length of the foot is naturally employed as a rude measure of length, it came to be applied to a fixed measure of length ; and as the foot is at the bottom of the body the name was extended by analogy to the foot of a mountain, or the feet of a table j by a further extension, any position, plan, reason, or argument on which we place ourselves and rely, is called the foot or footing. The same word also denotes soldiers who fight upon their feet, or infantry, and the measured part of a verse having a definite length. That these very different meanings are naturally connected with the ori- ginal meaning is evident from the fact that the Latin and Greek words for foot are subject to exactly similar series of ambiguities. It would be a long task to trace out completely the various and often contradictory meanings of the word fallow. Originally a fellow was whaty&//0/.r another, that is a companion ; thus it came to mean the other of a pair, as one shoe is the fellow of the other, or simply an equal, as when we say that Shakspeare "hath not a fellow/ 54 OF THE AMBIGUITY [LESS From the simple meaning of companion again it comes U) denote vaguely a person, as in the question "What fellow is that?" but then there is a curious confusion oi depreciatory and endearing power in the word ; when a man is called a mere fellow, or simply a fellow in a par- ticular tone of voice, the name is one of severe contempt , alter the tone of voice 01 the connected words in the least degree, and it becomes one of the most sweet and en- dearing appellations, as when we speak of a dear 01 good fellow. We may still add the technical meanings ol the name as applied in the case of a Fellow of a College, or of a learned society. Another good instance of the growth of a number oi different meanings from a single root is found in the word post. Originally a post was something posited, or placed firmly in the ground, such as an upright piece oi wood or stone ; such meaning still remains in the cases of a lamp-post, a gate-post, signal-post, &c. As a post would often be used to mark a fixed spot of ground, as in a mile-post, it came to mean the fixed or appointed place where the post was placed, as in a military post, the post of danger or honour, &c. The fixed places where horsea were kept in readiness to facilitate rapid travelling during the times of the Roman empire were thus called posts, and thence the whole system of arrangement for the con- veyance of persons or news came to be called the posts. The name has retained an exactly similar meaning to the present day in most parts of Europe, and we still use it in post-chaise, post-boy, post-horse and postillion. A system of post conveyance for letters having been organ- ised for about two centuries in England and other coun tries, this is perhaps the meaning most closely associated with the word post at present, and a number of expres- sions have thus arisen, such as post-office, postage, postal- guide, postman, postmaster, postal-telegraph, &c. Curi IV.] Of TERMS. oasly enough we now have iron letter-posts, in which th word post is restored exactly to its original meaning. Although the words described above were selected on account of the curious variety of their meanings, I do not hesitate to assert that che majority of common nouns possess various meanings in greater or less number. Dr Watts, in his Logic, suggests that the words book, bible, fish, house, and elephant, are univocal terms, but the reader would easily detect ambiguities in each of them, Thus fish bears a very different meaning in natural his- tory from what it does in the mouths of unscientific per- sons, who include under it not only true fishes, but shell- fish or mollusca, and the cetacea, such as whales and seals, in short all swimming animals, whether they have the character of true fish or not. Elephant, in a station- er's or bookseller's shop, means a large kind of paper instead of a large animaL Bible sometimes means any particular copy of the Bible, sometimes the collection of works constituting the Holy Scriptures. The. word man is singularly ambiguous ; sometimes it denotes man as distinguished from woman; at other times it is cer- tainly used to include both /sexes ; and in certain recent election cases lawyers were unable to decide whether the word man as used in the Reform Act of 1867 ought or ought not to be interpreted so as to include women. On other occasions man is used to denote an adult male as distinguished from a boy, and it also often denotes one who is emphatically a man as possessing a masculine character. Occasionally it is used in the same way as groom, for a servant, as in the proverb, u Like master, like man." At other times it stands specially tor a bus band. 3. Among ambiguous words we must thirdly distinguish those which derive their various meanings in a somewhat different manner, namely by analogy or real resemblance j6 THE AMBIGUITY OF TERMS. [LESS, n When we speak of a sweet taste, a sweet flower, a swee tune, a sweet landscape, a sweet face, a sweet poem, it is evident that we apply one and the same word to very different things ; such a concrete thing as lump-sugar can hardly be compared directly with such an intellectual existence as Tennyson's May Queen. Nevertheless if the word sweet is to be considered ambiguous, it is in a dif- ferent way from those we have before considered, because all the things are called sweet on account of a peculiar pleasure which they yield, which cannot be desciibed otherwise than by comparison with sugar. In a similai way, we describe a pain as sharp, a disappointment as bitter, a person's temper as sour, the future as bright or gloomy, an achievement as brilliant ; all these adjectives implying comparison with bodily sensations of the sim- plest kind. The adjective brilliant is derived from the French briller, to glitter or sparkle ; and this meaning it fully retains when we speak of a brilliant diamond, a brilliant star, &c. By what a subtle analogy is it that we speak of a brilliant position, a brilliant achievement, brilliant talents, brilliant style ! We cannot speak of a clear explanation, indefatigable perseverance, perspicuous style, or sore calamity, without employing in each of these expressions a double analogy to physical impressions, actions, or events. It will be shewn in the sixth Lesson that to this process we owe the creation of all names connected with mental feelings or existences. Read Watts' Logic, Chapter iv. Locke's Essay on tJu Human Understanding, Book III Chapters ix. and X. LESSON V. Qf THE TWOFOLD MEANING OF TERMS-. IN EXTENSION AND INTENSION. THERE is no part of the doctrines of Logic to which I would more urgently request the attention of the reader than to that which I will endeavour to explain clearly in the present Lesson. I speak of the double meaning which is possessed by most logical terms the meaning in extension, and the meaning in Intension. I believe that the reader who once acquires a thorough apprehen- sion of the difference of these meanings, and learns to bear it always in mind, will experience but little further difficulty in the study of logic. The meaning of a term in extension consists of the objects to which the term may be applied ; its meaning in intension consists of the qualities which are necessarily possessed by objects bearing that name. A simple example will make this distinction most apparent What is the meaning of the name "metal"? The first and most ob- vious answer is that metal means either gold, or silver, or iron, or copper, or aluminium, or some other of the 48 substances known to chemists, and considered to have a metallic nature. These substances then form the plain and common meaning of the name, which is the meaning in extension. But if it be asked why the name is applied to all these substances and these only, the answer must be Because they possess certain qualities which belong to the nature of metal. We cannot, therefore, know to what substances we may apply the name, or to what we 38 I WOFOLD MEANING OF TERMS may not, unless we know the qualities which are indis- pensable to the character of a metal Now chemists lay these down to be somewhat as follows: (i) A metal must be an element or simple substance incapable ol decomposition or separation into simpler substances by any known means. (2) It must be a good conductor ol heat and electricity. (3) It must possess a great and peculiar reflective power known as metallic lustre*. These properties are common to all metals, or nearly all metals, and are what mark out and distinguish a metal from other substances. Hence they form in a certain way the meaning of the name metal, the meaning in intension, as it is called, to distinguish it from the former kind of meaning. In a similar manner almost any other common name has a double meaning. "Steamship" denotes in exten- sion the Great Eastern, the Persia, the Himalaya, or any one of the thousands of steamships existing or which have existed; in intension it means "a vessel propelled by steam-power." Monarch is the name of Queen Vic- toria, Victor Emmanuel, Louis Napoleon, or any one of a considerable number of persons who rule singly over countries; the persons themselves form the meaning in extension ; the quality of ruling alone forms the intensive meaning of the name. Animal is the name in extension of any one of billions of existing creatures and of indefi nitely greater numbers of other creatures that have ex isted or will exist ; in intension it implies in all those creatures the existence of a certain animal life and sense, or at least the power of digesting food and exerting force, which are the marks of animal nature. * It is doubtfully true that all metals possess metallic lustre, and chemists would find it very difhcult to give any consistent explanation of their use of the name ; bat the statements in thf tort are sufficiently true to furnish an example. V.] IN EXTENSION AND INTENSION. 3^ It is desirable to state here that this distinction ol extension and intension has been explained by logi- cians under various forms of expression. It is the pe- culiar misfortune of the science of logic to have a super* fluity of names or synonyms for the same idea. Thus the intension of a term is synonymous with its comprehen- ilon, or connotation, or depth; while the extension is synonymous with the denotation or breadth. This may be most clearly stated in the form of a scheme: The extension, extent, /he intension, intent, breadth, denotation, do- depth, connotation, or im- main, sphere or application plication of a name con-> of a name consists of the sists of the qualities the individual things to which possession of which by those the name applies. things is implied. Of these words, denotation and connotation are employed chiefly by Mr J. S. Mill among modern logical writers, and are very apt for the purpose. To denote is to mark down, and the name marks the things to which it may be applied or affixed; thus metal denotes gold, silver, cop- per, Sec. To connote is to mark along with (Latin con, together; notare, to mark), and the connotation accord- ingly consists of the qualities before described, the pos- session of which is implied by the use of the name metaL When we compare different but related terms we may observe that they differ in the quantity of their extension and intensioa Thus the term element has a greater extension of meaning than metal, because it includes in its meaning all metals and other substances as well But it has at the same time less intension of meaning; for among the qualities of a metallic substance must be found the qualities of an element, besides the othei qualities peculiar to a metal. If again we compare the terms metal and malleable metal, it is apparent fhat tht 40 TWOFOLD MEANING OF TERMS [LESi latter term does not include the metals antimony, arsenic and bismuth, which are brittle substances. Hence mal- leable metal is a term of narrower meaning in extension than metal ; but it has also deeper meaning in intension, because it connotes or implies the quality of malleability in addition to the general qualities of a metal WhiU malleable metal is again a narrower term in extension because it does not include gold and copper ; and I can go on narrowing the meaning by the use of qualifying ad- jectives until only a single metal should be denoted by the term. The reader will now see clearly that a general law of great importance connects the quantity of extension and the quantity of intension, viz. As the intension of a term IB Increased the extension is decreased. It must not be supposed, indeed, that there is any exact proportion be- tween the degree in which one meaning is increased and the other decreased. Thus if we join the adjective red to metal we narrow the meaning much more than if we join the adjective white, for there are at least twelve times as many white metals as red. Again, the term white man includes a considerable fraction of the meaning of the term man as regards extension, but the term blind man only a small fraction of the meaning. Thus it is obvious that in increasing the intension of a terra we ma) decrease the extension in any degree. In understanding this law we must carefully discrimi- nate the cases where there is only an apparent increase of the intension of a term, from those where the increase is real If I add the term elementary to metal^ I shall not really alter the extension of meaning, for all the metals are elements; and the elementary metals are neither more nor less numerous than the metals. But then the intension of the term is really unaltered at the same time ; for the quality of an element is really found among thf v.j IN EXTENSION AND INTENSION. 4* qualities of metal, and it is superfluous to specify it ovei again. A quality which belongs invariably to the whole of a class of things is commonly called a property of the class (see Lesson xn.), and we cannot qualify or restrict a term by its own property. This is a convenient place to notice a distinction be- cween terms into those which are connotative and those which are non-connotative, the latter consisting of the terms which simply denote things without implying any knowledge of their qualities. As Mr Mill considers this distinction to be one of great importance, it will be well to quote his own words*: " A non-connotative term is one which signifies a sub- ject only, or an attribute only. A connotative term is one which denotes a subject, and implies an attribute. By a subject is here meant anything which possesses attributes. Thus John, or London, or England, are names which signify a subject only. Whiteness, length, virtue, signify an attribute only. None of these names, therefore, are connotative. But white, long, virtuous, are connotative. The word white denotes all white things, as snow, paper, the foam of the sea, &c., and implies, or, as^it was termed by the schoolmen, connotes the attribute whiteness. The word white is not predi- cated of the attribute, but of the subjects, snow, &c. ; but when we predicate it of them, we imply, or connote, that the attribute whiteness belongs to them "All concrete general names are connotative. The word man, for example, denotes Peter, James, John, and an indefinite number of other individuals, of whom, taken as a class, it is the name. But it is applied to them, be* cause they possess, and to signify that they possess, cer- w System of Logic ; Vol. I. p. 31, 6th ed. Book L Chap. II IS 43 TWOFOLD MEANING OF TERMS [LESS fain attributes. . . . What we call men, are the subjects, the individual Styles and Nokes ; not the qualities by which their humanity is constituted. The name therefore is said to signify the subjects directly, the attributes to- directly ; it denotes the subjects, and implies, or involves or indicates, or, as we shall say henceforth, connotes, the Attributes, It is a connotative name .... " Proper names are not connotative : they denote the individuals who are called by them ; but they do not indi- cate or imply any attributes as belonging to those indivi- duals. When we name a child by the name Paul, or a dog by the name Caesar, these names are simply marks used to enable those individuals to be made subjects of dis- course. It may be said, indeed, that we must have had some reason for giving them those names rather than any others ; and this is true ; but the name, once given, is independent of the reason. A man may have been named John, because that was the name of his father ; a town may have been named Dartmouth, because it is situ- ated at the mouth of the Dart. But it is no part of the signification of the word John, that the father of the per- son so called bore the same name ; nor even of the word Dartmouth to be situated at the mouth of the Dart. If sand should choke up the mouth of the river, or an earth quake change its course, and remove it to a distance from the town, the name of the town would not necessarily be changed." I quote this in Mr Mill's own words, because though it expresses most clearly the view accepted by Mr Mill and many others, it is nevertheless probably erroneous. The connotation of a name is confused with the etymo- k ogical meaning, or the circumstances which caused it to be affixed to a thing. Surely no one who uses the name England, and knows what it denotes, can be ignorant of the peculiar qualities and circumstances of the country, v ] IN EXTENSION AND INTENSION. 4 m and these form the connotation of the term. To any one who knows the town Dartmouth the name must imply ths possession of the circumstances by which that town is cha- racterised at the present time. If the river Dart should b< destroyed or removed, the town would so far be altered, and the signification of the name changed. The name urould no longer denote a town situated on the Dart, but one which was formerly situated on the Dart, and it would be by a mere historical accident that the form of the name did not appear suitable to the town. So again any proper dame such as John Smith, is almost without meaning until we know the John Smith in question. It is true that the name alone connotes the fact that he is a Teuton, and is a male ; but, so soon as we know the exact individual it denotes, the name surely implies, also, the peculiar fea- tures, form, and character, of that individual In fact, as it is only by the peculiar qualities, features, or circum- stances of a thing, that we can ever recognise it, no name could have any fixed meaning unless we attached to it, mentally at least, such a definition of the kind of thing denoted by it, that we should know whether any given thing was denoted by it or not. If the name John Smith does not suggest to my mind the qualities of John Smith, how shall I know him when I meet him? for he certainly does not bear his name written upon his brow *. This, however, is quite an undecided question; and as Mr Mill is generally considered the best authority upon the subject, it may be well for the reader provisionally to accept his opinion, that singular or proper names are con -connotative, and all concrete general names are con- rotative. Abstract names, on the other hand, can hardly * Further objections to Mr Mill's views on this point will be found in Mr Shedden's Elements if Logic. London, 1864 pp. H, &c. 44 TWOFOLD MEANING OF TERMS. [LESi possess connotation at all, for as they already denote thi attributes or qualities of something, there is nothing left which can form the connotation of the name. Mr Mill, indeed, thinks that abstract names may often be consi- dered connotative, as when the name fault connotes the attribute of hurtfulness as belonging to fault. But if fault is a true abstract word at all I should regard hurt fulness as a part of its denotation ; I am inclined to think that faultiness is the abstract name, and that fault is gene- rally used concretely as the name of a particular action or thing that is faulty, or possesses faultiness. But the sub- ject cannot be properly discussed here, and the reader snould note Mr Mill's opinion that abstract names are usually non-connotative, but may be connotative in some cases. The subject of Extension and Intension may be pur- sued in Hamilton's Lectures on Logic, Lect VIIL j or in Thomson's Laws of Thought, Sections 48 to 52. It is much noticed in Spalding's Logic (Ency- clopaedia Britannica, 8th ^io!^> LESSON VI. THE GROWTH OF LANGUAGE. WORDS, we have seen, become equivocal in at least three different ways by the accidental confusion of different words, by the change of meaning of a word by iti habitual association with other things than its original meaning, and by analogical transfer to objects of a similai nature. We must however consider somewhat more :k>ely certain changes in language which arise out of thf 7WE 1 GROWTH OF LANGUAGE. 45 last cause, and which are in constant progress. We can almost trace in fact the way in which language is created and extended, and the subject is to the logician one of a highly instructive and important character. There are two great and contrary processes which modify language *s follows: 1. Generalisation, by which a name comes to be applied to a wider class of objects than before, so thai the extension of its meaning is increased, and the inten- sion diminished. 2. Specialisation, by which a name comes to be re- stricted to a narrower class, the extension being decrease** and the intension increased. The first change arises in the most obvious manner, from our detecting a resemblance between a new object, which is without a name, and some well-known object To express the resemblance we are instinctively led to apply the old name to the new object. Thus we are well acquainted with ^/ajj, and, if we meet any substance having the same glassy nature and appearance, we shall be apt at once to call it a kind of glass ; should we often meet with this new kind of glass it would probably come to share the name equally with the old and original kind of glass. The word coal has undergone a change of this kind ; ori- ginally it was the name of charked or charred wood, which was the principal kind of fuel used five hundred years ago. As mineral coal came into use it took the name from the former fuel, which it resembled more nearly than any- thing else, but was at first distinguished as sea-coal or pit-coal. Being now far the more common of the two, it has taken the simple name, and we distinguish charred wood as charcoal, Paper has undergone a like change ; originally denoting the papyrus used in the Roman Em- pire, it was transferred to the new writing material made of cotton or linen rags, which was introduced at a quit* 46 THE GROWTH OF LANGUAGE. [LESS uncertain period. The word character is interesting OB account of its logical employment ; the Greek x a P LKT ^f denoted strictly a tool for engraving, but it became trans- ferred by association to the marks or letters engraved with it, and this meaning is still retained by the word when we speak of Greek characters, Arabic characters, i. e. figures IT letters. But inasmuch as objects often have natural marks, signs> or tokens, which may indicate them as well as artificial characters, the name was generalized, and now means any peculiar or distinctive mark or quality by which an object is easily recognised. Changes of this kind are usually effected by no parti- cular person and with no distinct purpose, but by a sort of unconscious instinct in a number of persons using the name. In the language of science, however, changes are often made purposely, and with a clear apprehension of the generalization implied. Thus soap in ordinary life is applied only to a compound of soda or potash with fat ; but chemists have purposely extended the name so as to include any compound of a metallic salt with a fatty substance. Accordingly there are such things as lime-soap and lead-soap, which latter is employed in making common diachylon plaster. Alcohol at first de- noted the product of ordinary fermentation commonly called spirits of wine, but chemists having discovered that many other substances had a theoretical composition closely resembling spirits of wine, the name was adopted for the whole class, and a long enumeration of different kinds of alcohols will be found in Dr Roscoe's lessons on chemistry. The number of known alcohols is likewis subject to indefinite increase by the progress of discovery. Every one of the chemical terms acid, alkali, metal, alloy, earth, ether, oil, gas, salt, may be shown to have under- gone great generalizations. In other sciences there is hardly a less supply of VI.] THE GROWTH OF LANGUAGE. 45 instances. A lens originally meant a lenticular shaped or double convex piece of glass, that being the kind oJ glass most frequently used by opticians. But as glasses of other shapes came to be used along with lenses, the name was extended to concave or even to perfectly flat pieces of glass. The words lever, plane, cone, cylinder, arc, conic section, curve, prism, magnet, pendulum, ray, light, and many others, have been similarly generalized* In common language we may observe that even proper or singular names are often generalized, as when in the time of Cicero a good actor was called a Roscius after an acior of preeminent talent. The name Caesar was adopted by the successor of Julius Caesar as an official name of the Emperor, with which it gradually became synonymous, so that in the present day the Kaisers of Austria and the Czars of Russia both take their title from Caesar. Even the abstract name Caesarism has been formed to express a kind of imperial system as established by Caesar. The celebrated tower built by a king of Egypt on the island of Pharos, at the entrance of the harbour of Alexandria, has caused lighthouses to be called phares in French, and pharos in obsolete English. From the celebrated Roman General Quintus Fabius Maximus any one who avoids bringing a contest to a crisis is said to pursue a Fabian policy. In science also singular names are often extended, as when the fixed stars are called distant suns, or the com- panions of Jupiter are called his moons. It is indeed one theory, and a probable one, that all general names were created by the process of generalization going on in the early ages of human progress. As the comprehension of general notions requires higher intellect than the appre- hension of singular and concrete things, it seems natural that names should at first denote individual objects, and ihould afterwards be extended to classes. We have a 48 THE GROWTH Ofr LANGUAGE. glimpse of this process in the case of the Australian native who had been accustomed to call a large dog Cadli, but when horses were first introduced into the country they adopted this name as the nearest description of a horse A very similar incident is related by Captain Cook of the natives of Otaheite. It may be objected, however, that a certain process of judgment must have been exerted before the suitability of a name to a particular thing could have been perceived, and it may be considered probable that specialization as well as generalization must have acted in the earliest origin of language much as it does at present. Specialization is an exactly opposite process to gene ralization and is almost equally important It consists in narrowing the extension of meaning of a general name, so that it comes to be the name only of an individual or a minor part of the original class. It is thus we are fur- nished with the requisite names for a multitude of new implements, occupations and ideas with which we deal in advancing civilization. The name physician is derived from the Greek et of metaphors ; as, in taking the reins of government, overturning the government, taking the bit between the teeth, the Government Whip, being heavily weighted, &c. No doubt it might be shewn that every other important occupation of life has furnished its corresponding stock of metaphors. It is easy to shew, however, that this process, besides going on consciously at the present day, must have acted throughout the history of language, and that we owe to it almost all, or probably all, the words expressive of re- fined mental or spiritual ideas. The very word spirit, now the most refined and immaterial of ideas, is but the Latin spiritus, a gentle breeze or breathing; and inspiration, esprit, or wit, and many other words, are due to this me- taphor. It is truly curious, however, that almost all the words in different languages denoting mind or soul imply the same analogy to breath. Thus, soul is from the Gothic root denoting a strong wind or storm ; the Latin words animus and anima are supposed to be connected with the Greek avffju>s, wind; ^X 1 ? * s certainly derived from V* to olow 5 *nw/*a, ^ or breath, is used in the New Testament for Spiritual Being ; and our word ghost has been asserted to have a similar origin. Almost all the terms employed in mental philosophy >r metaphysics, to denote actions or phenomena of mind, are ultimately derived from metaphors. Apprehension is the putting forward of the hand to take anything ; com- prehension is the taking of things together in a handful 42 fa THE GROWTH OF LANGUAGE. [LVS\ extension is the spreading out; intention, the bending to-, -xplication, the unfolding; application, the folding toj conception, the taking up together ; relation, the carrying back ; experience is the thoroughly going through a thing ; difference is the carrying apart ; deliberation, the weighing out ; interruption, the breaking between ; proposition, the placing before; intuition, the seeing into; and the list might be almost indefinitely extended. Our English iiame for reason, the understanding, obviously contains some physical metaphor which has not been fully ex- plained ; with the Latin intellect there is also a metaphor. Every sense gives rise to words of refined meaning ; sapience, taste, insipidity, gout, are derived from the sense of taste ; sagacity, from the dog's extraordinary power of smell ; but as the sense of sight is by far the most acute and intellectual, it gives rise to the larger part of lan- guage ; clearness, lucidity, obscurity, haziness, perspicuity, and innumerable other expressions, are derived from this sense. It is truly astonishing to notice the power which lan- guage possesses by the processes of generalization, speci- alization, and metaphor, to create many words from one single root. Prof. Max Miiller has given a remarkable instance of this in the case of the root spec, which means sight, and appears in the Aryan languages, as in the San- scrit spas, the Greek o-KCTrro/iat, with transposition of con- sonants, in the Latin specio, and even in the English spy. The following is an incomplete list of the words deve- loped from this one root ; species, special, especial, speci- men, spice, spicy, specious, speciality, specific, specializa- tion, specie (gold, or silver), spectre, specification, spec tacle, spectator, spectral, spectrum, speculum, specular, speculation. The same root also enters into composi- tion with various prefixes; and we thus obtain a series of words, suspect, aspect, circumspect, expect, inspect, *i.l THE GROWTH OF LANGUAGE. 53 prospect, respect, retrospect, introspection, conspicuous, perspicuity, perspective; with each 6f which, again, a number of derivatives is connected. Thus, from suspect, we derive suspicion, suspicable, suspicious, suspiciously, suspiciousness. I have estimated that there are in aU at least 246 words, employed at some period or other ir the English language which undoubtedly come from the one root spec. J. S. Mill's Logic, Book IV. Chap. i. 'On the Natural History of the Variations in the Meanings of Terms.' Archbishop Trench, On the Study of Words. Max M tiller, Lectures on the Science of Language. LESSON VIL LEIBNITZ ON KNOWLEDGE. IN treating of terms it is necessary that we should clearly understand what a perfect notion of the meaning of a term requires. When a name such as monarch, or civili- zation, or autonomy is used, it refers the mind to some thing or some idea, and we ought if possible to obtain a perfect knowledge of the thing or idea before we use the word; In what does this perfect knowledge consist ? What are its necessary characters? This is a question which the celebrated mathematician and philosopher Leibnitz attempted to answer in a small treatise or tract first published in the year 1684. This tract has been the ^asis of what is given on the subject in several recent Vrks on Logic, and a complete translation of the trad 54 LEIBNITZ ON KNOWLEDGE. \\ has been appended by Mr Baynes to his translation oi the Port Royal Logic. As the remarks of Leibnitz him- self are not always easy to understand, I will not confine myself to his exact words, but will endeavour to give the simplest possible statement of his views, according as chey have been interpreted by Dr Thomson or Sir W, Hamilton. Knowledge is either obscure or clear ; either confused or distinct; either adequate or inadequate; and lastly either symbolical or intuitive. Perfect knowledge must be clear, distinct, adequate and intuitive ; if it fails in any one of these respects it is more or less imperfect. We may, therefore, classify knowledge as in the following scheme Knowledge. Clear. Obscure. Distinct Confused Adequate. Inadequate. Intuitive. Symbolical. Perfect. A notion, that is to say our knowledge of a thing, is obsoure when it does not enable us to recognize the thing again and discriminate it from all other things. We have a clear notion of a rose and of most common flowers because we can recognise them with certainty, and do not confuse them with each other. Also we have a clear notion of any of our intimate friends or persons whom we habitually meet, because we recognise them whenever we see them with the utmost certainty and without hesita- tion. It is said that a shepherd acquires by practice a clear notion of each sheep of his nock, so as to enable him to single out any one separately, and a keeper of II.] LEIBNITZ OA KNOWLEDGE. 55 hounds learns the name and character of each hound, while other persons have only an obscure idea of the nounds generally, and could not discriminate one from the other. But the geologist cannot give a clear idea oi what sandstone, conglomerate, or schist, or slate, or trap rock consists, because different rocks vary infinitely in degree and character, and it is often barely possible tc say whether a rock is sandstone or conglomerate, schist or slate, and so on. In the lower forms of life the natu- ralist hardly has a clear notion of animal life, as distin- guished from vegetable life ; it is often difficult to decide whether a protophyte should be classed with animals or plants. Clear knowledge, again, is confused, when we cannot distinguish the parts and qualities of the thing known, and can only recognise it as a whole. Though any one instantly knows a friend, and could discriminate him from all other persons, yet he would generally find it impos- sible to say how he knows him, or by what marks. He could not describe his figure or features, but in the very roughest manner. A person unpractised in drawing, who attempts to delineate even such a familiar object as a horse or cow, soon finds that he has but a confused notion of its form, while an artist has a distinct idea of the form of every limb. The chemist has a distinct as well as a clear notion of gold ai d silver, for he can not only tell with certainty whether any metal is really gold or silver, but he can specify and describe exactly the qualities by which he knows it ; and could, if necessary, mention a great many other qualities as well We have a very dis- tinct notion of a chess-board, because we know it consists of 64 square spaces; and all our ideas of geometrical figures, such as triangles, circles, parallelograms squares, pentagons, hexagons, &c. are or ought to be perfectly dis- tinct. But when we talk of a constitutional gov****nnt 56 LEIBNITZ ON KNOWLEDGE [LESS or a civilised nation, we have only the vaguest idea oi irnat we mean. We cannot say exactly what is requisite to make a Government constitutional, without including aiso Governments which we do not intend to include; inJ so of civilized nations; these terms have neither dis- tinct nor clear meanings. It is to be remarked that no simple idea, such as that of red colour, can be distinct in the meaning here in- tended, because nobody can analyse red colour, or de- scribe to another person what it is. A person who haj been blind from birth cannot be made to conceive it ; and it is only by bringing an actual red object before the eye that we can define its character. The same is generally true of all simple sensations, whether tastes, smells, co- lours, or sounds; these then may be clearly known, but not distinctly r , in the meaning which Leibnitz gives to this word. To explain the difference which Leibnitz intended to denote by the names adequate and inadequate, is not easy. He says, "When everything which enters into a distinct notion is distinctly known, or when the last ana- lysis is reached, the knowledge is adequate, of which I scarcely know whether a perfect example can be offered the knowledge of numbers, however, approaches near to it" To have adequate knowledge of things, then, we must not only distinguish the parts which make up our notion of a thing, but the parts which make up those parts. For mstance, we might be said to have an adequate notion of a chess-board, because we know it to be made up of 64 squares, and we know each of those squat es distinctly, because each is made up of 4 equal right lines, joined at right angles. Nevertheless, we cannot be said to have a distinct notion of a straight line, because we cannot well define it. or resolve it into anything simpler. To be com. 4.J LEIBNITZ ON KNOWLEDGE. 7 pletely adequate, our knowledge ought to admit of analysis after analysis ad infinitum, so that adequate knowledge would be impossible. But, as Dr Thomson remarks, we may consider any knowledge adequate which carries thf analysis sufficiently far for the purpose in view. A me- chanist, for instance, has adequate knowledge of a ma- chine, if he not only know its several wheels and parts, out the purposes, materials, forms, and actions of those parts ; provided again that he knows all the mechanical properties of the materials, and the geometrical properties of the forms which may influence the working of the machine. But he is not expected to go on still further and explain why iron or wood of a particular quality is strong or brittle, why oil acts as a lubricator, or on what axioms the principles of mechanical forces are founded. Lastly, we must notice the very important distinction of symbolical and intuitive knowledge. From the original meaning of the word, intuitive would denote that which we gain by seeing (Latin, intueor, to look at), and any knowledge which we have directly through the senses, or by immediate communication to the mind, is called intuitive. Thus we may learn intuitively what a square or a hexagon is, but hardly what a chiliagon, or figure of i coo sides, is. We could not tell the difference by sight of a figure of 1000 sides and a figure of 1001 sides. Nor can we imagine any such figure completely before tht mind. It is known to us only by name or symbolically. All large numbers, such as those which state the velocity of light (186,000 miles per second), the distance of the sun (91,000,000 miles), and the like, are known tonis only by symbols, and they are beyond our powers of imagination. Infinity is known in a similar way, so that we can in an intellectual manner become acquainted with that o4 which our senses could never inform us. We speak alto 5 LEIBNITZ ON KNOWLEDGE. [LES& of nothing, of zero, of that which is self-contradictory, of the non-existent, or even of the unthinkable or incon- ceivable. although the words denote what can never bf realized in the mind and still less be perceived through the senses intuitively, but can only be treated in a merely symbolical way. In arithmetic and algebra we are chiefly occupied with symbolical knowledge only, since it is not necessary m working a long arithmetical question or an algebraical problem that we should realise to ourselves at each step the meaning of the numbers and symbols. We learn from algebra that if we multiply together the sum and difference of two quantities we get the difference of the squares ; as, in symbols which is readily seen to be true, as follows a + b a-t -ab-P In the above we act darkly or symbolically, using the letters a and b according to certain fixed rules, without knowing or caring what they mean ; and whatever mean- ing we afterwards give to a and b we may be sure the process holds good, and that the conclusion is true with- out going over the steps again. But in geometry, we argue by intuitive perception oi the truth of each step, because we actually employ a re- presentation in the mind of the figures in question, and satisfy ourselves that the requisite properties are really Dossessed by the figures. Thus the algebraical truth hown above in symbols may be easily proved to hold tru* HL] LEIBNITZ ON KNOWLEDGE. & of lines and rectangles contained under those Hnes, as a corollary of the 5th Prop, of Euclid's Second Book. Much might be said concerning the comparative ad- vantages of the intuitive and symbolical methods. The Utter is usually much the less laborious, and gives tht most widely applicable answers ; but the symbolical sel- dom or never gives the same command and comprehen- sion of the subject as the intuitive method. Hence the study of geometry is always indispensable in education, although the same truths are often more readily proved by algebra. It is the peculiar glory of Newton that he was able to explain the motions of the heavenly bodies by the geometric or intuitive method ; whereas the great- est of his successors, such as Lagrange or Laplace, have treated these motions by the aid of symbols. What is true of mathematical subjects may be applied to all kinds of reasoning ; for words are symbols as much as A, B, C t or x t y, *, and it is possible to argue with words without any consciousness of their meaning. Thus if I say that " selenium is a dyad element, and a dyad element is one capable of replacing two equivalents of hydrogen," no one ignorant of chemistry will be able to attach any meaning to these terms, and yet any one will be able to conclude that " selenium is capable of replacing two equivalents of hydrogen." Such a person argues in a purely symbolical manner. Similarly, whenever in com- mon life we use words, without having in mind at the moment the full and precise meaning of the words, we possess symbolical knowledge only. There is no worse habit for a student or reader to acquire than that of accepting words instead of a know- ledge of things. It is perhaps worse than useless to read a work on natural history about Infusoria, Foraminifera, Rotifera and the like, if these names do not convey cleai images to the mind. Nor can a student who has not 60 LEIBNITZ OX KNOWLEDGE. [LKSi mtnessed experiments, and examined the substances with his own eyes, derive any considerable advantage from works on chemistry and natural philosophy, where he will meet with hundreds of new terms which would be to him mere empty and confusing signs. On this account we should lose no opportunity of acquainting ourselves, by means of our senses, with the forms, properties and changes of things, in order that the language we employ may, as far as possible, be employed Intuitively, and we may be saved from the absurdities and fallacies into which we might otherwise fall. We should observe, in short, the advice of Bacon ipsis consuescere rtbus to accustom ourselves to things themselves. Hamilton's Lectures on Logic. Lect IX Barnes' Port Royal Logic. Part L Chap. 9, and Ap- pendix. LESSON VIIL KINDS OF PROPOSITIONS. A TERM standing alone is not capable of expressing truth; it merely refers the mind to some object or class of objects, about which something may be affirmed or denied, but about which the term itself does not affirm or deny any- thing. "Sun," "air, 7 * "table," suggest to every mind objects of thought, but we cannot say that " sun is true," or " air is mistaken," or " table is false." We must join words or terms into sentences or propositions before they can express those reasoning actions of the mind to which viii. 1 KINDS OF PROPOSITIONS. 61 truth or falsity may be attributed. " The sun is bright," "the air is fresh," "the table is unsteady," are statements which may be true or may be false, but we can certain!) entertain the question of their triith in any circumstances Now as the logical term was defined to be any comb ma tion of words expressing an act of simple apprehension. so a logical proposition is any combination of words expressing an act of judgment. The proposition is in short the result of an act of judgment reduced to the form of language. What the logician calls a proposition the grammarian calls a sentence. But though every proposition is a sen- tence, it is not to be supposed that every sentence is a proposition. There are in fact several kinds of sentences more or less distinct from a proposition, such as a Sen- tence Interrogative or Question, a Sentence Imperative or a Command, a Sentence Optative, which expresses a wish, and an Exclamatory Sentence, which expresses an emotion of wonder or surprise. These kinds of sentence may possibly be reduced, by a more or less indirect mode of expression, to the form of a Sentence Indicative, which is the grammatical name for a proposition ; but until this be done they have no proper place in Logic, or at least no place which logicians have hitherto sufficiently ex- plained. The name proposition is derived from the Latin wordy pro, before, and pono, I place, and means the laying 01 placing before any person the result of an act of judg- ment. Now every act of judgment or comparison must involve the two things brought into comparison, and every proposition will naturally consist of three parts" the two terms or names denoting the things compared, and the copula or verb indicating the connection between them, as it was ascertained in the act of judgment. Thus the proposition, " Gold is a yellow substance," expresses 62 KINDS OF PROPOSITIONS. [LESS, An agreement between gold and certain other substances previously called yellow in regard to their colour. Gold and yellow substance are evidently the two terms, and is the copula. It is always usual to call the first term of a proposl don the subject, since it denotes the underlying matter is it were (Latin, sub, under, and jactum, laid) about which something is asserted. The second term is called the predicate, which simply means that which is affirmed or asserted. This name is derived from the Latin prce- dicare, to assert, whence comes the French name predi- cateur, corrupted into our preacher. This Latin verb is not to be confused with the somewhat similar one pre- dlcere, which has the entirely different meaning to pre- dict or foretell. I much suspect that newspaper writers and others, who pedantically use the verb "to predi. cate," sometimes fall into this confusion, and really mean to predict, but it is in any case desirable that a purely technical term like predicate should not be needlessly introduced into common language, when there are so many other good words which might be used. This and all other technical scientific terms should be kept to their proper scientific use, and the neglect of this rule injures at once the language of common life and the language of science. Propositions are distinguished into two kinds, accord- ing as they make a statement conditionally or uncondi- ionally. Thus the proposition, "If metals are heated they are softened," is conditional, since it does not make an assertion concerning metals generally, but only in the circumstances when they become heated. Any circum- stance which must be granted or supposed before the assertion becomes applicable is a condition. Conditional propositions are of two kinds, Hypothetical and Disjunc- tive, but their consideration will be best deferred to a KINDS OF PROPOSITIONS. 6, subsequent Lesson (xix). Unconditional propositions are those with which we shall for some time be solely concerned, and these are usually called Categorical Pro- positions, from the Greek verb Kanjyopf'w (kategorM, to assert or affirm). The following diagram will conveniently represent the classification of sentences and propositions as far as we have yet proceeded : = Proposition \ * e j" c J" Hypothetical errogative \ Condltlonal \ Disjunctive. Sentence Imperative Optative . Exclamatory It is now necessary to consider carefully the several kinds of categorical propositions. They are classified according to quality and according to quantity. As re- gards quality they are either affirmative or negative ; as regards quantity they are either universal or particular. An affirmative proposition is one which asserts a cer- tain agreement between the subject and predicate, so that the qualities or attributes of the predicate belong to the subject The proposition, " gold is a yellow substance," states such an agreement of gold with other yellow sub- stances, that we know it to have the colour yellow, as well as whatever qualities are implied in the name sub- stance. A negative proposition, on the other hand, as- serts a difference or discrepancy, so that some at least of the qualities of the predicate do not belong to the sub- ject. * Gold is not easily fusible" denies that the qua- lity of being easily fused belongs to gold. Propositions are again divided according to quantity into universal and particular propositions. If the propo- sition affirms the predicate to belong to the whole of the wbject, it is an universal proposition, as in the example OF PROPOSITIONS. [ all metals are elements," which affirms that the quality of being undecomposable or of being simple in nature it true of all metals. But if we say " some metas are brit- tle," the quality of brittleness is affirmed only of some indefinite portion of the metals, and there is nothing in the proposition to make us sure that any certain metal is brittle. The name particular being derived from the diminutive of the Latin pars would naturally signify a small part, but in logic it must be carefully interpreted as signifying any part, from the smallest fraction up to nearly the whole. Particular propositions do not include cases where a predicate is affirmed of the whole or of none of the subject, but they include any between these limits. We may accordingly count among particular propositions all such as the following: A rery few metals are less dense than water. Most elements are metals. Many of the planets are comparatively small bodies. Not a few distinguished men have had distinguished sons. The reader must carefully notice the somewhat subtle point explained further on/:hat the particular proposition though asserting the predrcate only of a part j>f the sub- ject, does not deny it to be true of the whole. I Aristotle, indeed, considered that there were alto- gether four kinds of proposition as regards quantity, namely f Universal Proposition Singular I Indefinite. The singular proposition is one which has a Derm for its subject, as m Socrates was very wise London is a vast city. fill.] KINDS OF PROPOSITIONS. 6l But we may fairly consider that a singular proposition is an universal one ; for it clearly refers to the whole ol the subject, which in this case is a single individual thing. Indefinite or indesignate propositions are those which we devoid of any mark of quantity whatever, so that the form of words gives us no mode of judging whether the predicate is applicable to the whole or only part of the subject. Metals are useful, Comets are subject to the law of gravitation, are indefinite propositions. In reality, however, such propositions have no distinct place in logic at all, and the logician cannot properly treat them until the true and precise meaning is made apparent. The predicate must be true either of the whole or of part of the subject, so that the proposition, as it stands, is clearly incomplete ; but if we attempt to remedy this and supply the marks of quantity, we overstep the proper boundaries of logic and assume ourselves to be acquainted with the subject matter or science of which the proposi- tion treats. We may safely take the preceding examples to mean "some metals are useful" and " all comets are subject to the law of gravitation," but not on logical grounds. Hence we may strike out of logic altogether the class of indefinite propositions, on the understanding ;hat they must be rendered definite before we treat them. may observe, however, that in the following lessons I hallTrequeritly use propositions in the indefinite form as examplesr~bn the understanding that where no sign ol quantity appears, the universal quantity is to be assumed. It is probable that wherever a term is used aj one, it / _ ( ought to be interpreted as meaning the whole of i But however this may be, we need not recognize the inde- finite proposition as a distinct kind ; and singular propo- sitions having been resolved into universals, there remain nly the two kinds, Universal and Particular. Remembering now that there are two kinds of prupv 66 KINDS OF PROPOSITIONS. [LESS ition as regards quality, and two as regards quantity wt hall be able to form altogether four varieties, thus : f Universal (^"native A \Negative B Proposition < \ Negative The vowel letters placed at the right hand are sym- bols or abbreviated names, which are always used to denote the four kinds of proposition; and there will be no difficulty in remembering their meaning if we observe that A and I occur in the Latin verb affirmo, I affirm, and E and in nego, I deny. There will not generally be any difficulty in referring to its proper class any proposition that we meet with in writings. The mark of universality usually consists of some adjective of quantity, such as all, every, each, any, the whole; but whenever the predicate is clearly intended to apply to the whole of the subject we may treat the pro- position as universal. The signs of a particular proposi- tion are the adjectives of quantity, some, certain, a few, many, most, or such others as clearly indicate part ai Uast. The negative proposition is known by the adverbial particle not being joined to the copula ; but in the propo- sition Bj that is the universal negative, we frequently use the particle no or none prefixed to the subject. Thus, " no metals are compound," " none of the ancients were acquainted with the laws of motion," are familiar forms oJ the universal negative. The student must always be prepared too to meet with misleading or ambiguous forms of expression. Thus the proposition, " all the metals are not denser than water," might be taken as B or 0, according as we interpret it tt Via.] KINDS OP PROPOSITIONS. 6v mean "no metals are denser than water," or "not all the metals," &c., the last of course being the true sense The little adjective few is very subject to a subtle am- biguity of this kind ; for if I say "few books are at one* learned and amusing," I may fairly be taken to assert that a few books certainly are so, but what I really mean to draw attention to is my belief that "the greater num ber of books are not at once learned and amusing." A proposition of this kind is generally to be classed rathei as than I. The word some is subject to an exactly similar ambiguity between some but not all, and some at least, it may be all; the latter appears to be the correct interpretation, as shewn in the following lesson (p 79). As propositions are met with in ordinary language they are subject to various inversions and changes of the simple logical form. (1) It is not uncommon, especially in poetry, to find the predicate placed first, for the sake of emphasis 01 variety ; as in " Blessed are the merciful ;" " Comes some- thing down with eventide ;" " Great is Diana of the Ephe- sians." There is usually no difficulty in detecting sucli an inversion of the terms, and the sentence must then be reduced to the regular order before being treated in logic. (2) The subject may sometimes be mistaken for the predicate when it is described by a relative clause, stand- ing at the end of the sentence, as in " no one is free who is enslaved by his appetites." Here free is evidently the predicate, although it stands in the middle of the sentence, and "one who is enslaved by his appetites* is the real subject This proposition is evidently of the form R. Propositions are also expressed in various modes dif- fering from the simple logical order, and some of UM different kinds which arise must be noticed. 68 KINDS OF PROPOSITIONS. [l Exclusive propositions contain some words, such at only, alone, none but, which limit the predicate to the subject Thus, in "elements alone are metals," we arc informed that the predicate "metal" cannot be applied to anything except "elements," but we are not to understand that "all elements are metals." The same meaning is expressed by " none but elements are metals f or, again, by " all that are not elements are not metals f and this we shall see in the next lesson is really equivalent to "all metals are elements." Arguments which appear fallacious at first sight will often be found correct when they con- tain exclusive propositions and these are properly inter- preted. Exceptive propositions affirm a predicate of all the subject with the exception of certain defined cases, to which, as is implied, the predicate does not belong. Thus, " all the planets, except Venus and Mercury, are beyond the earth's orbit," is a proposition evidently equivalent to two, viz. that Venus and Mercury are not beyond the earth's orbit, but that the rest are. If the exceptions are not actually specified by name an exceptive proposi- tion must often be treated as a particular one. For il 1 say " all the planets in our system except one agree with Bode's law," and do not give the name of that one excep- tion, the reader cannot, on the ground of the proposition, assert of any planet positively that it does agree with Bode's law. Some propositions are distinguished as explicative 01 essential, because they merely affirm of their subject a predicate which is known to belong to it by all who can define the subject. Such propositions merely unfold what is already contained in the subject. "A parallelo- gram has four sides and four angles," is an explicative or essential proposition. " London, which is the capital oi England, is the largest city of Europe/' contains two pro- VIIL] KINDS OF PROPOSITIONS. 69 positions ; of which one merely directs our attention to a fact which all may be supposed to know, viz, that London is the capital of England. Ampllative propositions, on the other hand, join a new predicate to the subject Thus to those who do not know the comparative sizes of cities in Europe, the last example contains an ampliative proposition. The greater number of propositions are of this kind. Tautologoua or Truistlo propositions are those which merely affirm the subject of itself, and give no informa- tion whatever; as in, "whatever is, is;" "what I have written, I have written." It is no part of formal Logic to teach us how to inter- pret the meanings of sentences as we meet them in writ- ings ; this is rather the work of the grammarian and philologist Logic treats of the relations of the different propositions, and the inferences which can be drawn from them; but it is nevertheless desirable that the reader should acquire some familiarity with the real logical meaning of conventional or peculiar forms of expression, and a number of examples will be found at the end of the book, which the reader is requested to classify and treat as directed. In addition to the distinctions already noticed it has long been usual to distinguish propositions as they are pore or modal. The pure proposition simply asserts that the predicate does or does not belong to the subject, while the modal proposition states this cum modo, or with an intimation of the mode or manner in which the predicate belongs to the subject The presence of any adverb of time, place, manner, degree, &c^ or any expression equi- valent to an adverb, confers modality on a proposition. "Error is always in haste;" "justice is ever equal;" "a perfect man ought always to be conquering himself," are nples of modal propositions in this acceptation ol 70 KINDS OF PROPOSITIONS. [LEsa the name. Other logicians, however, have adopted a different view, and treat modality as consisting in the decree of certainty or probability with which a judgment is made and asserted. Thus, we may say, " an equilateral triangle is necessarily equiangular ;" " men are generally trustworthy f " a falling barometer probably indicates a coming storm ;" "Aristotle's lost treatises may possibly be recovered ? and all these assertions are made with a dif- ferent degree of certainty or modality. Dr Thomson is no doubt right in holding that the modality does not affect the copula of the proposition, and the subject could only be properly treated in a work on Probable Reason- ing. Many logicians have also divided proposition* ac- cording as they are true or false, and it might well seem to be a distinction of importance. Nevertheless, it is wholly beyond the province of the logician to consider whether a proposition is true or not in itself; all that he has to determine is the comparative truth of propositions that is, whether one proposition is true when another is. Strictly speaking, logic has nothing to do with a pro- position by itself; it is only in converting or transmuting certain propositions into certain others that the work of reasoning consists, and the truth of the conclusion is only so far in question as it follows from the truth of what we shall call the premises. It is the duty of the special sci- ences each in its own sphere to determine what are true propositions and what are false, and logic would be but mother name for the whole of knowledge could it take this duty on itself. See Mr Mill's System of Logic ; Book I. Chap. IT, which generally agrees with what is given above. Chap- ters V. and vi. contain Mr Mill's views on the Nature tad Import of Propositions, which subject may be furthei IX ] THE OPPOSITION OF PROPOSITIONS. 7, studied in Mr Mill's Examination of Sir W. Hamilton'* Philosophy, Chap. xvin. ; Hamilton's Lectures on Lcgic^ No. xiii.; and MansePs Prolegomena Logica, Chap. II.; but the subject is too metaphysical in character to be treated in this work. LESSON IX. THE OPPOSITION OF PROPOSITIONS. WE have ascertained that four distinct kinds of propo- sitions are recognized by logicians, the Universal affirm- ative, the Particular affirmative, the Universal negative, and the Particular negative, commonly indicated by the symbols A, I, E, 0. It is now desirable to compare toge- ther somewhat minutely the meaning and use of proposi- tions of these various kinds, so that we may clearly learn how the truth of one will affect the truth of others, or how the same truth may be thrown into various forms of ex- pression. The proposition A expresses the fact that the thing or el-ass of things denoted by the subject is included in, and forms part of the class of things denoted by the predicate. Thus " all metals are elements" means that metals form a part of the class of elements, but not the whole. Ai there are altogether 63 known elements, of which 48 are metals, we cannot say that all elements are metals. The proposition itself does not tell us anything about element* in general; it is not in fact concerned with elements, metals being the subject about which it gives us informa fa THE OPPOSITION |LKSi don. This is best indicated by a kind of diagram, first osed by the celebrated mathematician Euler, in his lettert to a German princess. In Fig. I, the metals are supposed to be enclosed in the small circle somewhat as sheep Slight be in a pinfold, this circle containing all the metals ind nothing else. The greater circle is supposed to con- tain in a similar manner all the elements and nothing else Now as the small circle is wholly within the larger one, it follows that all the metals must be counted aa Fig... dements, but of the part of the elements outside the circle of metals we know nothing from the proposition. The particular affirmative proposition I exactly resem- bles A in meaning, except that only part of the subject is brought into question. When I say that " some metals are brittle," I mean that of a collection of all the dif- ferent metals a few at least might be picked out which would be found to be brittle ; but the word some is ex- ceedingly indefinite, shewing neither the exact number ol brittle metals, nor how we are to know them from the others, unless indeed by trying whether they are brittle. This proposition will be properly represented in Eider's mode by two intersecting circles, one supposed to enclose all metals, and the other all brittle substances. Th mere fact of the two circles intersecting proves that som* OL] OF PROPOSITIONS. Fig... part of one class must coincide with some part of the other class, which is what the proposition is intended to express. Concerning the portions of the circles which dc not overlap the proposition tells us nothing. The universal negative proposition E denies the ex- istence of any agreement or coincidence between the sub- ject and predicate. Thus from " no metals are compound substances," we learn that no metal is to be found among compound substances, and it follows necessarily that no compound substance can be found among the metals. For were there a compound substance among the metals, there would evidently be one metal at least among the compound substances. This entire separation in thought of the two classes is well shewn in Killer's method by two disconnected circles. Fig. 3- | JMofe ] [ Comfowidt. | The leader will easily see that the proposition 8 f4 THE OPPOSITION distinguished from A and I, by the fact that it gives us some information concerning the whole of the predicate^ because we learn that none of the objects included in the predicate can be found among those included in the sub lect The affirmative propositions, on the other hand, warranted as in holding that the objects denoted by thi subject, or some particular part of them, were included in the predicate, but they give us no warrant for saying that any specified part of the predicate is in the subject Because we merely know that a substance is an element, we do not learn from the proposition " all metals are ele- ments" whether it is a metal or not. And from the pro- position " some metals are brittle," we certainly cannot ascertain whether any particular brittle substance is a netal. We must seek the information from other sources. But from "no metals are compounds" we learn of any compour 1 substance that it is not a metal, as well as of a metal *hat it is not a compound substance. The important difference above explained is expressed in technical language by saying that the proposition E distributes its predicate, whereas the affirmative proposi- tions A and I do not distribute their predicates. By dis- tribution of a term is simply meant taking it universally^ or referring to all parts of it; and as the validity of any argument or syllogism will usually depend upon the suffi- cient distribution of the terms occurring in it, too much mention cannot be paid to this point Judging from the examples we have had, it will be seen that the universal affirmative distributes its subject, bat not its predicate ; for it gives us some information concerning all metals, but not all elements. The parti- cular affirmative distributes neither subject nor predicate; for we do not learn anything from our example concern- 'ng all metals nor concerning all brittle substances. ^ut niversal negative distributes both subject and predi- OF PROPOSITIONS. 7$ cate, for we learn something of all metals and also of *A compound substances. The particular negative proposition will be found to distribute its predicate, but not its subject. When I say some metals are not brittle, I intentionally refer only to a part of the metals, and exclude them from the class of brittle substances ; but I cannot help at the same time referring to the whole of the brittle substances. If the metals in question coincided with any part of the brittle substances they could not be said to be excluded from the class. To exclude a thing from any space, as from a particular chamber of a house, it must not merely be removed from some part, but from any part, or from the whole of that space or chamber. Ruler's diagram for this proposition may be constructed in the same manner as for the proposition I as follows : Fig. 4. It is apparent that though part of the metals fall into the circle of brittle substances, yet the remaining portion are excluded from any part of the predicate. We may state the result at which we have now arrived in the following form : (Universal \ Affirmative A. ( Negative E. Particular i Affirmative * j Negative 0. Subject. Distributed Distributed Undistributed. Undistributed Predicate. Undistributed Distributed Undistributed Distributed 76 THE OPPOSITION [LESS We shall now discover with great ease the relations oi the four propositions, each to each, that is to say, the way m which they are opposed to each other. It is obvious that the truth of one proposition interferes more or less completely with the truth of another proposition having the same subject and predicate. If " all metals are ele- ments," it is impossible that "some metals are not ele- ments," and still more palpably impossible, so to say, that '* no metals should be elements." The proposition A, then, is inconsistent with both E and ; and, vice versd, E and are inconsistent with A. Similarly, B is inconsistent with A and X. But this important difference must be noted, that if A be false, is necessarily true, but E may or may not be true. If it is not true that " all men are sincere," it follows that " some men are not sincere," but it does not in the least follow that " no men are sincere," This difference is expressed by saying that A and are con- tradictory propositions, whereas A and B are called oon- fcrmzy propositions. It is plain that A and B, as in " all men are sincere" and "no men are sincere," represent the utmost possible contrariety of circumstances. ID order to prove the falsity of A, it is sufficient to establish the truth of 0, and it is superfluous, even if possible, to prove E ; similarly B is disproved by proving I, and il is superfluous to prove A. Any person who asserts a uni- versal proposition, either A or B, lays himself under the necessity of explaining away or disproving every single exception brought agaii t it An opponent may always restrict himself to the uch easier task of finding in- stances which apparent or truly contradict the univer- sality of the statement, \t if he takes upon himself tc firm the direct contrary, be is himself open to easy at- tack. Were it to be asserted, for instance, that "All Christians are more moral than Pagans," it would be asy to adduce examples showing that " Some Christians nt] of PROPOSITIONS. n are not more moral than Pagans," but it would be absurd to suppose that it would be necessary to go to the con- trary extreme, and shew that " No Christians are mow moral than Pagans." In short A is sufficiently and best disproved by 0, and E by I. It will be easily apparent that, vict versa, is disproved by A, and I by E ; nor is there, indeed, any other mode at all of disproving these particular propositions. When we compare together the propositions I and we find that they are in a certain sense contrary in na- ture, one being affirmative and the other negative, but that they are still consistent with each other. It is true both that " Some metals are brittle," for instance Anti- mony, Bismuth and Arsenic ; but it is also true that " Some metals are not brittle." And the reader will ob- serve that when I affirm " Some metals are elements," there is nothing in this to prevent the truth of " Some metals are not elements," although on other grounds we know that this is not true. The propositions I and are called subcontraries each of the other, the name con- noting a less degree of contrariety than exists between A and E. As regards the relation of A to I and B to 0, it is plain that the truth of the universal includes and necessitates the truth of the particular What may be affirmed or denied of all parts of a class may certainly be affirmed or denied similarly of some part of the class. From the truth of the particular we have no right to infer either the truth or falsity of the universal of the same quality. These pairs of propositions are called subalterns, i.e. one under the other (Latin sub under, and alter the other af two), or we may say more exactly that I and are respectively the subaltcrnates of A and B, each of which is a subalternans. THE OPPOSITION [Li Thlp relations of the propositions just described 11 clearly shown in the following scheme : Contraries ............... B rv I ............ Subcontraries It is so highly important to apprehend completely and readily the consistency or opposition of propositions, that I will put the matter in another form. Taking any two propositions having the same subject and predicate, they must come under one of the following statements : 1. Of contradictory propositions, one must be true and one false. 2. Of contrary propositions, both cannot be true, and both may be false. 3. Of subcontrary propositions, one only can be false, and both may be true. 4. Of subalterns, the particular is true if the universal be true ; but the universal may or may not be true when the particular is true. I put the same matter in yet another form in the fol- lowing table, which shows how the truth of each of A, I, and 0, affects the truth of each of the ethers. DL] OF PROPOSITIONS. A E I is is is is If A be true true false true false, B false true false true. I doubtful false true doubtful false doubtful doubtful true. It will be evident that from the affirmation of univer sals more information is derived than from the affirmation of particulars. It follows that more information can b derived from the denial of particulars than from the denial of universals, that is to say, there are less cases left doubtful, as in the above table. The reader may well be cautioned, however, against an ambiguity which has misled some even of the most eminent logicians. In particular propositions the adjec tive some is to be carefully interpreted as some, and there may or may not be more or all. Were we to interpret it as some, not more nor all, then it would really give to the proposition the force of I and combined. If I say " some men are sincere," I must not be taken as implying that " some men are not sincere ;" I must be understood to predicate sincerity of some men, leaving the character of the remainder wholly unaffected. It follows from this that, when I deny the truth of a particular, I must not be understood as implying the truth of the universal of the same quality. To deny the truth of " some men are mor- tal" might seem very natural, on the ground that not some but All men are mortal ; but then the proposition denied would really be some men are not mortal, i. e. not I. Hence when I deny that " some men are immortal" I mean that "no men are immortal ;" and when I deny that "some men are not mortal," I mean that "all men are mortal." It has long been usual to compare propositions af So OPPOSITION OF PROPOSITIONS. [LESS. IX regards the quality of the subject matter to which the) refer, and what is technically called the matter was dis- tinguished into three kinds, necessary, contingent, and im possible. Necessary matter consists of any subject it which the proposition A may be affirmed ; impossible ii which B may be affirmed. Any subject or branch of know- ledge in which universal statements cannot usually be made is called contingent matter, and it implies the truth of I and O. Thus "comets are subject to gravitation," though an indefinite or indesignate proposition v p. 65), may be interpreted as A, because it refers to a part of natural science where such general laws obtain. But "men are sincere" would be properly interpreted as par- ticular or X, because the matter is clearly contingent. The truth of the following statements is evident. In necessary matter A and I are true ; E and false. In contingent matter I and O are true ; A and E false. Inimpossible matter E and are true ; A and I false. In reality, however, this part of logical doctrine is thoroughly illogical, because in treating a proposition we have no right, as already explained (p. 70), to assume ourselves acquainted with the science to which it refers. Our duty is to elicit the exact consequences of any state- ments given to us. We must learn in logic to transform information in every possible way, but not to add extra- oms facts. LESSON X, CONVERSION OF PROPOSITIONS, AND IMMEDIATE INFERENCE. WE are said to infer whenever we draw one truth from another truth, or pass from one proposition to another. As Sir W. Hamilton says, Inference is "the carrying out into the last proposition what was virtually contained in the antecedent judgments." The true sphere of the science oj" logic indeed is to teach the principles on which this act of inference must be per- formed, and all the ^previous consideration of terms and propositions is only useful or pertinent so far as it assists us to understand the processes of inference. We have to consider in succession all the modes in which the same information may be moulded into differ- ent forms of expression often implying results of an apparently different character. Logicians are not agreed exactly as to what we may include under the name Inference, and what we should not. All would allow that there is an act of inference when we see drops oi abater on the ground and believe that it has rained. This is a somewhat complicated act of inference, which are shall consider in later lessons under the subject ol Induction. Few or none would say that there is an act of inference in passing from " The Duke of Cambridge is the Commander-in-chief," to "The Commander-in- chief is the Duke of Cambridge." But without paying much regard to the name of the process I shall in this la CONVERSION OF PROPOSITIONS, [LKS8 lesson point out all the ways in which we can from a ingle proposition of the forms A, E, I or 0, pass to anothei proposition. We are said to convert a proposition when we transpose its subject and predicate ; but in order thai the converse or converted proposition shall be inferred from the convertend, or tkat which was to be converted, we must observe two rules (i) the quality of the^pro- position f affirmative or negative^ must be preserved^ and (2) no term must be .distributed in the Converse unless it was atstr^igdin the Convertend. If in "all metals are elements" we were simply to transpose the terms, thus " all elements are metals," we imply a certain knowledge about all elements, whereas it has been clearly shewn that the predicate of A is un- distributed, and that the convertend does not really give us any information concerning all elements. All that we can infer is that "some elements are metals;" this converse proposition agrees with the rule, and the pro- cess by which we thus pass from A to I is called Con- vorsion by Lim^^on, or Per accidens. When the converse is a proposition of exactly the same form as the convertend the process is called simple MinvraHmi Thus from "some metals are brittle sub- stances" I can infer "some brittle substances are metals," as all the terms are here undistributed. Thus I is simply converted into I. Again, from " no metals are compounds," I can pass directly to "no compounds are metals," because these propositions are both in B, and all the terms are there- fore distributed. Euler's diagram (p. 73, Fig. 3) clearly shows, that if all the metals are separated from all the compounds, all the compounds are necessarily separate* from all the metals. The proposition is then simply converted into & *.] AND IMMEDIATE INFERENCE. S| But in attempting to convert the proposition w encounter a peculiar difficulty, because its subject is un- distributed; and yet the subject should become by con- version the predicate of a negative proposition, which distributes its predicate. Take for example the propo- sition, "some existing things are not material substances." By direct conversion this would become "all material substances are not existing things f which is evidently absurd. The fallacy arises from existing things being distributed in the converse, whereas it is particular in the convertend ; and the rules of the Aristotelian logic prevent us from inserting the sign of particular quantity before the predicate. The converse would be equally untrue and fallacious were we to make the subject par- ticular, as in " some material substances are not exist- ing things." We must conclude, then, that the propo- sition cannot be treated either by simple conversion or conversion by limitation. It is requisite to apply a new process, which may be called Conversion by Negation, and which consists in first changing ^Ke convertend into an affirmative proposition, and then converting it simply. If we attach the negation to the predicate instead of to the copula, the proposition becomes "some exist- ing things are immaterial substances," and, converting simply, we have "some immaterial substances are ex- isting things," which may truly be inferred from the con- rertend. The proposition 0, then, is only to be converted by this exceptional method of negation. Another process of conversion can be applied to the proposition A, and is known as conversion by contra- position. From " all metals are elements," it neces- sarily follows that "all not-elements are not metals.* If this be not at the first moment~apparent, a little re- flection will render it so, and from fig. 5 we see that if ill the metals be among the elements, whatever is not ele* 6-a 4 CONVERSION OF PROPOSITIONS, [LK ment, or outside the circle of elements, must also be outside the circle of metals. We may also prove the troth Fig. 5- f the contrapositive proposition in this way, if we may anticipate the contents of Lesson xxin.: If what is not element should be metal, then it must be an element by the original proposition, or it must be at once an ele- ment and not an element ; which is impossible accord- ing to the Primary Laws of Thought (Lesson XIV.), since nothing can both have and not have the same property. It follows that what is not-element must be not-metal Mistakes may readily be committed in contrapositive conversion, from a cause which will be more apparent in Lesson XXII. We are very liable to infer from a pro- position of the form "all metals are elements," that all Hot-metals are not-tUments, which is not only a false statement in itself, but is not in the least warranted by the original proposition. In fig. 5, it is apparent that because a thing lies outside the circle of metals, it does not necessarily lie outside the circle of elements, which is wider than that of metals. Nevertheless the mistake is often made in common life, and the reader will do well to remember that the process of conversion by contra- position consists only in taking the negative of the pre- dicate of the proposition A, as a new subject, and affirm- ing of it universally the negative of the old subject. t} AND IMMEDIATE INFERENCE. 83 Contrapositive conversion cannot be applied to UM particular propositions I and at all, nor to the propo- sition E, in that form ; but we may change B into A by attaching the negation to the predicate, and then the process can be applied. Thus "no men are perfect" may be changed into "all men are not-perfect," Le. 'are imperfect," and then we infer by contraposition " all not-imperfect beings are not-men." But not-im- perfect is really the same as perfect, so that our new proposition is really equivalent to " all perfect beings are not men," or " no perfect beings are men," (B) the sim- ple converse of the original proposition. There remain to be described certain deductions which may be drawn from a proposition without convert- ing its terms. They may be called immediate inferences, and have been very clearly described by Archbishop Thomson in his "Outline of the Necessary Laws of Thought "(pp. 156, &c.). Immediate Inference by Privative Conception consists in passing from any affirmative proposition to a negative proposition implied in it, or equivalent to it, or vice versa, in passing from a negative proposition to its correspond- ing affirmative. The following table contains a proposition of each kind changed by privative conception into an equivalent proposition : {A all metals are elements. B no metals are compounds. {E no men are perfect A all men are imperfect 1 1 some men are trustworthy. (0 some men are not untrustworthy. JO some men are not trustworthy. (1 some men are untrustworthy. The truth of any of the above can be clearly illustrated 16 CONVERSION OF PROPOSITIONS^ [LB8* by diagrams ; thus it will be apparent that if the whole circle of metals lies inside the circle of elements, no part can lie outside of that circle or among the compounds. Any of the above propositions may be converted, but the results will generally be such as we have already ob- tained. Thus the simple converse of " no metals are compounds" is " no compounds are metals," or " no not- elements are metals," the contrapositive of " all .metals are elements." From the last example we get also by simple conversion " some untrustworthy beings are men, 11 which is obviously the converse by negation, as before explained. Applying this kind of conversion to " some men are not untrustworthy," we have " some not-untrust- worthy beings are men." Lastly, from "all men arc imperfect" we may obtain through conversion by limita- tion, " some imperfect beings are men." Immediate Inference by added determinants consists in joining some adjective or similar qualification both to tne subject and predicate of a proposition, so as to ren- der the meaning of each term narrower or better detei'- mined. Provided that no other alteration is made the truth of the new proposition necessarily follows from the truth of the original in almost all cases. From "all metals are elements," we may thus infer that "all very heavy metals are very heavy elements." From " a comet is a material body" we infer " a visible comet is a visible material body." But if we apply this kind of inference too boldly we may meet with fallacious and absurd results. Thus, from "all kings are men," we might infer " all incompetent kings are incompetent men ;" but it does not at all follow that those who art incompetent as kings would be incompetent in othei positions. In this case and many others the qualifying adjective is liable to bear different meanings in the sub- ject and predicate ; but the inference will only be true a/ X.] AND IMMEDIATE INFERENCE. 9) necessity when the meaning is exactly the same In each case. With comparative terms this kind of inference prill seldom be applicable; thus from "a cottage is a building," we cannot infer "a huge cottage is a hug* building," since a cottage may be large when compared irith other cottages, but not with buildings generally. Immediate Inference by Complex Conception is closely iimilar to the last, and consists in employing the subject and predicate of a proposition as parts of a more com- plex conception. From " all metals are elements," I can pass to " a mixture of metals is a mixture of elements." From " a horse is a quadruped" I infer " the skeleton of a horse is the skeleton of a quadruped." But here again the reader must beware of applying the process where the new complex conception has a different meaning in the subject and predicate. Thus, from " all Protestants are Christians," it does not follow that "a majority of Protestants are a majority of Christians," nor that "the most excellent of the Protestants is the most excellent of the Christians." The student is recommended to render himself fami- liar with all the transformations of propositions, or im- mediate inferences described in this lesson ; and copious examples are furnished for the purpose. It is a good exercise to throw the same proposition through a series of changes, so that it comes out in its original form at last, and thus proves the truth of all the intermediate changes ; but should conversion by limitation have beec tued, the original universal proposition cannot be re- gained, but only the particular proposition corresponding io it On Immediate Inference, Archbishop Thomsons Outline of the Laws of Thought, 85 93. LESSON XL LOGICAL ANALYSIS OF SENTENCE* PROPOSITIONS as they are usually to be found in writ ten or spoken compositions seldom exhibit the simple form, the conjunction of a subject, copula, and predicate, whirh we have seen to be the proper logical construction. Not only is the copula often confused with the predicate, but several propositions may be combined into one gram matical sentence. For a full account of the analysis of sentences I shall refer to several excellent little works devoted to the subject ; but I will here attempt to give a sketch of the various ways in which a sentence may be constructed. So often is the copula united to the predicate in oidinary language, that the grammarian treats the propo- sition as composed of only two parts, the subject and predicate, or verb. Thus the proposition, "The sun rises," apparently contains nothing but a subject "the sun," and a predicate "rises;" but the proposition is really equivalent to "the sun is rising," in which the copula is distinctly shown. We shall, therefore, con- sider the verb or grammatical predicate as containing both copula and logical predicate. In Latin one single word may combine all the three parts of Che proposition, as in turn* " I am ? and the celebrated exclamation of Caesar Vtni, vuK, vici, " I came, I saw, I conquered," contain! three distinct and complete propositions in three words, These peculiar cases only arise, however, from the para if the proposition having been blended together and dis- LESS. XI.] ANALYSIS OF SENTENCES. 89 guised in one word ; and in the Latin sum, the letter m is a relic of the pronoun me, which is the real subject erf the proposition. If we had a perfect acquaintance with the Grammar of any language it would probably not con- tradict the logical view of a sentence, but would perhaps sxplain how the several parts of the complete proposition *ad become blended and apparently lost, just as the words will and not are blended in the colloquial " I wont.* 5 A grammatical sentence may contain any number of distinct propositions, which admit of being separated but which are combined together for the sake of brevity. .In the sentence, "Art is long and Time is fleeting," there are two distinct subjects, Art and Time, and two predicates, "long" and "fleeting," so that we have simply two propositions connected by the conjunction and. We may have however several distinct subjects with one and the same predicate ; as in " Thirty days hath September, April, June, and November. " In this well-known couplet the predicate " having thirty days " is placed first for the sake of emphasis, and there are four subjects, September, April, &c., of each of which it is affirmed. Hence these lines really contain four distinct propositions. Again , there may be one subject with a plurality of predicates, so that several different propositions are as- serted without the repetition of the subject and copula. Thus the sentence "Nitrogen is a colourless, tasteless, inodorous gas, lightly lighter than air," contains one subject only, Ni* bvgen, but four or five predicates ; it is plainly equiva- lent to "Nitrogen is colourless," "Nitrogen is tasteless," 11 Nitrogen is a gas," and so on. Lastly, we may have several subjects and 90 LOGICAL ANALYSIS [L predicates all combined in the same sentence, and with only one copula, so that each predicate is asserted ol each subject ; and a great number of distinct propositions are condensed into one brief sentence. Thus in the sen- tence, " Iron, Copper, Lead and Zinc are abundant, cheap and useful metals,*" we have evidently four subjects, and we may be said to have four predicates, "abundant," 44 cheap," "useful," and "metal" As there is nothing to prevent our applying each predicate to each subject the sentence really contains 16 distinct propositions in only 1 1 words ; thus " Iron is abundant," " Iron is cheap,' "Copper is abundant," "Copper is cheap," and soon. In the curious sentence, '* Hearts, tongues, figures, scribes, bards, poets, can- not think, speak, cast, write, sing, number, his love to Antony*," Shakspeare has united six subjects and six predicates, or verbs, so that there are, strictly speaking, six times six or thirty-six propositions. In all the cases above noticed the sentence is said to be compound, and the distinct propositions combined together are said to be coordinate with each other, that is of the same order or kind, because they do not depend upon each other, or in any way affect each other's truth. The abundance, cheapness, or utility of iron need not be stated in the same sentence with the qualities of cop- per, lead or zinc ; but as the predicates happen to be the same, considerable trouble in speaking or writing is saved by putting as many subjects as possible to the same set of predicates. It is truly said that brevity ts the soul of wit, and one of the great arts of compo- sition consists in condensing as many statements as possible into the fewest words, so long as the meaning ii tot confused thereby. Antony an* CUopatra, Act IIL Sc. t. M.] OF SENTENCES. 91 Propositions are however combined in a totally dif ferent manner when one proposition forms a part of the subject or predicate of the other. Thus in the sen- tence, "The man who is upright need not feai accusa- tion," there are two verbs, and two propositions, but one of these only describes the subject of the other; "who is upright " evidently restricts the application of the pre- dicate " need not fear accusation " to a part of the class " man. " The meaning of the whole sentence might be expressed in the form " The upright man need not fear accusation. " And it is clearly seen that the clause or apparent propo- sition is substituted for an adjective. Such a clause or proposition is called subordinate, because it merely as- sists in the formation of the principal sentence, and has no meaning apart from it ; and any sentence containing a subordinate clause is said to be complex. Almost any part of a sentence may thus be replaced by a subordinate clause. Thus in "Oxygen and Nitrogen are the gases which form the largest part of the atmosphere," there is a subordinate clause making part of the predicate, and the meaning might be expressed nearly as well in this way, " Oxygen and Nitrogen are the gases forming the largest part of the atmosphere." In the case of a modal proposition (see p. 69), or one which states the manner in which the predicate belongs to the subject, the mode may be expressed either by an adverb, or by a subordinate clause. "As a man lives so he dies" is such a proposition; for it means, "a man dies as he lives," and "as he lives" is equivalent to an adverb ; if he lives well, he dies well ; if he lives badly, he lies badly. Adverbs or adverbial clauses may also specify the time, place, or any other circumstance con ccrned in the truth of the main proposition. Assuming the reader to be acquainted with the gram 92 LOGICAL ANALYSTS [LES& matical terms used, we may thus state the parts of whicl the most complex sentence must consist The antject may consist of I A noun ; as in " The Queen reigns." 2. A pronoun ; as in " She reigns." 3. An adjective converted into a noun ; as in " H'kth are civilized." 4. A gerund ; as " Seeing is believing " 5. An infinitive ; as " To see is to believe." 6. A subordinate clause ; as " Who falls from virtut is lost" The subject may be qualified or restricted by combin ing with it an attribute which may be expressed in any ol the following ways : 1. An adjective; as, "Fresh air is wholesome." 2. A participle ; as " Falling stars are often seen." 3. A noun used as an adjective ; as " Iron ships are now much employed." 4. A noun and preposition ; as "ships of iron are now much employed." 5. A possessive case; as " Chathants son was the great minister Pitt." 6. A noun in apposition ; as " The Metropolis London is the most populous of cities." 7. A gerund or dative infinitive ; as, " The desire to ?o abroad is common in Englishmen." The predicate consists almost always of a verb, which often has some object or qualifying words; thus it may be 1. A simple tense of a complete verb ; as " The sun rite" 2. A compound tense ; as " The sun has risen? \. An incomplete verb and complement; as "Thf M* appMrs rough." W.J OF SENTENCES. 93 4. The verb "to be" and an adjective: as "Time i. fluting? 5. A verb with an object ; as " Warmth melts ice? 6. A verb with an adverbial; as "The snow fall* tkukly? The object of a verb is usually a noun or pronoun, but any other of the six kinds of expressions which may serve as a subject may also serve as an object. The adverbial qualifying a verb and expressing the manner, time, place, or other circumstance affecting the proposition may be 1. An adverb ; as " The days pass slowly." 2. A noun and preposition ; as " The resolution was passed by a large majority" 3. An absolute phrase ; as " The snow melts, the sun having risen." 4. A dative infinitive ; as " She stoops to conquer? 5. Any phrase equivalent to an adverb ; as " The divi- dends are paid twice a year." Various modes of exhibiting the construction of sen tences by symbols and names for the several parts have been invented ; but I belieye that by far the simplest and most efficient mode is to exhibit the construction in the form of a diagram. Any two or more parts of a. sentence >vhich are co-ordinate with each other, or bear the sarm relation to any other part, are written alongside eact jther, and coupled together by a bracket; thus the du 41 am, Iron i ( abundant, Copper I I cheap, Lead ( "* | useful Zinc I I metals, shows that there art- four co-ordinate subject* 94 LOGICAL ANALYSIS [Lisa and four coordinate predicates in the example previous!) taken, Whenever one part of a sentence is subordinate to another part it may be connected with it by a line drawi A any convenient direction. Thus the analysis of the Allowing sentence is readily shown by the diagram belov t : " No one who is a lover of money, a lover of pleasure, and a lover of glory, is likewise a lover of mankind ; but oaly he who is a lover of virtue." t a lover of money, who is / a lover of pleasure, | ( a lover of glory. one is not j a lovef of mankin ^ he only is ) I who is a lover of virtue. ?Ve see that the sentence is both compound and com- plex, that is to say it contains two principal coordinate propositions with a common predicate, " a lover of man- kind" The first proposition is negative and its subject is described by three subordinate clauses, while the second proposition is affirmative and has one subordinate clause. I conclude this somewhat lengthy lesson with the analysis of a few sentences, of which the first consist! rf some remarkably complex lines from a poem of Bur bidge: " He who metes, as we should mete, Could we His insight use, shall most approve, Not that which fills most space in earthly eyes, But what though Time scarce note it as he ; Fills, like this little daisy at my feet, Its function best of diligence in love." ,] OF SENTENCES. gft which fills most space in earthly eyt* ( not that He shaU most approve } bu( wfaa 1 | , - - - who metes its function of like this littlf as we should mete diligence in daisy at my | love feet, mid we His insight use. ^h fime scan* aote it as he flies. " Most sweet it is with unuplifted eyes To pace the ground, if path there be or none, While a fair region round the traveller lies Which he forbears again to look upon ; Pleased rather with some soft ideal scene, The work of fancy, or some happy tone Of meditation slipping in between, The beauty coming, and the beauty gone." WORDSWORTH It is most sweet f To pace the ground _ with unuplifted if path while a fair region ^ es there j be round the I ( or none traveller lies r&ch (region) he (the traveller) forbears to look upon {some soft ideal scene the work of fancy or some happy tone of meditation slipping in between the beauty coming and the beauty gone. ID the above sentence there is evidently one rubjetf 96 LOGICAL ANALYSIS [LESS. * to pace the ground," which by means ol the pronoun it is connected with the predicate most sweet. The main part ot the sentence however consists of three adverbials, expressing the manner and surrounding circumstances and the third adverbial is developed in a very complicated manner. The sentence is not compound, but is complei on account of four subordinate propositions. In the following sentence there is strictly but on< principal proposition, " We find," but this is only a mode of introducing the true purport of the sentence, " the two classes of intellectual operations have much that is differ ent, much that is common." ft When the notions with which men are conversant in the common course of life, which give meaning to theii familiar language and which give employment to theii hourly thoughts, are compared with the ideas on which exact science is founded, we find, that the two classes of intellectual operations have much that is different, much that is common." we find that the two classes (* f) I of intellectual j much that is different operations have ( much that is common When the notions * are compared . with which which give which give with the ideas f men are meaning employ- j conversant to their ment to on w ^ch in the familiar their hourly exact science is common language thoughts founded, course of life Here the two classes form a collective term, and hare two coordinate predicates rendering the sentence so far a compound one. The greater part of the sentence, how- ever, consists of a complicated subordinate sentence of XL] OF SENTENCES. 97 the nature of an adverbial, expressing the time or occa- sion when this is found to be the case. As a last example we take the sentence given below: "The law of gravitation, the most universal truth at which human reason has yet arrived, expresses not merely the general fact of the mutual attraction of all matter; not merely the vague statement that its influence decreases as the distance increases, but the exact numerical rate at which that decrease takes place ; so that when its amount is known at any one distance it may be exactly calculated for any other." at which human reason has yet arrired the most universal truth The law of gravitation expresses not merely the general fact of the mutual attraction of all matter not merely the vague statement that its influence decreases as the distance increases butt! numei at whi decrea pk te exact ical rate ch that se takes tee BO that its amount may be calculated for any other dis- | [tancc when it is known at any one distance. W. S. Dalgleish's Grammatical Analysis, or J. D. Morell's Analysis of Sentences. Alex. Bain's English Composition and RM*- torie, pp. 91 1 17, treats of construction of sentences. LESSON XII. THE PREDICABLES, DIVISION, AND DEFINITION IT is desirable that the reader, before proceeding further, should acquire an exact comprehension of the meaning of certain logical terms which are known as the Predicables, meaning the kinds of terms or attributes which can always be predicated of any subject These terms are five in number ; genus, species, difference, property, and acci- dent ; and when properly employed are of exceeding use and importance in logical science. It would neither be possible nor desirable in this work to attempt to give any idea of the various and subtle meanings which have been attributed to the predicables by ancient writers, and the most simple and useful view of the subject is what alone can be given here. Any class of things may be called a genus (Greek yW, race or kind), if it be regarded as made up of two or more species. " Element" is a genus when we con- sider it as divided into the two species "metallic and non-metallic." Triangle is a genus as regards the species acute-angled, right-angled, and obtuse-angled. On the other hand, a species is any class which is re- garded as forming part of the next larger class, so that the terms genus and species are relative to each othei, the genus being the larger class which is divided, and the species the two or more smaller classes into which the genus is divided. It is indispensable, however, to regard these expres- sions in the double meaning of extension and intension LESS, xii ] THE PREDICABLES, ETC. & From the explanation of these different meanings in Lesson V. it will be apparent that the extent of a genus or species is simply the number of individuals included in it, and there will always be fewer individuals in the species than in the genus. In extent the genus book in- cludes all books of whatever size, language, or contents j if divided in respect to size the species of book are folio, quarto, octavo, duodecimo, &c. ; and, of course, each oi these species contains much fewer individual books than the whole genus. In Intension the genus means, not the individual things contained in it, but the sum of the qualities com- mon to all those things, and sufficient to mark them out clearly from other classes. The species similarly means the sum of the qualities common to all the individuals forming part of the genus, and sufficient to mark them out from the rest of the genus, as well as from all other things. It is evident, therefore, that there must be more qualities implied in the meaning of the species than of the genus, for the species must contain all the qualities of the genus, as well as a certain additional quality or qualities by which the several species are distinguished from each other. Now these additional qualities form the difference, which may be defined as the quality or sum of qualities which mark out one part of a genus from the other part ot parts. The difference (Latin differentia, Greek flta- < which belongs to the whole of the genus, from the Specific Property, which belongs to the whole of a lowest species. Lastly, an accident (Latin atcidens, Greek break the following rule. 3. The definition must be exactly equivalent to tlu species defined, that is to say, it must be an expression the denotation of which is neither narrower nor wider than the species, so as to include exactly the same objects. The definition, in short, must denote the species, the whole species, and nothing but the species, and this may really be considered a description of what a definition is. 4. A definition must not be expressed in obscurejigura- tive or ambiguous language. In other words, the terms employed in the definition must be all exactly known, otherwise the purpose of the definition, to make us ac- quainted with the sufficient marks of the species, is obviously defeated. There is no worse logical fault than to define ignotum per ignotius, the unknown by the still more unknown. Aristotle's definition of the soul as * The Entelechy, or first form of an organized body which has potential life,' certainly seems subject to this objection. 5. And lastly, A definition must not be negative when U can be affirmative. This rule however is often not applicable, and is by no means always binding. Read Mr M ifl on the nature of Classification and th< ML] AND DEFINITION. in five Predicables, System of Logic, Book I. Chap VII. For ancient Scholastic Views concerning De finition, see Mansel's Artis Logica Rudiment* (Aldrich), App. Note C. LESSON XIII. PASCAL AND DESCARTES ON METHOD. IT may be doubted whether any man ever possessed a more acute and perfect intellect than that of Blaise Pascal He was born in 1623, at Clermont in Auvergne, and from his earliest years displayed signs of a remark- able character. His father attempted at first to prevent his studying geometry, but such was Pascal's genius and love of this science, that, by the age of twelve, he had found out many of the propositions of Euclid's first book without the aid of any person or treatise. It is difficult to say whether he is most to be admired for his mathe- matical discoveries, his invention of the first calculating machine, his wonderful Provincial Letters written against the Jesuits, or for his profound Pense*es or Thoughts, a collection of his reflections on scientific and religious topics. Among these Thoughts is to be found a remarkable fragment upon Logical method, the substance of which is also given in the Port Royal Logic. It forms the second article of the Pensees, and is entitled Reflexions sur la Geometric en general. As I know no composition in which perfection of truth and clearness of expression are more nearly attained, I propose to give in this lesson a free translation of the more important parts of this 112 PASCAL AND DESCARTES (.LESS fragment, appending to it rules of method from tht Port Royal Logic and from Descartes' celebrated Essaj on Method. The words of Pascal are nearly as follows. "The true method, which would furnish demonstrar tions of the highest excellence, if it were possible tc employ the method fully, consists in observing two prin- cipal rules. The first rule is not to employ any term of which we have not clearly explained the meaning; the second rule is never to put forward any proposition which we cannot demonstrate by truths already known ; that is to say, in a word, to define all the terms, and to prove aH thf propositions. But, in order that I may observe the rules of the method which I am explaining, it is neces- sary that I declare what is to be understood by Definition, "We recognise in Geometry only those definitions which logicians call Nominal Definitions, that is to say, only those definitions which impose a name upon things clearly designated in terms perfectly known ; and I speak only of those definitions." Their value and use is to clear and abbreviate dis- course by "expressing in the single name which we impose what could not be otherwise expressed but in several words ; provided nevertheless that the name im- posed remain divested of any other meaning which it might possess, so as to bear that alone for which we intend it to stand. "For example, if we need to distinguish among numbers those which are divisible into two equal parts, from those which are not so divisible, in order to avoid the frequent repetition of this distinction, we give a name to it in this manner : we call every number divisible into two equal parts an Even Number. " This is a geometrical definition, because after having clearly designated a thing, namely any numbei divisible feto two equal parts, we give it a name divesteU of every XIIL] ON METHOD. 113 other meaning, which it might have, in order to bestow upon it the meaning designated. " Hence it appears that definitions are very free, and that they can never be subject to contradiction, for there is nothing more allowable, than to give any name we wish to a thing which we have clearly pointed out. It is only necessary to take care that we do not abuse this liberty ol imposing names, by giving the same name to two differ- ent things. Even that would be allowable, provided that we did not confuse the results, and extend them from one to the other. But if we fall into this vice, we have a very sure and infallible remedy ; it is, to substitute men- tally the definition in place of the thing defined, and to hold the definition always so present in the mind, that every time we speak, for instance, of an even number, we may understand precisely that it is a number divisible into two equal parts, and so that these two things should be so combined and inseparable in thought, that as often as one is expressed in discourse, the mind may direct it- self immediately to the other. " For geometers and all who proceed methodically only impose names upon things in order to abbreviate discourse, and not to lessen or change the ideas of the things concerning which they discourse. They pretend that the mind always supplies the entire definition of the biief terms which they employ simply to avoid the con- fusion produced by a multitude of words. " Nothing prevents more promptly and effectively the insidious fallacies of the sophists than this method, which are should always employ, and which alone suffices to banish all sorts of difficulties and equivocations. " These things being well understood, J return to my explanation of the true method, which consists, as I said* in defining everything and proving everything. "Certainly this method would be an excellent one; 8 H4 PASCAL AND DESCARTES [Lisa were it not absolutely impossible. It is evident that th nrst terms we wished to define would require previous terms to serve for their explanation, and similarly the first propositions we wished to prove, would presuppose other propositions preceding them in our knowledge ; and thus it is clear that we should never arrive at the first terms or first propositions. "Accordingly in pushing our researches further and further, we arrive necessarily at primitive words which we cannot define, and at principles so clear, that we cannot find any principles more clear to prove them by. Thus it appears that men are naturally and inevitably incapa- ble of treating any science whatever in a perfect method $ but it does not thence follow that we ought to abandon every kind of method The most perfect method avail- able to men consists not in defining everything and de- monstrating everything, nor in defining nothing and de- monstrating nothing, but in pursuing the middle course of not defining things which are clear and understood by all persons, but of defining all others ; and of not proving truths known to all persons, but of proving all others. From this method they equally err who undertake to de- fine and prove everything, and they who neglect to do it in things which are not self-evident." It is made plain in this admirable passage that we can never by using words avoid an ultimate appeal to things, because each definition of a word must require one or more other words, which also will require defini- tion, and so on ad infinitum. Nor must we ever rerun, back upon the words already denned ; for if we define A by B, and B by C, and C by Z>, and then D by A, wt commit what may be called a circulus in definiettdo; a most serious fallacy, which might Jead us to suppose that we know the nature of A, B, C, and D, when we reall) nothing about them. XIII.] ON METHOD. 115 Pascal's views of the geometrical method were clearly rammed up in the following rules, inserted by him in the Port Royal Logic*. 1. To admit no terms in the least obscure or equivo- cal urithout defining them. 2. To employ in the definitions only terms perfectly known or already explained. 3. To demand as axioms only truths perfectly ev dent. 4. To prove all propositions which are at all obscure, by employing in their proof only the definitions which have preceded, or the axioms which have been accorded, or the propositions which have been already demonstrated, or the construction of the thing itself which is in dispute, when there may be any operation to perform. 5. Never to abuse the equivocation of terms by failing to substitute for them, mentally, the definitions which restrict and explain them. The reader will easily see that these rules are much more easy to lay down than to observe, since even geo- meters are not agreed as to the simplest axioms to assume, or the best definitions to make. There are many differ- ent opinions as to the true definition of parallel lines, and the simplest assumptions concerning their nature; and how much greater must be the difficulty of observing Pascal's rules with confidence in less certain branches of science. Next after Geometry, Mechanics is perhaps the most perfect science, yet the best authorities have been far from agreeing as to the exact definitions of such notions as force ', mass, moment, power, inertia, and the most different opinions are still held as to the simples! axioms by which the law of the composition of forces may be proved Nevertheless if we steadily bear in mind, i * Mr Spencei Baynes' Translation, p. 317. 8-* 116 PASCAL AND DECARTES [LESS studying each science, the necessity of defining ev, or things of which one is equal, but the other unequal to the same third thing, are unequal to each other. Lastly, A and E are two lines both un- equal to D and unequal to each other, whereas A and B are two lines both unequal to D but equal to each other; thus we plainly see that " two things both unequal to the same thing may or may not be equal to each other." From what precedes it will be apparent that all rea soning requires that there should be one agreement at least; if there be two agreements we may reason to a third agreement; if there be one agreement and one difference we may reason to a second difference ; but if there be two differences only we cannot reason to any conclusion whatever. These self-evident principles will in the next Lesson serve to explain some of the rules of the Syllogism. Logicians however have not confined themselves to the use of these Canons, but have often put the same truth into a different form in axioms known as the Dicta dt omni et nullo of Aristotle. This celebrated Latin phrase means " Statements concerning all and none," and the axiom, or rather pair of axioms, is usually given iv the following words : 124 THE LAWS OP THOUGHT. [LESJ Whatever is predicated of a term distributed whether affirmatively or negatively, may be predicated in Itiu manner of everything contained under it. Or more briefly : What pertains to the highet class pertains als* to tJu lower. This merely means, in untechnical language, that what may be said of all the things of any sort or kind may be said of any one or any cart of those things ; and, secondly, what may be denied of all the things in a class may be denied of any one or any part of them. What- ever may be said of " All planets " may be said of Venus, the Earth, Jupiter, or any other planet ; and, as they may all be said to revolve in elliptic orbits, it follows that this may be asserted of Venus, the Earth, Jupiter, or any other planet Similarly, accoruing to the negative part of the Dicta, we may deny that the planets are self- luminous, and knowing that Jupiter is a planet may deny that Jupiter is self-luminous. A little reflection would show that the affirmative Dictum is really the first of the Canons in a less complete and general form, and that the negative Dictum is similarly the second Canon. These Dicta in fact only apply to such cases of agreement be- tween terms as consist in one being the name of a smaller class, and another of the larger class containing it Lo- gicians have for the most part strangely overlooked the important cases in which one term agrees with another to the extent of being identical with it ; but this is a subject which we cannot fitly discuss here at any length. It is treated in my little work called The Substitution oj Similars*. Some logicians have held that in addition to the three tews which are called the Primary Laws of Thought, * Macmilkn and Co. 1869. XIV.] THE LAWS OF THOUGHT. ia* there is a fourth called " The Principle or Law of Sum cient Reason." It was stated by Leibnitz in the following words : Nothing happens without a reason why it should bi 96 rather than otherwise. For instance, if there be a pail tf scales in every respect exactly alike on each side and arith exactly equal weights in each scale, it must remain motionless and in equilibrium, because there is no reason why one side should go down more than the other. It is certainly a fundamental assumption in mechanical science that if a body is acted upon by two perfectly equal forces m different directions it will move equally between them, because there is no reason why it should move more to one side than the other. Mr Mansel, Sir W. Hamilton and others consider however that this law has no place in logic, even if it can be held self-evident at all ; and the question which appears open to doubt need not be dis- cussed here. I have so freely used the word axiom in this lesson that it is desirable to clear up its meaning as far as pos- sible. Philosophers do not perfectly agree about its deri- vation or exact meaning, but it certainly comes from the verb dio', which is rendered, to think worthy. It gene- rally denotes a self-evident truth of so simple a character that it must be assumed to be true, and, as it cannot be proved by any simpler proposition, must itself be taken aa the basis of reasoning. In mathematics it is clearly used 'n this sense. See Hamilton's Lectures on, Logic, Lectures 5 and & ^ LESSON XV THE RULES OF THE SYLLOGISM. SYLLOGISM is the common name for Mediate Inference, or inference by a medium or middle term, and is to Ix distinguished from the process of Immediate Inference, 01 inference which is performed without the use of any third or middle term. We are in the habit of employing a middle term or medium whenever we are prevented from comparing two things together directly, but can compare each of them with a certain third thing. We cannot compare the sizes of two halls by placing one in the other, but we can measure each by a foot rule or other suitable measure, which forms a common measure, and enables us to ascei tain with any necessary degree of accuracy their relative dimensions. If we have two quantities of cotton goods and want to compare them, it is not necessary to bring fhe whole of one portion to the other, but a sample is cut off, which represents exactly the quality of one portion, and, according as this sample does or does not agree with the other portion, so must the two portions of goods agree or differ. The use of a middle term in syllogism is closely pa- rallei to what it is in the above instances, but not exactly the same. Suppose, as an example, that we wish to ascertain whether or not "Whales are viviparous," and mat we had not an opportunity of observing the fact directly ; we could yet show it to be so if we knew that ** whales are mammalian animals," and that " ^11 ; XV.] THE RULES OF THE SYLLOGISM. i* t malian animals are viviparous." It would follow tha. " whales are viviparous ; " and so far as the inference if concerned it does not matter what is the meaning wt attribute to the words viviparous and mammalian. la this case " mammalian animal " is the middle term, The name Syllogism means the joining together ia Jiought of two propositions, and is derived from the Greek words Every syllogism contains three, at, and we compared A with B and C with Z>, we should either have no common medium at all between A and Z?, or we should require a second syllogism, so as first to compare A and C with B t and then A and D with C. The middle term may always be known by the fact ihat it does not occur in the conclusion. The major term is always the predicate of the conclusion, and the minor Una the subject. These terms are thus called because in the universal affirmative proposition (,A) the predicate is necessarily a wider or greater or major term than the subject ; thus in " all men are mortals," the predicate in- cludes all other animals as well as men, and i* obviously a major term or wider term than men. XV.] THE RULES OF THh SYLLOGISM. 129 Again, the syllogism necessarily consists of a premise called the major premise, in which the major and middle terms are compared together; of a minor premise which similarly compares the minor and middle terms ; and of a conclusion, which contains the major and minor terms jnly. In a strictly correct syllogism the major premise ilways stands before the minor premise, but in ordinary writing and speaking this rule is seldom observed ; and that premise which contains the major term still con- tinues to be the major premise, whatever may be its position. The third rule is a very important one, because many fallacies arise from its neglect. By the middle term being distributed once at least, we mean (see p. 74) that the whole of it must be referred to universally in one premise, if not both. The two propositions All Frenchmen are Europeans, All Russians are Europeans, do not distribute the middle term ar all, because Uiey are both affirmative propositions, which have (p. 75) undistributed predicates. It is apparent that French- men are one part of Europeans, and Russians anothei part, as shown in Killer's method In Fig. 6, so thai Ijo THE RULES OF THE SYLLOGISM. [LE*S there is no real middle term. Those propositions would equally allow of Russians being or not being Frenchmen j for whether the two interior circles overlap or not they are equally within the larger circle of Europeans Again the two propositions All Frenchmen are Europeans, All Parisians are Europeans, 10 not enable us to infer that all Parisians are French men. For though we know of course that all Parisians Fig. 7- are included among Frenchmen, the premises would allow of their being placed anywhere within the circle ol Europeans. We see in this instance that the premises and conclusion of an apparent argument may all be true and yet the argument may be fallacious. The part of the third rule which refers to an ambi guoms middle term hardly requires explanation. It has been stated (Lesson iv.) that an ambiguous term is one which has two different meanings, implying different con notations, and it is really equivalent to two different terms which happen to have the same form of spelling, so that they are readily mistaken for each other. Thus if we were to argue that because ** all metals are elements and XV.] THE RULES OF THE SYLLOGISM. I) brass is metal, therefore it is an element," we should bt committing a fallacy by using the middle term metal it two different senses, in one of which it means* the pure simple substances known to chemists as metals, and in the other a mixture of metals commonly called metal in the arts, but known to chemists by the name alloy. In many examples which may be found in logical books the ambiguity of the middle term is exceedingly obvious, but the reader should always be prepared to meet with cases where exceedingly subtle and difficult cases of ambiguity occur. Thus it might be argued that "what is right should be enforced by law, and that charity is right and should therefore be enforced by the law." Here it is evident that right is applied in one case to what the conscience approves, and in another case to what public opinion holds to be necessary for the good of society. The fourth rule forbids us to distribute a term in the conclusion unless it was distributed in the premises. As the sole object of the syllogism is to prove the conclusion by the premises, it is obvious that we must not make a statement concerning anything unless that thing was mentioned in the premises, in a way warranting the state- ment. Thus if we were to argue that " because many nations are capable of self-government and that nations capable of self-government should not receive laws from a despotic government, therefore no nation should receive laws from a despotic government," we should be clearly exceeding the contents of our premises. The minor term, many nations, was particular in the minor premise, and aiust not be made universal in the conclusion. The pre- mises do not warrant a statement concerning anything but the many nations capable of self-government. The above argument would therefore be fallacious and would b$ technically called an Ulicit process of the minor term, meaning that we have improperly treated the minor term o a 133 THE RULES OF THE SYLLOGISM. [LESi Such a breach of the fourth rule as is described above is exceedingly easy to detect, and is therefore very seldom committed. But an Ulldt process or improper treatment of the major term is more common because it is not so trans- parently fake. If we argued indeed that "because all Anglo-Saxons love liberty, and Frenchmen are not Anglo- Saxons, therefore they do not love liberty," the fallacy would be pretty apparent; but without a knowledge of logic it would not be easy to give a clear explanation of the fallacy. It is apparent that the major term loving liberty, is undistributed in the major premise, so that Anglo-Saxons must be assumed to be only a part of those who love liberty. Hence the exclusion of Frenchmen from the class Anglo-Saxons does not necessarily exclude them from the class who love liberty (see Fig. 8). The Fig. 8. conclusion of the false argument being negative distri- butes its predicate, the major term, and as this is un- distributed in the major premise we have an Illicit majox as we may briefly call this fallacy. The following is an obscurer example of the same fallacy ; " Few students t?.] THE RULES OF THE SYLLOGISM. 133 are capable of excelling in many branches of knowledge, ind such as can so excel are deserving of high commen- dation ;" hence " few students are deserving of high com* mendation." The little word " few " has here the doubit meaning before explained (p. 67), and means that "a few are, &c., and the rest are not" The conclusion is thus really a negative proposition, and distributes the major term "deserving of high commendation." But this major term is clearly undistributed in the major premise, which merely asserts that those who can excel in many branches of knowledge are deserving, but says or implies nothing about other students. The fifth rule is evidently founded on the principle noticed in the last lesson, that inference can only proceed where there is agreement, and that two differences or disagreements allow of no reasoning. Two terms, as the third Canon states, may both differ from a common term and yet may or may not differ from each other. Thus il Fig. 9. LEunptaauA we were to argue that Americans are not Europeans, and Virginians are not Europeans, we see that both termi iisagree with the middle term Europeans, and yet they 134 THE RULES OF THE SYLLOGISM ^LESB agree between themselves. In other cases the two nega- tire premises may be plainly true while it ^ ill be quite uncertain whether the major and minor terms agree or not Thus it is true, for t^ that "Colonists are not Europeans, and Americans are not Europeans," but this gives us no right to infer that Colonists are at are not Americans. The two negative premises are re- presented in fig. 9, by excluding the circles of Colonists and Americans from that of Europeans ; but this exclusion may still be effected whether Colonists and Americans coincide partially, or wholly, or not at all A breach oi this rule of the syllogism may be conveniently called the fallacy of negative premise*. It must not however be supposed that the mere occurrence of a negative particle (not or no) in a proposition renders it negative in the manner contemplated by this rule. Thus the argument " What is not compound is an element Gold is not compound ; Therefore Gold is an element," contains negatives in both premises, but is nevertheless ^id, because the negative in both cases affects the middle term, which is really the negative term not-compound. The truth of the sixth rule depends upon that of the Mjuom, that if two terms agree with a common third term they agree with each other, whence, remembering that a negative proposition asserts disagreement, it is evident that a negative conclusion could not be drawn from really affirmative premises. The corresponding negative axiom prevents our drawing an affirmative conclusion if either premise should be really negative. Only practice how- ever will enable the student to apply this and the preceding rules of the syllogism with certainty, since fallacy may be hidden and disguised by various form$ of expression. Numerous examples are given at the end of **.] THE RULES OF THE SYLLOGISM. 13 I the book by which the student may acquire facility in the analysis of arguments. The remaining rules of the syllogism, the ;th and 8th, are by no means of a self-evident character and are in fact corollaries of the first six rules, that is consequences *'hich follow from them. We shall therefore have to ihew that they are true consequences in a future Lesson. We may call a breach of the 7th rule a fallacy of parti- cular premises, and that of the 8th rule the fallacy of a universal conclusion from a particular premise^ but these fallacies may really be resolved into those of Illicit Process, or Undistributed Middle. Fcr many details concerning the Aristotelian and Scholastic Views of the Syllogism, and of Formal Logic generally, see the copious critical notes to Mans el's edition of Aldrich's Art is Logic* Rudi- nunta. 2nd Ed. Oxford. 1852. LESSON XVI. THE MOODS AND FIGURES OF THE SYLLOGISM. WE are now in full possession of those principles of rea- soning, and the rules founded upon them, by which a true syllogism may be known from one which only seems to be a true one, and our task in the present Lesson is to ascertain the various shapes or fashions in which a process of mediate inference or syllogism may be met with. We know that every syllogistic argument must contain three propositions and three distinct terms each occurring twice in those propositions. Each proposition !36 THE MOODS AND FIGURES [LKSti of the syllogism may, so far as we yet know, be eitha affirmative or negative, universal or particular, so that it is not difficult to calculate the utmost possible varieties of modes in which a syllogism might conceivably be con- structed. Any one of the four propositions A, E, I, or may in short be taken as a major premise, and joined with any one of the same form as a minor premise, and any one ol ^he four again may be added as conclusion. We should thus obtain a series of the combinations or modes of joining the letters A, E, I, 0, a few of which are here writ ten out: AAA AEA AIA AOA EAA EEA AAB AEB AIB AOE EAE EEE AAI AEI All AOI EAI EEI AAO ABO AIO AOO EAO Ac. It is obvious that there will be altogether 4x4x4 or 6/ such combinations, of which 23 only are gfven above The student can easily write out the remainder by carry- ing on the same systematic changes of the letters. Thus beginning with AAA we change the right-hand letter suc- cessively into E, I, and 0, and then do the same beginning with AEA instead ; after the middle letter has been carried through all its changes we begin to change the left-hand letter. With each change of this we have to repeat all the sixteen changes of the other letters, so that there will obviously be altogether 64 different conceivable modes /f arranging propositions into syllogisms. We call each of these triplets of propositions a mood or torrn of the syllogism (Latin modus, shape , and we have o consider how many of such forms can really be used in 'alid arguments, as distinguished from those which break ie or more of the rules of the syllogism. Thus the mood 4EA would break the 6th rule, that if one premise b negative the conclusion must be so too; AIE breaks the xvi. J OF THE SYLLOGISM. 137 converse part of the same rule, that a negative conclusion can only be proved by a negative premise ; while ERA, EEE &c., break the 5th rule, which prohibits our reasoning at all from two negative premises. Examples of any oJ these moods can easily be invented, and their falsity would be very apparent ; thus for AEA we might take All Austrians are Europeans, No Australians are Europeans ; Therefore, all Australians are Austrians. Many of the 64 conceivable moods are excluded by the 7th and 8'th rules of the syllogism. Thus ATA and BH break the rule, that if one premise be particular the con- clusion must be so also, while HA, 100, 010 and many others, break the rule against two particular premises. Some combinations of propositions may break more than one rule ; thus OOO has both negative premises and parti- cular premises, and OOA also violates as well the 6th rule. It is an admirable exercise in the use of the syl- logistic rides to write out all the 64 combinations and then strike out such as break any ru4e ; the task if pur sued systematically will not be so long or tedious aj might seem likely. It will be found that there are only twelve moods which escape exclusion, and may so far b considered good forms of reasoning, and these are AAA EAE IAI 010 AAI EAO (ESO) AEB EXO ABO AOO Of these however IEO will have shortly to oe rejected, because it will be found really to break the 4th rule, and involves Illicit process of the major term. There arc 138 THE MOODS AND FIGURES {LESS then, only eleven moods of the syllogism which are reall) valid; and we may thus account for the whole of th \ixty-four moods. Numbex Excluded by of mood* Negative premises, Rule 5 16 Particular premises 7 12 One negative premise 6 12 One premise particular,, 8 8 Negative conclusion 6 ~ 4 Illicit major 4. I Total excluded 53 Valid moods II Total 64 We have by no means exhausted as yet all the possible varieties of the syllogism, for we have only de- urmined the character, affirmative or negative, generai or particular of the propositions, but have not decided the ways in which the terms may be disposed in them. The major term must be the predicate of the conclusion, but it may either be subject or predicate of the majoi premise, and similarly the minor term or subject of the conclusion, may be either the subject or predicate of the minor premise. There thus arise four different ways, 01 as they are called Figures, in which the terms can b disposed. These four figures of the syllogism are shewi in the following scheme, taking X to denote the major term Y middle Z. minor lit Fig. 2nd Fig. 3rd Fig. 4th Fig, Major Premise YX XY YX XY Minor ZY ZY YZ YZ Conclusion ZX ZX ZX ZX XVI.] OF THE SYLLOGISM. 139 These figures must be carefully committed to memory, which will best be done by noting the position of the middle term. This term stands first as subject of the major premise in the ist Figure, second as predicate ic both premises of the 2nd Figure, first again as subject ol both premises in the 3rd Figure, and in an intermediate position in the 4th Figure. In the conclusion, of course, the major and minor terms have one fixed position, and when the middle term is once correctly placed in any figure we easily complete the syllogism. The reader will hardly be pleased to hear that each of the eleven valid moods will have to be examined in each of the four figures separately, so that there are 44 cases still possible, from which the valid syllogisms have to be selected. Thus the mood ABB in the first figure would be as follows : All K's are A?s, No Z*s are K*s; Therefore No Z's are X*s. rhis would break the 4th rule and be an Illicit Major, because X is distributed in the conclusion, which is a negative proposition, and not in the major premise. In the second figure it would be valid: All X J * are KX NoZ'sare K*s; Therefore No Z's are X 1 1 In the third figure it becomes All l"s are JT% No K's are Z's, No Z*s are A?$, and again breaks the 4th rule, as regards the major term, Lastly in the 4th figure it is valid, as the reader may easily satisfy himself. 140 TOT MOODS AND FIGURES LES* When all the valid moods are selected out of the 44 possible ones, there are found to be altogether 24, whicb are as follows: Valid Moods of th Syllogism. First Second Third Fourth Figure. Figure. Figure. Figure. AAA BAE AAI AAI BAB AEE IAI ABB All BIO All 1A1 no AGO EAO BAO OAO BIO [BAO] EIO [AEO] [AEO] ive of the above moods are set apart and enclosed in brackets, because though valid they are of little or no use. They are said to have a weakened conclusion, because the conclusion is particular when a general one might have been drawn. Thus AAI, in the first figure is represented by the example : All material substances gravitate, All metals are material substances ; Therefore some metals gravitate. It is apparent that the conclusion only states a part of the truth, and that in reality a/I metals gravitate. It is not actually an erroneous conclusion, because it must be carefully remembered (p. 77 that the affirming of a subaltern or particular proposition does not deny the corresponding general proposition. It is quite true that tome metals gravitate, and it must be true because all of them do so. Bat when we can as readily prove that ail 4o gravitate it is desirable to adpt this conclusion. If we agree with most logicians to overlook the ex- istence of the five syllogisms with weakened conclusions, XVI.] OF THE SYLLOGISM. 141 there will remain nineteen which are at once valid and useful. In the next lesson certain ancient mnemonic lines will be furnished by which alone it would be possible for most persons to carry in the memory these 19 combi nations ; but the reader will in the mean time be able to gather from the statement of the moods in p. 140 the truth of the following remarks concerning the peculiar character of each figure of the syllogism. The first figure is the only one which proves the pro- position A, or has A for its conclusion. It is the only figure, too, which can prove any one of the four proposi- tions A, B, I, 0. As regards the premises, it is especially important to note that the major premise is always universal (A or B), and the minor premise affirmative (A 01 I) : this peculiarity will be further considered in the next lesson. The second figure only proves negative conclusions (B or O), and the reason is easily apparent. As the middle term in this figure is the predicate of both premises it would necessarily be undistributed in both premises if these were affirmatives, and we should commit the fallacy exemplified in p. 137. It follows that one premise must be negative and of course one only, so that of the major and minor terms one must be included or excluded wholly from the middle, and the other at the same time excluded or included at least partially. To illustrate this we may take X y Y and Z to represent, as before, the major, mid- lie and minor terms of a syllogism, and the four moodi o< ihu figure are then SAB ABB noJTsare Y>*> aUJTsueFX ail Z's are K*s ; no Z's are K*; /. no Z's are AT's. .-. no Z's are X**, THE MOODS AND FIGURES [LESS AOO all AT's are K% some Z's are not K*s ; . some Z's are not A' ; s. BIO no AT's arc ^s, some Z's are Vs ; /. iome Z's are not X 's. The nature of the moods of the second figure is clearl ' shewn in the following figures : Fig. 10. (Cesare.) Fig. ii. (Camestret.) It wfll also be observed that in the second figure tk< aunor premise may be any of the four A, B, I, 0. The third figure only proves particulars (I or , and 't always has an affirmative minor premise (A or I). II also contains the greatest number of moods, since in nc case is the conclusion a weakened one. XTL] OF THE SYLLOGISM. 143 The fourth igure is usually considered unnatural and comparatively useless, because the same arguments cat oe more clearly arranged in the form of the first figure, which in some respects it resembles. Thus it proves all the propositions except A, namely, E, 1, 0, and i*s fir mood AAI, is in reality a weakened form of AAA in the first figure. Many logicians, including in recent times Sir W. Hamilton, have rejected the use of this figure altogether. It is evident that the several figures of the syllogism possess different characters, and logicians have thought that eav A/ LESSON XVIL REDUCTION OF THE IMPERFECT FIGURES OF THE SYLLOGISM. IN order to facilitate the recollection of the nineteen /alid and useful moods of the syllogism, logicians invented, at least six centuries ago, a most curious system of artificial words, combined into mnemonic verses, which may b XVII.] OF THE SYLLOGISM. 14) readily committed to memory. This device, however m genious, is of a barbarous and wholly unscientific cha- racter ; but a knowledge of its construction and use is stiD expected from the student of logic, and the verses arc Aerefore given and explained below. Barbara, Celarent, Darii^ /Vrwque, prioris; Cesare, Camestres, Festino, Baroko, secundae; Tertia, Darapti, Disamis, Datisi, Felapton, Bokardo, Ferison, habet ; Quarta insuper addit Bramantip, Camenes, Dimaris, Fesapo, Fresison. The words printed in ordinary type are real Latin words, signifying that four moods whose artificial names are Barbara, Celarent, Darii and Ferio, belong to the first figure ; that four others belong to the second ; six more to the third ; while the fourth figure moreover contains five moods. Each artificial name contains three vowels, which indicate the propositions forming a valid mood ; thus, Cte/ArEnt signifies the mood of the first figure, which has E for a major premise, A for the minor, and E for the conclusion. The artificial words altogether contain exactly the series of combinations of vowels shown in p. 140, excepting those in brackets. These mnemonic lines also contain indications of the mode in which each mood of the second, third and fourth figures can be proved by reduction to a corresponding mood of the first figure. Aristotle looked upon the first figure as a peculiarly evident and cogent form of argu- ment, the Dictum de omni et nvllo being directly ap- plicable to it, and he therefore called it the Perfect Figure. The fourth figure was never recognised by him, and it ii often called the Oalenian figure, because the celebrated Galen is supposed to have discovered it The second and third figures were known to Aristotle as the Imperfect which it was necessary to reduce to the first 146 THE IMPERFECT FIGURES [LSS figure by certain conversions and transpositions of UM premises, for which directions are to be found in the artificial words. These directions are as follows : - s indicates that the proposition denoted by the prc ceding vowel is to be converted simply. p indicates that the proposition is to be converted p*f (ucidttts, or by limitation. m indicates that the premises of the syllogism are tc be transposed, the major being made the minor of a new syllogism, and the old minor the new major. The m is derived from the Latin mutare, to change. JB, C, Z>, F, the initial consonants of the names, in- dicate the moods of the first figure, which are produced by reduction; thus Cesare, Camestres and Camenes are reducible to Celarent, Darapti, &c., to Darii, Fresison to Ferio and so on, k denotes that the mood must be reduced or proved by a distinct process called Indirect reduction, or rtd-uct* ad impossibiU) which will shortly be considered. Let us now take some syllogism, say in Canustrcs, and follow the directions for reduction. Let the example be All stars are self-luminous (i) All planets are not self-luminous (2) Therefore no planets are stars (3) The first s in Camestres shows that we are to convert simply the minor premise. The m instructs us to change the order of the premises, and the final s to convert the conclusion simply. When all these changes are made we obtain No self-luminous bodies are planets Converse of (a) All stars are self-luminous (i) T herefore no stars are planets Converse of (3) This, it will be found, is a syllogism in Celarent as Blight be known from the initial C in Camestres. OF THE SYLLOGISM. i> As another example let us take Fesapo, for instance : No fixed stars are planets, All planets are round bodies ; Therefore some round bodies are not fixev Stan. According to the directions in the name, we are ti convert simply the major premise, and by limitation the minor premise. We have then the following syllogism in Ferio: No planets are fixed stars, Some round bodies are planets ; Therefore some round bodies are not fixed stars. The reader will easily apply the same process of con- version or transposition to the other moods, according to the directions contained in their names, and the only moods it will be necessary to examine especially are Bramantip, Baroko and Bokardo. As an example of Bramantip we may take : All metals are material substances, All material substances are gravitating bodies; Therefore some gravitating bodies are metals. The name contains the letter m, which instructs us to transpose the premises, and the letter p, which denotes conversion by limitation ; effecting these changes wt have: All material substances are gravitating bodies, All metals are material substances ; Therefore some metals are gravitating bodies. This is not a syllogism in Barbara, as we might hav* expected, but is the weakened mood AAI of the first figure It is evident that the premises yield the conclusion * all metals are gravitating bodies," and we must take the letter p to indicate in this mood that the conclusion is weaker than it might be. In truth the fourth figure is M IO 2 148 THE IMPERFECT FIGURES [LES imperfect and unnatural in form, containing nothing but ill-arranged syllogisms, which would have been bettci itated in the first figure, that Aristotle, the founder of logical science, never allowed the existence of the figure at all. It is to be regretted that so needless an addition pras made to the somewhat complicated forms of the syllogism. Indirect reduction. The moods Baroko and Bokardo give a good deal of trouble, because they cannot be re- duced directly to the first figure. To show the mode of treating these moods we will take X, K, Zto represent the major, middle and minor terms of the syllogism, and Baroko may then be stated as follows : All X's are K>s, Some Z's are not V* ; Therefore Some Z's are not ^'s. Now if we convert the major premise by Contrapo- sition (p. 83) we have "all not-ys are not-A^'s," and, making this the major premise of the syllogism, we have All not- K*s are not X\ Some Z's are not- X 1 * ; Therefore Some Z's are not X's. Although both the above premises appear to be nega- tive, this is really a valid syllogism in Ferio, because two of the negative particles merely affect the middl* term (see p. 134), and we have therefore effected the re Auction of the syllogism. Bokardo, when similarly stated, is as follows ; Some y* are not A*'s, All rsareZ'f; Therefore Some Z's are not Jfs. OF THE SYLLOGISM. 149 To reduce this, convert the major pre/nise by nega- tion, and then transpose the premises. V/e have: All X's are Z's, Some not-A^s are K's; Therefore Some not-Jf's are Z's. This conclusion is the converse by negation of the lormer conclusion, the truth of which is thus proved by reduction to a syllogism in Darii. Both these moods, Baroko and Bokardo, may however be proved by a peculiar process of Indirect reduction, closely analogous to the indirect proofs often employed by Euclid* in Geometry. This process consists in supposing the conclusion of the syllogism to be false, and its con- tradictory therefore true, when a new syllogism can easily be constructed which leads to a conclusion contradictory of one of the original premises. Now it is absurd in logic to call in question the truth of our own premises, for the very purpose of argument or syllogism is to deduce a con- clusion which will be tiue when the premises are true. The syllogism enables us to restate in a nw form the in- formation which is contained in the premises, just as a machine may deliver to us in a new form the material which is put into it The machine, or rather the maker of the machine, is not responsible for the quality of the materials furnished to it, and similarly the logician is not responsible in the least for the truth of his premises, but only for their correct treatment. He must treat them, if he treat them at all, as true ; and therefore a conclusion which requires the falsity of one of our premises is alto- gether absurd. To apply this method we may take Baroko, at bo (ore: AllJTsare K's (l) Some 2Ts are not K's (2) Therefore Some Z's are not JCs (j> 150 THE IMPERFECT FIGURES [LSS& If this conclusion be not true then its contradictory, 'aflZ's are A'V must of necessity be regarded as true (pp. 76 79). Making this the minor premise of a new syllogism with the original major premise we have : All JTs are K>s (i) All Z*s are X's contradictory of 3) Htnce All Z*s are XV Now this conclusion in A, is the contradictory of our old minor premise in 0, and we must either admit one of our own premises to be false or allow that our original con- clusion is true. The latter is of course the alternative we choose, We treat Bokardo in a very similar manner ; Some X*s are not A?s (i) All K's are Z>s (2) Therefore Some Z's are not X*s (3) If this conclusion be not true then ' all Z*s are X*s ' must be true. Now we can make the syllogism : All Z*s are X*s Contradictory of (3) All rs area's (2) Hence All Fs are A"s. This conclusion is the contradictory of (i), the original major premise, and as this cannot be allowed, we must either suppose (2) the original minor premise to be false, which is equally impossible, or allow that our origina) conclusion is true. It will be observed that in both these cases of Indireci Reduction or Proof we use a syllogism in Barbara, which fact is indicated by the initial letters of Baroko and Bo- kardo. The same process of Indirect proof may ta applied to any of the other moods, but it is not usual to do so as the simpler process of direct or as it is often called ostensive reduction is sufficient ml Of THE SYLLOGISM. 151 It will be remembered that when in Lesson xv. /p. 135) we considered the rules of the syllogism, there were two supplementary rules, the 7th and 8th, concerning particu Ur premises, which were by no means of a self-evident character, and which require to be proved by the six more fundamental rules. We have now sufficiently advanced to consider this proof with advantage. The 7th rule forbids us to draw any conclusion from two particular pre- mises ; now such premises must be either n, 10, 01, or 00. Of these n contain no distributed term at all, so that the 3rd rule, which requires the middle term to be distributed, must be broken. The premises 00 evidently break the 5th rule, against negative premises. The conclusion oi the pair 10 must be negative by the 6th rule, because one premise is negative; the major term therefore will be distributed, but as the major premise is a particular affirmative it cannot be distributed without committing the fallacy of illicit process of the major, against rule 4. Lastly the premises 01 contain only one distributed term, the predicate of the major premise. But as the conclusion must be negative by rule 6th, the major term must be distributed : we oagnt to have then in the premises two distributed terms, one for the middle term, the other for the major term ; but as the premises contain only a single distributed term, we must commit the fallacy either ol undistributed middle or of illicit process of the major term, if we attempt to draw any conclusion at all We thus see that in no possible case can a pair of particular premises give a valid conclusion. The 8th rule of the syllogism instructs us that if one premise of a syllogism be particular the conclusion must Iso be p:\rticular. It can only be shown to be true by going over all the possible cases and observing that the six princip;d rules of the syllogism always require the conclusion ;c be particular. Suppose for inttance the 152 IRREGULAR AND COMPOUND ther side. This practice has not been very long dis- xmtinued at the University of Oxford, and is said to be till maintained in some continental Universities; but except in such school disputations it must be allowed that perfectly formal syllogisms are seldom employed. por, meaning heap. It is usually stated in this way : All A's are &s, All 0s are Cs, All Cs are /?$, All &s are ?* ; Therefore all A* s are .fs. The chain can be carried on to any length provided it is perfectly consecutive, so that each term except the first and last occurs twice, once as subject and once as predi- cate. It hardly needs to be pointed out that the sorites really contains a series of syllogisms imperfectly ex- pressed; thus First Syllogism. Second Syllogism. Last Syllogism, ^s are Cs, Cs are /7s, D > s are ^s, A's are ffs ; A's are (7s; A's are Z?'s ; .'. A's are Cs. /. A's are Z7s. .-. A's are E's. Each syllogism furnishes a premise to the succeeding one, of which it is therefore the prosyllogism, and any syllo- gism may equally be considered the episyllogism of that which precedes. In the above sorites all the premises were univenal and affirmative, but a sorites may contain one particular premise provided it be the first, and one negative premise provided it be the last The reader may easily assure himself by trial, that if any premise except the first wew xnil.] SYLLOGISMS. 157 particular the fallacy of undistributed middle would be committed, because one of the middle terms would be the predicate of one affirmative premise and the subject of another particular premise. If any premise but the last arere negative there would be a fallacy of illicit process o* the major term. It is not to be supposed that the forms of the syllogisir hitherto described are all the kinds of reasoning actuall) employed in science or common life. In addition to the hypothetical and disjunctive syllogisms and some othei forms to be described in succeeding lessons, there are really many modes of reasoning of which logicians hare not taken much notice as yet. This was clearly pointed out more than two hundred years ago by the writers ol the Port Royal Logic ) a work first printed in the year 1662, but which has been since reprinted very often and trans- lated into a great many languages. The book is named from a place near Paris where a small religious com- munity lived, of which the authors of the book, namely Arnauld and Nicole, and a contributor to it the great philosopher and mathematician Pascal, were the most celebrated members. The Port Royal Logic was to a considerable extent the basis of the well-known Watts' Logic, but the reader can now be referred to an admirable translation of the original work made by Professor Spencer Bayn*s, of St Andrew's. Many improvements of Logic may be found in thii work, such as the doctrine of Extension and Intension explained in Lesson V. In the 9th Chapter of the 3rd Part moreover it is wisely pointed out that "little pains are taken in applying the rules of the syllogism to reason- Ings of which the propositions are complex, though thii is often very difficult, and there are many arguments of this nature which appear bad, but which are nevertheless very good; and besides, the use of such reasonings if A A 158 IRREGULAR AND COMPOUND [LE88 much more frequent than that of syllogisms which art quite simple." Some examples are given of the complex syllogisms here referred to; thus: The sun is a thing insensible, The Persians worship the sun ; Therefore the Persians worship a thing insensible. Phis is an argument which cannot be proved by the rule* of the syllogism, and yet it is not only evidently true, but is an exceedingly common land of argument Another example is as follows : The Divine Law commands us to honour kings ; Louis XIV. is a long ; Therefore the Divine Law commands us to honour Louis XIV. The reader will also find that arguments which are really quite valid and syllogistic are expressed in language so that they appear to have four distinct terms and thus to break one of the rules of the syllogism. Thus if I say " Diamonds are combustible, for they are composed of carbon and carbon is combustible," there are four terms employed, namely, diamonds, combustible, composed ol carbon, and carbon. But it is easy to alter the construc- tion of the propositions so as to get a simple syllogism without really altering the sense, and we then have : What is composed of carbon is combustible ; Diamonds are composed of carbon ; Therefore diamonds are combustible. Examples are given at the end of the book of concise irguments, taken from Bacon's Essays and other writings, which the student can reduce to the syllogistic form by easy alterations ; but it should be clearly understood that these changes are of an extra-logical character, and belong more properlv to the science of language. JLVIII.J SYLLOGISMS. ift I may here explain that the syllogism and the soritei can be expressed either in the order of extension or that of intension. In regard to the number of individual things the noble metals are part of the metals, and thr metals are part of the elements ; but in regard to in- tension, that is to say the qualities implied in the names, element is part of metal, and metal is part of noble metal. So again in extension the genus of plants Anemone is part of the order Ranunculaceas, and this is part oi the great class Exogens; but in intension the cha- racter of Exogen is part of the character of Ranuncu- laceae, and this is part of the character of Anemone. Syllogistic reasoning is equally valid and evident in either case, and we might represent the two modes in ordinary language as follows : Extensive Syllogism. All Ranunculaceae are Exogens ; The Anemone is one of the Ranunculaceae ; Therefore the Anemone is an Exogen. Intensive Syllogism. All the qualities of Ranunculaceae are qualities of Anemone ; All the qualities of Exogen are qualities of Ranun- culaceae ; Therefore all the qualities of Exogen are qualities of Anemone. Any sorites can be similarly represented either in ex- .ension or intension. Concerning the Aristotelian doctrine of the Enthy meme, see Mansel's Aldrich, App. Note F, and Hamil- ton's Lectures on Logic, Lecture xx. Port Royal Logic^ translated by T. Spencer Baynes, 5th ed. Edinburgh l86l, cto . Of CONDITIONAL LESSON XIX. OF CONDITIONAL ARGUMENTS. i T will be remembered that when treating of proposition. ; But A is B\ Therefore C is Z>, is a constructive hypothetical syllogism. It must be carefully observed that the minor premise affirms the antecedent of the major premise, whence the argument is said to be of the modus ponens, or mood which posits or affirms. It is probably one of the most familiar and common kinds of argument. The form, If A is B t Cii D\ But C is not D ; PV Therefore A is not B. OF CONDITIONAL represents the corresponding Destructive hypothetic*. yUoglsm, also called the modus tollens, or the mood which removes the consequent. It must be carefully ob- served again that it is the consequent, not the antecedent. which is denied. The only rule which is requisite for testing the validity of such syllogisms embodies what we have observed above ; viz. tRat either the antecedent must be affirmed^ ^ 9r the consequent denied. If either part of this rule be broken, a serious fallacy will be committed. Thus the apparent argument, HA is*, CisZ?; But C is D ; ^$ - ^ ~^o~* * Therefore A is B, is really a fallacy which we may call the fallacy of affirm* ing the consequent, and its fallacious nature is readily un- derstood by reflecting that " A being B * is not stated to be the only condition on which C is D. It may happen that when is F, or G is ff, or under a hundred other circumstances, C is D, so that the mere fact of C being D is no sufficient proof that A is B. Thus, if a man's cha- racter be avaricious he will refuse to give money for usefui purposes ; but it does not follow that every person who refuses to give money for such purposes is avaricious. There may be many proper reasons or motives leading liim to refuse ; he may have no money, or he may con- sider the purpose not a useful one, or he may have more useful purposes in view. A corresponding fallacy arises from denying tk< 3*4+ M**4 as in the form If A is B, Cis D\ But A is not B ; Therefore C is not /X XIX] ARGUMENTS. 16* The error may be explained in the same way; foi as "A being B n is not stated to be the only condition of C being D, we may deny this one condition to be true but it is possible that the consequent may happen to be true for other reasons, of which we know nothing. Thus if a man is not avaricious we cannot conclude that he will be sure to give money whenever asked. Or take the fol- lowing example : "If the study of Logic furnished the mind with a multi- tude of useful facts like the study of other sciences, it would deserve cultivation; but it does not furnish the mind with a multitude of useful facts; therefore it does not deserve cultivation." This is evidently a fallacious argument, because the acquiring of a multitude of useful facts is not the only ground on which the study of a science can be recom- mended. To correct and exercise the powers of judgment and reasoning is the object for which Logic deserves to be cultivated, and the existence of such other purpose is ignored in the above fallacious argument, which evidently involves the denial of the antecedent. Although it is usual in logical works to describe the hypothetical proposition and syllogism as if they were different in nature from the categorical proposition and syllogism, yet it has long been known that the hypo- theticals can be reduced to the categorical form, and brought under the ordinary rules of the syllogism. As a general rule the hypothetical proposition can be readily converted into a universal affirmative proposition (A) of exactly the same meaning. Thus our instance, "If iron is impure, it is brittle," becomes simply "Impure iron is brittle." In making this alteration in a hypothetical syl- logism it will be found necessary to supply a new minoi term; thus m the case, II 2 <04 OF CONDITIONAL I If iron is impure it is brittle ; But it is impure ; Therefore it is brittle, ire have to substitute for the indefinite pronoun premises, and that there is no true middle term at al I. Often, however, the ambiguity is of a subtle and difficult character, so that different opinions may be held conct rning it. Thus we might argue : "He who hanns another should be punished. He who communicate* an infectious disease to another per- son harms him. Therefore he who communicates an infectious disease to another person should be punished." This may or may not be held to be a correct argument according to the kinds of actions we should consider to come under the term harm, according as we regard negli- gence or malice requisite to constitute harm. Mam 172 LOGICAL FALLACIES. fLESS difficult legal questions are of this nature, as for stance: Nuisances are punishable by law ; To keep a noisy dog is a nuisance ; To keep a noisy dog is punishable by law. The question here would turn upon the degree oi auisance which the law would interfere to prevent Oi again : Interference with another man's business is illegal; Underselling interferes with another man's business; Therefore underselling is illegal Here the question turns upon the kind of interference, and it is obvious that underselling is not the kind of in- terference referred to in the major premise. The Fallacy of Amphibology consists in an ambiguous grammatical structure of a sentence, which produces mis- conception. A celebrated instance occurs in the prophecy of the spirit in Shakspeare's Henry VI. : " The Duke yet lives that Henry shall depose," which leaves it wholly doubtful whether the Duke shall depose Henry, or Henry the Duke. This prophecy is doubtless an imitation of those which the ancient oracle of Delphi is reported to have uttered ; and it seems that this fallacy was a great resource to the oracles who were not confident in their own powers of foresight. The Latin language gives great scope to misconstructions, because it does not require any fixed order for the words of a sentence, and when there are two accusative cases with an infinitive verb, it may be difficult to tell except from the context which comes in regard to sense before the verb. The double meaning which may be given to "twice two and three" arises from amphibology ; it may be 7 or 10, according as we add the 3 after or before multiplying. In the careless construction of sentences it is often impossible to JOL] LOGICAL FALLACIES. 173 tell to what part any adverb or qualifying clause refers. Thus if a person says " I accomplished my business and returned the day after," it may be that the business was accomplished on the day after as well as the return ; but it may equally have been finished on the previous day. Any ambiguity of this kind may generally be avoided by a simple change in the order of the words; as for instance, * I accomplished my business, and, on the day after, returned." Amphibology may sometimes arise from con- fusing the subjects and predicates in a compound sentence, as if in "platinum and iron are very rare and useful metals " I were to apply the predicate useful to platinum and rare to iron, which is not intended. The word " re- spectively" is often used to shew that the reader is not at liberty to apply each predicate to each subject. The Fallacy of Composition is a special case of equivo- cation, arising from the confusion of an universal and a collective term. In the premises of a syllogism we may affirm something of a class of things distributively^ that is, of each and any separately, and then we may in the con- clusion infer the same of the whole put together. Thus we may say that " all the angles of a triangle are less than two right angles," meaning that any of the angles is less than two right angles ; but we must not infer that all the angles put together are less than two right angles. We must not argue that because every member of a jury is very likely to judge erroneously, the jury as a whole are also ver> likely to judge erroneously ; nor that because each of the witnesses in a law case is liable to give false or mis- taken evidence, no confidence can be reposed in the con- current testimony of a number of witnesses. It is by a fallacy of Composition that protective duties are still sometimes upheld. Because any one or any few trades which enjoy protective duties are benefited thereby, it u supposed that all trades at once might be benefited simi 174 LOGICAL FALLACIES. [LES& larly ; but this is impossible, because the protection of one trade by raising prices injures all others. The Fallacy of Division is the converse of the pre- ceding, and consists in using the middle term col- lectively in the major premise but distributively in th minor, so that the whole is divided into its parts. Thus it might be argued, "All the angles of a triangle are 'together) equal to two right angles; ABC is an angle of a triangle ; therefore ABC is equal to two right angles." Or again, " The inhabitants of the town consist of men, women and children of all ages ; those who met in the Guildhall were inhabitants of the town; therefore they consisted of men, women and children of all ages;" or, " The judges of the court of appeal cannot misinterpret the law; Lord A.B. is a judge of the court of appeal; therefore he cannot misinterpret the law." The Fallacy of Accent consists in any ambiguity arising from a misplaced accent or emphasis thrown upon some word of a sentence. A ludicrous instance is liable to occur in reading chapter xiii. of the First Book of Kings, verse 27, where it is said of the prophet "And he spake to his sons, saying, Saddle me the ass. And they saddled him." The italics indicate that the word him was supplied by the translators of the authorized version, but it may suggest a very different meaning. The Com- mandment "Thou shalt not bear false witness against thy neighbour " may be made by a slight emphasis of the voice on the last word to imply that we are at liberty to bear false witness against other persons. Mr De Morgan who remarks this also points out that the erroneous quoting of an author, by unfairly separating a word from its context or italicising words which were not intended to be italicised, gives rise to cases of this fallacy. It is curious to observe how many and various may be the meanings attributable to the same sentence according .] LOGICAL FALLACIES. 17$ as emphasis is thrown upon one word or another. Thui the sentence "The study of Logic is not supposed tt communicate a knowledge of many useful facts," may b made to imply that the study of Logic does communicate such a knowledge although it is not supposed to ; or that it communicates a knowledge of a few useful facts ; or that it communicates a knowledge of many useless facts. This ambiguity may be explained by considering that if you deny a thing to have the group of qualities A y B, C, D, the truth of your statement will be satisfied by any one quality being absent, and an accented pronunciation will often be used to indicate that which the speaker believes to be absent. If you deny that a particular fruit is ripe and sweet and well-flavoured, it may be unripe and sweet and well-flavoured; or ripe and sour and well-flavour- ed; or ripe and sweet and ill-flavoured; or any two or even all three qualities may be absent. But if you deny it to be ripe and sweet and well-flavoured, the denial would be understood to refer to the last quality. Jeremy Bentham was so much afraid of being misled by this fallacy of accent that he employed a person to read to him, as I have heard, who had a peculiarly monotonous manner of reading. The Fallacy of the Figure of Speech is the sixth and last of the semi-logical fallacies, and is of a very trifling character. It appears to consist in any grammatical mistake or confusion between one part of speech and an- other. Aristotle gravely gives the following instance : " Whatever a man walks he tramples on ; a man walki the whole day ; therefore he tramples on the day." Heft n adverbial phrase is converted into a noun object LESSON XXL MATERIAL FALLACIES. THE Material fallacies are next to be considered; and thefc. importance is very great, although it is not easy tfl illustrate them by brief examples. There are altogethet seven kinds of such fallacies enumerated by Aristotle and adopted by subsequent logicians, as follows : 1. The Fallacy of Accident 2. The Converse Fallacy of Accident. 3. The Irrelevant Conclusion. 4. The Petitio Principii. 5. The Fallacy of the Consequent or Non seqnitor. 6. The False Cause. 7. The Fallacy of Many Questions. Of these the two first are conveniently described to- gether. The fallacy of accident consists in arguing erro- neously from a general rule to a special case, where a certain accidental circumstance renders the rule inappli- cable. The converse fallacy consists in arguing from a special case to a general one. This latter fallacy is usu- ally described by the Latin phrase a ditto secundum quid ad dictum simpliciter, meaning " from a statement undei a condition to a statement simply or without that con- dition." Mr De Morgan has remarked in his very inte* resting Chapter on Fallacies* that we ought to add a third fallacy, which would consist in arguing from om tpecial case to another special case. * formal Lo&, Chapter XIIL LESS, xxi.] MATERIAL FALLACIES. 177 I will try by a few examples to illustrate these kinds of fallacy, but much difficulty is often encountered in saying to which of the three any particular example is best r* ferred. A most ancient example repeated in almost every logical hand-book is as follows : " What you bought yes terday you eat to-day ; you bought raw meat yesterday; therefore you eat raw meat to-day." The assertion in the conclusion is made of meat with the accidental quality of rawness added, where the first premise evidently speaks oi the substance of the meat without regard to its accidental condition. This then is a case of the direct fallacy. If it is argued again that because wine acts as a poison when used in excess it is always a poison, we fall into the converse fallacy. It would be a case of the direct fallacy of accident to infer that a magistrate is justified in using his power to forward his own religious views, because every man has a right to inculcate his own opinions. Evidently a magistrate as a man has the rights of other men, but in his capacity of a magistrate he is distinguished from other men, and he must not infer of his special powers in this respect what is only true of his rights as a man. For another instance take the following : " He who thrusts a knife into another person should be punished ; a surgeon in operating does so; therefore he should be ounished." Though the fallacy of this is absurdly manifest, it is not so manifest how we are to classify the error. We may for instance say that as a general rule whoever stabs or cuts another is to be punished unless it can be snewn to have been done under exceptional cir- cumstances, as by a duly qualified surgeon acting for the i?ood of the person. In this case the example belongs to the direct fallacy of accident. In another view we might interpret the first premise to mean the special case of thrusting a knife maliciously; to argue from that to thf 12 178 MATERIAL FALLACIES. [L case of a surgeon would be to infer from one special CAM to another special case. It is undoubtedly true that to give to beggars promote! mendicancy and causes evil ; but if we interpret this tc mean that assistance is never to be given to those whc solicit it, we fall into the converse fallacy of accident, inferring of all who solicit alms what is only true of those who solicit alms as a profession. Similarly it is a very good rule to avoid lawsuits and quarrels, but only as a general rule, since there frequently arise circumstances in which resort to the law is a plain duty. Almost all the difficulties which we meet in matters of law ana moral duty arise from the impossibility of always ascer- taining exactly to what cases a legal or moral rule does or does not extend ; hence the interminable differences of opinion, even among the judges of the land. The Third Material Fallacy is that of the Irrelevant Conclusion, technically called the Ignoratio EUnchi, or literally Ignorance of the Refutation. It consists in arguing to the wrong point, or proving one thing in such a manner that it is supposed to be something else that is proved. Here again it would be difficult to adduce con- cise examples, because the fallacy usually occurs in the course of long harangues, where the multitude of words and figures leaves room for confusion of thought and forgetfulness. This fallacy is in fact the great resource of those who have to support a weak case. It is not un- known in the legal profession, and an attorney for the defendant in a lawsuit is said to have handed to the barrister his brief marked, "No case ; abuse the plaintiffs attorney." Whoever thus uses what is known as trgumentum ad homintm, that is an argument which rests, not upon the merit of the case, but the character or position of those engaged in it, commits this fallacy. II a man is accused of a crime it is no answer to say thai KXL] MATERIAL FALLACIES. ift the prosecutor is as bad. If a great change in the law is proposed in Parliament, it is an Irrelevant Conclusion tc argue that the proposer is not the right man to bring it forward. Everyone who gives advice lays himself open to the retort that he who preaches ought to practise, or that those who live in glass houses ought not to throw stones. Nevertheless there is no necessary connection between the character of the person giving advice and the goodness of the advice. The argumentum ad populum is another form 01 Irrelevant Conclusion, and consists in addressing argu- ments to a body of people calculated to excite their feel- ings and prevent them from forming a dispassionate judgment upon the matter in hand. It is the great weapon of rhetoricians and demagogues. Petitlo Principil is a familiar name, and the nature of the fallacy it denotes is precisely expressed in the phrase begging the question. Another apt name for the fallacy is circulus inprobando, or "a circle in the proof." It con- sists in taking the conclusion itself as one of the premises of an argument Of course the conclusion of a syllogism must always be contained or implied in the premises, but only when those premises are combined, and are dis- tinctly different assertions from the conclusion. Thus in the syllogism, Bi*C, AkB, therefore A is C, the conclusion is proved by being deduced from two propositions, neither of which is identical with it ; but if the truth of one of these premises itself depends upott .he following syllogism, AisC, therefore A is B t 12 * tdo MATERIAL FALLACIES. [Uta it is plain that we attempt to prove a proposition by itself which is as reasonable as attempting to support a body upon itself. It is not easy to illustrate this kind of fal- lacy by examples, because it usually occurs in long argu- ments, and especially in wordy metaphysical writings We are very likely to fall into it however when we employ a mixture of Saxon and Latin or Greek words, so as to appear to prove one proposition by another which is really the same expressed in different terms, as in the following: "Consciousness must be immediate cognition of an object ; for I cannot be said really to know a thing unless my mind has been affected by the thing itself." In the use of the disjunctive syllogism this fallacy is likely to happen ; for by enumerating only those alterna- tives which favour one view and forgetting the others it is easy to prove anything. An instance of this occurs in the celebrated sophism by which some of the ancient Greek philosophers proved that motion was impossible. For, said they, a moving body must move either in the place where it is or the place where it is not ; now it is absurd that a body can be where it is not, and if it moves it can- not be in the place where it is; therefore it cannot move at alL The error arises in the assumption of a premise which begs the question; the fact of course is that the body moves between the place where it is at one moment and the place where it is at the next moment. Jeremy Bentham however pointed out that the use even of a single name may imply a Petitio Principii. Thus in a Church assembly or synod, where a discussion is taking place as to whether a certain doctrine should be condemned, it would be a Petitio Principii to argue thai the doctrine is heresy, and therefore it ought to be con- demned. To assert that it is heresy is to beg the question, because every one understands by heresy a doctrine which is to be condemned. Similarly in Parliament XXL] MATERIAL FALLACIES. iSi bill is often opposed on the ground that it is unconstitu- tional and therefore ought to be rejected; but as no precise definition can be given of what is or is not con- stitutional, it means little more than that the measure is distasteful to the opponent. Names which are used in this fallacious manner were aptly called by Bentham Question-begging Epithets. In like manner we beg the question when we oppose any change by saying that it is un-English. The Fallacy of the Consequent is better understood by the familiar phrase non sequitur. We may apply this name to any argument which is of so loose and inconsequent a character that no one can discover any cogency in it. It thus amounts to little more than the Assertion of a conclusion which has no connection with the premises. Prof. De Morgan gives as an example the following : " Episcopacy is of Scripture origin ; the Church of England is the only episcopal Church in Eng- land; ergo, the Church established is the Church that should be supported." By the Fallacy of the False Cause I denote that which has generally been referred to by the Latin phrase mm causa pro causd. In this fallacy we assume that one thing is the cause of another without any sufficient grounds. A change in the weather is even yet attributed to the new moon or full moon which had occurred shortly before, although it has been demonstrated over and over again that the moon can have no such effect. In former centuries any plague or other public calamity which fol- lowed the appearance of a comet or an eclipse was considered to be the result of it The Latin phrase post ko^ ergo propter hoc (after this and therefore in conse- quence of this) exactly describes the character of these fallacious conclusions. Though we no longer dread signs and omens, yet we often enough commit the fallacy; as 1 82 MATERIAL FALLACIES. [LESS. XXL when we assume that all the prosperity of England is the result of the national character, forgetting that the plenti- ful coal in the country and its maritime position have contributed to our material wealth. It is no doubt equally fallacious to attribute no importance to national character; and to argue that because England has in past centuries misgoverned Ireland all the present evils of Ireland art due to that misgovernment Lastly there is the somewhat trivial Fallacy of Manj Questions, which is committed by those who so combine two or three questions into one that no true answer can be given to them. I cannot think of a better example than the vulgar pleasantry of asking, " Have you left ofl beating your mother?" Questions equally as unfair are constantly asked by barristers examining witnesses in a court of justice, and no one can properly be required to answer Yes or No to every question which may be ad- dressed to him. As Aristotle says, " Several questions put as one should be at once decomposed into theii several parts. Only a single question admits of a single answer : so that neither several predicates of one subject nor one predicate of several subjects, but only one predi- cate of one subject, ought to be affirmed or denied ir a tingle answer." Read Prof, de Morgan's excellent and amusing Chaptei on Fallacies, Formal Logic, Ch. XHL iVhately's remarks on Fallacies, Elements Book IIL, are often very original anc* acute. LESSON XXII. THE QUANTIFICATION OF THE PREDICATE THE syllogism has been explained in the preceding three lessons almost exactly in the form in which it has been taught for more than two thousand years. Just as Geo- metry has been taught in the way and order first adopted by the ancient Greek writer Euclid, so Logic has been taught nearly as Aristotle taught it about the year 335 B.C. But within the last few years teachers have at last come to the conclusion in England that Euclid's ideas of Geometry are not as perfect as could be desired. During the last 30 or 40 years also it has been gradually made apparent that Aristotle's syllogism is not an absolutely perfect system of logical deduction. In fact, certain eminent writers, especially Sir William Hamilton, Pro- fessor De Morgan, Archbishop Thomson and Dr Boole, have shewn that we need to make improvements from the very basis of the science. This reform in Logic is called by the somewhat mys- terious name of the quantification of the predicate, but the reader who has found no insuperable difficulty io the preceding lessons need not fear one here. To quan- tify the predicate is simply to state whether the ivhoU of tkt part only of the predicate agrees t*ith or differ* from the subject. In this proposition, All metals are elements, " i4 THE QUANTIFICATION [LESS the subject is quantified, but the predicate is not; we know that all metals are elements, but the proposition does not distinctly assert whether metals make the whole of the elements or not. In the quantified proposition " All metals are some elements," the little word some expresses clearly that in reality the metals form only a part of the elements. Aristotle avoid- ed the use of any mark of quantity by assuming, as we have seen, that all affirmative propositions have a par- ticular predicate, like the example just given ; and that only negative propositions have a distributed or universal predicate. The fact however is that he was entirely in error, and thus excluded from his system an infinite number of affirmative propositions which are universal in both terms. It is true that "All equilateral triangles are all equiangular triangles," but this proposition could not have appeared in his system except in the mutilated form "All equilateral triangles are equiangular." Such a proposition as "London is the capital of England," or " Iron is the cheapest metal," had no proper place whatever in his syllogism, since both terms aie singular and identical with each other, and both are accordingly universal. As soon as we allow the quantity of the predicate to De stated the forms of reasoning become much simplified. We may first consider the process of conversion. In our lesson on the subject it was necessary to distinguish be- tween conversion by limitation and simple conversion. But now one single process of simple conversion is suffi cient for all kinds of propositions. Thus the quantified proposition of the form A, "All metals are some elements," XXIL] OF THE PREDICATE. 18$ is simply converted into " Some elements are all metals, * The particular affirmative proposition " Some metals are some brittle substances * becomes by mere transposition of terms " Some brittle substances are some metals." rhe particular negative proposition " Some men are not (any) trustworthy persons 11 is also converted simply into " Not any trustworthy persons are some men," though the result may appear less satisfactory in this form than in the affirmative form, as follows, " Some men are some not-trustworthy persons," converted simply into " Some not -trustworthy persons are some men." The universal negative proposition 2 is converted simply as before, and finally we have a new affirmative proposition universal both in subject and predicate ; as in "All equilateral triangles are all equiangular triangles," which may obviously be converted simply into "All equiangular triangles are all equilateral triangles." This doubly universal affirmative proposition is of most frequent occurrence; as in the case of all definitions and singular propositions ; I may give as instances "Honesty is the best policy," "The greatest truths are the simplest truths," "Virtue alone is happiness below," " Self-exaltation is the fool's paradise." When affirmative propositions are expressed in the quantified form all immediate inferences can be readily drawn from them by this one rule, that whatever we do with one term we should do with the other term. Thus from the doubly universal proposition, " Honesty is the best policy," we infer that "what is not the best policy is 1 86 THE QUANTIFICATION [LESS not honesty," and also " what is not honesty is not the best policy." From this proposition in fact we can draw two contraposltlves ; but the reader will carefully rememba that from the ordinary unquantified proposition A we can only draw one contrapositive (see p. 84). Thus U "metals arc elements" we must not say that "what are not metals are not elements." But if we quantify the predicate thus, "All metals are some elements," we may infer that " what are not metals are not some elements." Immediate inference by added determinant and complex conception can also be applied in either direction to quantified propositions without fear of the errors noticed in pp. 86-7. It is clear that in admitting the mark of quantity before the predicate we shall double the number of propositions which must be admitted into the syllogism, because the predicate of each of the four propositions A, E, I, may be either universal or particular. Thus we arrive at a list of eight conceivable kinds of propositions, which are stated in the following table. U All X is all Y. j I Some X is some K I Affirmative A All A" is some K. | propositions. Y Some X is all K E No X is (any) Y. ) Some X is not some Y. I Negative 1 No X is some K | propositions. Some X is no Y. The letters X and Y are used to stand for any subject uid predicate respectively, and the reader by substituting various terms can easily make propositions of each kind. The symbolic letters on the left-hand side were proposed by Archbishop Thomson as a convenient mode of reter XXIL] OF THE PREDICATE. ife ring to each of the eight propositions, and are ver> suitably chosen. The doubly universal affirmative pro- position is called U ; the simple converse of A is called f ; the Greek letter i\ (Eta, e) is applied to the proposi- tion obtained by changing the universal predicate of I into a particular predicate ; and the Greek (Omega, d) is applied to the proposition similarly determined from 0, All these eight propositions are employed by Sir W. Ha- milton, but Archbishop Thomson considers that two of them, n and , are never really used. It is remarkable that a complete table of the above eight propositions was given by Mr George Bentham in a work called Outline of a New System of Logic, published in 1827, several years previous to the earliest of the logical publications of 3ir W. Hamilton. But Mr Bentham considered that some rf the propositions are hardly to be distinguished from others; as T from A, of which it is the simple converse; or ij from 0. The employment even of the additional two proposi- tions U and Y introduced by Thomson much extends the list of possible syllogisms, making them altogether 62 in number, without counting the fourth figure, which is not employed by Hamilton and Thomson. When the whole eight propositions are admitted into use we are obliged to extend the list of possible syllogisms so as to contain 12 affirmative and 24 negative moods in each of the three first figures. The whole of these moods are conveniently stated in the table on.the next page, given by Archbishop Thomson at p. 188 of his Laws of Thought. Sir W. Hamilton also devised a curious system of notation for exhibiting all the moods of the syllogism in a dear manner. He always employed the letter M to denote the middle term of the syllogism, and the two letters C and F (the Greek capital letter Gamma) for the two terms appearing in the conclusion. The copula of th THE QUANTIFICATION \\XSL TaMe of Moods of Ou Syllogism. FIRST FIGURE. SECOND Fio. THIRD FIOUR*. Aftnn, Neg. Affirm. , Neg. Affirm. Nef. 1 UUU EUE UUU EUE UUU EUE UEE UEE UEF ti AYI ,Y YYI OY AAI 7 A A0 YO- A,* iii AAA ,A, YAA OA, AYA A,, Y, , AO*, iv YYY OYO AYY ,YO YAY OAO YOO AOO Y,0 V All ,I YII Gin All ,1. * IYI Y! IYI IY! IAI ^A! Til UYY EYO UYY EYO UAY EAO UOO UOO u,o riii AUA ,u, YUA OU, AUA ,u, AE, YE, AE, ix UAA EAE UAA EAE UYA EYE u,, u,, UOi? X YUY ouo AUY ,UO YUY OUO YEE AEE YEE xi UII EIO UII EIO UII EIO xii IUI Tul IUI Tul IUI Tul IE, I IE, IE, proposition was indicated by a line thickened towards the subject ; thus C BB_ M means that " Cis M.* To indicate the quantity of the terms Hamilton inserted a ttHl/1 OF THE PREDICATE. 189 colon (:) between the term and the copula when the quantity is universal, and a comma (,) when the quantity is particular. Thus we readily express the following affirmative propositions. C : m ,M All Cs are some JTs (A) C : mm\ :M All C's are all J/'s (V) C , , M Some C's are some J/'s (I) and so on. Any affirmative proposition can be converted into the corresponding negative proposition by drawing a stroke through the line denoting the copula, as in the following C : MM+- '-M No C is any M (M) C , w+ - : M Some C is not any M (0) C , MM^ , M Some (7 is not some M () Any syllogism can be represented by placing M the middle term in the centre and connecting it on each side with the other terms. The copula representing the con- clusion can then be placed below; Barbara is expressed as follows Tke negative mood Celarent is similarly Genre in the second figure is thus represented- Sir W. Hamilton also proposed a new law or uprenw Anon of the syllogism by which the validity of an! forms 190 THE QUANTIFICATION [LESS, of the syllogism might be tested. This was stated in tht following words : "What worse relation of subject and predicate subsists between either of two terms and a common third term, with which both are related, and one at least positively so that relation subsists between these two terms themselves." By a worse relation, Sir William means that a negative relation is worse than an affirmative and a particular than a universal. This canon thus expresses the rules that if there be a negative premise the conclusion must be nega- tive, and if there be a particular premise the conclusion must be particular. Special canons were also developed for each of the three figures, but in thus rendering the system complex the advantages of the quantified form of proposition seem to be lost. Prof. De Morgan also discovered the advantages of the quantified predicate, and invented a system differing greatly from that of Sir W. Hamilton. It is fully ex- plained in his Formal Logic, The Syllabus of a new System of Logic, and various important memoirs on the Syllogism in the Transactions of the Cambridge Philo- sophical Society. In these works is also given a com- plete explanation of the " Numerically Definite Syllogism." Mr De Morgan pointed out that two particular premises may often give a valid conclusion provided that the actual quantities of the two terms are stated, and when added together exceed the quantity of the middle term. Thus if the majority of a public meeting vote for the first resolution, and a majority also vote for the second, it follows necessarily that some who voted for the first voted also for the second. The two majorities added togethei exceed the whole number of the meeting, so that they could not consist of entirely different people. They may indeed consist of exactly the same people ; but all that we can deduce from the premises is that the excess of the xxil.] OF THE PREDICATE. 19* two majorities added together over the number of the meeting must have voted in favour of each resolution This kind of inference has by Sir W. Hamilton been said to depend on ultra-total distribution ; and the name of Plurative Propositions has been proposed for all those irhich give a distinct idea of the fraction or number of the subject involved in the assertion. T. Spencer Baynes, Essay on the new Analytic oj Logical Forms; Edinburgh, 1850. Prof. Bowen's Treatise on Logic or the Laws of Pure Thought, Cambridge, U. S. 1866 (Triibner and Co.) gives a full and excellent account of Hamilton's Logic. LESSON XXIII. BOOLE'S SYSTEM OF LOGIC. IT would not in the least be possible to give in an ele- mentary work a notion of the system of indirect inference first discovered by the late Dr Boole, the Professor of Mathematics at the Queen's College, Cork. This system was founded as mentioned in the last lesson upon the Quantification of the Predicate, but Dr Boole regarded Logic as a branch of Mathematics, and believed that he could arrive at every possible inference by the principles of algebra. The process as actually employed by him is very obscure and difficult ; and hardly any attempt to introduce it into elementary text-books of Logic has yet been made. I have been able to arrive at exactly the same results 192 BOOLE'S SYSTEM OF LOGIC. [LESS. AS Dr Boole without the use of any mathematics; arui though the very simple process which I am going to describe can hardly be said to be strictly Dr Boole's logic, it is yet very similar to it and can prove everything that Dr Boole proved. This Method of Indirect Inference is founded upon the three primary Laws of Though! stated in Lesson xiv., and the reader who may have thought them mere useless truisms will perhaps be sui- prised to find how extensive and elegant a system of deduction may be derived from them. The law of excluded middle enables us to assert that anything must either have a given quality or must have it not Thus if iron be the thing, and combustibility the quality, anyone must see that "Iron is either combustible or incombustible." This division of alternatives may be repeated as often as we like. Thus let Book be the class of things to be di- vided, and English and Scientific two qualities. Then any book must be either English or not English; again an English book must be either Scientific or not Scientific, and the same may be said of books which are not English Thus we can at once divide books into four classes Books, English and Scientific. Books, English and not-Scientific. Books, not-English and Scientific. Books, not-English and not- Scientific. This is what we may call an exhaustive division of the class Books; for there is no possible book which does not fall into one division or other of these four, on account of the simple reason, that if it does not fall into ainy of the three first it must fall into the last. The pro- cess can be repeated without end, as long as any new circumstance can be suggested as the ground of division. Thus we might divide each class again according -as the AXIII.] BOOLE'S SYSTEM OF LOGIC. 193 books are octavo or not octavo, bound or unbound, pub- lished in London or elsewhere, and so on. We shall call Ihis process of twofold division, which is really the pro- ;ess of Dichotomy mentioned in p. 107, the development tf a term, because it enables us always to develope the itxnost number of alternatives which need be considered. As a general rule it is not likely that all the alterna- tives thus unfolded or developed can exist, and the nexf point is to ascertain how many do or may exist The Lavi of Contradiction asserts that nothing can combine con- tradictory attributes or qualities, and if we meet with any term which is thus self-contradictory we are authorized at once to strike it out of the list Now consider our old example of a syllogism : Iron is a metal ; All metals are elements ; Therefore iron is an element We can readily prove this conclusion by the indirect method. For if we develope the term iron, we have foui alternatives; thus- Iron, metal, element Iron, metal, not-element Iron, not- metal, element Iron, not-metal, not-element But if we compare each of these alternatives with the premises of the syllogism, it will be apparent that several of them are incapable of existing. Iron, we are informed, is a metal Hence no class of things "iron, not-metal" can exist Thus we are enabled by the first premise to strike out both of the last two alternatives which combine iron and not-metal. The second alternative, again, com- bines metal and not-element ; but as the second premise informs us that "all metals are elements," it must b struck out There remains, then, only one alternative 194 BOOLE'S SYSTEM OF LOGIC. [LESS which is capable of existing if the premises be true, and a j there cannot conceivably be more alternatives than those considered, it follows demonstratively that iron occurs only in combination with the qualities of metal and ele- ment, or, in brief, that it is an element We can, however, prove not only the ordinary syllo- gistic conclusion, but any other conclusion which can b< drawn from the same premises ; the syllogistic conclusion is in fact only one out of many which can usually be ob- tained from given premises. Suppose, for instance, that we wish to know what is the nature of the term or class not-element, so far as we can learn it from the premises just considered. We can develope the alternatives of this term, just as we did those of iron, and get the following Not-element, iron, metal. Not-element, iron, not-metaL Not-element, not-iron, metal. Not-clement, not-iron, not-metal Compare these combinations as before with the premises. The first it is easily seen cannot exist, because all metals are elements ; for the same reason the third cannot exist ; the second is likewise excluded, because iron is a metal and cannot exist in combination with the qualities of not- metaL Hence there remains only one combination to represent the class desired namely, Not-element, not-iron, not-metaL Thus we learn from the premises that every not-ele eaent is not a metal and is not iron. As another example of this kind of deductive procesi I will take a case of the Disjunctive Syllogism, in the ne gatire mood, as follows : A fungus is either plant or animal , A fungus is not an animal ; Therefore it is a plant ttlil.] BOOL&S SYSTEM OF LOGIC. ifc Now if we develope all the possible ways in which responding sides proportional and angles not equal (abcD\ or neither their corresponding angles equal noi corresponding sides proportional (abed)" In performing this method of inference it is soon seeu to proceed in a very simple mechanical manner, and th< only inconvenience is the large number of alternatives 01 combinations to be examined. I have, therefore, devised several modes by which the labour can be decreased; the simplest of these consists in engraving the series of 1 6 combinations on the opposite page, which occur over and over again in problems, with larger and smaller sets, upon a common writing slate, so that the excluded ones may be readily struck out with a common slate pencil, and yet the series may be employed again for any future logical question. A second device, which I have called the "Logical abacus," is constructed by printing the letters upon slips of wood furnished with pins, contrived so that any part or class of the combinations can be picked out mechanically with very little trouble ; and a logical problem is thus solved by the hand, rather than by the head More recently however I have reduced the system to a completely mechanical form, and have thus embodied the whole of the indirect process of inference in what may be called a Logical Machine. In the front of the machine are seen certain moveable wooden rods carrying the set of 16 combinations of letters which are seen on the preceding page. At the foot are 21 keys like those of a piano; eight keys towards the left hand are marked with the letters A, a, B y b, C, f t D, d, and are intended to represent these terms when occurring in the subject of a proposition. Eight other keys towards the right hand represent the same letters or terms when oc- curring in the predicate. The copula of a proposition if 200 BOOLES SYSTEM OF LOGIC. [LI* represented by a key in the middle of the series the ful stop by one to the extreme right, while there are two othei keys which serve for the disjunctive conjunction or t ac cording as it occurs in subject or predicate. Now if the letters be taken to stand for the terms of a syllogism or any other logical argument, and the keys of the instru- ment be pressed exactly in the order corresponding to the words of the premises, the 16 combinations will be so selected and arranged thereby that at the end only the possible combinations will remain in view. Any question can then be asked of the machine, and an infallible answei will be obtained from the combinations remaining. The internal construction of the machine is such, therefore, as actually to perform the work of inference which, in Dr Boole's system, was performed by a very complicated mathematical calculation. It should be added, that there is one remaining key to the extreme left which has the effect of obliterating all previous operations and restoring all the combinations to their original place, so that the machine is then ready for the performance of any new problem. An account of this logical machine may be found in the Proceedings of the Royal Society for Jan. 2oth, 1870, the machine having on that day been exhibited in action to the Fellows of the Society. The principles of the method }f inference here described are more completely stated in The Substitution of Similar s*, and the Pure Logic\, which [ published in the years 1869 and 1864. I may add, that :hr first-named of these works contains certain views as >o the real nature of the process of inference which I do ' Ttu Substitution of Similars, the true Principle of Reason- nf, derrved from a modification of Aristotle's Dictum. Mac- aulLui and Co. 1869. t Purt Lofic, or the Logic of Quality apart from Quantity >&* Reward Stanford, Charing Cross. XXIIL] BOOLE^S SYSTEM OF LOGIC. aoi not think it desirable to introduce into an elementary work like the present, on account of their speculative character. The process of inference, on the other hand, which I ha/c derived from Boole's system is of so self-evident a charac- ter, and is so clearly proved to be true by its reduction to a mechanical form, that I do not hesitate to bring it to the reader's notice. George Boole, Mathematical Analysis of Logic, 1847 An Investigation of the Laws of Thought. Londor Walton and Maberly, 1854. LESSON XXIV. ON METHOD, ANALYSIS AND SYNTHESIS. IT has been held by many writers on Logic that, in addi- tion to the three parts of logical doctrine which treat successively of Terms, Propositions and Syllogisms, there was a fourth part, which treats of method. Just as the doctrine of Judgment considers the arranging of terms and their combination into Propositions, and the doc- trine of Syllogism considers the arranging of propositions that they may form arguments, so there should in like manner be a fourth part, called Method, which should govern the arrangement of syllogisms and their combina- tion into a complete discourse. Method is accordingly defined as consisting in such a disposition oj the parts o) m discourse that the whole may be most easily intelligible. The celebrated Peter Ramus, who perished in the massacre of St Bartholomew, first proposed to make method in this manner a part of the science of Logic ; but 2t* ON METHOD, ANALYSIS [LESS it may well be doubted whether any definite set of rulej or principles can be given to guide us in the arrangement of "arguments. Every different discourse must consist oi arguments arranged in accordance wilh the peculiar nature af the subject ; and no general rules can be given for treat- ing things which are infinitely various in the mode of treat- ment required. Accordingly the supposed general rules of method are no better than truisms, that is, they tell us nothing more than we must be supposed to know before- hand Thus, we are instructed in composing any dis- course to be careful that 1. Nothing should be wanting or redundant. 2. The separate parts should agree with each other. 3. Nothing should be treated unless it is suitable to the subject or purpose. 4. The separate parts should be connected by suit- able transitions. But it is evident that the whole difficulty consists in deciding what is wanting or redundant, suitable or con- sistent. Rules of this kind simply tell us to do what we ought to do, without defining what that is. There exist nevertheless certain general modes of treating any subject which can be clearly distinguished, and should be well understood by the logical student. Logic cannot teach him exactly how and when to use each kind of method, but it can teach him the natures and powers of the methods, so that he will be more Ukeiy to use them rightly. We must distinguish, 1. The method of discovery, 2. The method of instruction. The method of discovery is employed in the acquisi- tion of knowledge, and really consists in those processes af inference and induction, by which general truths ara ascertained from the collection and examination of par- AND SYNTHESIS. 205 ticular facts. This method will be the subject of most ol our remaining Lessons. The second method only applies when knowledge has already been acquired and express- ed in the form of general laws, rules, principles or truths jo that we have only to make ourselves acquainted will hese and observe the due mode of applying them tt particular cases, in order to possess a complete acquaint ance with the subject A student, for example, in learning Latin, Greek, French, German, or any well-known language, receives a complete Grammar and Syntax setting forth *he whole of the principles, rules and nature of the language. He receives these instructions, and takes them to be true on the authority of the teacher, or the writer of the book; and after rendering them familiar to his mind he has nothing to do but to combine and apply the rules in read- ing or composing the language. He follows, in short, the method of Instruction. But this is an entirely differ- ent and opposite process to that which the scholar must pursue who has received some writings in an unknown language, and is endeavouring to make out the alpha- bet, words, grammar, and syntax of the language. He possesses not the laws of grammar, but words and sen- tences obeying those laws, and he has to detect the laws if possible by observing their effects on the written language. He pursues, in short, the method of discovery consisting in a tedious comparison of letters, words, and phrases, such as shall disclose the more frequent combi- nations and forms in which they occur. The process would be a strictly inductive one, such as I shall partially exemplify in the Lessons on Induction ; but it is far more difficult than the method of Instruction, and depends to a great extent on the happy use of conjecture and hypothesis which demands a certain skill and inventive ability. Exactly the same may be said of the investigation o( so4 ON METHOD, ANALYSIS [LESi natural things and events. The principles of mechanic* ol the lever, inclined plane, and other Mechanical Powers or the Laws of Motion, seem comparatively simple and obvious as explained to us in books of instruction. But the early philosophers did not possess such books ; they had only the Book of Nature, in which is set forth not the laws but the results of the laws, and it was only after the most patient and skilful investigation, and after hundreds of mistakes, that those laws were . ascertained It is very easy now to understand the Copernican system of Astronomy, which represents the planets as revolving round the sun in orbits of various magnitude. Once know- ing the theory we can readily see why the planets have such various movements and positions, and why they sometimes stand still ; it is easy to see, too, why in ad- dition to their own proper motions they all go round the earth apparently every day in consequence of the earth's diurnal rotation. But all these changes were exceedingly puzzling to the ancients, who regarded the earth as stand- ing still. The method of discovery thus begins with facts ap- parent to the senses, and has the difficult task of detecting those universal laws or general principles which can only be comprehended by intellect. It has been aptly said that the method of discovery thus proceeds front thing* better known to us, or our senses (nobis notiord), to those which are more simple or better known in nature (notiora nature). The method of Instruction proceeds in the opposite direction, beginning with the things notiora nature, and proceeding to show or explain the things nobis notiora. The difference is almost like that between hiding and seeking. He who has hidden a thing knows where to find it; but this is not the position of a discoverer, who has no clue except such as he may meet in his own diligent and sagacious search. AND SYNTHESIS. 903 Closely corresponding to the distinction between the methods of Discovery and Instruction is that between the methods of Analysis and Synthesis. It is very im- portant indeed that the reader should clearly apprehend the meanings of these terms in their several applications. Analysis is the process of separating a whole into its parts, and synthesis the combination ot parts into a whole. The analytical chemist, who receives a piece ol mineral for examination, may be able to separate com- pletely the several chemical elements of which it is composed and ascertain their nature and comparative quantities ; this is chemical analysis. In other cases the chemist mixes together carefully weighed quantities of certain simple substances and combines them into a new compound substance ; this is chemical synthesis. Logical analysis and synthesis must not be confused with the physical actions, but they are nevertheless actions ol mind of an analogous character. In logical synthesis we begin with the simplest possible notions or ideas, and combine them together. We have the best possible example in the elements of Geometry In Euclid we begin with certain simple notions of points, straight lines, angles, right angles, circles, &c. Putting together three straight lines we make a triangle ; joining to this the notion of a right-angle, we form the notion of a right-angled triangle. Joining four other equal lines at right angles to each other we gain the idea of a square, and if we then conceive such a square to be formed upon each of the sides of a right-angled triangle, and reason from the necessary qualities of these figures, we discover that the two squares upon the sides containing the right angle must together be exactly equal to the square upon the third side, as shewn in the 47th Proposition ol Euclid's first book. This is a perfect instance ol com* oining simple ideas into more complex ones. 06 ON METHOD, ANALYSIS [LESS We have often, however, in Geometry to pursue the opposite course of Analysis. A complicated geometrical figure may be given to us, and we may have, in order to prove the properties which it possesses, to resolve it inte its separate parts, and to consider the properties of thost parts each distinct from the others. A similar distinction between the analytical and syn thetic methods can be traced throughout the natura sciences. By keeping exact registers of the appearance and changes of the weather we may readily acquire at immense collection of facts, each such recorded fact implying a multitude of different circumstances occurring together. Thus in any storm or shower of rain we have to consider the direction and force of the wind ; the tem- perature and moistness of the air ; the height and forms ol the clouds; the quantity of rain which falls, or the light- ning and thunder which occur with it. If we proceed by analysis only to explain the changes of the weather we should have to try resolving each storm or change of weather into its separate circumstances, and comparing each with every other to discover what circumstances usually go together. We might thus ascertain no doubt with considerable certainty what kinds of clouds, and what changes of the wind, temperature, moisture, &c^ usually precede any kind of storm, and we might even in time give some imperfect explanation of what takes place in the atmosphere. But we might also apply with advantage the syn- thetical method. By previous chemical investigations we know that the atmosphere consists mainly of the two fixed gases, oxygen and nitrogen, with the vapour of irater, the latter being very variable in quantity. We can try experimentally what takes place when portions of such air of various degrees of moistness are com- pressed or allowed to expand, or are mixed togethei, at XXIV.] AND SYNTHESIS. so? often happens in the atmosphere. It is thus discovered that whenever moist air is allowed to expand cloud is produced, and it may be drops of rain. Dr Hut- ton, too, found that whenever cold moist air is mixed with warm moist air cloud is again produced. We can safely argue from such small experiments to what takei place in the atmosphere. Putting together synthetically, from the sciences of chemistry, mechanics, and electricity,, all that we know of air, wind, cloud and lightning, we are able to explain what takes place in a thunder-storm far more completely than we could do by merely observing directly what happens in the storm. We are here how- ever anticipating the methods of inductive investigation, which we must consider in the following lessons. It will appear that Induction is equivalent to analysis, and that the deductive kinds of reasoning which we have treated in prior lessons are of a synthetic character. It has been said that the synthetic method usually corresponds to the method of instruction and the analytic method to that of discovery. But it may be possible to discover new truths by synthesis and to teach old ones by analysis. Sir John Herschel in his well-known Out- lines of Astronomy partially adopts the analytic method; he supposes a spectator in the first place to survey the appearances of the heavenly bodies and the surface of the earth, and to seek an explanation; he then leads him through a course of arguments to show ;hat these appearances really indicate the rotundity of the earth, its revolution about its own axis and round the sun, and its subordinate position as one of the smaller planets of the solar system. Mr Norman Lockyer's Elementary Lessons in Astronomy is a clear example of the synthetic method of instruction ; for he commences by describing the sun, the centre of the system, and successively adds the planet! and other members of the system, until at last we havf 208 ON METHOD, ANALYSIS [LESi the complete picture ; and the reader who has temporarily received everything on the writer's authority, sees that the description corresponds with the truth. Each method, it must be allowed, has its own advantages. It must be carefully observed that the meaning oi Analysis, and therefore that of synthesis, varies according as we look to the intension or extension of terms. To divide or analyse a class of things in extension I must add a quality or difference. Thus I divide the class organism when I add the quality vegetable p , and separate vegetable organism from what is not vegetable. Analysis in exten- sion is therefore the same process as synthesis in inten^ Ion ; and vice versd, whenever I separate or analyse a group of qualities each pan belongs to a larger class oi things in extension. When I analyse the notion vegetable organism, and regard the notion organism apart from vegetable, it is apparent that I really add the whole class of animal organisms to the class I am considering so that analysis in intension is synthesis in extension. The reader who has well considered the contents of Lessons V. and XII. will probably see that this connection of the two processes is only a re-statement of the law, (p. 40), that "as the intension of a term is increased the extension is decreased." To express the difference between knowledge derived deductively and that obtained inductively the Latin phrases a priori and a posteriori are often used. By A priori reasoning we mean argument based on truthi previously known ; A posteriori reasoning, on the contrary, proceeds to infer from the consequences of a general truth what that general truth is. Many philosophers con- sider that the mind is naturally in possession of certain laws or truths which it must recognise in every act of thought ; all such, if they exist, would be a priori truths. It cannot be doubted, for instance, that we must always KXIV.] AND SYNTHESIS. 209 recognise m thought the three Primary Laws of Thought considered in Lesson xiv. We have there an a priori knowledge that "matter cannot both have weight and be without weight," or that "every thing must be either self- hiininous or 'not self-lutninous." But there is no law of thought which can oblige us to think that matter has weight, and luminous ether has not weight ; that Jupitei and Venus are not self-luminous, but that comets are to some extent self-luminous. These are facts which are no doubt necessary consequences of the laws of nature and the general constitution of the world ; but as we are not naturally acquainted with all the secrets of creation, we have to learn them by observation, or by the a posteriori method. It is not however usual at the present time to restrict the name a priori to truths obtained altogether without recourse to observation. Knowledge may originally be of an a posteriori origin, and yet having been long in possession, and having acquired the greatest certainty, it may be the ground of deductions, and may then be said to give a priori knowledge. Thus it is now believed by all scientific men that force cannot be created or destroy- ed by any of the processes of nature. If this be true the force which disappears when a bullet strikes a target must be converted into something else, and on a priori grounds we may assert that heat will be the result. It is true that we might easily learn the same truth a posteriori, by picking uv portions of a bullet which has just struck a target ana observing that they are warm. But there is a great advantage in a priori knowledge ; we can often apply it in cases where experiment or observation would be difficult. If I lift a stone and then drop it, the most delicate instruments could hardly show that the stone was heated by striking the earth ; yet on a priori grounds I know that it must have been so, and can easily calcu- 14 210 PERFECT INDUCTION AND (LESS late the amount of heat produced. Similarly we know, without the trouble of observation, that the Falls of Ni- agara and all other waterfalls produce heat This is fairly an instance of a priori knowledge because no one that I have heard of has tried the fact or proved it a pos* ieriorij nevertheless the knowledge is originally founded on the experiments of Mr Joule, who observed in certain well-chosen cases how much force is equivalent to a certain amount of heat. The reader, however, should take care not to confuse the meaning of ct priori thus explained with that given to the words by the philoso- phers who hold the mind to be in the possession of know- ledge independently of all observation. It is not difficult to see that the a priori method is equivalent to the synthetic method (see p. 205) considered in intension, the a posteriori method of course being equi- valent to the analytic method. But the same difference is really expressed in the words deductive and inductive; and we shall frequently need to consider it in the following lessons. For general remarks upon Method see the Port Roytl Logic, Part iv. LESSON XXV. PERFECT INDUCTION AND THE INDUCTIVE SYLLOGISM. Wl have in previous lessons considered deductive rea- soning, which consists in combining two or more genera] propositions synthetically, and thus arriving at a con- clusion which is a proposition or truth of less generality xxv.] THE INDUCTIVE SYLLOGISM. 211 than the premises, that is to say, it applies to fewer indi- vidual instances than the separate premises from which it was inferred. When I combine the general truth that " metals are good conductors of heat," with the truth that "aluminium is a metal," I am enabled by a syllogism in the mood Barbara to infer that " aluminium is a good con- ductor of heat." As this is a proposition concerning one metal only, it is evidently less general than the premise, which referred to all metals whatsoever. In induction, on the contrary, we proceed from less general, or even from individual facts, to more general propositions, truths, or, as we shall often call them, Laws of Nature. When it is known that Mercury moves in an elliptic orbit round the Sun, as also Venus, the Earth, Mars, Jupiter, &c., we are able to arrive at the simple and general truth that " all the planets move in elliptic orbits round the sun." This is an example of an inductive process of reasoning. It is true that we may reason without rendering our conclusion either more or less general than the premises, as in the following : Snowdon is the highest mountain in England or Wales. Snowdon is not so high as Ben Nevis. Therefore the highest mountain in England or Wales is not so high as Ben Nevis. Again : Lithium is the lightest metal known. Lithium is the metal indicated by one bright red line in the spectrum*. Therefore the lightest metal known is the metal indicated by a spectrum of one bright red line. In these examples all the propositions are singular propositions, and merely assert the identity of singular * Roscoe's Lessons in Elementary Chemistry, p. 199. 142 fli PERFECT INDUCTION AND terms, so that there is no alteration of generality. Each conclusion applies to just such an object as each of the premises applies to. To this kind of reasoning the apt name of Traduction has been given. Induction is a much more difficult and more important rind of reasoning process than Traduction or even Deduc- tion ; for it is engaged in detecting the general laws or uniformities, the relations of cause and effect, or hi short ail the general truths that may be asserted concerning the numberless and very diverse events that take place in the natural world around us. The greater part, if not, as some philosophers think, the whole of our knowledge, is ultimately due to inductive reasoning. The mind, it is plausibly said, is not furnished with knowledge in the form of general propositions ready made and stamped upon it, but is endowed with powers of observation, com- parison, and reasoning, which are adequate, when well educated and exercised, to procure knowledge of the world without us and the world within the human mind. Even when we argue synthetically and deductively from simple ideas and truths which seem to be ready in the mind, as in the case of the science of geometry, it may be that we have gathered those simple ideas and truths from previous observation or induction of an almost unconscious kind This is a debated point upon which I will not here speak positively ; but if the truth be as stated, Induction will be the mode by which all the materials of knowledge are brought to the mind and analysed. Deduction wiL then be the almost equally important process by which the knowledge thus acquired is utilised, and by which new Inductions of a more complicated character, as we shall lee, are rendered possible. An Induction, that is an act of Inductive reasoning, is called Perfect when all the possible cases or instances to which the conclusion can refer, have been examined and THE INDUCTIVE SYLLOGISM. aij enumerated in the premises. If, as usually happens, it is impossible to examine all cases, since they may occur at future times or in distant parts of the earth or othei regions of the universe, the Induction is called Imperfect The assertion that all the months of the year are of less length than thirty-two days is derived from Perfect In- duction, and is a certain conclusion because the calendar is a human institution, so that we know beyond doubt how many months there are, and can readily ascertain that each of them is less than thirty-two days in length. But the assertion that all the planets move in one direction round the sun, from West to East, is derived from Imper- fect Induction ; for it is possible that there exist planets more distant than the most distant-known planet Nep- tune, and to such a planet of course the assertion would apply. Hence it is obvious that there is a great difference between Perfect and Imperfect Induction. The latter includes some process by which we are enabled to make assertions concerning things that we have never seen or examined or even known to exist But it must be care- fully remembered also that no Imperfect Induction can give a certain conclusion. It may be highly probable or nearly certain that the cases unexamined will resemble those which have been examined, but it can never be certain. It is quite possible, for instance, that a new planet might go round the sun in an opposite direction to the other planets. In the case of the satellites belonging to the planets more than one exception of this kind has been discovered, and mistakes have constantly occurred b science from expecting that all new cases would exactly resemble old ones. Imperfect Induction thui gives only a certain degree of probability or likelihood that all instances will agree with those examined. Per- fect Induction, on the other, hand, gives a necessary and BI4 PERFEC1 INDUCTION AND certain conclusion, but it asserts nothing beyond whai ras asserted in the premises. i Mr Mill, indeed, differs from almost all other logicians in holding that Perfect Induction is improperly called Cnduction, because it does not lead to any new knowledge. He defines Induction as inference from the known to ttu unknown, and considers the unexamined cases which are apparently brought into our knowledge as the only gain from the process of reasoning. Hence Perfect Induction seems to him to be of no scientific value whatever, be- cause the conclusion is a mere reassertion in a briefer form, a mere summing up of the premises. I may point out, however, that if Perfect Induction were no more than a process of abbreviation it is yet of great importance, and requires to be continually used in science and common life. Without it we could never make a comprehensive statement, but should be obliged to enumerate every par- ticular. After examining the books in a library and finding them to be all English books we should be unable to sum up our results in the one proposition, "all the books in this library are English books ;" but should be required to go over the list of books every time we desired to make any one acquainted with the contents of the library. The fact is, that the power of expressing a great number of particular facts in a very brief space is essential to the pro- gress of science. Just as the whole science of arithmetic consists in nothing but a series of processes for abbreviat- ing addition and subtraction, and enabling us to deal with a. great number of units in a very short time, so Perfect Induction is absolutely necessary to enable us to deal with ft great number of particular facts in a very brief space. It is usual to represent Perfect Induction in the form of an Inductive Syllogism, as in the following instance . Mercury, Venus, the Earth, &c., all move round the sue from West to East xxv.j THE INDUCTIVE SYLLOGISM. tl Mercury, Venus, the Earth, &c., are all the known Planet* Therefore all the known planets move round the sun from West to East. This argument is a true Perfect Induction because the conclusion only makes an assertion of all known planets which excludes all reference to possible future discoveries and we may suppose that all the known planets have been enumerated in the premises. The form of the argument appears to be that of a syllogism in the third figure, namely Darapti, the middle term consisting in the group of the known planets. In reality, however, it is not an ordinary syllogism. The minor premise states not that Mercury, Venus, the Earth, Neptune, &c., are contained among the known planets, but that they are those planets, or are identical with them. This premise is then a doubly universal proposition of a kind (p. 184 7) not re- cognised in the Aristotelian Syllogism. Accordingly we may observe that the conclusion is a universal proposi- tion, which is not allowable in the third figure of the syl- logism. As another example of a Perfect Induction we may take January, February, December, each contain leit than 32 days. January December are all the months of the year. Therefore all the months of the year contain kss than 32 days. Although Sir W. Hamilton has entirely rejected the notion, it seems worthy of inquiry whether the Inductive Syllogism be not really of the Disjunctive form of Sylla gism. Thus 1 should be inclined to represent the last example in the form: A month of the year is either January, or February, M March or December but January has less n6 PERFECT INDUCTION AND [LE8& than 32 days ; and February has less than 32 days ; and so on until we come to December, which has less than 52 days. It follows clearly that a month must in any case have less than 32 days; for there are only 12 possible case*, tnd in each case this is affirmed. The fact is that the major premise of the syllogism on the last page is a compound sentence with twelve subjects, and is therefore equivalent to twelve distinct logical propositions. The minor premise is either a disjunctive proposition, as I have represented it, or something quite different from anything we have elsewhere had. From Perfect Induction we shall have to pass to Im- perfect Induction ; but the opinions of Logicians are not in agreement as to the grounds upon which we are war- ranted in taking a part of the instances only, and con- cluding that what is true of those is true of all. Thus if we adopt the example found in many books and say This, that, and the other magnet attract iron ; This, that, and the other magnet are all magnets ; Therefore all magnets attract iron, we evidently employ a false minor premise, because this, that, and the other magnet which we have examined, cannot possibly be all existing magnets. In whatevei form we put it there must be an assumption that the mag- nets which we have examined are a fair specimen of all magnets, so that what we find in some we may expect in ail Archbishop Whately considers that this assumption thould be expressed in one of the premises, and he repre ents Induction as a Syllogism in Barbara as follows : That which belongs to this, that, and the other magnet. belongs to all ; Attracting iron belongs to this, that, and the other ; Therefore it belongs to all XXV.] THE INDUCTIVE SYLLOGISM 11} But though this is doubtless a correct expression of th Assumption made in an Imperfect Induction, it does not in the least explain the grounds on which we are allowed to make the assumption, and under what circumstances such an assumption would be likely to prove true. Some writers have asserted that there is a Principle called the Uniformity of Nature, which enables us to affirm that what has often been found to be true of anything will continue to be found true of the same sort of thing. It must be observed, however, that if there be such a principle it is liable to exceptions; for many facts which have held true up to a certain point have afterwards been found not to be always true. Thus there was a wide and unbroken induction tending to show that all the Satellites in the planetary system went in one uniform direction round their planets. Nevertheless the Satellites of Uranus when discovered were found to move in a retrograde direction, or in an opposite direction to all Satellites previously known, and the same peculiarity attaches to the Satellite of Neptune more lately discovered. We may defer to the next lesson the question of the varying degree of certainty which belongs to induction in the several branches of knowledge. The advanced student may consult the following with idvantage : Mansel's Aldrich, Appendix, Notes G and H. Hamilton's Lectures on Logic, Lecture xvii., and Appen dix VI I., On Induction and Example, VoL II., p. 358. J. S, Mill's System, of Logic, Book III. Chap. 2, Of Induction* improperly so-t*U*4> LESSON XXVI. GEOMETRICAL AND MATHEMATICAL INDUO TION, ANALOGY AND EXAMPLE. IT is now indispensable that we should consider with great care upon what grounds Imperfect Induction is founded. No difficulty is encountered in Perfect Indue tion because all possible cases which can come under the general conclusion are enumerated in the premises, so that in fact there is no information in the conclusion which was not given in the premises. In this respect the In- ductive Syllogism perfectly agrees with the general prin- ciples of deductive reasoning, which require that the in- formation contained in the conclusion should be shown only from the data, and that we should merely unfold, or transform into an explicit statement what is contained in the premises implicitly. In Imperfect Induction the process seems to be of a wholly different character, since the instances concerning which we acquire knowledge may be infinitely more numerous than those from which we acquire the know- ledge. Let us consider in the first place the process of Geometrical Seasoning which has a close resemblance to inductive reasoning. When in the fifth proposition of the first book of Euclid we prove that the angles at the base of an isosceles triangle are equal to each other, it is done by taking one particular triangle as an example. A figure is given which the reader is requested to regard as having two equal sides, and it is conclusively proved thai if the sides be really equal then the angles opposite to those sides must be equal also. But Euclid says nothing about other isosceles triangles ; he treats one single triangle as a sufficient specimen of all isosceles triangles, AND EXAMPLE. 21* and we are asked to believe that what is true of that if true of any other, whether its sides be so small as to be only visible in a microscope, or so large as to reach to the furthest fixed star. There may evidently be an infinite number of isosceles triangles as regards the length of the equal sides, and each of these may be infinitely varied by increasing or diminishing the contained angle, so that the number of possible isosceles triangles is infinitely infinite ; ind yet we are asked to believe of this incomprehensible number of objects what we have proved only of one single specimen. This might seem to be the most extremely Imperfect Induction possible, and yet every one allows that it gives us really certain knowledge. We do know with as much certainty as knowledge can possess, that if lines be conceived as drawn from the earth to two stars equally distant, they will make equal angles with the line joining those stars ; and yet we can never have tried the experiment The generality of this geometrical reasoning evidently depends upon the certainty with which we know that all isosceles triangles exactly resemble each other. The pro- position proved does not in fact apply to a triangle unless it agrees with our specimen in all the qualities essential to the proof. The absolute length of any of the sides or the absolute magnitude of the angle contained between any of them were not points upon which the proof de- pended they were purely accidental circumstances ; hence we are at perfect liberty to apply to all new cases of an isosceles triangle what we learn of one case. Upon a similar ground rests all the vast body of certain know- ledge contained in the mathematical sciences not only All tV geometrical truths, but all general algebraical trutlb. It was shown, for instance, in p. 58, that ii a and b be two quantities, and we multiply together their sum and difference, we get the difference of thf tic INDUCTION, ANALOGY [LESA squares of a and b. However often we try this It will b< found true ; thus if a = 10 and b = 7, the product of tht sum and difference is 17 x 3 = 51; the squares of th> quantities are 10 x 10 or 100 and 7 x 7 or 49, the dirfer ence of which is also 51. But however often we tried tbt rule no certainty would be added to it; because whei proved algebraically there was no condition which re tricted the result to any particular numbers, and a and b might consequently be any numbers whatever This generality of algebraical reasoning by which a pro- perty is proved of infinite varieties of numbers at once, is one of the chief advantages of algebra over arithmetic There is also in algebra a process called Mathematical Induction or Demonstrative Induction, which shows the powers of reasoning in a very conspicuous way. A good example is found in the following problem : If we take the first two consecutive odd numbers, i and 3, and add them together the sum is 4, or exactly twice two; if we take three such numbers 1+3 + 5, the sum is 9 or exactly three times three; if we take four, namely i + 3 + 5 + 7 the sum is 1 6, or exactly four times four; or generally, if we take any given number of the series, 1+3 + 5 + 7 + ... the sum is equal to the number of the terms multiplied by itself. Anyone who knows a very little algebra can prove that this remarkable law is universally true, as follows Let n be the number of terms, and assume for a moment that this law is true up to n terms, thus 1+3 + 5 + 7 + ...... +(2- 1)='. Now add 2 + 1 to each side of the equation. It fol lows that But the last quantity * a + 2 + I is just equal to (n + 1) . 10 that if the law is true for n terms it is true also for -*-] We are enabled to areue from each single case of AND EXAMPLE. 221 the law to the next case ; buv we have already shown that k is true of the first few cases, therefore it must be true 01 all. By no conceivable labour could a person ascertain by trial what is the sum of the first billion odd numbers, and yet symbolically or by general reasoning we know with certainty that they would amount to a billion billion, and aeither more nor less even by a unit. This process oi Mathematical Induction is not exactly the same as Geo- metrical Induction, because each case depends upon the last, but the proof rests upon an equally narrow basis of experience, and creates knowledge of equal certainty and generality. Such mathematical truths depend upon observation of a few cases, but they acquire certainty from the per- ception we have of the exact similarity of one case to another, so that we undoubtingly believe what is true of one case to be true of another. It is very instructive to contrast with these cases certain other ones where there is a like ground of observation, but not the same tie of similarity. It was at one time believed that if any integral number were multipled by itself, added to itself and then added to 41, the result would be a prime number, that is a number which could not be divided by any other in tegral number except unity ; in symbols, **+ .r+4i = prime number. This was believed solely on the ground of trial and experience, and it certainly holds for a great many values of x. Thus when x is successively made equal to the aumbers in the first line below, the expression **+ r + 41 jures the values in the second line, and they are all primf ismbers : 01234$ 6 789 10 4 43 47 S3 61 71 3 97 113 3' *$' No reason however could be given why it should 1*2 INDUCTION, ANALOGY always be true, and accordingly it is found that the rule does not always hold true, but fails when x = 40. Then we have 40x40-1-40 + 41 = 1681, but this is clearly equal to 41 x 40 + 41 or 41 x 41, and is not a prime number. In that branch of mathematics which treats of the peculiar properties and kinds of numbers, other proposi- tions depending solely upon observation have been as- serted to be always true. Thus Fermat believed that 2 a 4- 1 always represents a prime number, but could not give any reason for the assertion. It holds true in fact until the product reaches the large number 4294967297, which was found to be divisible by 641, so that the gene- rality of the statement was disproved. We find then that in some cases a single instance proves a general and certain rule, while in others a very great number of instances are insufficient to give any certainty at all; all depends upon the perception we have of similarity or identity between one case and another. We can perceive no similarity between all prime numbers which assures us that because one is represented by a certain formula, also another is; but we do find such similarity between the sums of odd numbers, or between isosceles triangles. Exactly similar considerations apply to inductions in physical science. When a chemist analyses a few grains of water and finds that they contain exactly 8 parts of oxygen and I of hydrogen for 9 parts of water, he feels warranted in asserting that the same is true of all pure water whatever be its origin, and whatever be the part of the world from which it comes. But if he analyse a piece of granite, or a sample of sea-water from one part of the tforld, he does not feel any confidence that it will resem- ble exactly a piece of granite, or a sample of sea-watei from another part of the earth ; hence he does not venture to assert of all granite or sea-water, what he finds true of xxvi.] AND EXAMPLE. i&\ single sample. Extended experience shows that gra- nite is very variable in composition, but that sea-water is rendered pretty uniform by constant mixture of currents. Nothing but experience in these cases could inform us how far we may assert safely of one sample what we nave ascertained of another. But we have reason to believe that chemical compounds are naturally fixed and invari- able in composition, according to Dalton's laws of com- bining proportions. No a priori reasoning from the principles of thought could have told us this, and we only learn it by extended experiment But having once shown it to be true with certain substances we do not need to repeat the trial with all other substances, because we have every reason to believe that it is a natural law in which all chemical substances resemble each other. It is only necessary then for a single accurate analysis of a given fixed compound to be made in order to inform us of the composition of all other portions of the same substance. It must be carefully observed however that all indue, UOOB in physical science are only probable, or that if cer- tain, it is only hypothetical certainty they possess. Can I be absolutely certain that all water contains one part of hydrogen in nine ? I am certain only on two con- ditions : 1. That this was certainly the composition of the sample tried. 2. That any other substance I call water exactly resembles that sample. But even if the first condition be undoubtedly true, I cannot be certain of the second. For how do I know what is water except by the fact of its being a transparent liquid, freezing into a solid and evaporating into steam, possessing a high specific heat, and a number of other distinct properties ? But can I be absolutely certain that every liquid possessing all these properties is water? 134 INDUCTION, ANALOGY [LESi Practically I can be certain, but theoretically I cannot Two substances may have been created so like each ithei that we should never yet have discovered the difference ; we might then be constantly misled by assuming of the one what is only true of the other. That this should evei happen with substances possessing the very distinct quali ties of water is excessively improbable, but so far is it from being impossible or improbable in other cases, that it has often happened. Most of the new elements dis- covered in late years have, without doubt, been mistaken previously for other elements. Caesium and Rubidium had been long mistaken for each other, and for Potassium, before they were distinguished by Bunsen and Kirchhofl by means of the spectroscope. As they are new known to be widely distributed, although in small quantities, it is certain that what was supposed to be Potassium in many thousands of analyses was partly composed of different substances. Selenium had probably been confused with Sulphur, and there are certain metals for instance, Rho- dium, Ruthenium, Indium, Osmium, and Beryllium Yttrium, Erbium, Cerium, Lanthanum, and Didymium Cadmium and Indium which have only recently been distinguished. The progress of science will doubtless show that we are mistaken in many of our identifications, and various difficulties thus arising will ultimately be ex- plained. Take again a very different case of induction. Are we certain that the sun will rise again to-morrow morning as it has risen for many thousand years, and probably f )i some hundred million years? We are certain only on thii condition or hypothesis, that the planetary system proceeds to-morrow as it has proceeded for so long. Many causes may exist which might at any moment defeat all our calculations ; our sun is believed to be a variable star, and for what we know it might at any moment suddenly HXVL] AND EXAMPLE. n\ explode or flare up, as certain other stars have been oi> served to do, and we should then be all turned into thin luminous vapour in a moment of time. It is not at all impossible that a collision did once occur in the planet- ary system, and that the minute planets or asteroids are the result Even if there is no large meteor, comet or sther body capable of breaking up the earth by collision, yet it is probable that the sun moves through space at the rate of nearly 300 miles per minute, and if some other star should meet us at a similar rate the consequences would be inconceivably terrible. It is highly improbable however that such an event should come to pass even in the course of a million years. The reader will now see that no mere Imperfect In- duction can give certain knowledge ; all inference proceeds upon the assumption that new instances will exactly re- semble old ones in all material circumstances ; but in natural phenomena this is purely hypothetical, and we may constantly find ourselves in error. In Mathematical Induction certainty arose from the cases being hypotheti- cal in their own nature, or being made so as exactly to correspond with the conditions. We cannot assert that any triangle existing in nature has two equal sides or two equal angles, and it is even impossible in practice that any two lines or angles can be absolutely equal. But it is nevertheless true that if the sides are equal the angles are equal. All certainty of inference is thus relative and hypothetical. Even in the syllogism the certainty of the conclusion only rests on the hypothesis of certainty in the premises. It is probable, in fact, that all reasoning reduces kself to a single type that what is true of one thing will be true of another thing, on condition of there being an exact resemblance between them in all material circum- stances. The reader will now understand with ease the nature aa6 INDUCTION, ANALOGY [LISS of reasoning by analogy. In strictness an analogy is not an identity of one thing with another, but an identity ol relations. In the case of numbers 7 is not identical with 10 nor 14 with 20, but the ratio of 7 to 10 is identical with the ratio of 14 to 20, so that there is an analogy between these numbers. To multiply two by two is not the same thing as to construct a square upon a line two units long ; but there is this analogy that there will be just as many units of area in the square as there are units in the product of two by two. This analogy is so evident that we fearlessly assert a square mile to consisi of 1760 x 1760 square yards without any trial of the truth. In ordinary language, however, analogy has come to mean any re- semblance between things which enables us to believe of one what we know of the other. Thus the planet Mars possesses an atmosphere, with clouds and mist closely resembling our own ; it has seas distinguished from the land by a greenish colour, and polar regions covered with snow. The red colour of tne planet seems to be due to the atmosphere, like the red colour of our sunrises and sunsets. So much is similar in the surface of Mars and the surface of the Earth that we readily argue there must be inhabitants there as here. All that we can certainly say however is, that if the circumstances be really similar, and similar germs of life have been created there as here, there must be inhabitants. The fact that many circumstances are similar increases the probability. But between the Earth ind the Sun the analogy is of a much fainter character ; we speak indeed of the sun's atmosphere being subject to storms and filled with clouds, but these clouds are heated probably beyond the temperature of our hottest furnaces ; if they produce rain it must resemble a shower of melted iron ; and the sun-spots are perturbations of so tremend- ous a sue and character, that the earth together with XXVI. J AND EXAMPLE. 33) half-a-dozen of the other planets could readily be swal- lowed up in one of them*. It is plain then that there is little or no analogy between the Sun and the Earth, anc we can therefore with difficulty form a conception of any- '.hing going on in a sun or star. Argument from analogy may be defined as direct inductive inference from one instance to any similai instance. It may, as Mr Mill says, be reduced to the following formula : "Two things resemble each other in one or mort respects ; a certain proposition is true of the one ; there- fore it is true of the other." This is no doubt the type o( all reasoning, and the certainty of the process depends entirely upon the degree of resemblance or identity be- tween the cases. In geometry the cases are absolutely identical in all material points by hypothesis, and no doubt attaches to the inference ; in physical science the identity is a question of probability, and the conclusion is HI a like degree probable. It should be added that Mr Mill considers Geometrical and Mathematical Induction not to be properly called Induction, for reasons of which the force altogether escapes my apprehension ; but the reader will find his opinions in the 2nd chapter of the 3rd book of his System of Logic, One form of analogical or inductive argument consists in the constant use of examples and Instances. The best way to describe the nature of a class of things is to pie- >ent one of the things itself, and point out the pr< perties which belong to the class as distinguished from those peculiar to the thing. Throughout these Lessons, as throughout every work on Logic, instances of propositions, of compound or complex sentences, of syllogisms, &c. 5 arc continually used, and the rtader is asked to apply to alJ * Lockyer's EUmtniary Lessons in Astronomy, 108. 15 a 128 OB SEX VA T1ON [] similar cases what he observes in the examples given It is assumed that the writer selects such examples as truly exhibit the properties in question. While all inductive and analogical inferences rest upon the same principles there are wide differences be- tween the sources of probability. In analogy we have two cases which resemble each other in a great many proper- ties, and we infer that some additional property in one is probably to be found ia the other. The very narrow basis of experience is compensated by the high degree of similarity. In the processes more commonly treated under the name Induction, the things usually resemble each other only in two or three properties, and we require to have more instances to assure us that what is true of of these is probably true of all similar instances. The less, in short, the intension of the resemblance the greater must be the extension of our inquiries. We proceed to the ordinary processes of Induction in the following Lessons. Mr Mill's System of Logic, Book ill. Chap. XX. Oj Analogy. Mans el's Aldruh, App. Note H. On Example and Analogy. LESSON XXVII. OBSERVATION AND EXPERIMENT. ALL knowledge, it may be safely said, must be ultimately founded upon experience, which is but a general name foi the various feelings impressed upon the mind at any period of its existence. The mind never creates entirely new knowledge independent of experience, and all that th reasoning powers can do is to arrive at the full meaning XXvii.1 AND EXPERIMENT. 2*3 of the facts which are in our possession. In previous centuries men of the highest ability have held that the mind of its own power alone could, by sufficient cogita- tion, discover what things outside us should be, and wou.d be found to be on examination. They thought that we were able to anticipate Nature by evolving from the human mind an idea of what things would bt made by the Creator. The celebrated philosopher Des cartes thus held that whatever the mind can clearl) conceive may be considered true; but we can conceive the existence of mountains of gold or oceans of fresh water, which do not as a fact exist Anything that we can clearly conceive must be conformable to the laws oi thought, and its existence is then not impossible, so far as our intellect is concerned; but the forms and sizes and manners in which it has pleased the Creator to make things in this or any other part of the universe, cannot possibly be anticipated by the exceedingly limited wisdom of the human mind, and can only be learnt by actual ex- amination of existing things. In the latter part of the I3th century the great Roger Bacon clearly taught in England the supreme importance of experience as the basis of knowledge ; but the same doctrine was also, by a curious coincidence, again upheld in the I7th century by the great Chancellor Francis Bacon, after whom it has been called the Baconian Phi- losophy. I believe that Roger Bacon was even a greater man than Francis, whose fame is best known ; but the words in which Francis Bacon proclaimed the importance of experience and experiment must be ever memorable. In the beginning of his great work, the Novum Organum, oi New Instrument, he thus points out our proper position as learners in the world of nature. "Man, the Servant and Interpreter of Nature, can de and understand as much as he has observed concerning t* OBSERVATION [LESt, the order of nature in outward tilings or in the mind more, he can neither know nor do." The above is the first of the aphorisms or paragraph* with which the Novum Organum commences. In the second aphorism he asserts that the unaided mind car, effect little and is liable to err ; assistance in the form of a definite logical method is requisite, and this it was the purpose of his New Instrument to furnish. The 3rd and 4th aphorisms must be given entire ; they are : " Human science and human power coincide, because ignorance of a cause deprives us of the effect For nature is not conquered except by obedience ; and what we dis- cover as a cause by contemplation becomes a rule IE operation." "Man can himself do nothing else than move natural bodies to or from each other ; nature working within ac- complishes the rest." It would be impossible more clearly and completely to express the way in which we discover science by inter- preting the changes we observe in nature, and then turn our knowledge to a useful purpose in the promotion of the arts and manufactures. We cannot create and we cannot destroy a particle of matter ; it is now known that we cannot even create or destroy force ; nor can we really alter the inner nature of any substance that we have to deal with. All that we can do is to observe carefully how one substance by its natural powers acts upon another substance, and then by noving them together at the right time we can effect our object; as Bacon says, "Nature working within does the rest." Had it not been the nature of heat when applied to water to develope steam possessing elastic power, it is needless to say that th steam-engine could never have been made, so that the invention of the steam-engine arose from observing the utility of the force of steam, and employing it accordingly xxvii. j AND EXPERIMENT. 33. It is in this sense that Virgil has proclaimed bin happ) who knows the causes of things Felix qui potuit rerum cognosccre ca*s*s t and that Bacon has said, Knowledge is Power. So fat is we have observed how things happen in nature, and OB arhat occasion particular effects are brought to pass, we are enabled to avoid or utilise those effects as we may desire, not by altering the natures of things, but by allowing them in suitable times and circumstances to manifest their own proper powers. It is thus, as Tenny- son has excellently said, that we " Rule by obeying Nature's Powers." Inductive logic treats of the methods of reasoning by which we may successfully interpret nature and learn the natural laws which various substances obey in different circumstances. In this lesson we consider the first requi- site of induction, namely, the experience or examination of nature which is requisite to furnish us with facts. Such experience is obtained either by observation or experiment. To observe is merely to notice events and changes which are produced in the ordinary course of nature, without being able, or at least attempting, to control or vary those changes. Thus the early astronomers observed the mo- tions of the sun, moon and planets among the fixed stars, and gradually detected many of the laws or periodical returns of those bodies. Thus it is that the meteorologist observes the ever-changing weather, and notes the height af the barometer, the temperature and moistness of the lir, the direction and force of the wind, the height and character of the clouds, without being in the least able to govern any of these facts. The geologist again is gene- nerally a simple observer when he investigates the nature and position of rocks. The zoologist, the botanist, and zj2 OBSER VA TION [LES* the mineralogist usually employ mere observation when they examine animals, plants, and minerals, as they aw met with in their natural condition. In experiment, on the contrary, we vary at our will the combinations of things and circumstances, and then observe the result. It is thus that the chemist discovers the composition of water by using an electric current to separate its two constituents, oxygen and hydrogen. The mineralogist may employ experiment when he melts twc Of three substances together to ascertain how a particular mineral may have been produced. Even the botanist and zoologist are not confined to passive observation ; for by removing animals or plants to different climates and dif- ferent soils, and by what is called domestication, they may try how far the natural forms and species are capable of alteration. It is obvious that experiment is the most potent and direct mode of obtaining facts where it can be applied. We might have to wait years or centuries to meet acci- dentally with facts which we can readily produce at any moment in a laboratory ; and it is probable that most of the chemical substances now known, and many exces- sively useful products, would never have been discovered at all by waiting till nature presented them spontaneously to our observation. Many forces and changes too may go on in nature constantly, but in so slight a degree as to escape our senses, and render some experimental means necessary for their detection. Electricity doubtless ope- rates in every particle of matter, perhaps at every mo- ment of time ; and even the ancients could not but notice its action in the loadstone, in lightning, in the Aurora Borealis, or in a piece of rubbed amber (electrum). But in lightning electricity was too intense and dangerous; in the other cases it was too feeble to be properly under- stood. The science of electricity and magnetism could KXVIL] AND EXPERIMENT. 233 only advance by getting regular supplies of electricity from the common electric machine or the galvanic bat- tery, and by making powerful electro-magnets. Most i) not all the effects which electricity produces must go on in nature, but altogether too obscurely for observation. Experiment, again, is rendered indispensable by the fact that on the surface of the earth we usually meet sub- stances under certain uniform conditions, so that we could never learn by observation what would be the nature of such substances under other conditions. Thus carbonic acid is only met in the form of a gas, proceeding from the combustion of carbon ; but when exposed to extreme pressure and cold, it is condensed into a liquid, and may even be converted into a snow-like solid sub- stance. Many other gases have in like manner been liquefied or solidified ; and there is reason to believe that every substance is capable of taking all the three forms of solid, liquid and gas, if only the conditions of temperature and pressure can be sufficiently varied. Mere observation of nature would have led us, on the contrary, to suppose that nearly all substances were fixed in one condition only, and could not be converted from solid into liquid and from liquid into gas. It must not be supposed however that we can draw any precise line between observation and experiment, and say where the one ends and the other begins. The dif- ference is rather one of degree than of kind; and all we can say is that the more we vary the conditions artificially the more we employ experiment. I have said that me- teorology is a science of nearly pure observation, but if we purposely ascend mountains to observe the rarefaction and cooling of the atmosphere by elevation, or if we make balloon ascents for the same purpose, like Gay Lussac and Glaisher, we so vary the mode of observation as almost to render it experimental. Astronomers again a 34 OBSER VA TION may almost be said to experiment instead of merely ob- serving when they simultaneously employ instruments ai far to the north, and as far to the south, upon the earth's surface as possible, in order to observe the apparent dif- ference of place of Venus when crossing the sun in a transit, so as thus to compare the distances of Venus and the sun with the dimensions of the earth. Sir John Herschel has excellently described the dif ference in question in his Discourse on the Study of Na- tural Philosophy*. " Essentially they are much alike, and differ rather in degree than in kind ; so that perhaps the terms passive and active observation might bettei express their distinction; but it is, nevertheless, highly important to mark the different states of mind in inqui- ries carried on by their respective aids, as well as their different effects in promoting the progress of science. In the former, we sit still and listen to a tale, told us, per- haps obscurely, piecemeal, and at long intervals of time, with our attention more or less awake. It is only by after rumination that we gather its full import ; and often, when the opportunity is gone by, we have to regret that our attention was not more particularly directed to some point which, at the time, appeared of little moment, but of which we at length appreciate the importance. In the latter, on the other hand, we cross-examine our witness, and by comparing one part of his evidence with the other, while he is yet before us, and reasoning upon it in his presence, are enabled to put pointed and searching ques- tions, the answer to which may at once enable us to make up our minds. Accordingly it has been found invariably, that in those departments of physics where the pheno- mena are beyond our control, or into which experimental enquiry, from other causes, has not been carried, th* pro XXVIL] AND EXPERIMENT. 23 i gress of knowledge has been slow, uncertain and irregu, lar ; while in such as admit of experiment, and in which mankind have agreed to its adoption, it has been rapid sure, and steady." Not uncommonly, however, nature has, so to speak made experiments upon a scale and for a duration witk which we cannot possibly compete. Thus we do not need to try the soil and situation which suits any given plant best ; we have but to look about and notice the habitat 01 situation in which it is naturally found in the most flou- rishing condition, and that, we may be sure, indicates the result of ages of natural experiment. The distances oi the fixed stars would probably have been for ever un- known to us did not the earth by describing an orbit with a diameter of 182,000,000 miles make a sort of experimen- tal base for observation, so that we can see the stars in very slightly altered positions, and thus judge their dis- tances compared with the earth's orbit*. Eclipses, tran- sits, occultations and remarkable conjunctures of the pla- nets, are also kinds of natural experiments which have often been recorded in early times, and thus afford data of the utmost value. Logic can give little or no aid in making an acute or accurate observer. There are no definite rules which can be laid down upon the subject. To observe well is an art which can only be acquired by practice and training ; and it is one of the greatest advantages of the pursuit of the Natural Sciences that the faculty of clear and steady ob- servation is thereby cultivated. Logic can however give us this caution, which has been well pointed out by Mr Mill to discriminate accurately between what we really do observe and what we only infer from the facts observed. So long as we only record and describe what our sensei See Lockyer't Elementary Leuons in Astronomy, Not XLVI, XLVII. *3<> OBSERVATION [LESS have actually witnessed, we cannot commit an error ; but the moment we presume or infer anything we are liable tc mistake. For instance, we examine the sun's surface with a telescope and observe that it is intensely bright except where there are small breaks or circular openings in the surface with a dark interior. We are irresistibly led to the conclusion that the inside of the sun is colder and darker than the outside, and record as a fact that we saw the dark interior of the sun through certain openings in its luminous atmosphere. Such a record, however, would involve mistaken inference, for we saw nothing but dark spots, and we should not have done more in observ- ation than record the shape, size, appearance and change of such spots. Whether they are dark clouds above the luminous surface, glimpses of the dark interior, or, as is now almost certainly inferred, something entirely different from either, can only be proved by a comparison of many unprejudiced observations. The reader cannot too often bear in mind the cau- tion against confusing facts observed with inferences from those facts. It is not too much to say that nine-tenths of what we seem to see and hear is inferred, not really felt Every sense possesses what are called acquired percep- tions, that is, the power of judging unconsciously, by long experience, of many things which cannot be the objects oi direct perception. The eye cannot see distance, yet we constantly imagine and say that we see things at such and such distances, unconscious that it is the result of judgment. As Mr Mill remarks, it is too much to say " I saw my brother." All I positively know is that I saw some one who closely resembled my brother as fat as could be observed It is by judgment only I can assert he was my brother, and that judgment may possi- Dly be wrong. Nothing is more important in observation and expert xxvii.] AND EXPERIMENT. j# ment than to be uninfluenced by any prejudice or theory in correctly recording the facts observed and allowing to them their proper weight. He who does not do so will almost always be able to obtain facts in support of an opinion however erroneous. Thus the belief still existi with great force in the majority of uneducated persons, that the moon has great influence over the weather. The changes of the moon, full, new and half moon, occur four times in every month, and it is supposed that any change may influence the weather at least on the day preceding or following that of its occurrence. There will thus be twelve days out of every 28 on which any change of wea- ther would be attributed to the moon, so that during the jear many changes will probably be thus recorded as favourable to the opinion. The uneducated observer is struck with these instances and remembers them care- fully, but he fails to observe, or at least to remember, that changes of weather often occur also when there is no change of the moon at all The question could only be decided by a long course of careful and unbiassed observation in which all facts favourable or unfavour- able should be equally recorded. All observations which have been published negative the idea that there can be any such influence as the vulgar mind attributes to the moon. But it would at the same time be an error to suppose that the best observer or experimentalist is he who holds no previous opinions or theories on the subject he inves- tigates. On the contrary, the great experimentalist is he who ever has a theory or even a crowd of theories or ideas upon his mind, but is always putting them to the test of experience and dismissing those which are false. The number of things which can be observed and experimented on are infinite, and if we merely set to work to record facts without any distinct purpose, our records will havf 138 OBSERVATION. frv. no value. We must have some opinion cr some the- ory to direct our choice of experiments, and it is more probable that we hit upon the truth in this way than merely by haphazard. But the great requisite of the true philosopher is that he be perfectly unbiassed and abandon every opinion as soon as facts inconsistent with it are observed. It has been well said by the celebrated Turgot, that " the first thing is to invent a system ; the second thing is to be disgusted with it;" that is to say, we ought to have some idea of the truth we seek, but should im- mediately put it to a severe trial as if we were inclined to distrust and dislike it rather than be biassed in its favour. Few men probably have entertained more false theories than Kepler and Faraday ; few men have discovered or established truths of greater certainty and importance. Faraday has himself said that " The world Httle knows how many of the thoughts and theories which have passed through the mind of a scientific investigator, have been crushed in silence and secrecy by his own severe criticism and adverse examina- tion ; that in the most successful instances not a tenth of the suggestions, the hopes, the wishes, the preliminary tonclusions have been realized *." The student is strongly recommended to read Sir J. Herschel's Preliminary Discourse on the Stud) if Natural Philosophy (Lardner's Cabinet Cycle- padia\ especially Part II. Chaps. 4 to 7, concerning Observation, Experiment, and the Inductive Pro cesses generally. AMtrn Cultttrt, edited by Yoram, p. *. [M*cmiUM KXY1U.] METHODS OF INDUCTION LESSON XXVIII. METHODS OF INDUCTION. WE have now to consider such methods as can be laid! down for the purpose of guiding us in the search for gene- ral truths or laws of nature among the facts obtained by observation and experiment. Induction consists in infer- ring from particulars to generals, or detecting a general truth among its particular occurrences. But in physical science the truths to be discovered generally relate to the connection of cause and effect, and we usually caU them laws of causation or natural laws. By the Gam* of an event we mean the circumstances which must have preceded in order that the event should happen, Nor is it generally possible to say that an event has one single cause and no more. There are usually many different things, conditions or circumstances necessary to the pro- duction of an effect, and all of them must be considered causes or necessary parts of the cause. Thus the cause of the loud explosion in a gun is not simply the pulling of the triggei, which is only the last apparent cause or jocaslon of the explosion ; the qualities of the powder; the proper form of the barrel ; the existence of some re- sisting charge ; the proper arrangement of the percussion cap and powder ; the existence of a surrounding atmo- sphere, are among the circumstances necessary to the loud report of a gun : any of them being absent it would not have occurred. The cause of the boiling of water again is not merely the application of heat up to a certain degree of tempera- MO METHODS OF INDUCTION. ture, but the possibility also of the escape of the vapour when it acquires a certain pressure. The freezing ol water similarly does not depend merely upon the with drawal of heat below the temperature of o Centigrade. It is the work of Induction then to detect those circum- stances which uniformly will produce any given effect and as soon as these circumstances become known, we have a law or uniformity of nature of greater or less gene- rality. In this and the following Lessons I shall often have tc use, in addition to cause and effect, the words antecedent and consequent, and the reader ought to notice their meanings. By an antecedent we mean any thing, condi- tion, or circumstance which exists before or, it may be, at the same time with an event or phenomenon. By a con- sequent we mean any thing, or circumstance, event, or phenomenon, which is different from any of the antecedents and follows after their conjunction or putting together. It does not follow that an antecedent is a cause, because the effect might have happened without it. Thus the sun's light may be an antecedent to the burning of a house, but not the cause, because the house would burn equally well in the night A necessary or indispensabU antecedent is however identical with a cause, being thai without which the effect would not take place. The word phenomenon will also be often used. It means simply anything which appears, and is therefore observed by the senses ; the derivation of the word from the Greek word Qaivoprvov, that which appears, exactly corresponds to its logical use. The first method of Induction is that which Mr Mill has aptly called the Method of agreement. It depends upon the rule that "If two or more instances of the phe- nomenon under investigation have only one circumstance in common, the circumstance in which alone all the in- *X/m.] METHODS OF INDUCTION. 341 stances agree, is the cause (or effect) of the given pheno menon." The meaning of this First Canon of inductive inquiry might, I think, be more briefly expressed by saying that the sole invariable antecedent of a phenomenon it probably its cause. To apply this method we must collect as many in- stances of the phenomenon as possible, and compare together their antecedents. Among these the causes will lie, but if we notice that certain antecedents are present or absent without appearing to affect the result, we conclude that they cannot be necessary antecedents. Hence it is the one antecedent or group of antecedents always present, when the effect follows, that we consider the cause. For example, bright prismatic colours are seen on bub- bles, on films of tar floating upon water, on thin plates of .mica, as also on cracks in glass, or between two pieces of glass pressed together. On examining all such cases they seem to agree in nothing but the presence of a very thin layer or plate, and it appears to make no appreciable difference of what kind of matter, solid, liquid, or gaseous, the plate is made. Hence we conclude that such colours are caused merely by the thinness of the plates, and this conclusion is proved true by the theory of the interference of light. Sir David Brewster beautifully proved in a similar way that the colours seen upon Mother-of-pearl are not caused by the nature of the substance, but by the form of the surface. He took impressions of the Mother- >f-pearl in wax, and found that although the substance was entirely different the colours were exactly the same. And it was afterwards found that if a plate of metal had a surface marked by very fine close grooves, it would havt iridescent colours like those of Mother-of-pearl. Henoe it is evident that the form of the surface, which it th only invariable antecedent or condition requisite for thi production of the colours, must be their cause ID JA2 METHODS OF INDUCTION. [] The method of agreement is subject to a serious difficulty, called by Mr Mill the Plurality of C&mea, con- sisting in the fact that the same effect may in different instances be owing to different causes. Thus if we in- quire accurately into the cause of heat we find that it is produced by friction, by burning or combustion, by elec- tricity, by pressure, &c. ; so that it does not follow that il there happened to be one and the same thing present in all the cases we examined this would be the cause. The second method of induction which we will now consider is free from this difficulty, and is known as the Method of Difference. It is stated in Mr Mill's Second Canon as follows : " If an instance in which the phenomenon under inves- tigation occurs, and an instance in which it does not occur, have every circumstance in common save one, that one occurring only in the former; the circumstance in which alone the two instances differ, is the effect, or the cause, or an indispensable part of the cause, of the phe- nomenon." In other words, we may say that the antecedent which is invariably present when the phenomenon follows, and invariably absent when it is absent, other circumstance! remaining the same, is the cause of the phenomenon in those circumstances. Thus we can clearly prove that friction is om cause of heat, because when two sticks are rubbed together they become heated; when not rubbed they do not become heated. Sir Humphry Davy showed that even two pieces of ice when rubbed together in a vacuum produce heat as shown by their melting, and thus completely demon strated that the friction is the source and cause of the heat. We prove that air is the cause of sound being communicated to our ears by striking a bell in the re* ceirtr of an air-pump, as Hawksbee first did in 1705, and xxvmj METHODS OF INDUCTION. 143 then observing that when the receiver is full of air w* hear the bell ; when it contains little or no air we do not hear the bell. We learn that sodium or any of its compounds produces a spectrum having a bright yellow double line by noticing that there is no such line in the spectrum of light when sodium is not present, but that il the smallest quantity of sodium be thrown into the flame or other source of light, the bright yellow line instantly appears. Oxygen is the cause of respiration and life, because if an animal be put into a jar full of atmospheric ah*, from vw-hich the oxygen has been withdrawn, it soon becomes suffocated. This is essentially the great method of experiment and its utility mainly depends upon the precaution of only varying one circumstance at a time, all other circuit* stances being maintained just as they were. This if expressed in one of the rules for conducting experiments given by Thomson and Tait in their great treatise on Natural Philosophy, Vol. I. p. 307, as follows: " In all cases when a particular agent or cause is to be studied, experiments should be arranged in such a way as to lead if possible to results depending on it alone ; or, if this cannot be done, they should be arranged so as to increase the effects due to the cause to be studied til] these so far exceed the unavoidable concomitants, that the latter maybe considered as only disturbing, not essen- tially modifying the effects of the principal agent" It would be an imperfect and unsatisfactory experi- ment to take air of which the oxygen has been converted into carbonic acid by the burning of carbon, and argue that, because an animal dies in such air, oxygen is the cause of respiration Instead of merely withdrawing the oxygen we have a new substance, carbonic acid, present, which is quite capable of killing the animal by its own poisonous properties. The animal in fact would be suffo 16 2 144 METHODS OF INDUCTION. [Lisa, cated even when a considerable proportion of oxygen remained, so that the presence of the carbonic acid is a disturbing circumstance which confuses and vitiates the -xperiment It is possible to prove the existence, and even to mea- sure the amount of the force of gravity, by delicately sus- pending a small ball about the size of a marble and then suddenly bringing a very heavy leaden ball weighing a ton or more close to it The small ball will be attracted and set in motion; but the experiment would not be of the least value unless performed with the utmost precaution. It is obvious that the sudden motion of the large leaden ball would disturb the air, shake the room, cause currents in the air by its coldness or warmth, and even occasion electric attractions or repulsions; and these would pro- bably disturb the small ball far more than the force of gravitation. Beautiful instances of experiment according to this method are to be found, as Sir John Herschel has pointed out, in the researches by which Dr Wells discovered the cause of dew. If on a clear calm night a sheet or othei covering be stretched a foot or two above the earth, so as to screen the ground below from the open sky, dew will be. found on the grass around the screen but not beneath it As the temperature and moistness of the air, and other circumstances, are exactly the same, the open sky must be an indispensable antecedent to dew. The same expe- riment is indeed tried for us by nature, for if we make observations of dew during two nights which differ in no- thing but the absence of clouds in one and their presence in the other, we shall find that the clear open sky is requi- site to the formation of dew. It may often happen that we cannot apply the method of difference perfectly by varying only one circumstance it a time. Thus we cannot, generally speaking, try the METHODS OF INDUCTION 145 qualities of the same substance in the solid and liquid condition without any other change of circumstances, be- cause it is necessary to alter the temperature of the sub- stance in order to liquefy or solidify it The temperature might thus be the cause of what we attribute to the liquid or solid condition. Under such circumstances we have to resort to what Mr Mill calls the Joint method of agree- ment and difference, which consists in a double applica- tion of the method of agreement, first to a number of instances where an effect is produced, and secondly, to a number of quite different instances where the effect is not produced. It is clearly to be understood, however, that the negative instances differ in several circumstances from the positive ones ; for if they differed only in one circumstance we might apply the simple method of differ- ence. Iceland spar, for instance, has a curious power of rendering things seen through it apparently double. This phenomenon, called double refraction, also belongs to many other crystals ; and we might at once prove it to be due to crystalline structure could we obtain any transpa- rent substance crystallized and uncrystallized, but subject to no other alteration. We have, however, a pretty satis- factory proof by observing that uniform transparent un- crystallized substances agree in not possessing double refraction, and that crystalline substances, on the other hand, with certain exceptions which are easily explained, agree in possessing the power in question. The principle of the Joint method may be stated in the following rule, which is Mr Mill's Third Canon : "If two or more instances in which the phenomenon occurs have only one circumstance in common, while two or more instances in which it does not occur have nothing in common save the absence of that circumstance ; the circumstance in which alone the two sets of instances (always or invariably) differ, is the effect, or the cause 246 METHODS OF INDUCTION. [LESS or an indispensable part of the cause, of the pheno menon." I have inserted the words in parentheses, as without them the canon seems to me to express exactly the oppo- site of what Mr Mill intends. It may facilitate the exact comprehension of these m dwctive methods if I give the following symbolic repre- sentation of them in the manner adopted by Mr Mill Let A, B, C, D, E^ &c., be antecedents which may b variously combined, and let a, 6, c y d, e, &c., be effects following from them. If then we can collect the following sets of antecedents and effects Antecedents. Consequents. ABC *bc ADE tuU AFG *fg AHK akk we may apply the method of agreement, and little doubt will remain that A, the sole invariable antecedent, is the ause of a. The method of difference is sufficiently represented by- Antecedents. Consequents. ABC abc BC be Here while B and C remain perfectly unaltered we find that the presence or absence of A occasions the presence or absence of a, of which it is therefore the cause, in the presence of B and C. But the reader may be cautioned against thinking that this proves A to be the cause of a under all circumstances whatever. Tie Joint method of agreement and difference is similars represented bv JLXVIII.J METHODS OF INDUCTION. 349 Antecedents. Consequents. ABC abc ADE me obstructive power in the space passed through, the nature of which is not yet understood. Mill's System of Logic, Book in. Chap. 10, Oftiu. Plurality of Causes; a*d of the Intermixture oj Effects. LESSON XXX. EMPIRICAL \ND DEDUCTIVE METHODS. WE have hitherto treated of Deduction and Induction as I they were entirely separate and independent methods. In reality they are frequently blended or employed alter- nately in the pursuit of truth ; and it may be said that all the more important and extensive investigations of science rely upon one as much as upon the other. It is probably the greatest merit in Mr Mill's logical writings that he points out the entire insufficiency of what is called the Baconian Method to detect the more obscure and difficult Laws of nature. Bacon advised that we should always begin by collecting facts, classifying them according to their agreement and difference, and gradually gathering *rom them laws of greater and greater generality. He protested altogether against "anticipating nature, "that is forming our own hypotheses and theories as to what the laws of nature probably are, and he seemed to think that systematic arrangement of facts would take the place of l6 EMPIRICAL AND DEDUCTIVE juts* all other methods. The reader will soon see that the progress of Science has not confirmed his opinions. When a law of nature is ascertained purely by induc- tion from certain observations or experiments, and has no other guarantee for its truth, it is said to be an empirical law. As Mr Mill says, "Scientific inquirers give the name of Empirical Laws to uniformities which observation 01 experiment has shown to exist, but on which they hesitate to rely in cases varying much from those which have been actually observed, for want of seeing any reason why such a law should exist" The name is derived from the Greek word efwmpio, meaning experience or trial In- stances of such laws are abundant We learn empiri- cally that a certain strong yellow colour at sunset, or an unusual clearness in the air, portends rain ; that a quick pulse indicates fever; that horned animals are always ruminants; that quinine affects beneficially the nervous system and the health of the body generally ; that strych- nine has a terrible effect of the opposite nature : all these are known to be true by repeated observation, but we can give no other reason for their being true, that is, we cannot bring them into harmony with any other scientific facts ; nor could we at all have deduced them or antici- pated them on the ground of previous knowledge. The connection between the sun's spots, magnetic storms, auroras, and the motions of the planets mentioned in the last Lesson, is perhaps the most remarkable known instance of an empirical induction ; for no hint has yet been given of the way in which these magnetic influences are exerted throughout the vast dimensions of the planet- ary system. The qualities of the several alloys of metals are also good instances of empirical knowledge. No one can tell before mixing two or three metals for the first time in any given proportions what the qualities of the mixture will be that brass should be both harder xxx.] METHODS. 257 and more ductile than either of its constituents, copper and zinc ; that copper alloyed with the very soft metal tin should make hard and sonorous bell-metal ; that a certain mixture of lead, bismuth, tin and cadmium, should melt with a temperature (65 cent.) far below that of boiling water*. However useful may be empirical knowledge, it is yet of slight importance compared with the well-connected and perfectly explained body of knowledge which con- stitutes an advanced and deductive science. It is in fact in proportion as a science becomes deductive, and enables us to grasp more and more apparently uncon- nected facts under the same law, that it becomes perfect. He who knows exactly why a thing happens, will also know exactly in what cases it will happen, and what dif- ference in the circumstances will prevent the event from happening. Take for instance the simple effect of hot water in cracking glass. This is usually learnt empiri- cally. Most people have a confused idea that hot water has a natural and inevitable tendency to break glass, and that thin glass, being more fragile than other glass, will be more easily broken by hot water. Physical science, how- ever, gives a very clear reason for the effect, by showing that it is only one case of the general tendency of heat to expand substances. The crack is caused by the success- ful effort of the heated glass to expand in spite of the colder glass with which it is connected. But then we shall see at once that the same will not be true of thin glass vessels ; the heat will pass so quickly through that the glass will be nearly equally heated ; and accordingly chemists habitually use thin uniform glass vessels to hold or boil hot liquids without fear of the fractures which would be sure to take place in thick glass vessels or bottles. The history of science would show conclusively that * Roscoe's Lessons in Elementary Chemistry, p. 175. j$8 EMPIRICAL AND DEDUCTIVE [L deduction was the clue to all the greatest discoveries, Newton, after Galileo the chief founder of experimen- tal philosophy, possessed beyond all question the great- est power of deductive thought which has ever been enjoyed by man. It is striking indeed to Compare his results in optics with those in chemistry or alchemy. It is not generally known that Newton was really an alche- mist, and spent days and nights in constant experiments in his laboratory, trying to discover the secret by which metals could be transmuted into gold. But in these re- searches all was purely empirical, and he had no clue to guide him to successful experiments. A few happy guesses given in his celebrated Queries are all the result of this labour. But in the science of Optics it was quite otherwise ; here he grasped general laws, and every ex- periment only led him to devise and anticipate the results of several others, each more beautiful than the lost. Thus he was enabled to establish beyond all doubt the founda- tions of the science of the Spectrum, now bearing such wonderful results. Some persons may suppose that Newton, living shortly after Bacon, adopted the Baconian method, but I believe that there is no reference to Bacon in Newton's works; and it is certain that he did not employ the method of Bacon. The Prineipia, though containing constant appeals to experiment and observa- tion, is nevertheless the result of a constant and sustained effort of deductive mathematical reasoning. What Mr Mill has called the Deductive Method, but which I think might be more appropriately called the 3omblned or Complete Method, consists in the alternate use of induction and deduction. It may be said to have three steps, as follows : 1. Direct Induction. 2. Deduction, or, as Mr Mill calls it, Ratiocination, 5. Verification. XXX. J METHODS. 349 The first process consists in such a rough and simpU appeal to experience as may give us a glimpse of the lawi which operate, without being sufficient to establish theii truth. Assuming them as provisionally true, we then proceed to argue to their effects in other cases, and a further appeal to experience either verifies or negatives the truth of the laws assumed. There are, in short, two appeals to experience connected by the kitermediate us of reasoning. Newton, for instance, having passed a ray of sun-light through a glass prism found that it was spread out into a series of colours resembling those of the rainbow. He adopted the theory that white light was actually com- posed of a mixture of different coloured lights, which became separated in passing through the prism. He saw that if this were true, and he were to pass an isolated ray of the spectrum, for instance, the yellow ray, through a second prism, it ought not to be again broken up into different colours, but should remain yellow whatever was afterwards done with it. On trial he found this to be the case, and afterwards devised a succession of similar con- firmatory experiments which verified his theory beyond all possible doubt. It was no mere accident that led Pascal to have a barometer carried up to the top of the mountain Puy dc Dome in France. Galileo, indeed, became acquainted by accident with the fact that water will not rise in an ordi- nary pump more than 33 feet, and was thus led to assert that the limited weight of the atmosphere caused it to rise. Torricelli, reasoning from this theory, saw that mercury, which is fourteen times as heavy as water, should not rise more than one -fourteenth part of the dis- ance, or about 29 or 30 inches. The experiment being tried verified the theory. It was the genius of Pascal, however, which saw that the experiment required to b raried in another way by carrying the mercurial barome 173 jto EMPIRICAL AND DEDUCTIVE [LESS ter to the top of a mountain. If the weight of the atmo sphere were really the cause of the suspension of the mer- cury, it ought to stand lowei on the mountain than below, because only the higher parts of the atmosphere pressed upon the mountain. The success of the experiment com pletely verified the original hypothesis. The progress ol the experimental sciences mainly depends upon the mod* in which one experiment thus leads to othen, and dis- closes new facts, which would in all probability have nevei come under our notice had we confined ourselves to the purely Baconian method of collecting the facts first and performing induction afterwards. The greatest result of the deductive method is no less than the theory of gravitation, which makes a perfect instance of its procedure. In this case the preliminary induction consisted, we may suppose, in the celebrated fall of the apple, which occurred while Newton was sitting in an orchard during his retirement from London, on account of the Great Plague. The fall of the apple, we are told, led Newton to reflect that there must be a power tending to draw bodies towards the earth, and he asked himself the question why the moon did not on that account fall upon the earth. The Lancashire astronomer Horrocks suggested to his mind another fact, namely, that when a stone is whirled round attached to a string, it exerts a force upon the string, often called centrifugal force. Hor- rocks remarked that the planets in revolving round the sun must tend in a similar way to fly off from the centre Newton was acquainted with Horrocks' views, and was thus possibly led to suppose that the earth's attractive force might exactly neutralise the moon's centrifugal tendency, so as to maintain that satellite in constant rotation. But it happened that the world was in possession ol certain empirical laws concerning the motions of the pi* xxx.] METHODS. jfa nets, without which Newton could scarcely have proceeded further. Kepler had passed a lifetime in observing the Heavenly bodies, and forming hypotheses to explain their motions. In general his ideas were wild and unfounded, but the labours of a lifetime were rewarded in the esta- blishment of the three laws which bear his name, and describe the nature of the orbits traversed by the planets, and the relation between the size of such orbit and. the time required by the planet to traverse it. Newton wai able to show by geometrical reasoning that if one body revolved round another attracted towards it by a force decreasing as the square of the distance increases, it would necessarily describe an orbit of which Kepler's laws would be true, and which would therefore exactly resemble the orbits of the planets. Here was a partial verification of his theory by appeal to the results of experience. But several other philosophers had gone so far in the investi- gation of the subject It is Newton's chief claim to ho- nour, that he carried on his deductions and verifications until he attained complete demonstration. To do this it was necessary first of all to show that the moon actually does fall towards the earth just as rapidly as a stone would if it were in the same circumstances. Using the best information then attainable as to the distance of the moon, Newton calculated that the moon falls through the space of 13 feet in one minute, but that a stone, if elevated so high, would fall through 15 feet Most men would have considered this approach to coincidence as a proof of his theory, but Newton's love of certain truth rendered him different even from most philosophers, and the dis- crepancy caused him to lay " aside at that time any fur- ther thoughts of this matter." It was not till many years afterwards (probably 15 or 1 6) that Newton, hearing of some more exact data from wkkh he could calculate the distance of the moon, 962 EMPIRICAL AND DEDUCTIVE [LESS was able to explain the discrepancy. His theory of gra- vitation was then verified so far as the moon was con- cerned ; but this was to him only the beginning of a long course of deductive calculations, each ending in a verifica- tion. If the earth and moon attract each other, and also the sun and the earth, similarly there is no reason why the sun and moon should not attract each other. Newton followed out the consequences of this inference, and showed that the moon would not move as if attracted by the earth only, but sometimes faster and sometimes slower. Comparisons with Flamsteed's observations of the moon showed that such was the case. Newton argued again, that as the waters of the ocean are not rigidly attached to the earth, they might attract the moon, and be attracted in return, independently of the rest of the earth. Certain daily motions would then be caused thereby exactly resembling the tides, and there were the tides to verify the fact It was the almost superhuman power with which he traced out geometrically the consequences of his theory, and submitted them to repeated comparison with experience, which constitutes his preeminence over all philosophers. What he began has been going on ever since. The places of the moon and planets are calculated for each day on the assumption of the absolute truth of Newton's law of gravitation. Every night their places are observed as far as possible at Greenwich or some other observatory; comparison of the observed with the predicted place is always in some degree erroneous, and if coincident would be so only by accident. The theory is never proved com- pletely true, and never can be ; but the more accurately the results ef the theory are calculated, and the more perfect the instruments of the astronomer are rendered, the more close is the correspondence. Thus the rude observations f Kepler and the few slight facts which worked on New XXX.J METHODS. j6 ton's mind, were the foundation of a theory which yielded indefinite means of anticipating new facts, and by con- stant verification, as far as human accuracy can go, has been placed beyond all reasonable doubt. Were space available it might be shown that all otha great theories have followed nearly the same course. The undulatory theory of sound was in fact almost verified by Newton himself, though when he calculated from it the velocity of sound there was again a discrepancy, which only subsequent investigation could explain. This theory no doubt suggested the corresponding theory of light, which when followed out by Young, Fresnel, and others, always gave results which were ultimately in harmony with observation. It even enabled mathematicians to anticipate results which the most ardent imagination could hardly have guessed, and which mere haphazard experiment might never have revealed. Dalton's laws ol equivalent proportions in chemistry, if not his atomic theory, were founded on experiments made with the simplest and rudest apparatus, but results deduced from them are daily verified in the nicest processes of modern chemical analysis. The still more modern theory of the Conservation of Energy, which had been vaguely antici- pated by Bacon, Rumford, Montgolfier, Seguin, Mayer and possibly others, was by Mr Joule brought to the test of experimental verification in some of the most beautiful and decisive experiments which are on record. It will be long before scientific men shall have traced out all the consequences of this grand principle, but its correspond' ence with fact already places it far beyond doubt It will now be apparent, I think, that though observa- tion and induction must ever be the ground of all certain knowledge 01 nature, their unaided employment could never have led to the results of modern science. He who merely collects and digests facts will seldom acquire a 204 EXPLANATION, TENDENCY, [LESS comprehension of their laws. He who frames a theor> and is content with his own deductions from it, like Des- cartes, will only surprise the world with his misused genius ; but the best student of science is he who with a copious store of theories and fancies has the highest power of foreseeing their consequences, the greatest dili- ^en.e. in comparing them with undoubted facts, and the greatest candour in confessing the ninety-nine mistakes he has made in reaching the one true law of nature. LESSON XXXI. EXPLANATION, TENDENCY, HYPOTHESIS, THEORY, AND FACT. IN the preceding Lessons I have used several expressions of which the meaning has not been defined. It will now be convenient to exemplify the use of these terms, and tc arrive as far as possible at a clear understanding of their proper meanings. Explanation is literally the making plain or clear, so that there shall be nothing uneven or obscure to inter- rupt our view. Scientific explanation consists in harmo- nizing fact with fact, or fact with law, or law with law, so that we may see them both to be cases of one uniform law of causation. If we hear of a great earthquake in >ome part of the world and subsequently hear that a eighbouring volcano has broken out, we say that the .arthquake is thus partially explained. The eruption ihows that there were great forces operating beneath the ;arth's surface, and the earthquake is obviously an effect Df such causes. The scratches which may be plainly seen upon the surface of rocks in certain parts of Wales and Cumberland, are explained by the former existence of gla- tiers in those mountains ; the scratches exactly hanroniif xxxi.] HYPOTHESIS, THEORY, AND FACT. 265 with the effects of glaciers now existing in Switzerland, Greenland, and elsewhere. These may be considered - planattons of fact by fact. A fact may also be explained by a general law ol nature, that is the cause and mode of its 'production may be pointed out and shown to be the same as operates in many apparently different cases. Thus the cracking ol glass by heat was- explained (p. 257) as one result of the universal law that heat increases the dimensions of solid bodies. The trade-winds are explained as one case of the general tendency of warm air to rise and be displaced by cold and dense air. The very same simple laws of heat and mechanics which cause a draught to flow up a chimney when there is a fire below, cause winds to blow from each hemisphere towards the equator. At the same time the easterly direction from which the winds come is explained by the simplest laws of motion ; for as the earth rotates from west to east, and mores much more rapidly at the equator than nearer the poles, the air tends to preserve its slower rate of motion, and the earth near the equator moving under it occasions an apparent motion of the wind from east to west. There are, according to Mr Mill, three distinct ways in which one law may be explained by other laws, or brought into harmony with them. The first is the case where there are really two or more separate causes in action, the results of which are combined or added together, homogeneously. As was before explained, homogeneous Intermixture of effects (p. 252) means that the joint effect is simply the sum of tht separate effects, and is of the same kind with them. GUI last example of the trade-winds really comes under thii case, for we find that there is one law or tendency which causes winds to blow from the arctic regions toward* tht equator, and a second tendency which causes then to blo 266 EXPLANATION, TENDENCY, [LESS from east to west These tendencies are combined to gether, and cause the trade- winds to blow from the North- Ernst in the northern hemisphere, and from the South- East in the southern hemisphere. The law according to which ihe temperature of the air is governed in any part of the earth is a very complicated one, depending partly on the law by which the sun's heating power is governed, partly on the power of the earth to radiate the heat away into space, but even more perhaps on the effect of currents of air or waier in bringing warmth or carrying k away. The path of a cannon-ball or other projectile is deter- mined by the joint action of several laws ; firstly, the simple law of motion, by which any moving body tends to move onward at an uniform rate in a straight line ; secondly, the law of gravity, which continually deflects the body towards the earth's surface ; thirdly, the resist- ance of the air, which tends to diminish its velocity. The reader will perhaps have noticed the frequent use of the word tendency, and I have repeatedly spoken of a cause as tending to produce its effect If the joint and homogeneous action of causes has been clearly explained, it will now be clear that a tendency means a cause which will produce an effect unless there be opposite causes, which, in combination with it, counteract and disguise that effect Thus when we throw a stone into the air the attractive power of the earth tends to make it fall, out the upward motion we have impressed upon it disguises th< result for a certain time. The interminable revolving motion of the moon round the earth is the result of two balanced tendencies, that towards the earth, and that tc proceed onward in a straight line. The laws of motior. and gravity are such that this balance must always be preserved ; if the moon by any cause were brought nearei to the earth its tendency to fly off would be increased, and would exceed the effect of gravity until it had regained XXXI. J HYPOTHESIS, THEORY, AND FACT. its proper distance. A tendency then is a cause may or may not be counteracted. In the second case of explanation an effect is shown to be due, not to the supposed cause directly, but to an Intermediate effect of t&at cause. Instead of A being thf ca-ise of C, it is found that A is the cause of /?, and b the :ause of C, so that B constitutes an Intermediate link, This explanation may seem to increase the complexity of the matter, but it really simplifies it ; for the connection of A with B may be a case of a familiar and simple law, and so may that of B with C ; whereas the law that A pro- duces C may be purely empirical and apparently out of harmony with everything else. Thus in lightning it seems as if electricity had the power of creating a loud explosion ; but in reality electricity only produces heat, and it is the heat which occasions sound by suddenly expanding the air. Thus thunder comes into harmony with the sound of artillery, which is also occasioned by the sudden expansion of the heated gases emitted by the powder. When chlorine was discovered it was soon found to have a strong power of bleaching, and at the present day almost all bleaching is done by chlorine instead of the sun, as formerly. Inquiry showed however that k was not really the chlorine which destroyed colour, but that oxygen is the intermediate and active agent Chlorine decomposes water, and taking the hydrogen leaves the oxygen in a state of great activity and ready to destroy the organic colouring matter. Thus a number of facts are harmonized ; we learn why dry chlorine does not bleach, and why there are several other substances which resemble chlorine in its bleaching power, for instance, ozone, peroxide of hydrogen, sulphurous acid, and a pecu- liar oxide of vanadium, lately discovered by Dr Roscoe. It would be impossible to understand the effect at all un- less we knew that it is probably due to active oxygen 01 068 EXPLANATION, TENDENCY, [ucsa ozone in all the cases, even in the old method of bleach- ing by exposure to the sun *. The third and much more important case of e planation is where one law is shown to be a ease of more general law. As was explained in Lesson xxiv. we naturally discover the less general first, and gradually penetrate to the more simple but profound secrets o< naturt. It has often been found that scientific men were in possession of several well-known laws without perceiv- ing the bond which connected them together. Men, for instance, had long known that all heavy bodies tended to fall towards the earth, and before the time of Newton it was known to Hooke, Huyghens, and others, that some force probably connected the earth with the sun and moon. It was Newton, however, who clearly brought these and many other facts under one general law, so that each fact or less general law throws light upon every other. The science of Electricity now harmonizes a vast series of partial laws and facts between which it was a truly difficult task to discover any resemblance. The chief properties of the magnet had been fairly known since the time of Gilbert, the physician of Queen Elixa- beth ; common frictional electricity was carefully stu- died by Otto von Guericke, Epinus, Coulomb, and others ; Galvanism was elaborately investigated almost as soon as Galvani and Volta discovered the fact that the che- mical action of one substance on another may produce electricity. In the early part of this century there were three distinct sciences, Magnetism, Electricity and Gal- vanism ; now there is but one science. Oersted at Copenhagen gave in 1819 the first link between them, by pointing out that an electric current may cause move- taeats in a compass-needle. Ampere and Faraday worked Watts' Diftitnoy of Chtmistry, VoL I. p. foi. xxxi.] HYPOTHESIS, THEORY, AND FACT 269 out the complicated relations of the three sciences, com- prehending them finally in a wider science, which may be called Electro-magnetism, or we may perhaps conveniently generalize the name Electricity so as to comprehend all the phenomena connected with it A number of minor laws and detached facts are com- prehended and explained in the theory now generally accepted, that heat, electricity, light, and in fact all the phenomena of nature, are but manifestations in different forms of one same kind of energy. The total amount of energy existing in the universe is held to be fixed and un- alterable, like the quantity of matter ; sometimes it is disguised by affecting only the insensible molecules; at other times it is seen to produce palpable mechanical effects, as in the fall of a stone, or the expansion of steam. Now it had been previously known, ever since the time of the Greeks, that a simple lever, although greatly altering the character of force by making its action slower or faster, does not alter its amount, because the more intense the force the slower and more limited is its action. In modern times a similar truth was proved of every kind of machine ; and it was recognised that, apart from friction, no kind of mechanism either creates or destroys energy. It had been independently recognised that electricity produced in the galvanic battery was exactly proportional to the amount of chemical action, and that almost any one of the forces named could be converted into any one of the others. All such facts are now comprehended under one general theory, the details of which are being gradually rendered more certain and accurate, but th< main principle of which is that a certain amount of me- chanical energy is equal to a certain amount of heat, a certain amount of electricity, of chemical action, or even of muscular exertion. The word hypothesis is much used ia connection will iTO EXPLANATION, TENDENCY, [l the subject we are discussing, and its meaning must be considered. It is derived from the Greek words wr, under, and 6kw, placing, and is therefore exactly synony- mous with the Latin word supposttio, a placing under whence our common word supposition. It appears tc mean in science the imagining of some thing, force 01 cause, which underlies the phenomena we are examining and is the agent in their production without being capable of direct observation. In making a hypothesis we assert the existence of a cause on the ground of the effects observed, and the probability of its existence depends upon the number of diverse facts or partial laws that we are thus enabled to explain or reduce to harmony. To be of any value at all a hypothesis must harmonize at least two different facts. If we account for the effects of opium by saying with Moliere that it possesses a dormitive power, or say that the magnet attracts because it has a magnetic power, every one can see that we gain nothing. We know neither more nor less about the dormitive or magnetic power than we do about opium or the magnet. But if we suppose the magnet to attract because it is occupied by circulating currents of electricity the hypo- thesis may seem a very improbable one, but is valid, because we thus draw a certain analogy between a magnet and a coil of wire conveying electricity. Such a coil of wire attracts other coils exactly in the way that one mag- net attracts another ; so that this hypothesis enables us to harmonize several different facts. The existence of intense heat in the interior of the earth is hypothetical in so far as regards the impossibility of actually seeing and measuring the heat directly, but it harmonizes so many facts derived from different sources that we can hardly doubt its existence. Thus the occurrence of hot springs and volcanoes are some facts in its favour, though they might be explained on other grounds ; the empirical lav bjects; when, in fact, we consider a thing only as a member of a class, we are said to genermlixe. If, aftei studying the properties of the circle, we proceed to those jf the ellipse, parabola and hyperbola, it is soon found that the circle is only one case of a whole class of curves called the conic sections, corresponding to equations o< the second degree; and I generalize when I regard cer- tain of the properties of the circle as shared by many other curves. Dr Whewell added to the superabundance of terms to express the same processes when he introduced the ex- pression Colligation of ficU. Whenever two things are found to have similar properties so as to be placed in the same class they may be said to be connected together. We connect together the places of a planet as it moves round the sun, when we conceive them as points upon a common ellipse. Whenever we thus join together pre- viously disconnected facts, by a suitable general notion or hypothesis, we are said to colligate them, Dr Whewell adds that the general conceptions employed must be (i) clear, and (2) appropriate ; but it may well be ques- tioned whether there is anything really different in theat processes from the general proceat of natural classifi catioi rhich we hare considered. LESSON XXXIIL REQUISITES OF A PHILOSOPHICAL LANGUAGE. AMONG the subsidiary processes requisite to the successful prosecution of inductive reasoning must be placed the construction of a suitable language. It is in fact impos- sible to over-estimate the importance of an accurate and copious language in any science; and the study of things would be almost useless without names to denote those things and record our observations concerning them. It is easily apparent, indeed, that language serves three distinct and almost independent purposes : 1. As a means of communication. 2. As a mechanical aid to thought 3. As an instrument of record and reference. In its first origin language was used chiefly if not exclu- sively for the first purpose. Savage tribes exist in great numbers at the present day who seem to accumulate no knowledge. We may even say that the lower animals often possess some means of communication by sounds or natural signs which constitute language in the first sense, though they are incapable of reasoning by general notions. Some philosophers have held that it fs impossible to carry on reasoning without the use of language. The true nominalist went so far as to say that there are no such things as general notions, and that general names therefore constitute all that is general in science and tSS REQUISITES OF A [LESt, reasoning. Though this is no doubt false (see p. 13), b must nevertheless be allowed that unless general ideas were fixed and represented by words, we could nevej attain to sustained thought such as we at present enjoy The use of language in the second purpose is doubtless indispensable in a practical point of view, and reasoning may almost be considered identical with the correct us< of words. When language is used solely to assist reason- ing there is no need that the meaning of each word should be uxed ; we might use names, as the letters x, y, *, a, by c, &c., are used in algebra to denote any quantity that happens to occur in a problem. All that is requisite is never to confuse the meaning attributed to a word in one argument with the different meaning attributed in another argument Algebra may, in fact, be said to con- sist of a language of a very perfect kind adapted to the second purpose only, and capable of leading a person to the solution of a problem in a Symbolical or mechanical manner. Language, as it is furnished to us ready made by the habitual growth of centuries, is capable of fulfilling all three purposes, though by no means in a perfect manner. As words possess a more or less fixed customary meaning we can not only reason by their aid, but communicate our thoughts or record them ; and it is in this last respect we have now to treat the subject The multitude of facts required for the establishment of a science could not be retained in the memory witt sufficient accuracy. Hence an indispensable subsidiar) of induction is the means of describing and recording GUI observations. Thus only can knowledge be accumulated, so that each observer shall start with the advantage oj knowing what has been previously recorded and proved It will be necessary then to consider the mode in which language serves for the registration of facts, and to investi xxxm.J PHILOSOPHICAL LANGUAGE, 289 gate the requisite qualities of a philosophical -anguagc suitable to the needs of science. As an Instrument of record language must evidently possets two principal requisites : 1. Precision or definiteness of meaning. 2. Completeness. A name is worse than useless unless, when used to record a fact, it enables MS to ascertain what was the nature of the fact recorded. Accuracy and precision is then a more important quality of language than abun- dance. The want of an appropriate word will seldom give rise to actual error and fallacy ; it will merely oblige us to employ a circumlocutory phrase or else leave the fact unrecorded. But it is a self-evident convenience that whenever a thing, notion, or quality has often to be refer red to there should be a name appropriated to the purpose, and there ought only to be one name. Let us consider in succession what must be the character of a precise and complete language. It may not previously have struck the reader, but it is certainly true, that description is impossible without the assertion of resemblance between the fact described and some other fact We can only describe a thing by giving it a name ; but how can we learn the meaning of that name ? If we describe the name by other names we only have more names of which the meanings are required, We must ultimately learn the meanings, not from namei but from things which bear those names. If anyone were ignorant of the meaning of blue he could not be in- formed but by reference to something that excited in him the sensation of blueness, and had he been blind from birth he could not acquire any notion of what blueness was. There are indeed a number of words so familiar to us from childhood that we cannot tell when or how we learnt their meanings, though it must have been by refer 19 JOX> RI*.U,UlSlTt<.S OF A I LESS encc to things. But when we come to the more precise use of names we soon have to make fresh reference to physical objects. Then we should describe the several kinds of blue colour as sky-blue, azure-blue, indigo- blue, cobalt-blue ; green colour we likewise distinguish as sea- green, olive-green, emerald-green, grass-green, &<"fc T (5) The Chinese are industrious. 7. Why it it desirable in controversy .,, retute a state- ment by its contradictory and not by its contrary 2 ' X. Ctnvtrrie* **d Immediate Inftrtitc*. 1. Define inference and conversion. 2. What are converse and convertend propositions? J. State the rules of valid conversion, 4. Name all the kinds of conversion. 5. By what process do we pass from each of tfee fol- lowing propositions to the next ? (1) No knowledge is useless. (2) No useless thing is knowledge, (3) All knowledge is noMiseless. ** 3 ' I (4) All knowledge is useful (5) What is not useful is not knowledge l (6) What is useless is not knowledge, - (7) No knowledge is useless. & Gire the logical opposites of the following prop* *^ *ft sition, and the converse of its contradictory ? ^ *\\*~" ' * He cannot become rich who will not labour." p Apply negative conception to the proposition " Afi men are falUble * then convert and show thai . the result is the contrapositive of the original QUESTIONS AND&XERCISES. fc. Classify the propositions subjoined into the foui following groups: ,^y-~~" . Those which can be inferred from (i). \*~r t. Those from which (i) can be inferred. "*r c. Those which do not contradict (i), but be inferred frem it J. Those which contradict (i). p (i) AH just acts are expedient acts. (2) No expedient acts are unjust, - *"-- (3) No J ust acts a* 6 inexpedient &, ^ (4) All inexpedient acts are unjust (5) Some unjust acts are inexpedient c) (6) No expedient acts are v*s (7) Some inexpedient CvS are unjust (8) All expedient acts are just (9) No inexpedient acts are just (10) All unjust acts are inexpedient (n) Some inexpedient acts are just acts. (12) Some expedient acts are just (13) Some just acts are expedient .'14) Some unjust acts are expedient LlSSONS VIII. IX. and X. Examples of Propotittou. The reader is desired to ascertain the logical character 3f each of the following propositions; he is to state ol ?ach whether it is affirmative or negative, universal, par- ticular, singular or indefinite, pure or modal, exclusive or exceptive, &c. ; when irregularly stated he is to reduce the proposition to the simple logical order; he is then to convert the proposition, and to draw immediate inference* from it by any process which may be applicable. (1) All birds are feathered. (2) No reptiles are feathered. (l) Fixed stars are self-luminous. QUESTIONS AND EXERCISE*. (4) Perfect happiness is impossible. (5) Life every man holds dear. (6) Every mistake is not a proof of ignorance. (7) Some of the most valuable books are seldom read (8) He jests at scars who never felt a wound (9) Heated metals are softened. (10) Not one of the Greeks at Thermopylae escaped. (11) Few are acquainted with themselves. (12) Whoso loveth instruction loveth knowledge, '13) Nothing is harmless that is mistaken for a virtue ^14) Some of our muscles act without volition. (15) Metals are all good conductors of heat (16) Fame is no plant that grows on mortal soil. (17) Only the brave deserve the fair. (18) No one is free who doth not command himself. (19) Nothing is beautiful except truth. (20) The wicked shall fall by his own wickedness. (21) Unsafe are all things unbecoming. (22) There is no excellent beauty that hath not some strangeness in the proportion. (23) It is a poor centre of a man's actions, himself. (24) Mercy but murders, pardoning those that mi. (25) I shall not all die. (Non omnis morutr.) (26) A regiment consists of two battalions. (27) Tis cruelty to load a falling man. (28) Every mistake is not culpable. (29) Quadrupeds are vertebrate anim^ (30) Not many of the metals are brittle. (31) Many are the deserving men who are unfortunate (32; Amalgams are alloys of mercury. (33) One kind of metal at least is liquid. (34) Talents are often misused. 35) Some parallelograms have tAep- adjoining equal (36) Britain is an island. (37) Romulus and Remus were twins. QUESTIONS AND EXERCISES. 90$ (38) A man's a num. (39) Heaven is all mercy. (s are X's. Some Z's are not X's. No Z's are X's. (3) No fish suckles its young ; The whale suckles its young ; Therefore the whale is no fish. 6. Deduce conclusions from the following pifmiiM and state to what mood the syllogism belongs. (1) Some amphibious animals are mammalian. All mammalian animals are vertebrate. (2) All planets are heavenly bodies. No planets are self-luminous. (5) Mammalian animals are quadrupeds. No birds are quadrupeds. (4) Ruminant animals are not predacious. The lion is predacious. QUESTIONS AND EXERCISES. jn jr. Invent examples to show that false premises may give true conclusions. 8. Supply premises to the following conclusions : - (1) Some logicians are not good reasoners. (2) The rings of Saturn are material bodies. (3) Party government exists in every democracy. (4) All fixed stars obey the law of gravitation. LESSON XVII.~77k Syllogism; Reductum. I. State and explain the mnemonic lines Barbara, Celarent, &c. S. Construct syllogisms in each of the following moods, taking X, Y, Z, for the major, middle, and minor terms respectively, and show how to reduce them to the first figure : Cesare, Festino, Darapti, Datisi, Ferison, Camenes, Fesapo. 3. What is the use of Reduction ? 4. Prove that the following premises cannot give a universal conclusion E I, I A, O A, I E. 5. Prove that the third figure must have an affirmative minor premise, and a particular conclusion. 6. Reduce the moods Cesare and Camenes by the Indirect method, or Reductio ad Impossible. LESSON XVIII. Irregular and Compound Syllogisms. 1. Describe the meaning of each of the terms En- thymeme, Prosyllogism, Episyllogism, Epichei- rema, Sorites. au Make an example of a syllogism in which there arc two prosyllogisms. 3. Construct a sorites of four premises and resolve it into distinct syllogisms. 4. What are the rules to which a sorites most conform/ i SI* QUESTIONS AND EXERCISES. $. Th reader is requested to analyse the following arguments, to detect those which are false, and to ascertain the rules of the syllogism which they break ; if the argument appears valid he is to ascertain the figure and mood to which it belongs, to stete it in correct logical form, and then if it b in an imperfect figure to prove it by reduction to the first figure. The first six of the examples should be arranged both in the extensive and intensive orders. ^ _ 1. None but mortals are men. t C^-IA^-CA*^ Monarchs are nielL A Therefore inon^rchs are mortals. A !/* Personal deformity is an affliction of nature. Disgrace is not an affliction of nature. Therefore personal deformity is not disgrace. 8. Some statesmen are also authors ; for such are Mr Gladstone, Lord Derby, Lord Russell, and Sir G. C Lewis. C This explosion must have been occasioned by gun- powder; for nothing else would have possessed sufficient force. ft. Every man should be moderate; for excess will cause diseasej,:^ , dju^^o -^t*^^^c* i. Blessed are the mercifuljior they shall obtain mercy. T. As almost all the organs of the body have a known use, the spleen must have some use. ft Cogito, ergo sum. (I think, therefore I exist) . Some speculative men are unworthy of trust ; for they are unwise, and no unwise man can be trusted. jO. No idle person can be a successful writer of his- tory; therefore Hume, Macaulay, Hallam and Grote must have been industrious. QUESTIONS AND EXERCISES. 313 11. Who spareth the rod, hateth his child; the parent who loveth his child therefore spareth not Uw rod. 11. Comets must consist of heavy matter; for other wise they would not obey the law of gravitation IS. Lithium is an element ; for it is an alkali-pro ducing substance, which is a metal, which is an element 14. Rational beings are accountable for their actions; brutes not being rational, are therefore exempt from responsibility. 15 A singular proposition is a universal one; for it applies to the whole of its subject It. Whatever tends to withdraw the mind from pur- suits of a low nature deserves to be promoted ; classical learning does this, since it gives us a taste for intellectual enjoyments ; therefore it deserves to be promoted. 17. Bacon was a great lawyer and statesman ; and as he was also a philosopher, we may infer that any philosopher may be a great lawyer and statesman, 18. Immoral companions should be avoided ; but some immoral companions are intelligent persons, so that some intelligent persons should be avoided. 19. Mathematical study undoubtedly improves the reasoning powers ; but, as the study of logic is not mathematical study, we may infer that it doei not improve the reasoning powers, to Every candid man acknowledges merit in a rival every learned man does not do so; therefore every learned man is not candid. LESSON XIX. Conditional Argument*. I. What are the kinds of conditional proportion*, and by what signs can you recognise them? JI4 QUESTIONS AND EXERCISES. r What are the rules of the hypothetical syllogism ? y, To what categorical fallacies do breaches of the rules correspond? 4. Select from the following such as are valid argv ments, and reduce them to the categorical form explain the fallacious reasoning in the others (1) Rain has fallen if the gftrnnl^is wet; but th ^~\ ground is not wet ; therefore rain has not fallen (2) If rain has fallen, the ground is wet ; but rain has not fallen ; therefore the ground is not wet. (3) The ground is wet, if rain has fallen ; the ground is wet ; therefore rain has fallen. (4) If the ground is wet, ram has fallen ; but rain has fallen ; therefore the ground is wet N.B. In these as in other logical examples the student must argue only from the premises, and not from any other knowledge of the subject-matter. 5. Show that the canons of syllogism (p. 121) may be stated indifferently in the hypothetical or categorical form. 6. State the following in the form of a Disjunctive 01 Dilcmmatic argument, and name the kind to which it belongs. If pain is severe it will be brief; and if it last long it will be slight; therefore it is to be patiently borne. LESSONS XX. and XXI Fallacies. i Classify fallacies. t. Explain the following expressions : A dicto secundum quid ad dictum simpliciter ; igno- ratio elenchi ; argumentum *d hominem ; art> mentum ad populum ; petitio principii ; circului in probando; non sequitur; post hoc ergc propter hoc QUESTIONS AND EXERCISES. ji$ j. What is arguing in a circle; and what is a qne* tion-begging epithet? 4. What differences of meaning may be produced ta the following sentence by varying the accent? M Newton's discovery of gravitation is not generally believed to have been at all anticipated by several philosophers in England and Holland." 5. Point out the misinterpretations to which the fol- lowing sentences might be liable. (l) He went to London and then to Brighton by the express train. (l) Did you make a long speech at the meeting? (3) How much is five times seven and nine? MISCELLANEOUS EXAMPLES. LESSONS IX. to XXI. (Continued from p. 313.) The following examples consist partly of true and partly of false arguments. The reader is requested to treat them as follows : 1. If the example is not in a simple and complete logical form, to complete it in the form which appears most appropriate. 2. To ascertain whether it is a valid or fallacious argument 3. To assign the exact name of the argument or fal- lacy as the case may be. 4. If a categorical syllogism, to reduce it to the first figure. 5. If a hypothetical syllogism, to state it in the cate gorical form. ^-^tl. Elementary substances alone are metals. Iron ii a metal ; therefore it is an elementary substance ,( QUESTIONS AND EXERCISES.. 2> r A To Athenians could have been Hdots ; for all thi HelJts were slaves, and all Athenians were free ' St. Aristoue must have been a man of extraordinary A $ ^industry; for only such a man could have pro * duced his works. vrv4^H*ccxA* AXV*. -c^^-^ui- 4. Nothing is better than wisdom; dry bread is ' better than nothing ; therefore dry bread is better- than wisdom. IB Pitt was not a great and useful minister; for though he would have been so had he came out Adam Smith's doctrines of Free Trade, he did not carry out those doctrines. St. Only the virtuous are truly noble; some who are called noble are not virtuous; therefore some who are called noble are not truly noble. ST. Ireland is idle and therefore starves ; she starves, and therefore rebels. SS. No designing person ought to be trusted; en- 1 gravers are by profession designers; therefore C. ^ they ought not to be trusted. SS, Logic as it was cultivated by the schoolmen **'^ proved a fruitless study ; therefore Logic as it is ^^^ cultivated at the present day must be a fruitless ^ ^* study likewise. ft. Is a stone a body? Yes. Then is not an animal a body? Yes. Are you an animal ? I think so. v ^ Ergo, you are a stone, being a body. Lucian. SI If ye were Abraham's children, ye would do the works of Abraham. John viii. 39. SI He that is of God heareth God's words : ye there fore hear them not, because ye are not of GoTI^ \^ John viil 47. 99. Mahomet was a wise lawgiver; for he studied the character of his people, Q y \ t/ / \ QUESTIONS AND EXERCISES. 317 4. Every one desires virtue, because every one desires happiness. 15. His imbecility of character might have been in- ferred from his proneness to favourites ; for all weak princes have this failing. De Morgan. St. He is brave \vho conquers his passions ; he who resists temptation conquers his passions; so that he who resists temptation is brave. IT Suicide is not always to be condemned; for it is but voluntary death, and this has been gladly embraced by many of the greatest heroes oV antiquity. 8. Since all metals are elements, the most rare of all the metals must be the most rare of all (he elements. tt. The express train alone does not stop at this sta- tion ; and as the last train did not stop it must have been the express train. 40. Peel's remission of taxes was beneficial ; the taxes remitted by Peel were indirect; therefore the remission of indirect taxes is beneficial 41. Books are a source both of instruction and amuse- ment ; a table of logarithms is a book ; there- fore it is a source both of instruction and amuse* inert. 41. All desires are not blameable ; all desires are liable to excess ; therefore some things liable to excess are not blameable. 41 Whosoever intentionally kills another should suffer death ; a soldier, therefore, who kills his enemy should suffer death. 44. Projectors are unfit to be trusted; this man has formed a project; therefore he is unfit to be trusted. 4A. Few towns in the United Kingdom have more thai i* QUESTIONS AND EXERCISES. 300,000 inhabitants ; and as all such towns ought to be represented by three members in Parlia- ment, it i? evident that few towns ought to hav three representatives. *4. All the works of Shakspeare cannot be read in a day; therefore the play of Hamlet, being on of the works of Shakspeare, cannot be read ir a day. T. In moral matters we cannot stand still ; therefore he who does not go forward is sure to fall behind. *8. The people of the country are suffering from famine ; and as you are one of the people of the country you must be suffering from famine. *9. Those substances which are lighter than water can float upon it ; those metals which can float upon it are potassium, sodium, lithium, &c. ; therefore potassium, sodium, lithium, &c., are lighter than water. 60. The laws of nature must be ascertained by De- duction, Traduction or Induction ; but the former two are insufficient for the purpose ; therefore the laws of nature must be ascertained by In- duction. 61. A successful author must be either very industrious or very talented ; Gibbon was very industrious, therefore he was not very talented. 62. You are not what I am ; I am a man ; therefore you are not a man. t The holder of some shares in a lottery is sure to gain a prize ; and as I am the holder of somi shares in a lottery I am sure to gain a prize. t. Gold and silver are wealth ; and therefore the diminution of the gold and silver in the country by exportation is the diminution of the wealth of the country. QUESTIONS AND EXERCISES. 319 M. Over credulous persons ought never to be believed and as the Ancient Historians were in man) instances over credulous they ought never to bi believed. It, Some mineral compounds are not decomposed by heat ; all organic substances are decomposed b> heat ; therefore no organic substances are mi- neral compounds. 5T. Whatever schools exclude religion are irreligious , Non-sectarian schools do not allow the teaching of religious creeds ; therefore they are irreligious. 58. Night must be the cause of day ; for it invariably precedes it. 59. The ancient Greeks produced the greatest master- pieces of eloquence and philosophy ; the Lace- daemonians were ancient Greeks ; therefore they produced the greatest masterpieces of eloquence and philosophy. 0. All presuming men are contemptible; this man, therefore, is contemptible ; for he presumes to believe his opinions are correct. 1. If a substance is solid it possesses elasticity, and so also it does if it be liquid or gaseous ; but all substances are either solid, liquid or gaseous; therefore all substances possess elasticity. II If Parr's life pills are of any value those who take them will improve in health ; now my friend who has been taking them ha improved in health j therefore they are of value. t. He who calls you a man speaks truly ; ne who calli you a fool calls you a man ; therefore he who calls you a fool speaks truly. 4. Who is most hungry eats most ; who eats least is most hungry ; therefore who eats least eats most W. What produces intoxication should be prohibited j 310 QUESTIONS SiND EXERCISES. the use of spirituous liquors causes intoxication therefore the use of spirituous liquors should be prohibited. f . What we eat grew in the fields ; loaves of bread are what we eat ; therefore loaves of bread grew in the fields. /. If light consisted of material particles it would possess momentum ; it cannot therefore consist of material particles, for it does not possess momentum. 18 Everything is allowed by law which is morally right ; indulgence in pleasures is allowed by law ; therefore indulgence in pleasures is morally right. t. AH the trees in the park make a thick shade ; this is one of them, therefore this tree makes a thick shade* TO. All visible bodies shine by their own or by re- flected Hght. The moon does not shine by its own, therefore it shines by reflected light ; but the sun shines by its own light, therefore it canncx shine by reflected light Tl. Honesty deserves reward ; and a negro is a fellow- creature ; therefore, an honest negro is a fellow- creature deserving of reward. Tl. Nearly all the satellites revolve round their planet! from west to east ; the moon is a satellite; there- fore it revolves round its planet from west to east Tl. Italy is a Catholic country and abounds in beg gars; France is also a Catholic country, and therefore abounds in beggars. ft. Every law is either useless or it occasions hurt U some person ; now a law that is useless ought to be abolished ; and so ought every law that occa- sions hurt; therefore every law ought to bi abolished. QUESTIONS AND EXERCISES. jai ff. The end of a thing is its perfection ; death is the end of life ; therefore death is the perfection oi life. ft When we hear that all the righteous people are happy, it is hard to avoid exclaiming, What ! are all the unhappy persons we see to be thought unrighteous? f T. I am offered a sum of money to assist this person in gaining the office he desires; to assist a person is to do him good, and no rule of morality forbids the doing of good ; therefore no rule oi morality forbids me to receive the sum of money for assisting the person. TS. Ruminant animals are those which have cloven feet, and they usually have horns; the extinct animal which left this foot-print had a cloven foot; therefore it was a ruminant animal and had horns. Again, as no beasts of prey are rumi- nant animals it cannot have been a beast of prey Tt. We must either gratify our vicious propensities, or resist them; the former course will involve us in sin and misery; the latter requires self- denial; therefore we must either fall into sin and misery or practise self-denial. ft. The stonemasons are benefited by the masons' union ; the bricklayers by the bricklayers' union ; the hatmakers by the hatmakers' union; in short, every trade by its own union ; therefore it is evident that if all workmen had unions all workmen would be benefitted thereby. L Every moral aim requires the rational means of attaining it ; these means are the establishment of laws ; and as happiness is the moral aim of man it follows that the attainment of happ requires the establishment of laws. ai |22 QUESTIONS AND EXERCISES. 83 He that can swim needs not despair to fly ; for tc swim is to fly in a grosser fluid, and to fly is to swim in a subtler. St. The Helvetii, if they went through the country of the Sequani, were sure to meet with various difficulties ; and if they went through the Roman province, they were exposed to the danger of opposition from Caesar; but they were obliged to go one way or the other ; therefore they were cither sure of meeting with various difficulties, or exposed to the danger of opposition from Caesar. De Bello Gallico, lib. I. 6. 84. Riches are for spending, and spending for honour and good actions; therefore extraordinary ex pense must be limited by the worth of the occa- sion. Bacon. 85. If light is not refracted near the surface of the moon, there cannot be any twilight; but if the moon has no atmosphere light is not refracted near its surface; therefore if the moon has no atmosphere there cannot be any twilight %i The preservation of society requires exchange; whatever requires exchange requires equitable valuation of property ; this requires the adoption of a common measure ; hence the preservation of society requires the adoption of a common measure. 87 The several species of brutes being created to prey upon one another proves that the human species were intended to prey upon them. The more correct the logic, the more certainly the conclusion will be wrong if the premises are false. Therefore where the premises are wholly uncertain, the best logician is the 'east safo guide. QUESTIONS AND EXERCISES. 333 8. If our rulers could be trusted always to look to the best interests of their subjects, monarchy would be the best form of government ; but they cannot be trusted; therefore monarchy is not the best form of government 30, If men were prudent, they would act morally for their own good ; if benevolent, for the good oi others. But many men will not act morally, cither for their own good, or that of others ; such men, therefore, are not prudent or benevolent. 01. He who bears arms at the command of the magis- trate does what is lawful for a Christian; the Swiss in the French service, and the British in the American service, bore arms at the command of the magistrate ; therefore they did what was lawful for a Christian. Whately. W. A man that hath no virtue in himself ever envieth virtue in others ; for men's minds will either feed upon their own good or upon others' evil ; and who wanteth the one will prey upon the other. Bacon. 98. The object of war is durable peace; therefore soldiers are the best peace-makers. 94. Confidence in promises is essential to the inter- course of human life ; for without it the greatest part of our conduct would proceed upon chance. But there could be no confidence in promises, if men were not obliged to perform chem ; the obli- gation, therefore, to perform promises is essential to the same ends and in the same degree. If the majority of those who use public-houses are prepared to close them, legislation is unne- cessary ; but if they are not prepared for such a measure, then to force it on them by outside pressure is both dangerous and unjust. fl a t*4 QUESTIONS AND EXERCISES. f . He who believes himself to be always in the rigL, in his opinion, lays claim to infallibility ; you always believe yourself to be in the right in you: opinion ; therefore you lay claim to infallibility Whately. tT. If we never find skins except as the teguments of animals, we may safely conclude that animals cannot exist without skins. If colour cannot exist by itself, it follows that neither can any- thing that is coloured exist without colour. So, if language without thought is unreal, thought without language must also be so. . No soldiers should be brought into the field who are not well qualified to perform their part ; none but veterans are well qualified to perform their part; therefore none but veterans should be brought into the field. Whately. ft. The minimum visibile is the least magnitude which can be seen ; no part of it alone is visible, and yet all parts of it must affect the mind in order that it may be visible ; therefore, every part of it must affect the mind without being visible. lOf. The scarlet poppy belongs to the genus Papaver, of the natural order Papaveraceae ; which again is part of the subclass Thalamiflorae, belonging to the great class of Dicotyledons, Hence the scarlet poppy is one of the Dicotyledons. til Improbable events happen almost every day ; but what happens almost every day is a very pro- bable event ; therefore improbable events art very probable events. Whately. UtSSON XXI 1. Quantification of the PrtduaU. What does the quantification of the predicate mean! QUESTIONS AND EXERCISES. 335 S. Assign to each of the following propositions ttt proper symbol, and the symbol of its converse (1) Knowledge is power. (2) Some rectangles are all squares. (3) Only the honest ultimately prosper. (4) Princes have but their titles for their glories. (5) In man there is nothing great but mind. (6) The end of philosophy is the detection of unity, j. Draw all the contrapositive propositions and imme- diate inferences you can from the following pro- positions : (1) London is a great city. (2) London is the capital of England. (3) All ruminant animals are all cloven-footed ani- mals. (4) Some members of parliament are all the minis- ters. 4. Write out in Hamilton's notation the moods Baroko Darapti, Felapton, Bokardo. LESSON XXIII. BooUs System of Logic. I. Apply this system of inference to prove the syl- logisms on p. 141, in Cesare, and Camestres. 5. Show that if all A's are not B's, then no ffs art A's ; and that if all A's are all B's, then all not jfs are all not ffs. V Develope the term substance, as regards the tenm vegetable, animal, organic; then select the com* binations which agree with these premises : " What is vegetable is not animal but is or- ganic ; what is animal is organic." Test the validity of this argument : " Good always triumphs, and vice always fails ; therefore the victor cannot be wrong, nor *he vanquished right." 16 QUESTIONS AND EXERCISES. 5. It is known of a certain class of things that (l) Where the quality A is, B is not. (a) Where B is, and only where B is, C and D are What can we infer from these premises ol the class of things in which A is not pre- sent but C is present ? 4 II all X's are ^s; all *s are C's; all Cs are /7s shew that all X's are /7s, and that all not /7s are not X's. LESSON XXIV. Method. I. What is the supposed position of method accord- ing to former logical writers, and what are the rules of method? a. Explain the expressions ncbis notiora, and notion* natures. 3. Of what kind is the usual method of instruction ? 4. Prove that analysis in extension is synthesis in in-; tension, using some of the series of terms in Question 6, Lesson V. as illustrations. 5. Explain the exact meanings of the expressions L priori and a posteriori knowledge. 6. To which kind belongs our knowledge of the fol- lowing facts ? (i) The light of the stars takes a long time to reach us. (a) Vaccination is a preservative against small-pox (5) A meteor becomes heated in passing through the air. (4] There must be either some inhabitants or TM inhabitants upon Jupiter. LESSON XXV. Perfect Induction. k Define and distinguish Deduction, Induction, and Traduction. QUESTIONS AND EXERCISES. 31} 2. Find an instance of reasoning in Traduction. ^. Distinguish Perfect and Imperfect Induction. 4. How does Mr Mill define Induction, and what if his opinion of Imperfect Induction? 5. What is the use of Perfect Induction? 6 Construct some instances of the inauctive syllo- gism, and show that they may be thrown into a disjunctive form. LESSON XXVI. Induction, Analogy and ExampU. X. From what circumstance arises the certainty and generality of reasoning in geometry? 2. Find other instances of certain and general reason* ing concerning the properties of numbers. 3. Why are inductive conclusions concerning prime numbers uncertain and not general? 4. Why is a single instance sometimes sufficient to warrant a universal conclusion, while in other cases the greatest possible number of concurring in- stances, without any exception, is not sufficient to warrant such a conclusion? 5. What are the strict and ordinary meanings of the word analogy? 6. Explain the use of Examples. 7. Explain exactly the difference between analogical argument and ordinary induction. LESSON XXVll.Ohtrvatum and Experiment. I. What is the false method of Science against which Bacon protested? a. Explain the exact meaning of Bacon's assertions, that man is the Servant and Interpreter of Nature, and that Knowledge is Power. j, How does experiment differ from observation? |28 QUESTIONS AND EXERCISES. 4. Classify the sciences according as they empto) passive observation, expenment, or both. 5. Name the chief points in whir.h experiment is superior to mere observation. 6 What is the principal precaution needful in obser- vation ? 7. Explain how it is possible to anticipate nature an j yet establish all conclusions upon the results of experience. LESSONS XXVI 1 1. and XXIX. Methods of Induction. 1. Define exactly what is meant by a cause of an event, and distinguish cause, occasion, antece- dent. 2. Point out all the causes concerned in the following phenomena : (1) The burning of a fire. (2) The ordinary growth of vegetables. (3) The cracking of a glass by hot water. 3. State and explain in your own words Mr Mill's first three Canons of Inductive Method. 4. Point out exactly how the Joint Method differs from the simple Method of Difference. 5. Give some instances of simple experiments fulfil* ling completely the conditions of the Method ol Difference. i What can you infer from the following instance!? Antecedents. Consequents. ABDE stqp BCD qsr BFG vq* ADE tsf HK ^r?w ABFG ABE ~....f QUESTIONS AND EXERCISES. 329 f. (l) Friction alters the temperature of the bodies rubbed together. (2) The sun is supposed to move through space. (3) A ray of light passing into or out of a dense medium is deflected. Point out the successive questions which would have to be decided in the investigation of the above phenomena. 8. Find some simple instances of the homogeneous and heterogeneous intermixture of effects, and of the methods of concomitant variations and residues. 9. Since 1842 there has been a great reform of the British tariff, and a great increase of British trade. Does this coincidence prove that the first circumstance is the cause of the second? JO. Supposing us to be unacquainted with the causes of the following phenomena, by what methods should we investigate each ? (1) The connection between the barometer and the weather. (2) A person poisoned at a meal (3) The connection between the hands of a clock. (4) The effect of the Gulf-stream upon the climate oi Great Britain. LESSON XXX. Empirical and Deductive Mttkods 1. Define Empirical Law, and find a few additional instances of such laws. 2. What are the three steps of the Deductive Method / 3. Trace some of the successive steps in the progress of the theory of gravitation, showing that it was established by this method. QUESTIONS AND EXERCISES LESSON XXXI. Explanation, Ac. .. What do you mean by the explanation of a fact ? 1. State the three ways in which a law of nature maj be explained, and suggest some additional in* stances of each case. j, Define tendency. Do all causes consist only of tendencies, or can you find examples to the con- trary? f. Give a definition of hypothesis. How may a valid be distinguished from an invalid hypothesis ? f. What place does hypothesis hold in the Deductive Method? 6. Explain the ambiguities of the words theory and fact. LESSON XXXII. Classification i. Define classification, and give the derivation of the word. i. What do you mean by important characters in classification ? 3. State Dr Whewell's criterion of a good natural arrangement 4. Distinguish between a natural and artificial system of classification. 5. What do you mean by a characteristic quality ? Is it always an important quality ? 6. Define abstraction, generalization, and colligation of facts. 7. What are the characters of a notion properly ab tracted? LlSSON XXXIII. Requisites of a Philosophical Language. L What are the three purposes for which we use language? QUESTIONS AND EXERCISES. 331 ft. What are the two chief requisites of a philosophical language ? 3. By what considerations should we be guided in choosing between a new and old scientific term? 4. Distinguish a Descriptive Terminology and a No- menclature ; separate the following terms ac- cording as they belong to one or the other: Rose, Rosaceae, Rose-like, Potassium, Alkaloid, Ruminant Animal, Ruminating, Ruby, Ruoy red What does Mr Mill mean by the expression N tall Kind? INDEX, 4*D OONCISI VOCABULARY OF LOGICAL AND TERMS. Abacus, the logical, 190 abucissio Infiniti (the cutting off of the infinite or negative part,, the process by which we determine the position of an object in a system of classes, by successive comparison and rejection of those classes to which it does not belong. Absolute terms, i.t. non-relative terms, 25 ; sometimes used as name of non-connotative terms, 41 Abstract terms, ao, 43 Abstraction, a8< Accent, fallacy of, 174 accident, fallacy of, 176 ; toe pre- di cable, 103 Accidental definition is a defi- nition which assigns the properties of a species, or the accidents of an individual ; it is more commonly called a Dttcription. Acquired perceptions. 236 Added determinants, inference by, 86 Adeq , equate knowledge, 56 A dicto secundnm quid, fcc., fallacy of, 176 Adjectives, ai Adverbial*, 93 Affirmative proposition*, 63 Algebraic reasoning, 58, aig Ambiguity of all, to ; of MM/, 70 of many old terms, t^i ; of term* In Political Economy, 292 Ambiguous middle term, 190, 171 Amphibology, fallacy of, IT* Ampliative proportion*. 6g Analogue, a thing analogous to VMIH. other thing. analysis, method <* M* Analogy, the cause of ambiguity 35, 50 ; reasoning by, 3268 Analytics, (rw AjwAvruti,) the titli given in the second century to por- tions of the Organon, or LogicaJ Treatises of Aristotle ; they were distinguished as the Prior and Pos- terior Analytics. Analytic syllogism, a syllogism in which the conclusion is placed first, the premises followirf as the reasons. See Synthttit. ^ylkgism the distinction is unimportant Antecedent, of a hypothetical pit position, 160 ; of an event, 340 Anticipation of nature, 239 Antinomy (d^rl, against ; Kopos, law), the opposition of one law or roll to another. Kant. A posteriori knowledge, aot A priori knowledge, 208 Arbor Porphyriana, see Trtt of Porphyry. Argument, (Latin, a*?iu, from tipybc, clear, manifest,) the process ol reasoning, the shewing or proving that which is doubtful by that whicfi is known. See Inference. The mid- dle term of a syllogism is sometime! called specially the argument. Argumentum a fortiori, an argument in which we prove thai the case in question is more strouf or probable than one already aon- ceded to be sufficiently so. Argnmentnm ad hominem, 178 Argnmentnm ad Judiciuao, ' to the common MOM INDEX. 333 Argnmenttun ad ignoranti- Canons of syllogism, 1212; Hamil am, an argument founded MI the ignorance of adversaries. ton's supreme Canon, 189 Canons of Mill's Inductive Methods, Ar&omentum ad populum, First, 340 ; Second, 242 ; Third, 945; 179 Fourth, 252; Fifth, 249 Argnmentum ad verecun- diam, an appeal to our respect for Categorematic words, 18 Categorical propositions, 63 some great authority. Categories, the ntmm* ffttfra, at Ajg-umentu-m ex concesso, most extensive classes into whack a proof derived from a proposition things can be distributed ; they an already conceded. ten in number, as follows : Aristotle's Dicta, 123 Ovo-ia, Substance ; Hwrov, Quan- Art and Science, distinction of, 7 tity ; notov, Quality ; Hpd n, Re- Artificial Classification, 284 lation ; ttottlv, Action ; Ud, Habit or condition. Association of ideas, (cusocia, to accompany; socius, a companion,) Everything which can be affirmed must come under one or other of these the natural connection existing in highest predicates, which were de- the mind between impressions which scribed in the first treatise of Aris- have previously coexisted, or which totle's Orfanon, called the Catego- are similar. Any idea tends to bring ries. into the mind its associated ideas, in accordance with the two great laws Cause, meaning of, 239 Aristotle distinguished four kind* of association, the Law of Conti- of causes for the existence of a thing guity, and the Law of Similarity. i. The Material Cause, the sub- Assumption, (assumo, to take for stance or matter composing it ; 2. granted,) any proposition taken as the basis of argument ; in a special The Formal Cause, the pattern, type or design, according to which it is sense, the minor premise of a cate- shaped ; 3. The Efficient Cause, the gorical syllogism. Attribute, (attribuo, to ive or force employed in shaping it ; 4. The Final Cause, the end, motive ascribe to,) a quality or circumstance which may be affirmed (or denied) or purpose of the work. Chance, ignorance of the causes of a thing; opposed to Sttbstance^ which see. which are in action ; see Probability. Character, derivation of the word, Attribute in grammar, 92 46 Attributive term, i. e. ConntUtivt Characteristics, 385 CirculUS in definiendo, no, 114 Axiom, defininition of, 1*5 Ctrculus in probando, 179 Clearness of knowledge, 54 Baconian method, 955; Philoso- Cognition, (cognosce, to know,) phy. 229 knowledge, or the action of mind i* Barbara, Celarent, &c., 145 Begging the Question, 179 acquiring knowledge. Colligation of Facts, Dr WhewelTi Belief, assent to a proposition, ad- expression for the mental union at mitting of any degree of strength, from the slightest probability to the fullest certainty ; see Probability. facts by some suitable conception, see 286 Collective terms, 19 Bentham, George, new system of Combined or complete method oi Logic, 187 Boole, George, his system of Logic, QI, hi* Laws of Thought, 197; kis logical works, 201 investigation, 258 Comparison, 'com, together; far, equal or like, the action of mind bj which we judge whether two object* 334 INDEX. of thought are the same or different Coixseq nence , th* connection D& in certain points. See Judgment. twetc antecedent and consequent Compatible terms are those which, but often cued ambiguously for tfa* hough distinct, are not contradic- Utter jory, and Gin therefore be affirmed Consequent of a hypothetical pro of the same subject ; as " large " and ' position, 161 " heavy ; " " bright-coloured " and | Consequent or effect of a caow " nauseous." 240 Jomplex conception, inference by, 87 Soviplex sentence, 91 ; syllogism, Consequent, fallacy of the, 181 Conservation of energy, 963, 269 Consilience of Inductions, toe 158 agreement of inductions derived Composition of Causes, the from different and independent seriei principle which is exemplified in all cases in which the joint effect of of facts, as when we learn the mo- tion of the earth by entirely different several causes : s identical with the modes of observation and reasoning. sum of their separate effects. J. S. Whtwell. Mill. See pp. 253. 265 Composition, fallacy of, 173 Consistency of propositions, 78 Consistent terms, see cam^atiiU Compound sentence, 90 terms. Comprehension of terms, see /- faMMK Computation, 127 Contingent, [conti*g, to touch,) that which may or may not happen ; opposed to the necessary and im- Concept, that which is conceived, the result of the act of conception ; possible. Contingent matter, 80 nearly synonymous with general no- Continuity, Law of, the principle tion, idea, thought. that nothing can pass from one ex- Conception (ctm, together; <*/*, to take). An ambiguous term, mean- treme to another without passing through all the intermediate degrees; ing properly the action of mind in which it takes several things toge- motion, for instance, cannot be instan- taneously produced or destroyed. ther, so as to form a general notion ; Contradiction, Law of, 117, 193 or again, in which it forms " a men- Contradictory terms, 24, 1x9; tal image of the several attributes propositions. 76 given in any word or combination of Contraposition, conversion by, words." Mtuutl. 83, 1 86 Conceptualists, 13 Conclusion of syllogism, 15, xj; weakened, 140 Concrete terns, so Converse fallacy of accident, 17* Conversion of propositions, 82 85 ; with quantified predicate, 184 Convertend, 82 Conditional propositions, 62, 160 Confusion of words, ambiguity Coordinate propositions, 90 Copula, 16 from, 31 Conjugate words, those which come Corollary, a proposition which fol- lows immediately from another which from the same root or stock, as has been proveoL ktumm^ k*f**H, knrwt*gly t know- Correction of observations, 253 ledge. Correlative terms. a< Connotation of terms, 30, 41 ; Criterion V^-nioioV. from KIKVW. to ought to be exactly fixed, 990 Consciousness, the immediate judge), any fact, rule, knowledge, or means requisite to the formation knowledge which the mind kas of of a judgment which shall decide a its sensations and thoughts, and, in doubtful question. general, of all its present operation*. RfuL Cross division, 105 Ooa*etary CoroOwy. Data, {plural of datum, that wMct INDEX. is given, ) the facts or assertions from which an inference is to be drawn. Deduction and Induction, 212 Deductive or combined method, 258, 272 De facto, what actually or really happens : opposed to dt jure, what ought to happen by law or right. Definition, the logical process, 109, 1 1 ; of logic, i Degree, terms expressing, 24; ques- tions of, 120 Demonstration, (demonstr*, to point out,) strictly the pointing out the connection between premises and conclusion. The trm is more ge- nerally used for any argument or reasoning regarded as proving an asserted conclusion. A demonstra- tion is either Direct or Indirect. In the latter case we prove the conclu- sion by disproving Us contradictory, or she wing that the conclusion cannot be supposed untrue. Demonstrative Induction, 220 Do Morgan's logical discoveries and writings, 100 Denotation of terms, 39 Depth of a notion, see Intension, Derivatives from the root tftc, sight, 52 Descartes on Method, 116, 220 Description, vet Accidental Defi- nition. Descriptive terminology* *& Destructive dilemma, 168; hypo- thetical syllogism, 1624 Desynonymization of terms, 49 Determination, the distinguishing of parts of a genus by reunion of the genus and difference. See Division. Development of a teim, 193 Diagrams, of sentences, 93 7 ; of syllogisms, 129133, 142; of pro- positions, 72 75 Dialectic (WAXTUOI -n*vq, the art of discourse, from atoAeyr#u, to discourse). The original name of Logic, perhaps invented by Plato; ako used to denote the Logic of Probable Matter (Aristotle), the light use of Reason and Language, the Science of Beiag; >t it tfaua highly ambiguous term. Dichotomy, division by, 107, 193 Dicta de omnl et nuDo, 123 Difference, the predicabU, 99 Differentiation of terms, 49 Dilemma, 167 Disbelief, the state of nind in whicl we are fully persuaded that som< opinion is not true. J. S. Mill. Ii is equivalent to belief it. the contra dictory opinion or assertion, and i> not to be confused with Dvi*bt, wind see. Discourse, or reasoning, 15 Discovery, method of, 202 Disjunctive, propositions, 62, 160; syllogism, 166, 194 Distinct knowledge, 55 Distribution of terms, 19, 74 3, 82, 129 Division, logical, 105 ; metaphysical 108 ; fallacy of, 174 Doubt, (dubito, to go two ways,) the state of mind in which we hesitate between two or more inconsistent opinions. See Disbelief. Drift of a proposition, the varying meaning which may be attributed to the same sentence according to ac- centuation. See Fallacy of accent, 1745 the doctrine of those who consider that all knowledge is derived merely from experience. Empirical Law, 156 Enthymeme, 153 Epicheirema, 155 Episyllogism, 155 Equivocal terms, 29 Equivocation, 30; causes of, 31; fallacy of, 171 Essence, (gssentia, from ftse, to be,; " the very being of anything, where- by it is what it is." Locks. It is an ancient scholastic word, which can not be really defined, and should b* banished from use. Essential propositions, 68 Euler's diagrams. 72 5, 129 133, 142 Evidence. (/, and vtdtrt, to see, literally th* seeing of anything The word now means any facts ap- prehended by the mind and mad the ground* of knowledge and belief INDEX. Examples, use of, **j Exceptive proportion*. 68 Excluded middle, law of, 117, 119, 192 Exclusive proposition*, 68 Exhaustive division, 107, 192 Experience, 228 Experimentum crucis, an ex- periment which decides between two rival theories, and shews which is to be adopted, as a finger-post shews which of two roads is to be taken. Explanation, of facts, 364 ; of laws, 265 Explicative propositions, 68 Exposita, a proposition given to be treated by some logical process. Extension and intension, 37, *o8 Extensive Syllogism, 159 Extremes of a proposition, are its ends or terms, the subject and predi- Fact, 175 Fallacy, purely logical, 170; semi- logical, 170 175 ; material, 176 182 ; in hypothetical syllogism, i6a ; in dilemma, 168 False causa, fallacy of, 181 False propositions, 70 Figure of speech, fallacy of, 175 Figures of the syllogism, 138 ; their uses, 143 Form and matter of thought, A Fundaxnentum divisionisjios Fundamentum relation! a, the ground of relation, i.e. the series of events or circumstances which es- tablish a relation between two cor- relative terms. Fundamental principles of syllo- gism, 121 Gtalenian, or 41* figure of the syl- logism, 145 Seneral notions, 43 ; terms. 18 Generalization, 286 ; of TJL*S. Generic property, 102 3enu, 98 : gencralissimum, too Geometrical reasoning jS, ti8; Pascal on, 115 Grammatical predicate, 88 ; tence, 89 Gravitation, theory of, 6c tation, 187 ichel, I Hamilton, Sir W. Method rf No Herschel, Sir ] , on active passive observation, 234 Heterogeneous, 101 ; ture of effects, 252 Homogeneous, 101 ; intermktmt of effects, 253, 265 Homologue, whatever is kernel* Homo fomology, a special term for tha analogy existing between parts of different plants and animals, as be- tween the wing of a bird and the fore leg of a quadruped, or between the scales of a fish and the feathers of a bird. Homonymous terms, 30 Hypothesis, 269, 270 Hypothetical propositions, 6,x6o, syllogism, 161 a Idea (i'a, fUoc, image), a term used ambiguously, but generally equiva- lent to thought, notion, concept Defined by Locke as "Phantasm notion, species, or whatever it is which the mind can be employed about in thinking." To have an id*a of a thing is to think of that thing. Identity, kw of, 1178 Idol (clftwAor, lof, image), Bacou's figurative name for the sources of error ; he enumerated four kinds ; Idols of the Tribe, which affect all people ; Idols of the Cave, which are peculiar to an individual ; of the Forum, which arise in the inter- course of mem ; of the Theatre, which proceed from the systems of philoso- phers. Ignoratio Elenohi, 178 Illation (iliatum, past particrpl* of i*Jero, to bring ini. See Inftrntc*. Illative, that which can be inferred. Illicit process, ( the minor tors, 131 ; of the major teem, 13*, 139 Immediate inference, 837 Imperfect figures of the syllo gism, 145 Imperfect Induction, 913 Impossible matter, 80 Inconsistent terms imply qualitiei which cannot coexist ia the taxM INDEX. 337 Inconsistent propositions, 74 Indefinite propositions, 65 Indefinite or infinite term, is a ne- Leibnitz on Knowledge, 53 Lemma (Xa^di/w, to take or a sume), a proposition, a premis* gative term which only marks an granted ; in geometry, a preliminary object by exclusion from a class. tn design ate propositions. See In- proposition. Limitation, conversion by, 82, 87 definite propositions. Logic, derivation of name, 6 Indirect demonstration. See De- Logical abacus, slate and machine^ monstration. 199 indirect inference, method of, Logomachy, 29* Indirect reduction of the syllo- Lowest specie*, 100 gism, 146, 148 9. Miachine, the logical, 199 Individual, what cannot be divided without losing its name and distinc- Major, term, 128 ; premise, 129 Many questions, fallacy of, x8s tive qualities, although generally capable of physical division or par- Material fallacies, 170, 176 Mathematical induction, 220 tition, which see. Matter of thought, 4 ; of proposi- Induction, 212 tions, 80 Inductive syllogism, 211, 214 Inference, defined, 8x ; immediate, Matter is defined by J. S. Mill as " the external cause to which w 85 87 ; mediate, 126 ascribe our sensations," or as Per- Infima species, xoo manent Possibility of Sensation. Innate ideas, see a priori trutks,*<& Mediate inference, 126 Inseparable accident, 103 Instances, use of, 227 Membra dividentia, the parts into which a class is divided; tk Intension and extension of terms, constituent species of a genus. 37, 99, 208 ; law of relation, 40 Intensive syllogism, 159 Metaphor, 50 Metaphysical division, 108 Intention, first and second, a dis- tinction between terms thus defined Metaphysics (ra fxrra ra v 41, 102, 109 Propositions, 10, 16 , several kinds of, 60 ; affirmative and negative, 63 ; categorical, 63; conditional, 62, 160; disjunctive, 62, 160 ; essential or ex- plicative, 68; exclusive, exceptive, 68 hypothetical, 62, 162 ; indefinite or indesignate, 65; modal, 69, 91; opposition of, 78 ; particular, 63^6, 72, 79 : pure, 69 ; plurative, 191 ; ir- regular, 67 ; quality and quantity of, 63 Proyllogim, 155 Prcndoutt* genus, ioi Quantification of predicate, 183 Quantity of propositions, 63; qoc* tior.s of quantity, 120 Quateroio terminorum, 190 Ramean tree, see Tn f P*r pkyry Ratiocination, a name quivale* to Syllogism or Deduction, adoptei by I. S. Mill. Realism, 13 Reason (ratio, from rtor, to think) a term of wide and ambiguous mean ing ; it has sometimes been special!] used to denote the minor premise o> a syllogism. Reasoning, or discourse, 15 Record, language as instrument of, Reductio ad absurdum or a* impossible, an indirect demonstra tion founded upon the impossibilit3 of a contradictory supposition, 146 Reduction of the syllogistic figures. 145 ; f hypothetical to categorical syllogisms, 163 5 Relation (relatum, past participle of re/era, to bear back), any con- nection in thought or fact betwee* two things, si Relative terms, 25 Residual phenomena, 254 Residues, method of, 252 Rules of the syllogism, 137 Scholastic Philosophy, a g neral name for the systems of philo- sophy taught during the middle age} from the 9th to the i6th century, flourishing chiefly in the i3th and 1 4th centuries. The subject wai chiefly the logic of Aristotle, varied with theology, metaphysics, gram- mar, or rhetoric. Second Intention, m Intention, Secundi adjacentis, of the se cond adjacent, an expression in in* correct Latin, applied to a gram- matical sentence or proposition COP taining only two parts, the subjeo and verb, without a distinct copola Self-contradictory terms, 143 Semilogical fallacies, 171 Sentence, grammatical. 61. lQ Separable a 4* INDEX. Slgniflcates of term ure things denoted or signified by it Similars, substitution of, 124, 200 Simple, apprehension, n ; conver- sion, 82, 184 Singular, terms, 18 ; propositions, 64 Sophism (cro^icr^a, from awpia, wis- dom), a false argument; the name often implies that a false argument is consciously used for deception. Sorites, 156 Specialization of names, 45, 48 Species, in logic, 98; in natural history, 101 Subaltern, propositions, 77; genera and species, 100 Subalternans, subaltern- ates, 77 Sub contrary Propositions, 77 Subject of a proposition, 62, 93 Subjective, that which belongs to the thinking subject, the ego, or mind engaged in thought ; opposed to objective, which see. Subordinate propositions, 91 Substance (sub, under ; stans from start, to stand), that which underlies and bears phenomena or attributes ; strictly speaking it is either mind or matter, but it is more commonly used in the material sense. Substitution of similars, 124, 200 Subsumption [sub, under ; sutno, to take or put), a name used by Sir W. Hamilton for the minor premise of a syllogism, because it brings or nt&runtes a special case under the rule expressed in the major premise Subsumptipn of a law is Mr Mill's expression for the third mode of explaining a law by shewing it to be a particular case of a more ge- neral law, 268 Sufficient Reason, Principle or Law of, 125 Bui generis, 101 Summum genus, TOO Sumption (sumo, to take), Sir W. Hamilton's name for the major pre- mise of a syllogism. Supposition, 170 Syllogism, 10, 127; inductive.au, t)yrnolical knowledge, 57 Syncategorematic words, il Synthesis, 205 Synthetic syllogism, a syllo gism in which the conclusion stand* last ; see A nalytic syllogism. System, ((rv'tm^a, from (rwtVnm*. to put together), a connected body 01 knowledge. Tacit premise, 153 Tautologous propositions, 69 Tendency, 266 Terminology, 293 Terms, 10, 16, 17 Tertii adjacentts, of the third adjacent, an expression in incorrect Latin, applied to a grammatical sen- tence or proposition in which the subject, copula and predicate, are all distinctly stated. Theory (9t