PLEASE HANDLE WITH CARE University of Connecticut Libraries -h0 ^^3^ 0/ ^ '^' .r^JxV BOOK 160. J53 c. 1 JEVONS # ELEMENTARY LESSONS IN LOGIC 3 T1S3 DDDDMTOD T PLEASE NOTE It has been necessary to replace some of the original pages m this book with photocopy reproductions because of damage or mistreatment by a previous user Replacement of damaged materials is both expensive and time-consuming. Please handle this volume with .are so that information will not be lost to future readers. ^If^ r". ^°' ^^'P'"9 '° Preserve the University's esearch collections, " I, LEMENTARY LESSONS IN LOGIC: DEDUCTIVE AND INDUCTIVE. 'TH COPIOUS QUESTIONS AND EXAMPLES, AND A VOCABULARY OF LOGICAL TERMS. BY W. STANLEY JEVONS, LL.D. (EDINB.), M-A. (LOND.), F.R.S. SontJon : MACMILLAN AND CO. A2TB NEW YOBK. 1893 \The Bight of Translation is reserved.] First Edition i 1870. Reprinted 1871, 1872, 1S74, 1875, 1877, 1878, 1880, 1881, 1882, 1883, 1884, 1885, 1886, 1889, i8go, 1893. 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 latural 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. vi PRE FA CE. Logic is not only an exact science, but is the most 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 t(ic 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 are selected from the examination papers of the Oxford, . PREFACE. vii ^ T.ondon, and Edinburgh Universities. In my own classes T have constantly found that the working and solution of logical questions, the examination of argu- ments and the detection of fallacies, is a not less * 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 have 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 t logicians generally agree, than discussing the points in which there is a difference of opinion. The recent logical discoveries of Sir W. Hamilton, Mr George Bentham, Prof de Morgan, and especially the late Prof 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 WTiewell 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 m.ost 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 his reading of the subject treated, so that this little volume may serve as a guide to a more extended course of study. ^ Hampstead, November^ 1876. * TABLE OF CONTENTS. LESSON PAGE I. Definition and Sphere of the Science i II. The Three Parts of Logical Doctrine 9 TERMS. III. Terms, and their various Kinds i6 IV. Of the Ambiguity of Terms 27 V. Of the twofold meaning of terms — in Extension and Intension 37 VI. The Growth of Language 44 VII. Leibnitz on Knowledge 53 PROPOSITIONS. Kinds of Propositions 60 The Opposition of Propositions 71 Conversion of Propositions, and Immediate In- ference 81 Logical Analysis of Sentences 88 The Predicables, Division, and Definition 98 Pascal and Descartes on Method jw X TABLE OF CONTENTS. SYLLOGISM. { LESSON >AGK J XIV. The Laws of Thought 117 XV. The Rules of the Syllogism 126 XVI. The Moods and Figures of the Syllogism 135 XVIL Reduction of the Imperfect Figures 144 XVIII. Irregular and Compound Syllogisms 152 / XIX. Of Conditional Arguments 160 FALLACIES. XX. Logical Fallacies 169 . XXI. Material Fallacies 176 n RECENT LOGICAL VIEWS. XXII. The Quantification of the Predicate 183 XXIII. Boole's System of Logic 191 ' METHOD. XXIV. Of Method, Analysis, and Synthesis 201 v. INDUCTION. ^ • / XXV. Perfect Induction and the Inductive Syllogism 210 XXVI. Geometrical and Mathematical Induction, Ana- logy, and Example 210. XXVII. Observation and Experiment 228, XXVIII. Methods of Induction 239J XXIX. Methods of Quantitative Induction 24; [ TABLE OF CONTENTS. xi LESSON PAGE XXX. Empirical and Deductive Methods 255 XXXI, Explanation, Tendency, Hypothesis, Tlieory and Fact 264 SUBSIDIARIES OF INDUCTION. XXXII. Classification, and Abstraction 276 XXXIII. Requisites of a Philosophical Language 287 Questions and Exercises 296 Examples of Terms 297 — 299 Examples of Propositions 303 Examples of Arguments 312, 315 Index ,. 332 > t*W.^\ > ^ N*>^?^S . ^ 5:A ^v L^^ixJ" a^Mi.^ (L<^ikZ^cc<< INTRODUCTION. LESSON I. DEFINITION AND SPHERE OF THE SCIENCE. ' Logic may be most briefly defined as the Science of 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 it 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 not 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 anyway 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 * I 2 DEFINITION AND SPHERE [less. by the objects treated in the Science. The science of 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 treated in physical astronomy. In chemistry the law of equivalent proportions de- , scribes 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 arrive at natural laws. But there may be, and are, many things so fickle, complicated, and " — tmcertain, 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 ^ and reason. Thus if two things are identical with a third I.] OF THE SCIENCE. 3 common thing they are identical with each other. This 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 its meaning. 2. That they think in accordance with it whatever may be the subject about which they are thinking. Thus if the things considered are — London, The Metropohs, 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 with the planets;" it is identical only with one of the I — 2 4 DEFINITION AND SPHERE [less. / planets, and we therefore say that " it is a planet." Simi- larly 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 us to argue similarly concerning the following ; — / Iron, : Metals, Elementary substances. Iron is one of the metals, and metals are elements or 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- > pies 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 it \ 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 clearly apprehend the meaning of the expression " necessary forms of thought." A form is something which may remain uniform and unaltered, while the matter thrown into that form may be ^ varied. Medals struck from the same dies' have exactly the same form, but they may be of various matter, as I.] OF THE SCIENCE. 5 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 ojarjtwo latter arguments, as follows : — - / (Y) •, . ' (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 F, 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 em- / 6 DEFINITION AND SPHERE [less. ployed in another science, yet whatever the particular form may be, it must be logical, and must conform to the 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, ver}' 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 Xoyop, 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 later , Greek writers on reasoning were led to call their science * Except Philology, which is differently formed, and means the love or study of words ; the name of this science, if formed upon the same plan, would be logology. I.] OF THE SCIENCE. 7 enia-Tri^r] XoyiKi], or logical Science ; also rex^r] XoytKi^, or logical art The adjective Xoyt<7, being used alone, soon 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 teaches us^ 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 tasis 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 13th century, and called logic the Science of Sciences, called it also the Art of Arts, expressing fully its preeminence. Others have thus definCvd it — " Logic is the art of direct- ing the reason aright in acquiring the knowledge of things, for the instruction both of ourselves and others." Dr Isaac Watts, adopting tiiis view of logic, called his well-known work "the Art of Thinking." 8 DEFINITION AND SPHERE [less. It may be fairly said however that Logic has more 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 migh - 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 "i 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 who II.] OF THE SCIENCE. 9 can say that the nature and procedure of this intellect is not almost the highest and most interesting subject of study in which we can engage? In vain would any one deny the truth of the favourite aphorism of Sir W. Hamilton — In the world there is nothing great but man. 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 stiictly 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 are suitably described before we proceed to the actual forms of argument. If we examine a simple argument such as that given in the last lesson, thus — Iron is a metal, Every metal is an element, Therefore Iron is an element, — lo THE THREE PARTS OF [less. we see that it is made up of three statements or asser- tions, and that each of these contains, besides minor words, two nouns substantive or names of things, and the verb " is." In short, two names, or terms, when connected by a verb, make up an assertion or proposition; and three such propositions make up an argument, called in this case a syllogism. Hence it is natural and conve- nient first to describe terms, as the simplest parts ; next to proceed to the nature and varieties of propositions constructed out of them, and then we shall be in a posi- tion to treat of the syllogism as a whole. Such accord- ingly are the three parts of logical doctrine. But though we may say that the three parts of logic are concerned with terms, propositions, and syllogisms, it may be said with equal or greater truth that the acts of mind indicated by those forms of language are the real subject of our consideration. The opinions, or rather perhaps the expressions, of logicians have varied on this point. Archbishop Whately says distinctly that logic is entirely conversant about language ; Sir W. Hamilton, Mr Mansel, and most other logicians treat it as concerned with the acts or states of mind indicated by the words ; while Mr J. S. Mill goes back to the things themselves concerning which we argue. Is the subject of logic, then, language, thought, or objects.? 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 II.] LOGICAL DOCTRINE. ii 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 of ' things as they are; the state of mind within us must _ correspond with the state of thmgs without us whenever an 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. s The first of these, Simple Apprehension, is the act of mind 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 other things, and apprehettsion the taking hold by the mind. Thus the name or term Iron instantaneously makes the 12 THE THREE PARTS OF [less. mind think of a strong and very useful metal, but does not tell us anything about it, or compare it with any thing else. The words sun, Jupiter, Sirius, St PauVs Cathe- dral, are also terms which call up into the mind certain well-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 in relative position to each other, whereas two or three other bodies may be seen which resemble Jupiter in giving a steady light, and also in changing their place from night to night among the fixed stars. I have now by the action of judgment formed in my mind the general notion oi Jixed stars^ by II.] LOGICAL DOCTRINE. 13 bringing together mentally a number of objects which agree ; while from several other objects I have formed the general notion oi planets. Comparing the two general 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 philosophers 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- tion is the knowledge in the mind of the common pro- perties or resemblances of the things embraced under 14 THE THREE PARTS OF [less. 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 the fixed stars. It should be added, however, that there are many, including Sir \V. 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- perties, so that whatever is known of some may be in- ferred and asserted of others. It is in the application of such knowledge that we ; employ the third act of mind called discourse or reason- ing, by which from certain judgments we are enabled, , without any new reference to the real objects, to form a iiew judgment. If we know that iron comes under the II.] LOGICAL DOCTRINE. 15 general notion of metal, and that this notion comes under the still wider notion of element, then without further examination of iron we know that it is a simple unde- composable substance called by chemists an element. Or if from one source of information we learn that Neptune 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 the 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 prcB^ be- fore, and mitio, 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- i6 TERMS, AND THEIR [less. plete the doctrine of Logic. It is at any rate certain however that this fourth part is much inferior in import- ance and distinctness to the preceding three ; and all that will be said of it is to be found in Lesson xxiv. LESSON III. TERMS, AND THEIR VARIOUS KINDS. It has been explained in the preceding lesson that every 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 of the class dictionaries with the class of useful books con- sisting in the fact that the class of dictionaries fonns 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 may 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," the first term requires nine words for its expression, and the second term, four words (equal to each other) ; and there 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, termitius) 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 ;" 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 nouii 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 verb, can form the second term or predicate of a propo- sition. It may indeed be questioned whether an adjec- tive can ever form a term alone; thus in "Dictionaries are useful," it may be said that the substantive things or books is understood in the predicate , the complete sen- 2 i8 TERMS, AND THEIR [less. tence 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 adjectivej^.and ^ certain parts of a verb, are called categorematic words, from the Greek word Kar-qyopea), to assert or predicate. Those parts of speech, on the other hand, such as prepositions, adverbs, conjunctions, &c., which can only- form parts of names or terms are called syncategorematic words, because they must be used wi^/i other words in order to compose terms (Greek a-vv, with, and Karrjyopea)). Of syncategorematic words we need not take further notice except so far as they form part of categorematic terms. We have now to consider the various kinds and pecu- liarities of terms, so as to gain a clear idea of what they mean. Terms are first of all distinguished into singidar or individual, and gejieral or common terms, this being a very obvious division, but one of much importance. A Singular term is one which can denote only a single ob- ject, so long at least as it is used in exactly the samei meaning ; thus the Emperor of the French, the Atlantic Ocean, St Paul's, William Shakspeare, the most pre- cious of the metals, are singular terms. All proper names belong to this class ; for though John Jones is the name of many men, yet it is used not as meaning any of these men, but some single man — it has, in short, a different meaning in each case, just as London, the name of our capital, has no connexion in meaning with London in Canada. General terms, on the contrary, are applicable in the same sense equally to any one of an indefinite number of objects which resemble each other in certain qualities. Thus metal is a general name because it may be applied V- ^ c5 . <^ . /?> , III.] VARIOUS KINDS. 19 indifferently to gold, silver, copper, tin, aluminium, or any of about fifty known substances. It is not the name of any one of these more than any other, and it is in fact applied to any substance which possesses metallic lustre, which cannot be decomposed, and which has certain other qualities easily recognised by chemists. Nor is the number of substances in the class restricted; for as new kinds of metal are from time to time discovered they are added to the class. Again, while IMars, Jupiter, Saturn, &c., are singular terms, since each can denote only a single planet, the term planet is a general one, being applicable to as many bodies as may be discovered tq revolve round the sun as the earth does. We must carefully avoid any confusion between ge? neral and collective terms. By a collective term we mean the name of a number of things when all joined together as one whole ; like the soldiers of a regiment, the men of a jury, the crew of a vessel : thus a collective term is the name of all, but not of each. A general term, on the other hand, is the name of a number of things, but of each of them separately, or, to use the technical expression, distributively. Soldier, jurj'man, sailor, are the general names which may belong to John Jones, Thomas Brown, &c., but we cannot say that John Jones is a regiment, Thomas Brown a jury, and so on. The distinction is exceedingly obvious when thus pointed out, but it nf^y present itself in more obscure forms, and is then likely to produce erroneous reasoning, as will be pointed out in Lesson xx. It is easy to see that we must not divide terms into those which are general and those which are collective, because it will often happen that the same term is both general and collective, according as it is regarded. Thus, library is collective as regards the books in it, but is general as regards the great num- ber of different hbraries, private or public, which exist. 2 — 2 20 TERMS, AND THEIR [less. Regiment is a collective term as regards the soldiers which compose it, but general as regards the hundred different regiments, the Coldstream Guards, the High- land regiment, the Welsh Fusiliers, and the rest, which compose the British standing army. Army, again, is a collective whole, as being composed of a number of regi- ments organized together. Year is collective as regards the months, weeks, or days of which it consists, but is general as being the name either of 1869 or 1870, or any period marked by a revolution of the earth round the sun. We have not always in the English language suffi- cient means of distinguishing conveniently between the general and collective use of terms. In Latin this dis' tinctive use was exactly expressed by oviiies^ meaning all distributively, and aincti meaning all taken together, a contracted form of conjiincti (joined together). In English all men may mean a7iy jnan or all men together. Even the more exact word every is sometimes misused, as in the old proverb, ' Every little makes a mickle,' where it is obvious that every little portion cannot by itself make much, but only when joined to other httle portions. A second important distinction between terms is that of concrete terms and abstract terms ; and it cannot be better described than in the words of Mr Mill, by saying that a concrete name is the name of a thing, the abstract name is the name of a quality, attribute, or circumstance of a thing. Thus red house is the name of a physically- existing thing, and is concrete; redfiess is the name of one quality of the house, and is abstract. The word abstract means drawn from (Latin, abstractus, from abs- trahere, to draw away from), and indicates that the quality redness is thought of in the mind apart from all the other qualities which belong to the red house, or other red object. But though we can think of a quality by itself, we cannot suppose that the quality can exist physically III.] VARIOUS KINDS. 21 apart from the matter in which it is manifest to us. Red- ness means either a notion in the mind, or it means that in red objects which excites the notion. T he reader shnnld rar pfnHy n^gpryp that adiectlves are con crete^ not abstract . If we say that a book is use- ful, iris to the book we apply the adjective useful, and usefulness is the abstract noun which denotes the quahty ; similarly, the adjectives eqtial, grateful, reverent, ratio- nal, are the names of things, and the corresponding abs- tract nouns are equality, g7'atitude, reve7'e7ice, rationality. This distinction will become more apparent in reading Lesson v. It is a good exercise to try and discover pairs of cor- responding concrete and abstract names ; thus animal has animality ; miser, miserliness ; old, agedness, or old age ; substance, substantiality ; soap, soapiness ; shrub, shrubbiness ; and so on. But it by no means follows that an abstract word exists for each concrete ; table hardly has an abstract tabularity ; and though ink has inkiness, we should not find the abstract of pen. It is by the accidents of the history of language that we do or do not possess abstract names ; and there is a constant tendency to in- vent new abstract words in the progress of time and science. Unfortunately concrete and abstract names are fre- quently confused, and it is by no means always easy to distinguish the meanings. Thus relation properly is the abstract name for the position of two people or things to each other, and those people are properly called relatives (Latin, relativus, one who is related). But we constantly speak now of relations, meaning the persons themselves ; and when we want to indicate the abstract relation they have to each other we have to invent a new abstract name relationship. Nation has long been a concrete term, though from its form it was probably abstract at 22 TERMS, AND THEIR [LESS. first ; but so far does the abuse of language now go, especially in newspaper writing, that we hear of a tiation- ality meaning a nation, although of course if nation is the concrete, nationality ought to be the abstract, mean- ing the quality of being a nation. Similarly, action, mtentiojt, exteftsion, conception, and a multitude of other properly abstract names, are used confusedly for the corre- sponding concrete, namely, act, intent, extent, concept, &c. Production is properly the condition or state of a person who is producing or drawing something forth ; but it has now become confused with that which is produced, so that we constantly talk of the productions of a country, meaning the products. The logical terms, Proposition, Deduction, Induction, Syllogism, are all properly abstract words, but are used concretely for a Proposition, a De- duction, an Induction, a Syllogism ; and it must be al- lowed that logicians are nearly as bad as other people in confusing abstract and concrete terms. Much injury is done to language by this abuse. Another very obvious division of terms is between those which are positive, and those which are negative. The difference is usually described by saying that posi- tive terms signify the existence or possession of a quality, as in grateful, metallic, organic, etc., while the correspond- ing negatives signify the absence of the same qualities as in ungrateful, non-metallic, inorganic. The negative terms may be adjectives as above, or substantives, con- crete or abstract ; thus ingratitude, inequality, incon- venience are abstract negative terms; and individuals, unequals, &c. are concrete negatives. We usually consider as negative terms any which have a negative prefix such as not, non, un, in, &c. ; but there are a great many terms which serve as negatives without possessing any mark of their negative character. Darkness is the negative of light or lightness, since it means the absence of light; III.] VARIOUS KINDS. 23 compound is the negative of element, since we should give the name of compound to whatever can be deconi' posed, and element is what cannot be decomposed ; theo- retically speaking every term has its corresponding nega- tive, but it by no means follows that language furnishes the term ready-made. Thus table has the corresponding adjective tabular, but there is no similar negative tnitahi- larj one man may be called a bookworm, but there is no negative for those who are not bookworms, because no need of the expression has been felt. A constant process of invention of new negative terms goes on more rapidly perhaps than is desirable, for when an idea is not often referred to it is better to express it by a phrase than add to the length of the dictionary by a new-created word. It would seem that in many cases a negative term implies the presence of some distinct quality or fact. Thus incoiivenience doubtless implies the absence of conveniejice, but also the presence of positive trouble or pain occasioned thereby. Unhappiness is a negative term, but precisely the same notion is expressed by the positive term misery. The negative of healthy is un- healthy, but the positive term sickly serves equally well. It thus appears to be more a matter of accident than anything else whether a positive or negative term is used to express any particular notion. All that we can really say is that every positive term necessarily implies the possibihty of a corresponding negative term, which is the name of all those things to which the positive name cannot be applied. Whether this term has been invented or not is an accident of language: its existence may be assumed in logic. The reader may be cautioned against supposing that every term appearing to be of a negative character on account of possessing a negative prefix is really so. The participle unloosed certainly appears to be the negative of 24 TERMS, AND THEIR [LESS. loosed; but the two words mean exactly the same thing, the prefix ini not being really the negative ; invaluable, again, means not what is devoid of value, but what is so valuable that the value cannot be measured; and a shameless action can equally be called by the positive \ term, a shameficl action. Other instances might no ' doubt be found. Great care should be taken to avoid confusing terms which express the presence or absence of a quality with those which describe its degree. Less is not the negative oi greater because there is a third alternative, equal. The true negative di greater is not-greater, and this is equiva- lent to either equal or less. So it may be said that dis- agreeable is not the simple negative of agreeable, because there may be things which are neither one nor the other, but are indifferent to us. It would not be easy to say offhand whether every action which is not honest is dis- honest, or whether there may not be actions of an inter- mediate character. The rule is that wherever the question is one of degree or quantity a medium is possible, and the subject belongs rather to the science of quantity than to simple logic ; where the question is one of the presence or absence of a quality, there cannot be more than two alternatives, according to one of the Primary Laws of Thought, which we will consider in Lesson XIV. ,In the case of quantity we may call the extreme terms ;opposites; thus less is the opposite of greater, disagreeable of agreeable ; in the case of mere negation we may call the terms negatives or contradictories, and it is really indifferent in a logical point of view which of a pair of contradictory terms we regard as the positive and which as the negative. Each is the negative of the other. Logicians have distinguished from simple negative terms a class of terms called privative, such as blind, dead, S^c. Such terms express that a thing has been III.] VARIOUS KINDS, 25 deprived of a quality which it before possessed, or was capable of possessing, or usually does possess. A man may be born blind, so that he never did see, but he pos- sesses the organs which would have enabled him to see , except for some accident. A stone or a tree could not , have had the faculty of seeing under any circumstances. No mineral substance can properly be said to die or to be dead, because it was incapable of life ; but it may be ► called uncrystallized because it might have been in the form of a crystal. Hence we apply a privative term to anything which has not a quality which it was capable of having ; we apply a negative term to anything which has not and could not have the quality. It is doubtful however whether this distinction can be properly carried out, and it is not of very much importance. It is further usual to divide terms according as they are relative or absolute, that is, non-relative. The adjective ^ absolute means whatever is " loosed from connection with anything else" (Latin ab, from, and solutus, loosed); whereas relative means that which is carried in thought, at least, into connection with something else. Hence a , relative term denotes an object which cannot be thought of without reference to some other object, or as part of a larger whole. A father cannot be thought of but in rela- tion to a child, a monarch in relation to a subject, a shep- ■• herd in relation to a flock ; thus father, monarch, and / shepherd are relative terms, while child, subject, and / flock are the correlatives (Latin con, with, and relativus), ' or those objects which are necessarily joined in thought with the original objects. The very meaning, in fact, of father is that he has a child, of monarch that he has subjects, and of shepherd that he has a flock. As ex- ^ amples of terms which have no apparent relation to any- thing else, I may mention water, gas, tree. There does not seem to me to be anything so habitually associated 26 TERMS, AND THEIR [less. with water that we must think of it as part of the same idea, and gas, tree, and a multitude of other terms, also denote objects which have no remarkable or permanent relations such as would entitle the terms to be called rela- tives. They may therefore be considered absolute or non-relative terms. The fact, however, is that everything must really have relations to something else, the water to the elements of which it is composed, the gas to the coal from which it is manufactured, the tree to the soil in which it is rooted. By the very laws of thought, again, no thing or class of things can be thought of but by separating them from other existing things from which they differ. I cannot use the term mortal without at once separating all existing or conceivable things into the two groups mortal and immortal; metal, element, organic substance, and every other term that could be mentioned, would necessarily imply the existence of a correlative negative term, non- metallic, compound, inorganic substance, and in this respect therefore every term is undoubtedly relative. Logicians, however, have been content to consider as; relative terms those only which imply some peculiar and striking kind of relation arising from position in time 6r space, from connexion of cause and effect, &c. ; and it is in this special sense therefore the student must use the distinction. The most important varieties of terms having been explained, it is desirable that the reader should acquire a complete familiarity with them by employing the exercises at the end of the book. The reader is to determine con- cerning each of the terms there given : — i. Whether it is a categorematic or syncategore- matic term. 2. Whether it is a general or a singular term. 3- Whether it is collective or distributive. III.] VARIOUS KINDS. 27 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 that 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 choose 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 monarchs 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 Names, System of Logic Book I. chap. 2. LESSON IV. OF THE AMBIGUITY OF TERMS. There 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 words. 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 28 OF THE AMBIGUITY [less. sinj^le clear meaning and one meaning only, and when- ever two or more meanings are unconsciously confused together, we inevitably commit a logical fallacy. If, for instance, a person should argue that " punishment is an evil," and according to the principles of morality "no 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 poor, 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 in 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 of the dangerous nature of the tools he has to use in all IV.] OF TERMS. 29 communications and arguments. Hence I must ask much attention to the contents of this Lesson. Terms are said to be univocal when they can suggest to the mind no more than one single definite meaning. They are called equivocal or ambiguous when they have two or more diiTerent meanings. It will be observed, however, that a term is not equivocal because it can be apphed 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 in technical and scientific language. Steam-engine, gas- ometer, railway train, permanent way, and multitudes of such technical names denoting distinct common objects, are sufficiently univoca]. 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, and 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 30 OF THE AMBIGUITY [less. of terms equally precise and certain in meaning. (See Lesson XXXIII.) The names of individual objects, build- ings, events, or persons, again, are usually quite certain and clear, as in Julius Caesar, William the Conqueror, the first Napoleon, Saint Peter's, Westminster Abbey, the ^ Great Exhibition of 185 1, and so on. But however numerous may be the univocal terms which can be adduced, still the equivocal terms are asto- nishingly common. They include most of the nouns and , adjectives 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 again homoiiyniotis, from the Greek o\xoi^ same, and ovofia, 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 of i 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- vocal words, according as they are — ^' 1. Equivocal in sound only. 2. Equivocal in spelling only. 3. Equivocal both in sound and spelling. The first two classes are comparatively speaking of very slight 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 words rein, rain and reign, might and mite, &c. Owing partly IV.] OF TERMS. 31 to defects of pronunciation mistakes are not unknown 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 — 1. From the accidental confusion of different words. 2. 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 mcdiiuii or mediocre, from the French vioyen and the Latin mediiis, connected with the Anglo-Saxon viid^ or middle J or it denotes what is low-minded and base, being then derived from the Anglo-Saxon Gem(e?te, which means " that belonging to the moene or many," whatever in short is vulgar. The verb to 7nea7i. 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 rente prendre, to return), or a tear, the result of the action of 32 OF THE AMBIGUITY [less. rending^ this word being of Anglo-Saxon origin and one of the numerous class beginning in ror ivr, which imitate more or less perfectly the sound of the action which they denote. Pounds from the Latin poiidiis, a weight, is con- fused with poinid, in the sense of a village pinfold for , cattle, derived from the Saxon pyndati^ to pen up. Fell, a mountain, is a perfectly distinct word from fell, a skin , or hide; Sind ptilse, a throb or beating, and pt^lse, peas, beans, or potage, though both derived from the Greek or 4 Latin, are probably quite unconnected words. It is curious that gm, 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 07ie, derived from an Aryan root, through the Latin tmns, and the in- determinate pronoun, one (as in ^'' otie ought to do oji^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 meaning 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, k,», the House means either the chamber in which the mem- bers meet, or it means the body of members who happen t to be assembled in it at any time. Similarly, the word chiirch originally denoted the building {KvpiaKov, 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 Acts xiv. 23 ; or it means any body of persons holding IV.] OF TERMS. 33 the same opinions and connected in one organization, as in the Anglican, or Greek, or Roman Catholic Church ; 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 associated with the act of worship as to be often called the church /^r ^;ir^//^;/r^. It is quite evident moreover that the word entirely differs in meaning according as it is used by a member of the Anglican, Greek, 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 TTouy, TToSos- ; 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 ; 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 fellow. Originally a fellow was wh^t follows 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," 3 34 OF THE AMBIGUITY [less. From the simple meaning of companion again it comes to denote vaguely a person, as in the question "What fellow is that?" but then there is a curious confusion of 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 or 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 or good fellow. We may still add the technical meanings of 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 of 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 of 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 horses 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. 35 ously enough we now have iron letter-posts, in which the 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 the 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 jna7i as possessing a masculine character. Occasionally it is used in the same way as groom, for a servant, as in the proverb, " Like master, like man." At other times it stands specially for a hus- 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. 3—2 36 THE AMBIGUITY OF TERMS. [less. iv. When we speak of a sweet taste, a sweet flower, a sweet 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 described otherwise than by comparison with sugar. In a similar 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 b?-illiant is derived from the French brillery 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. hoc\ of gravitatio7t, 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 ^^ so?ne metals are useful" and ^^ all cornets 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 ^that they must be rendered definite before we treat them. I may observe, however, that in the following lessons I •( shall frequently use propositions in the indefinite form as examples, on the understanding that where no sign of , quantity appears, the universal quantity is to be assumed. It is probable that wherever a term is used alone, it ought to be interpreted as meaning the whole of its class. 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 only the two kinds, Universal and Particular. Remembering now that there are two kinds of propo- 5 66 KINDS OF PROPOSITIONS. [less. sition as regards quality, and two as regards quantity, we shall be able to form altogether four varieties, thus : — Proposition Universal 1^,^^"^.^^^^^ ^ \_ Negative E Particular/Affirmative I L 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 affinno, 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, nonej 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 at least. The negative proposition is known by the adverbial particle not being joined to the copula; but in the propo- sition E, that is the universal negative, we frequently use the particle no or none prefixed to the subject Thus, " no metals are compound," " 7ione of the ancients were acquainted with the laws of motion," are familiar forms of 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 E or 0, according as we interpret it to "" viiL] KINDS OF PROPOSITIONS. 67 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 once 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 '■Hhe greater Clum- ber of books are not at once learned and amusing." A proposition of 'this kind is generally to be classed rather as than I. The word some is subject to an exactly similar ambiguity between some but not all, and some at least, it 7nay be all; the latter appears to be the coiTect 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. (i) It is not uncommon, especially in poetry, to find the predicate placed first, for the sake of emphasis or 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 such 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 E, Propositions are also expressed in various modes dif- fering from the simple logical order, and some of the different kinds which arise must be noticed. 5-2 68 KINDS OF PROPOSITIONS. [less. Exclusive propositions contain some words, such" as only^ alo7te, 7i07ie but, which limit the predicate to the subject. Thus, in " elements alone are metals," we are 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;" or, again, by " all that are not elements are not metals ;" 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 if I 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 or 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 of England, is the largest city of Europe," contains two pro- . VIII.] 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. AmpUative 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. Tautologous or Truistic 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 pure or modaL The pure proposition simply asserts that the predicate does or does not belong to the subject, while the modal proposition states this cmn 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 examples of modal propositions in this acceptation of 70 KINDS OF PROPOSITIONS. [less. the name. Other logicians, however, have adopted a different view, and treat modality as consisting in the degree of certainty or probability with which a judgment is made and asserted. Thus, we may say, " an equilateral triangle is 7iecessarily equiangular ;" " men are generally trustworthy;" "a falling barometer /r •" J5 J> 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 be derived from the denial ^f particul^^than from the denial of universals, that i»b say, the^^^less cases left doubt/til, as in the above^)le. ^||h The reader may weJ^K cautioned, however, against an ambiguity which tjSMpiisled some even of the most eminent logicians. In ^ittticular propositions the adjec- tive some is to be carefully interpreted as sofne, and there may or may not be inore or all. Were we to interpret it as some, not more nor all^sj^ii it would really give to the proposition the force of f an^b 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 sotne 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 as 8o OPPOSITION OF PROPOSITIONS, [less. ix. regards the quality of the subject matter to which they 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 in which the proposition A may be affirmed ; impossible in which E 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 0. Thus "comets are subject to gravitation," though an indefinite or indesignate proposition (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 I, 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 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- neous 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 of 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 of water on the ground and believe that it has rained. This is a somewhat complicated act of inference, which we shall consider in later lessons under the subject of 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 6 Vij vp 82 CONVERSION OF PROPOSITIONS, [less. lesson point out all the ways in which we can from a single proposition of the forms A, E, I or 0, pass to another proposition. We are said to convert a proposition when we transpose its subject and predicate; but in order that the converse or converted proposition shall be inferred from the convertend, or that which was to be converted, we must observe two rules (i) the quality of the pro- position (affirmative or negative) must be preserved, and (2) no ter7n must be distributed in the Converse unless it was distributed in 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- version by Limitation, or Per accidens. When the converse is a proposition of exactly the same form as the convertend the process is called simple conversion. 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 E, 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 separated from all the metals. The proposition E is then simply converted into E. X.] AND I M MEDIA TE INFERENCE. 83 But in attempting to convert the proposition we 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 ;" 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 the 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 iinmaterial substances," and, converting simply, we have — "some immaterial substances are ex- isting things," which may truly be inferred from the con- vertend. 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 all the metals be among the elements, whatever is not ele- 6—2 84 CONVERSION OF PROPOSITIONS, [less. ment, or outside the circle of elements, must also be outside the circle of metals. We may also prove the truth of the contrapositive proposition in this way, if we may anticipate the contents of Lesson XXIII.: — 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 xxil. We are very liable to infer from a pro- position of the form "all metals are elements," that all not-metals are not-elenients, 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. X.] AND IMMEDIATE INFERENCE. 85 Contrapositive conversion cannot be applied to the particular propositions I and at all, nor to the propo- sition E, in that form ; but we may change E 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," i.e. "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," (E) 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. It is also called Obversion. The following table contains a proposition of each kind changed by privative conception into an equivalent proposition : jA all metals are elements. )E no metals are compounds. JE no men are perfect. (A all men are imperfect. fl some men are trustworthy. (0 some men are not untrustworthy. JO some men are not trustworthy. (I some men are untrustworthy. The truth of any of the above can be clearly illustrated 86 CONVERSION OF PROPOSITIONS, [less. 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,'' 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 are 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 the subject and predicate of a proposition, so as to ren- der the meaning of each term narrower or better deter- mined. Provided that no other alteration is rnade 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 inf^ 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 are incompetent as kings would be incompetent in other 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 of X.] AND IMMEDIATE INFERENCE. Z7 necessity when the meaning is exactly the same in each case. With comparative terms this kind of inference will seldom be applicable ; thus from " a cottage is a building," we cannot infer "a huge cottage is a huge building," since a cottage may be large when compared with other cottages, but not with buildings generally. Immediate Inference by Complex Conception is closely similar 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 been used, the original universal proposition cannot be re- gained, but only the particular proposition corresponding to it. On Im7nediate Inference, Archbishop Thomson, Outline of the Laws of Thought, \\ 85 — 92. LESSON XL LOGICAL ANALYSIS OF SENTENCES. 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, which 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 ordinary 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 the proposition, as in su7n,, "I am ;" and the celebrated exclamation of Cassar, Veni, vicfi, vici, " I came, I saw, I conquered," contains three distinct and complete propositions in three words. These peculiar cases only arise, however, from the parts of the proposition having been blended together and dis- LESS. XL] ANALYSIS OF SENTENCES. 89 guised in one word ; and in the Latin stan, the letter m is a reHc of the pronoun me, which is the real subject of ^ 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 explain how the several parts of the complete proposition had become blended and apparently lost, just as the words will and not are blended in the colloquial " I wont." 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, slightly lighter than air," contains one subject only, Ni- trogen, but four or five predicates ; it is plainly equiva- lent to "Nitrogen is colourless," "Nitrogen is tasteless," " Nitrogen is a gas," and so on. Lastly, we may have several subjects and several 90 LOGICAL ANALYSIS [less. predicates all combined in the same sentence, and with only one copula, so that each predicate is asserted of 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," "cheap," "useful,'' and "metal." As there is nothing to prevent our applying each predicate to each subject the sentence really contains i6 distinct propositions in only II 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 distmct propositions combined together are said to be coordinate with each other, that is of the same order or rank, 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 is 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 is not confused thereby. * Antony and Cleopatra, Act III. Sc. a. XI.] 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 fear 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 dies badly. Adverbs or adverbial clauses may also specify the time, place, or any other circumstance con- cerned in the truth of the main proposition. ^ Assuming the reader to be acquainted with the gram- 92 LOGICAL ANALYSTS [less. matical terms used, we may thus state the parts of which the most complex sentence must consist. The subject may consist of — 1. A noun ; as in " The Qiieeti reigns." 2. A pronoun ; as in " She reigns." 3. An adjective converted into a noun ; as in " Whites are civiHzed." 4. A gerund ; as " Seeing is believing." 5. An infinitive ; as " To see is to believe." 6. A subordinate clause ; as " Who falls from virtue is lost." The subject may be qualified or restricted by combin- ing with it an attribute which may be expressed in any of the following ways : 1. An adjective ; as, '''■Fresh air is wholesome." 2. A participle ; as " Fallijig 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 " ChathanHs 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 go 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 rises^^ 2. A compound tense ; as " The sun has risenP 3. An incomplete verb and complement ; as " The sea seems rouo;hP XI.] OF SENTENCES. 93 4. The verb " to be" and an adjective : as " Time is fleetingP 5. A verb with an object ; as " Warmth tnelts iceP 6. A verb with an adverbial; as "The snow falls thickly" 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 tHajority^ 3. An absolute phrase; as "The snow melts, the sun havifig 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 yearP Various modes of exhibiting the construction of sen- tences by symbols and names for the several parts have been invented ; but I believe 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 sen- tence which are co-ordinate with each other, or bear the same relation to any other part, are written beside each other, and coupled together by a bracket ; thus the dia- gram,— Iron I r abundant. Copper I I cheap. Lead j ^^^ j useful Zinc J I metals, clearly shows that there are four co-ordinate subjects. 94 LOGICAL ANALYSIS [less. and four co-ordinate predicates in the example previously taken. Whenever one part of a sentence is subordinate to another part it may be connected with it by a line drawn in any convenient direction. Thus the analysis of the following sentence is readily shown by the diagram below it :— "No one who is a lover of money, a lover of pleasure, and a lover of glory, is likewise a lover of mankind ; but only he who is a lover of virtue." {a lover of money, a lover of pleasure, a lover of glory, one is not , . , a lover of mankind, he only is I who is a lover of virtue. We 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 consists of 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 flies — Fills, like this little daisy at my feet. Its function best of diligence in love." XI.] OF SENTENCES. 95 which fills most space in earthly eyes I , -_ , „ ( not that He shall most approve j ^^^ ^^^^ ^^^^ ^^^^ who metes its function of like this little as we should mete diligence in daisy at my I love feet, could we His insight use. *T T~^- ' ^ -^ ^ though Time scarce note 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 I To pace the ground with unuplifted if path while a fair region eyes ^^^^ j be round the | i or none traveller lies | , ^ 1 which (region) he (the traveller) forbears to look upon ' f some soft ideal scene pleased ) , r— — — ' rather with ) the work of fancy ( or some happy tone of meditation sHpping in between the beauty coming and the beauty gone. In the above sentence there is evidently one subject o6 LOGICAL ANALYSIS [LESS. " to pace the ground," which by means of the pronoun //, is connected with the predicate most sweet. The main part of 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 complex on account of four subordinate propositions. In the following sentence there is strictly but one 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." " When the notions with which men are conversant in the common course of life, which give meaning to their familiar language and which give employment to their 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 I operations have ( much that is common When the notions ^ are compared , with the ideas f I on which exact science is founded. with which which give which give men are meaning employ- conversant to their ment to in the familiar their hourly common language thoughts course of life Here the two classes form a collective term, and have two coordinate predicates rendering the sentence so far a compound one. The greater part of the sentence, how- ever consists of a comphcated 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 mcreases, 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 arrived I 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 I as the distance increases but the exact numerical rate I at which that decrease takes place so that its amount may be calculated for any other dis- I [tance when it is known at any one distance. W. S. Dalgleish's Grammatical Analysis^ or J. D. Morell's Analysis of Se7ite7ices. Alex. Bain's English Compositioji and Rhe- toric^ pp.91 — Ii7j 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 yeVoff, 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 other, 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. 99 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 ; if divided in respect to size the species of book are folio, quarto, octavo, duodecimo, &c. ; and, of course, each of 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 species, 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 or parts. The difference (Latin differetitia^ Greek hia- <^opa) cannot have any meaning except in intension, * and when we use all the terms wholly in intension we may say that the difference added to the ge?ius makes the species. Thus if "building" be the genus, and we add the differ- ence "used for a dwelling," we get the species "house." If we take "triangle" as the genus, it means the sum of the qualities of " three-sided rectilineal figure ;" if we add the quahty of "havmg two sides equal," we obtain the , species " isosceles triangle." 7—2 loo THE PREDICABLES, DIVISION, [LESS. It will easily be seen that the same class of things may be both a genus and a species at the same time, ac- cording as we regard it as divided into smaller classes or forming part of a larger class. Thus triangle, which is a genus as regards isosceles triangle, is a species as re- gards right-lined geometrical figures. House is a species of building, but a genus with respect to mansion, cottage, villa, or other kinds of houses. We may, in fact, have an almost interminable chain of genera and species, each class being a species of the class next above it, and a genus as regards that next below. Thus the genus Bri- tish subject has the species Born in the United Kingdom, Colonial-born, and Naturalised. Each of these becomes a genus as regards the species male and female; each species again may be divided into adult and minor, edu- cated, uneducated, employed in some occupation or un- employed, self-maintaining, maintained by friends, or pauper; and so on. The subdivision may thus proceed until we reach a class of so restricted extent, that it cannot be divided except into individuals; in this case the species is called the lowest species or inflma species. All the intermediate genera and species of the chain are called subaltern (Latin sub, under, and alter^ the other of two), because they stand one under the other. If there be a genus which is not regarded as a species, that is as part of any higher genus, it is called the summum genus, the highest genus, or genus generalissimum^ the most general genus. It is questionable whether we can thus ' set any limit to the chain of classes. The class British subject is certainly not an absolute su7ninum genus, since it is but a species of man, which is a species of animal, living being, portion of the earth, substance, and so on. If there were any real summum genus it would probably be " Being," or " Thing," or " Object con- ceivable ;" but we may usefully employ the term to signify XII.] AND DEFINITION. loi the highest class of things comprehended in any science or classification. Thus "material substance" is the sum- mum genus examined in the science of chemistry; "in- habitant of the United Kingdom" is the summum genus enumerated and classified in the British census. Logi- cal terms are only a species of words or phrases, but they are the summum genus as regards logic, which has no- thing to do with the various parts of speech and the relations of words, syllables, and letters, examined by grammarians. Several very useful expressions have been derived from the words genus and species. When a thing is so peculiar and unlike other things that it cannot easily be brought into one class with them, it is said to be sui generis, or of its own genus ; thus the rings of Saturn are so different from anything else among the heavenly bodies that they may fairly be called sici generis. In zoology, the Ornithorhynchus, or Australian Duck-bill, the Amphi- oxus, and some other animals, are so peculiar that they may be called stci generis. When a substance is the same in all its parts, or when a number of things are all alike, we say that they are Jiomogeneotis (Greek oiiU., like, yevoff, kind), that is of the same nature ; otherwise they may be called heterogeneous (Greek erepo^, other). It is necessary to distinguish carefully the purely lo- gical use of the terms genus and species from their pecu- liar use in natural history. A species is there a class of plants and animals supposed to have descended from common parents, and to be the narrowest class possessing a fixed form ; the genus is the next higher class. But if we accept Darwin's theory of the origin of species, this definition of species becomes entirely illusory, since dif- ferent genera and species must have according to this theory descended from common parents. The species then denotes a merely arbitrary amount of resemblance I02 THE PREDICABLES, DIVISION, [less. which naturalists choose to fix upon, and which it is not possible to define more exactly. This use of the term, then, has no connection whatever with the logical use, according to which any class of things whatever is a species, provided it is regarded as part of a wider class or genus. The fourth of the Predicables is Property (Latin pro- prium, Greek Xbiov, own), which it is hardly possible to define in a manner free from objection and difficulty, but which may perhaps be best described as any quality which is common to the whole of a class, but is not neces- sary to mark out that class from other classes. Thus it is a property of the genus "triangle" to have the three in- ternal angles equal to two right angles; this is a very remarkable circumstance, which is always true of tri- angles, but it is not made a part of the genus, or is not employed in defining a triangle, because the possession of three straight sides is a sufficient mark. The properties of geometrical ^gures are very numerous; the Second Book of Euclid is occupied in proving a few properties of rect- angles ; the Third Book similarly of circles. As we com- monly use the tertn property it may or may not belong to ■ other objects as well as those in question; some of the properties of the circle may belong also to the ellipse ; some of the properties of man, as for instance the power of memory, or of anger, may belong to other animals. Logicians have invented various subtle divisions of pro- perties, but it will be sufficient to say that a peculiar pro- perty is one which belongs to the whole of a class, and to that class only, as laughter is supposed to belong only to mankind ; the property of containing the greatest space in a line of given length is peculiar to circles. When a pro- perty is not peculiar, it may belong to other classes of objects as well as that of which it is called the property. We may further distinguish the Generic Property, or that XII.] AND DEFINITION 103 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 accidens, Greek o-vh^c^t}- Kos) is any quality which may indifferently belong or not belong to a class, as the case may be, without affecting the other qualities of the class. The word means that which /a/Is or happens by chance, and has no necessary connection with the nature of a thing. Thus the absolute size of a triangle is a pure accident as regards its geometrical properties; for whether the side of a triangle be ^ of an inch or a million miles, what- ever Euclid proves to be true of one is true of the other. The birthplace of a man is an accident concerning him, as are also the clothes in which he is dressed, the position in which he rests, and so on. Some writers distinguish se- parable and inseparable accidents. Thus the clothes in which a man is dressed is a separable accident, because they can be changed, as can also his position, and many other circumstances ; but his birthplace, his height, his Christian name, &c., are inseparable accidents, because they can never be changed, although they have no neces- sary or important relation to his general character. As an illustration of some part of the scheme of clas- sification described under the name of Predicables, I may here give, as is usual in manuals of Logic, the Tree of Porphyry, a sort of example of classification invented by one of the earliest Greek logicians, named Porphyrius. I have simplified the common form in which it is given by translating the Latin names and omitting superfluous words. In this Tree we observe a succession of genera and species — Substance, Body, Living Being, Animal and Man. Of these Substance is the sum7nic7n genus, because it is not regarded as a species of any higher class ; Man I04 THE PREDICABLES, DIVISION, [less, i^ is the iiifima species, because it is a class not divided in- to any lower class, but only into individuals, of whom it is Substance, r Socrates, Plato, and others. usual to specify Socrates and Plato. Body, Living Being, and Animal are called subaltern genera and species, be- cause each is a species as regards the next higher genus, and a genus as regards the next lower species. The qualities imphed in the adjectives Corporeal, Animate, Sensible {i.e. capable of feeling) and Rational are the successive differences which occasion a division of each genus into species. It will be evident that the negative parts of the genera, namely Incorporeal Substance, In- XII.] AND DEFINITION. 105 animate Body, &c., are capable of subdivision, which has not been carried out in order to avoid confusing the figure. Logical division is the name of the process by which we distinguish the species of which a genus is composed. Thus we are said to divide the genus " book " when we consider it as made up of the groups foHo, quarto, octavo, duodecimo books, &c., and the size of the books is in this case the ground, basis, or principle of division, commonly called the Fimdamentum Divisionis. In order that a quality or circumstance may be taken as the basis of division, it must be present with some and absent with others, or must vary with the different species comprehended in the genus. A generic property of course, being present in the whole of the genus, cannot serve for the purpose of divi- sion. Three rules may be laid down to which a sound and useful division must conform : 1. The constituent species must exclude each other. 2. The constituent species must be equal when add- ed together to the genus. 3. The division must be founded upon one principle or basis. It would be obviously absurd to divide books into folio, quarto, French, German and dictionaries, because these species overlap each other, and there may be French or German dictionaries which happen to be quarto or folio and belong to three different species at once. A division of this kind is said to be a Cross Division, because there is morfe than one principle of division, and the seve- ral species in consequence cross each other and produce confusion. If I were to divide rectilineal figures into tri- angles, parallelograms, rectangles and polygons of more than four sides, I should commit all the possible faults in one division. The species parallelogram and rectangle do not exclude each other, since all rectangles must be [o6 THE PREDICABLES, DIVISION^ [less. parallelograms ; the constituent species are not altogether equal to the genus rectilineal figure, since irregular four- sided figures which are not parallelograms have been omitted ; and there are three principles of division, namely the number of sides, the directions of those sides, and the angles contained. But when subdivision is employed, and each of the species is considered as a genus which may be subjected to a further separation, a new principle of division may and in fact must be employed each time. Thus I can divide rectilineal figures according to the three principles mentioned above : Rectilineal Figure 3 sides 4 sides more than 4 sides Triangle Quadrilateral Polygon I ' \ 1 with parallel sides without parallel Parallelogram sides Trapezium. Here the principles of division are the number of their sides, and in the case of four-sided figures their paral- lelism. Triangles do not admit of division in this second respect We may make a new division of parallelograms, adopting the equality of sides and the size of the angles as the principles ; thus : Parallelogram . ' ; ' ; — T"! adjoining sides adjoining sides equal not equal right- not right- right- not right- angled angled angled angled Square Rhombus Oblong Rhomboid. The most perfect divisions in a logical point of view are produced by continually dividing each genus into two XII.] AND DEFINITION. 107 species by a difference, of which an example has been given in the Tree of Porphyry. This process is called Dichotomy (Greek hlxa, in two ; reTu/o), to cut) ; it is also called Exhaustive Division because it always of necessity obeys the second rule, and provides a place for every possible existing thing. By a Law of Thought to be con- sidered in the next Lesson, every thing must either have a quahty or not have it, so that it must fall into one or other division of the genus. This process of exhaustive division will be shewn to have considerable importance in Lesson XXII I., but in practice it is not by any means always necessary or convenient. It would, for instance, produce a needlessly long classification if we divided rec- tilineal figures thus : Rectilineal figure 3-sided not 3-sided Triangle , : , 4-sided not 4-sided Quadrilateral 5-sided not 5-sided Pentagon &c. As we know beyond all doubt that every figure must have 3, 4, 5, 6, or more sides, and no figure can belong to more than one group, it is much better at once to enume- rate the parts as Triangle, Quadrilateral, Pentagon, Hexa- gon, &c. Again, it would be very awkward if we divided the counties of England into Middlesex and not-Middle- sex; the latter into Surrey and not-Surrey; the latter, again, into Kent and not-Kent. Dichotomy is useless, and even seems absurd in these cases, because we can observe the rules of division certainly in a much briefer division. But in less certain branches of knowledge our divisions can never be free from possible oversight unless they proceed by dichotomy. Thus, if we divide the popula- tion of the world into three branches, Aryan, Semitic, and io8 THE PREDICABLES, DIVISION, [less. Turanian, some race might ultimately be discovered which is distinct from any of these, and for which no place has been provided ; but had we proceeded thus — Man Aryan not-Aryan Semitic not-Semitic Turanian not-Turanian, it is evident that the new race would fall into the last group, which is neither Aryan, Semitic, nor Turanian. All the divisions of naturalists are liable to this inconvenience, If we divide Vertebrate Animals into Mammalia, Birds, Reptiles, and Fish, it may any time happen that a new form is discovered which belongs to none of these, and therefore upsets the division. A further precaution required in Division is not to proceed from a high or wide genus at once to a low or narrow species, or, as the phrase is, divisio non faciat saltum (the division should not make a leap). The species should always be those of the proximate or next higher genus ; thus it would obviously be inconvenient to begin by dividing geometrical figures into those which have parallel sides and those which have not; but this principle of division is very proper when apphed to the proximate genus. Logical division must not be confused with physical division or Partition, by which an individual object, as a tree, is regarded as composed of its separate parts, root, trunk, branches, leaves, &c. There is even a third and distinct process, called Metaphysical Division, which con- sists in regarding a thing as an aggregate of qualities, and separating these in thought ; as when we discriminate the form, colour, taste, and smell of an orange. Closely connected with the subject of this Lesson is XII.] AND DEFINITION. 109 the process of Logical Definition, by which we determine the common quahties or marks of the objects belonging to any given class of objects. We must give in a defini- tion the briefest possible statement of such qualities as are sufficient to distinguish the class from other classes, and determine its position in the general classification of conceptions. Now this will be fulfilled by regarding the class as a species, and giving the proximate genus and the difference. The word genus is here used in its inten- sive meaning, and denotes the qualities belonging to all of the genus, and sufficient to mark them out ; and as the difference marks out the part of the genus in question, we get a perfect definition of the species desired. But we should be careful to give in a definition no superfluous marks ; if these are accidents and do not belong to the whole, the definition will be improperly narrowed, as if we were to define Quadrilateral Figures as figures with four equal sides ; if the superfluous marks belong to all the things defined they are Prope?'ties, and have no effect upon the definition whatever. Thus if I define parallelo- grams as " four-sided rectilineal figures, with the opposite sides equal and parallel, and the opposite angles equal," I have added two properties, the equality of the opposite sides and angles which necessarily follow from the paral- lelism of the sides, and only add to the complexity of the definition without rendering it more precise. There are certain rules usually given in logical works which express the precautions necessary in definition. 1. A definition should state the essejitial attributes of the species defified. So far as any exact meaning can be given to the expression "essential attributes," it means, as explained above, the proximate genus and difference. 2. A definition must not co7itain the na7ne defined. For the purpose of the definition is to make the species known, and as long as it is not known it cannot sei-ve to no THE PREDICABLES, DIVISION, [less. make itself known. When this rule is not observed, there is said to be ' circulus in deji^tiendo^ or ' a circle in defin- ing,' because the definition brings us round again to the very word from which we started. This fault will usually be committed by using a word in the definition which is really a synonym of the name defined, as if I were to define "Plant" as "an organized being possessing vege- table life," or elements as simple substances, vegetable being really equivalent to plant, and simple to elementary. If I were to define metals as " substances possessing me- tallic lustre," I should either commit this fault, or use the term metallic lustre in a sense which would admit other substances, and thus break the following rule. 3. The definition must be exactly equivalent to the 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 obscure^figura- tive or a7nbiguous laiiguage. In other words, the tenns 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,^ definition must not be iiegative where it can be affirmative. This rule however is often not applicable, and is by no means always binding. Read Mr Mill on the nature of Classification and the XII.] AND DEFINITION. in five Predicables, System of Logic, Book I. Chap. VII. For ancient Scholastic Views concerning De- finition, see Mansel's Artis Logics Rudimenta (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 Pensees 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 Geovietrie 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 the Port Royal Logic, and from Descartes' celebrated Essay on Method. The words of Pascal are nearly as follows. **The true method, which would furnish demonstra- tions of the highest excellence, if it were possible to 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 all the 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 number divisible into two equal parts, we give it a name divested of every XIII.] 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 of 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 brief terms which they employ simply to avoid the con- fusion produced by a multitude of words. " N othing prevents more promptly and effectively the insidious fallacies of the sophists than this method, which we should always employ, and which alone suffices to banish all sorts of difficulties and equivocations. " These things being well understood, I 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 TT4 PASCAL AND DESCARTES [less. were it not absolutely impossible. It is evident that the first 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 return back upon the words already defined ; for if we define A by B^ and B by C, and C by Z>, and then Z) by ^, we commit what may be called a circulus in definiendo; a most serious fallacy, which might lead us to suppose that we know the nature of ^, B^ C, and Z>, when we really know nothing about them. XIII.] ON METHOD. 115 Pascal's views of the geometrical method were clearly summed 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 without defining them. 2. To employ in the definitions only terms perfectly known or already explained. 3. To demand as axioms only truths perfectly evi- 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 zs force, mass, fnoment, power, inertia, and the most different opinions are still held as to the simplest axioms by which the law of the composition of forces may be proved Nevertheless if we steadily bear in mind, in * Mr Spencer Bajoies' Translation^ p. 317. 8—2 ii6 PASCAL AND DECARTES [less. studying each science, the necessity of defining every term as far as possible, and proving each proposition which can be proved by a simpler one, we shall do much to clear away error and confusion. I also wish to give here the rules proposed by the celebrated Descartes for guiding the reason in the attain- ment of truth. They are as follows : — 1. Never to accept anything as true, which we do not clearly know to be so ; that is to say, carefully to avoid haste or prejudice, and to comprise nothing more in our judgments than what presents itself so clearly and distinctly to the mind that we cannot have any room to doubt it. 2. To divide each difficulty we examine into as many parts as possible, or as may be required for resolv- ing it. 3. To conduct our thoughts in an orderly manner, commencing with the most simple and easily known objects, in order to ascend by degrees to the knowledge of the most complex. 4. To make in every case enumerations so complete, and reviews so wide, that we may be sure of omitting nothing. These rules were first stated by Descartes in his ad- mirable Discowse on Method^ in which he gives his reflec- tions on the right mode of conducting the reason, and searching for truth in any of the sciences. This little treatise is easily to be obtained in the original French, and has also been translated into English by Mr Veitch*. The reader can be strongly advised to study it. Always to observe the rules of Descartes and Pascal, or to know whether we in every case observe them properly, is im- * Published at Edinburgh in 1 850. XIII.] ON METHOD. 117 possible, but it must nevertheless be valuable to know at what we ought to aim. Read Locke's brief Essay on the Conduct of the Un- derstandings which contains admirable remarks on the acquirement of exact and logical habits of thought. LESSON XIV. THE LAWS OF THOUGHT. Before the reader proceeds to the lessons which treat of the most common forms of reasoning, known as the syllogism, it is desirable that he should give a careful attention to the very simple laws of thought on which all reasoning must ultimately depend. These laws describe the very simplest truths, in which all people must agree, and which at the same time apply to all notions which we can conceive. It is impossible to think correctly and avoid evident self-contradiction unless we observe what are called the Tliree Primary Laws of Tliouglit, which may be stated as follows : 1. The Law of Identity. Whatever is, is. 2. The Law of Contradiction. NotMng can both be and not be. 3. The Law of Excluded Middle. Everything must either be or not be. Though these laws when thus stated may seem ab- surdly obvious, and were ridiculed by Locke and others on that account, I have found that students are seldom able to see at first their full meaning and imponance. It will be pointed out in Lesson XXI 1 1, that logicians b^ve ii8 THE LAWS OF THOUGHT, [less. overlooked until recent years the very simple way in which all arguments may be explained when these self-evident laws are granted ; and it is not too much to say that the whole of logic will be plain to those who will constantly use these laws as the key. The first of the laws may be regarded as the best definition we can give of identity or sameness. Could any one be ignorant of the meaning of the word Identity, it would be sufficient to inform him that everything is identical witli itself. The second law however is one which requires more consideration. Its meaning is that nothing can have at the same time and at the same place contradic- tory and inconsistent qualities. A piece of paper may be blackened in one part, while it is white in other parts; or it may be white at one time, and afterwards become black; but we cannot conceive that it should be both white and black at the same place and time. A door after being open may be shut, but it cannot at once be shut and open. Water may feel warm to one hand and cold to another hand, but it cannot be both warm and cold to the same hand. No quality can both be present and absent at the same time ; and this seems to be the most simple and general truth which we can assert of all things. It is the very nature of existence that a thing cannot be otherwise than it is ; and it may be safely said that all fallacy and error arise from unwittingly reason- ing in a way inconsistent with this law. All statements or inferences which imply a combination of contradictory qualities must be taken as impossible and false, and the breaking of this law is the mark of their being false. It can easily be shewn that if Iron be a metal, and every metal an element, Iron must be an element or it can be nothing at all, since it would combine qualities which are inconsistent (see Lesson xxiii). XIV.] THE LAWS OF THOUGHT. 119 The Law of Excluded Middle is much less self-evident than either of the two preceding ones, and the reader will not perhaps see at the first moment that it is equally important and necessary with them. Its meaning may be best explained by saying that it is impossible to men- tion any thing and any quality or circumstance, without allowing that the quality or circumstance either belongs to the thing or does not belong. The name of the law expresses the fact that there is no third or middle course ; the answer must be Yes or No. Let the thing be rock and the quality hard; then rock must be either hard or not-hard. Gold must be either white or not white j a line must be either straight or not straight ; an action must be either virtuous or not virtuous. Indeed when we know nothing of the terms used we may never- theless make assertions concerning them in accordance with this law. The reader may not know and in fact chemists m.ay not really know with certainty, whether vanadium is a metal or not a metal, but any one knows that it must be one or the other. Some readers may not know what a cycloid is or what an isochronous curve is ; but they must know that a cycloid is either an isochro- nous curve or it is not an isochronous curve. This law of excluded middle is not so evident but that plausible objections may be suggested to it. Rock, it may be urged, is not always either hard or soft, for it may be half way between, a little hard and a little soft at the same time. This objection points to a distinction which is of great logical importance, and when neglected often leads to fallacy. The law of excluded middle affirmed nothing about hard and soft^ but only referred to hard and not-hard J if the reader chooses to substitute soft for not-hard he falls into a serious confusion between opposite terms and contradictory terms. It is quite possible that a thmg may be neither hard nor soft, being half way I20 THE LAWS OF THOUGHT. [less. between ; but in that case it cannot be fairly called hard, so that the law holds true. Similarly water must be either warm or not-warm, but it does not follow that it must be warm or cold. The alternative not-warm evi- dently includes all cases in which it is cold besides cases where it is of a medium temperature, so that we should call it neither warm nor cold. We must thus carefully distinguish questions of degree or quantity from those of simple logical fact. In cases where a thing or quality may exist to a greater or less extent there are many alter- natives. Warm water, for, instance may have any tempe- rature from 70° perhaps up to 120°. Exactly the same question occurs in cases of geometrical reasoning; for Euclid in his Elements frequently argues from the self- evident truth that any line must be either greater than, equal to, or less than any other line. While there are only two alternatives to choose from in logic there are three in Mathematics ; thus one line, compared with another, may be — {greater greater i ^ not-greater...j ••••••^^^"^=^1 (Mathematics. Another and even more plausible objection may be raised to the third law of thought in this way. Virtue being the thing proposed, and tria7igular the quality, the Law of Excluded Middle enables us at once to assert that virtue is either triangular or not -triangular. At first sight it might seem false and absurd to say that an immaterial notion such as virtue should be either triangular or not, because it has nothing in common with those material substances occupying space to which the notion of figure belongs. But the absurdity would arise, not from any falseness in the law, but from misinterpretation of the expression net-triangular. If in saying that a thing is XIV.] THE LAWS OF THOUGHT. 121 "not triangular" we are taken to imply that it has some figure though not a triangular figure, then of course the expression cannot be applied to virtue or anything im- material. In strict logic however no such implied mean- ing is to be allowed, and not-triangular will include both things ^vhich have figure other than triangular, as well as things which have not the properties of figure at all; and it is in the latter meaning that it is applicable to an im- material thing. These three laws then being universally and neces- sarily true to whatever things they are applied, become the foundation of reasoning. All acts of reasoning pro- ceed from certain judgments, and the act of judgment consists in comparing two things or ideas together and discovering whether they agree or differ, that is to say whether they are identical in any qualities. The laws of thought inform us of the very nature of this identity with which all thought is concerned. But in the operation of discourse or reasoning we need certain additional laws, or axioms, or self-evident truths, which may be thus stated : 1. Two terms agreemg- with o?ie and the same third term agree with each other. 2. Two terms of which one agrees and the other does not agree with one and the same third term^ do not agree with each other. These self-evident truths are commonly called the Canons or Fundamental Principles of Syllogism, and they are true whatever may be the kind of agreement in ques- tion. The example we formerly used (p. 3) of the a- greement of the terms "Most useful metal" and "cheapest metal" with the third common term " Iron," was but an instance of the first Canon, and the agreement con- sisted in complete identity. In the case of the " Earth," the " Planets," and " Bodies revolving in elliptic orbits," 122 THE LAWS OF THOUGHT, [less. the agreement was less complete, because the Earth is orxly one of many Planets, and the Planets only a small portion of all the heavenly bodies, such as Satellites, Comets, Meteors, and Double-Stars which revolve in such orbits. The second of the Canons applies to cases where there is disagreement or difference, as in the following example : Venus is a planet. Planets are not self-luminous. Therefore Venus is not self-luminous. The first of these propositions states a certain agree- ment to exist between Venus and planet, just as in the previous case of the Earth, but the second proposition states a disagreement between Planet and self-luminous bodies; hence we infer a disagreement between Venus and self-luminous body. But the reader will carefully observe that frotn two disagreemetits we can never infer anythifig. If the following were put forth as an argu- ment it would be evidently absurd : — Sirius is not a planet. Planets are not self-luminous. Therefore Sirius is not self-luminous. Both the premises or propositions given are true, and yet the conclusion is false, for all the fixed stars are self-luminous, or shine by their own light. We may, in fact, state as a third Canon that — 3. Two terms both disagreeing with 07te and the same third term 7nay or may not agree with each other. Self-evident rules, of an exactly similar nature to these three Canons, are the basis of all mathematical reasoning, and are usually called axioms. Euclid's first axiom is that "Things which are equal to the same thing are equal to one another ;" and whether we apply it to the length of lines, the magnitude of angles, areas, solids, numbers, XIV.] THE LAWS OF THOUGHT. 123 degrees, or anything else which admits of being equal or unequal, it holds true. Thus if the lines A and B are each equal to C it is evident that each is equal to the other. Euclid does not give axioms corresponding to the second and third Canons, but they are really used in Geometry. Thus if ^ is equal to B, but D is not equal to B, it follows that A is not equal to Z>, 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 an axiom called the Dictum de ofnjti et nullo of Aristotle. This celebrated Latin phrase means " Statement concerning all and none," and the axiom, or rather pair of axioms, is usually given in the following words : 124 THE LAWS OF THOUGHT. [less. Whatever is predicated of a term distributed whether affirmatively or negatively^ may be predicated in like manner of everything contained under it. Or more briefly : What pertains to the higher class pertains also to the 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 part 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, according 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 of Similars'*. Some logicians have held that in addition to the three laws which are called the Primary Laws of Thought, * Macmillan and Co. 1869. XIV.] THE LAWS OF THOUGHT. 125 there is a fourth called " The Principle or Law of Suffi- cient Reason." It was stated by Leibnitz in the following words : Nothing happens without a reason why it should be so rather than otherwise. For instance, if there be a pair of scales in every respect exactly alike on each side and with 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 in 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 a^i6u>, 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 as the basis of reasoning. In mathematics it is clearly used in this sense. See Hamilton's Lectures on Logic, Lectures 5 and 6. 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 be distinguished from the process of Immediate Inference, or 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 ascer- 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 the 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- rallel 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 that 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 "all mam- XV.] THE RULES OF THE SYLLOGISM. 127 malian animals are viviparous." It would follow that "whales are viviparous;" and so far as the inference is concerned it does not matter what is the meaning we attribute to the words viviparous and mammalian. In this case " mammalian animal " is the middle term. The name Syllogism means the joining together in thought of two propositions, and is derived from the Greek words o-yi/, together, and Xoyos, thought. It is thus exactly the equivalent of the word Co7nputatio7i, which means thinking together (Latin con, together, puto, to think), or reckoning. In a syllogism we so unite in thought two premises, or propositions put forward, that we are enabled to draw from them or infer, by means of the middle term they contain, a third proposition called the conclusion. Syllogism may thus be defined as the act of thought by which from two given propositions we proceed to a third proposition, the truth of which neces- sarily follows from the truth of these given propositions. When the argument is fully expressed in language it is usual to call it concretely a syllogism. The special rules of the syllogism are founded upon the Laws of Thought and the Canons considered in the previous Lesson. They serve to inform us exactly under what circumstances one proposition can be inferred from two other propositions, and are eight in number, as follov/s : — 1. Every syllogism has three and only three terms. These terms are called the major term, the minor term, and the middle term. 2. Every syllogism contains three, and only three propositions. These propositions are called the major premise, the minor premise, and the conclusion. 3. The middle term must be distributed once at leasts and must not be ambiguous. 128 THE RULES OF THE SYLLOGISM, [less. 4. No ter7n inust be distribiited in the conclusion which was not distributed in one of the premises. 5. From negative premises nothing can be inferred. 6. If one pretnise be negative^ the conclusion must be negative; and vice versa, to prove a negative con- elusion one of the premises must be negative. From the above rules may be deduced two subor- dinate rules, which it will nevertheless be convenient to state at once. 7. From two particular pretnises no conclusion can be drawn. 8. If 07ie premise be particular^ the conclusion must be particular. All these rules are of such extreme importance that it will be desirable for the student not only to acquire a perfect comprehension of their meaning and truth, but to commit them to memory. During the remaindejr of this lesson we shall consider their meaning and force. As the syllogism consists in comparing two terms by means of a middle term, there cannot of course be less than three terms, nor can there be more ; for if there were four terms, say A^ B, C, D, and we compared A with B and C with D, we should either have no common medium at all between A and D, or we should require a second syllogism, so as first to compare A and C with B, and then A and D with C. The middle term may always be known by the fact that it does not occur in the conclusion. The major term is always the predicate of the conclusion, and the minor term 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 is obviously a major term or wider terra than men. XV.] THE RULES OF THE 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 rhinor premise which similarly compares the minor and middle terms ; and of a conclusion, which contains the major and minor terms only. In a strictly correct syllogism the major premise always 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 at all, because they are both affirmative propositions, which have (p. 75) undistributed predicates. It is apparent that French- men are one part of Europeans, and Russians another part, as shown in Euler's method in Fig. 6, so that Y\z. 6. 130 THE RULES OF THE SYLLOGISM, [less. there is no real middle term. Those propositions would equally allow of Russians being or not being Frenchmen ; 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, do 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 of 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 amM- guous middle term hardly requires explanation. It has been stated (Lesson IV.) that an ambiguous term is one which has two different meanings, implymg 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, 131 brass is metal, therefore it is an element," we should be committing a fallacy by using the middle term metal in 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, ma7iy stations, was particular in the minor premise, and must not be made universal in the conclusion. The pre- mises do not warrant a statement concerning anything but the viany nations capable of self-government. The above argument would therefore be fallacious and would be technically called an illicit process of the minor term, meaning that we have improperly treated the minor term. 9—2 132 THE RULES OF THE SYLLOGISM, [less. Such a breach of the fourth rule as is described above is exceedingly easy to detect, and is therefore very seldom committed. But an illicit process or improper treatment of the major term is more common because it is not so trans- parently false. 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 major as we may briefly call this fallacy. The following is an obscurer example of the same fallacy;—" Few students t/W?^y ('S^^^^^ ^^4^^^^ XV.] THE RULES OF THE SYLLOGISM. 133 are capable of excelling in many branches of knowledge, and 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 double meaning before explained (p. (i'j), 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 if Fig. 9. \ Colonists we were to argue that Americans are not Europeans, and Virginians are not Europeans, we see that both terms disagree with the middle term Europeans, and yet they 134 THE RULES OF THE SYLLOGISM, [less. agree between themselves. In other cases the two nega- tive premises may be plainly true while it will be quite uncertain whether the major and minor terms agree or not. Thus it is true, for instance, that "Colonists are not Europeans, and Americans are not Europeans," but this gives us no right to infer that Colonists are or 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 of this rule of the syllogism may be conveniently called the fallacy of negative premises. It must not however be supposed that the mere occurrence of a negative particle {7iot 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 valid, because the negative in both cases affects the middle term, which is really the negative term not-compoimd. The truth of the sixth rule depends upon that of the axiom, 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 forms of expression. Numerous examples are given at the end of XV.] THE RULES OF THE SYLLOGISM. 135 the book by which the student may acquire faciHty in the analysis of arguments. The remaining rules of the syllogism, the 7th and 8th, are by no means of a self-evident character and are in fact corollaries of the first six rules, that is consequences which follow from them. We shall therefore have to shew that they are true consequences in a future Lesson. We may call a breach of the 7th rule 2. fallacy of parti- cular premises^ and that of the 8th rule the fallacy of a universal conclusion from a particular pre7nise^ but these fallacies may really be resolved into those of Illicit Process, or Undistributed Middle. For many details concerning the Aristotelian and Scholastic Views of the Syllogism, and of Formal Logic generally, see the copious critical notes to Mansel's edition of Aldrich's Artis Logiccz Rudi- menta. 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 136 THE MOODS AND FIGURES [less. of the syllogism may, so far as we yet know, be either 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 four as a minor premise, and any one of the 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 AAE AEE AIE AOE EAE EEE AAI AEI All AOI EAI EEI AAO AEO AIO AOO EAO &c. It is obvious that there will be altogether 4x4x4 or 64 such combinations, of which 23 only are given 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 of arranging propositions into syllogisms. We call each of these triplets of propositions a mood or form of the syllogism (Latin modus , shape), and we have to consider how many of such forms can really be used in valid arguments, as distinguished from those which break one or more of the rules of the syllogism. Thus the mood AEA would break the 6th rule, that if one premise be negative the conclusion must be so too : AIE breaks the XVI.] OF THE SYLLOGISM. 137 converse part of the same rule, that a negative conclusion can only be proved by a negative premise ; while EEA, EEE &c., break the 5th rule, which prohibits our reasoning at all from two negative premises. Examples of any of 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 8th rules of the syllogism. Thus AIA and EIE break the rule, that if one premise be particular the con- clusion must be so also, while IIA, 100, 010 and many others, break the rule against two particular premises. Some combinations of propositions may break more than one rule ; thus 000 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 rules to write out all the 64 combinations and then strike out such as break any rule ; the task if pur- sued systematically will not be so long or tedious as might seem likely. It will be found that there are only twelve moods which escape exclusion, and may so far be considered good forms of reasoning, and these are AAA EAE lAI OAO AAI EAO (lEO) AEE EIO AEO All AOO Of these however EEO will have shortly to be rejected, because it will be found really to break the 4th rule, and involves Illicit process of the major term. There are, 138 THE MOODS AND FIGURES {less. then, only eleven moods of the syllogism which are really valid; and we may thus account for the whole of the'' sixty-four moods. Number Excluded by of moods. 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 11 Total 64 We have by no means exhausted as yet all the possible varieties of the syllogism, for we have only de- termined the character, affirmative or negative, general 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 major 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, or as they are called Figures, in which the terms can be disposed. These four figures of the syllogism are shewn in the following scheme, taking X to denote the major term Y middle „ Z minor „ I St 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 Jirst as subject of the major premise in the ist Figure, second 2iS predicate in both premises of the 2nd Figure, yfn-/ again as subject of 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 AEE in the first figure would be as follows : All K's are JTs, No Z's are K's ; Therefore No Z^s are X^s. This 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 ^'s are F's, NoZ'sareK's; Therefore No Z's are X^s. In the third figure it becomes All K's are JTs, No K's are Z's, No Z's are ^s, 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. I40 THE MOODS AND FIGURES [less. When all the valid moods are selected out of the 44 possible ones, there are found to be altogether 24, which are as follows: Valid Moods of the Syllogism. First Second Third Fourth Figure. Figure. Figure. Figure. AAA EAE AAI AAI £A£ A££ lAI AEE All EIO All lAI EIO AOO EAO EAO OAO EIO [AAI] [EAO] EIO [EAO] [AEO] [AEOJ Five 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 all metals gravitate. It is not actually an erroneous conclusion, because it must be carefully remembered (p. ']']) that the affirming of a subaltern or particular proposition does not deny the corresponding general proposition. It is quite true that • some metals gravitate, and it must be true because all of them do so. But when we can as readily prove that all do gravitate it is desirable to adopt 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 peculiai 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, E, I, 0. As regards the premises, it is especially important to note that the major premise is always universal (A or E), and the minor premise affirmative (A or I) : this peculiarity will be further considered in the next lesson. The second figure only proves negative conclusions ,(E or 0), 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^ Kand Z" to represent, as before, the major, mid- dle and minor terms of a syllogism, and the four moods of this figure are then EAE AEE no X\ are I^s, all X's are Vs, all Z's are F's ; no Z's are K's ; •', no Z's are Jf 's. .-. no Z's are ^'s. 142 THE MOODS AND FIGURES [LESS. EIO no ^'s are K's, some Z's are F's ; *. some Z's are not ^'s. AOO all X's are F^s, some Z's are not Vs ; .'. some Z's are not AT's. The nature of the moods of the second figure is clearly shewn in the following figures : Fig. lo. (Cesare.) Fig. II. (Camestres.) Fig. \^. (Festino.) It will also be observed that in the second figure the minor premise may be any of the four A, E, I, 0. The third figure only proves particulars (I or 0), and it always has an affirmative minor premise (A or I). It also contains the greatest number of moods, since in no case is the conclusion a weakened one. XVI.] OF THE SYLLOGISM. 143 The fourth figure is usually considered unnatural and comparatively useless, because the same arguments can be 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, I, 0, and its first 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 each figure was best suited for certain special pur- poses. A German logician, Lambert, stated these pur- poses concisely as follows :— "The first figure is suited to the discovery or proof of the properties of a thing ; the second to the discovery or proof of the distinctions be- tween things; the third to the discovery or proof of in- stances and exceptions ; the fourth to the discovery, or exclusion, of the different species of genus." It may be added that the moods Cesare and Cames- tres are often used in disproving a statement, because they give a universal negative conclusion, founded upon the exclusion of one class from another. Thus if any one were still to assert that light consists of material particles it might be met by the following syllogism : " Material particles communicate impetus to whatever they strike. Light does not communicate impetus to whatever it strikes ; Therefore light is not material particles." The moods Baroko and Festino are less used, but allow of a particular conclusion being established. When we wish however to establish objections or 144 THE IMPERFECT FIGURES [less. exceptions to a general statement, whicli is indeed the natural way of meeting it, we employ the third figure. ^ The statement that "all metals are solids" would at once be disproved by the exception viercury^ as follows : Mercury is not solid, Mercury is a metal ; Therefore some metal is not solid. Were any one to assert that what is incomprehensible , cannot exist, we meet it at once with the argument that Infinity is incomprehensible, but that infinity certainly ♦ exists, because we cannot otherwise explain the nature of a curve line, or of a quantity varying continuously ; there- i- fore something that is incomprehensible exists. In this case" even one exception is sufficient entirely to negative the proposition, which really means that because a thing is incomprehensible it cannot exist. But if one incom- prehensible thing does exist, others may also; and all authority is taken from the statement. According to the Aristotelian system the third figure must also be employed whenever the middle term is a ' singular term, because in Aristotle's view of the subject a ^ singular term could not stand as the predicate of a pro- position. LESSON XVII. REDUCTION OF THE IMPERFECT FIGURES OF THE SYLLOGISM. In order to facilitate the recollection of the nineteen valid 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 be XVII.] OF THE SYLLOGISM, 145 readily committed to memory. This device, however in- genious, is of a barbarous and wholly unscientific cha- racter ; but a knowledge of its construction and use is still expected from the student of logic, and the verses are therefore given and explained below. Barbara, Celarent, Darii, Fertoo^ue, prions; CesarCj Cajnesires, Fesiino, Baroko, secundas; Tertia, Darapti, Disamis, Daiisi, Felapton, Bokardo, Ferisojt, habet ; Quarta insuper addit Brajnantip, Caiiieiies, Dwia?'is, FesapOj 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, CY.lkrY.nt 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 oniiii et niUlo being directly ap- plicable to it, and he therefore called it the Perfect Figure. The fourth figure was never recognised by him, and it is often called the Galenian figure, because the celebrated Galen is supposed to have discovered it. The second and third figures were known to Aristotle as the Imperfect Figures, which it was necessary to reduce to the first 146 THE IMPERFECT FIGURES [less. , figure by certain conversions and transpositions of the 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 pre- ceding vowel is to be converted simply. p indicates that the proposition is to be converted per accidens, or by limitation. m indicates that the premises of the syllogism are to 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 imitare^ to change. , B, C, Z>, F, the initial consonants of the names, in- dicate the moods of the first figure, which are produced -t- 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 reductio ad iinpossibile, which will shortly be considered. Let us now take some syllogism, say in Camestres, 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 ;;z instructs us to change the order of the premises, and the final s to convert the conclusion simply. When all these changes are made t we obtain No self-luminous bodies are planets Converse of (2) All stars are self-luminous (i) Therefore no stars are planets Converse of (3) This, it will be found, is a syllogism in Celarent, as might be knoWn from the initial C in Camestres. xviL] OF THE SYLLOGISM. 147 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 fixed stars. According to the directions in the name, we are to 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 ?n, which instructs us to transpose the premises, and the letter p, which denotes conversion by limitation ; effecting these changes we 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 have 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 so 10 — 2 / 148 THE IMPERFECT FIGURES [less. imperfect and unnatural in form, containing nothing but ill-arranged syllogisms, which would have been better stated 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 was made to the somewhat complicated forms of the syllogism. The two peculiar moods called 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^ V, Z to represent the major, middle and minor terms of the syllogism, and Baroko may then be stated as follows : All ^'s are K's, Some Z^s are not F's ; Therefore Some Z's are not ^'s. Now if we convert the major premise by Contrapo- sition (p. 83) we have " all not- F's are not J^'s," and, making this the major premise of the syllogism, we have All not- F's are not X's, Some Z's are not- F's ; Therefore Some Z's are not ^'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 middle term (see p. 134), and we have therefore effected the re- duction of the syllogism. Bokardo, when similarly stated, is as follows : Some F's are not ^'s, All F's are Z's; Therefore Some Z's are not X'^, XVII.] OF THE SYLLOGISM. 149 To reduce this, convert the major premise by nega- tion, and then transpose the premises. We have: All K's are Z's, Some not-X's are F's; Therefore Some not-^'s are Z's. This conclusion is the converse by negation of the former 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 true when the p7'emises are trtte. The syllogism enables us to restate in a new form the in- formation which is contained in the premises, just as a m.achine 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, as be- fore: All ^s are F's (i) Some Z's are not K's (2) Therefore Some Z's are not ^s (3) ISO THE IMPERFECT FIGURES [less. If this conclusion be not true then its contradictory, *a.\lZ's are ^'s' 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 ^s are F's (i) All Z's are X's contradictory of (3) Hence All Z's are F's. 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 F's are not ^s (i) All K's are Z's (2) Therefore Some Z's are not ^'s (3) If this conclusion be not true then 'all Z's are Jf's' must be true. Now we can make the syllogism : All Z's are ^'s Contradictory of (3) AllK'sareZs (2) Hence All K's are ^'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 original conclusion is true. It will be observed that in both these cases of Indirect 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 be 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. XVII.] 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- lar 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 II contain no distributed teim 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 of 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 ought 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 of 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 also be particular. It can only be shown to be true by going over all the possible cases and observing that the six principal rules of the syllogism always require the conclusion to be particular. Suppose for instance the 152 IRREGULAR AND COMPOUND [less. premises are A and I ; then they contain only one dis- tributed term, the subject of A, and this is required for the middle term by rule 3. Hence the minor term cannot be distributed without breaking rule 4, so that the con- clusion must be the proposition I. The premises AO would contain two distributed terms, the subject of A and the predicate of 0; but if we were to draw from them the conclusion E, the major and minor terms would require to be distributed, so that the middle term would remain undistributed against rule 3. The reader can easily prove the other cases such as EI by calculating the number of distributed terms in a similar manner: it will always be found that there are insufficient terms distributed in the premises to allow of a universal conclusion. LESSON XVIII. IRREGULAR AND COMPOUND SYLLOGISMS. It may seem surprising that arguments which are met with in books or conversation are seldom or never thrown into the form of regular syllogisms. Even if a complete syllogism be sometimes met with, it is generally employed in mere affectation of logical precision. In former cen- turies it was, indeed, the practice for all students at the Universities to take part in public disputations, during which elaborate syllogistic arguments were put forward by one side and confuted by precise syllogisms on the other side. This practice has not been very long dis- continued at the University of Oxford, and is said to be still maintained in some continental Universities ; but except in such school disputations it must be allowed that perfectly formal syllogisms are seldom employed. XVIII.] SYLLOGISMS. 153 In truth, however, it is not syllogistic arguments which are wanting; wherever any one of the conjunctions, therefore, because, for, since, hence, inasmuch as, conse- quently occurs, it is certain that an inference is being drawn, and this will very probably be done by a tine syllogism. It is merely the complete statement of the premises and conclusion, which is usually neglected be- cause the reader is generally aware of one or other of the premises, or he can readily divine what is assumed; and it is tedious and even offensive to state at full length what the reader is already aware of. Thus, if I say "atmo- spheric air must have weight because it is a material substance," I certainly employ a syllogism ; but I think it quite needless to state the premise, of which I clearly assume the truth, that " whatever is a material substance has weight." The conclusion of the syllogism is the first proposition, viz. "atmospheric air has weight." The middle term is "material substance," which does not occur in the conclusion; the minor is "atmospheric air," and the major, "having weight." The complete syllogism is evi- dently : All material substances have weight. Atmospheric air is a material substance ; Therefore atmospheric air has weight. This is in the very common and useful mood Barbara. A syllogism when incompletely stated is usually called an enthsrmeme, and this name is often supposed to be derived from two Greek words {Iv, in, and 6v\io^, mind), so as to signify that some knowledge is held by the mind and is supplied in the form of a tacit, that is a silent or understood premise. Most commonly this will be the major premise, and then the enthymeme may be said to be of the First Order. Less commonly the minor premise is unexpressed, and the enthymeme is of the Second 154 IRREGULAR AND COMPOUND [less. Order. Of this nature is the following argument: " Comets must be subject to the law of gravitation ; for this is true of all bodies which move in elliptic orbits." It is so clearly implied that comets move in elliptic orbits, that it would be tedious to state this as the minor premise in a complete syllogism of the mood Barbara, thus : All bodies moving in elliptic orbits are subject to the law of gravitation ; Comets move in elliptic orbits ; Therefore comets are subject to the law of gravitation. It may happen occasionally that the conclusion of a syllogism is left unexpressed, and the enthymeme may then be said to "belong to the Third Order. This occurs in the case of epigrams or other witty sayings, of which the very wit often consists in making an unexpressed truth ap- parent. Sir W. Hamilton gives as an instance of this kind of enthymeme the celebrated epigram written by Porson the English scholar upon a contemporary German scholar : " The Germans in Greek Are sadly to seek ; Not five in five score, But ninety-five more ; All, save only Hermann, And Hermann's a German." It is evident that while pretending to make an exception of Hermann, the writer ingeniously insinuates that since he is a German he has not a correct knowledge of Greek. The wonderful speech of Antony over the body of Caesar, in Shakspeare's greatest historical play, contains a series of syllogistic arguments of which the conclusions are suggested only. Even a single proposition may have a syllogistic force if it clearly suggest to the mind a second premise which . XVIII.] SYLLOGISMS. 155 thus enables a conclusion to be drawn. The expression of Home Tooke, "Men who have no rights cannot justly complain of any wrongs," seems to be a case in point ; for there are few people who have not felt wronged at some time or other, and they would therefore be likely to argue, whether upon true or false premises, as follows : Men who have no rights cannot justly complain of any wrongs; We can justly complain; Therefore we are not men who have no rights. In other words, we have rights. Syllogisms may be variously joined and combined together, and it is convenient to have special names for the several parts of a complex argument. Thus a syllo- gism which proves or furnishes a reason for one of the premises of another syllogism is called a Prosyllogism ; and a syllogism which contains as a premise the conclu- sion of another syllogism is called an Episyllogism. Take the example : All ^'s are A% And all Cs area's; Therefore all Cs are A's. But all Z^'s are Cs; Therefore All Us are A's. This evidently contains two syllogisms in the mood Bar- bara, the first of which is a Prosyllogism with respect to the second, while the second is an Episyllogism with respect to the first. The peculiar name Epicheirema is given to a syllogism when either premise is proved or supported by a reason implying the existence of an imperfectly expressed pro- syllogism ; thus the form, 156 IRREGULAR AND COMPOUND [less. All ^'s are A's, for they are P's, And all Cs are ^'s, for they are Qs ; Therefore all C's are A^s, is a double Epicheirema, containing reasons for both premises. The reader will readily decompose it into three complete syllogisms of the mood Barbara. A more interesting form of reasoning is found in the chain of syllogisms commonly called the Sorites, from the Greek word acopos, meaning /lea^p. It is usually stated in this way : All ^'s are B's, All ^'s are C's, All C's are Ds, All Ds are E's ; Therefore all A^s are £'s. 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 Syllo.gism. ^'s are C's, C's are Z>'s, Z^'s are -£"'s, A's are ^'s ; A''s are C's ; A's are Z>'s ; .-. ^'s areC's. .'. ^'s are Z>'s. .: 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 universal 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 were > XVIII.] 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 were negative there would be a fallacy of illicit process of the major term. It is not to be supposed that the forms of the syllogism hitherto described are all the kinds of reasoning actually employed in science or common life. In addition to the hypothetical and disjunctive syllogisms and some other forms to be described in succeeding lessons, there are really many modes of reasoning of which logicians have not taken much notice as yet. This was clearly pointed out more than two hundred years ago by the writers of 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 Baynes, of St Andrew's. Many improvements of Logic may be found in this 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 this 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 is 158 IRREGULAR AND COMPOUND [less. much more frequent than that of syllogisms which are . 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. This is an argument which cannot be proved by the rules of the syllogism, and yet it is not only evidently true, but is an exceedingly common kind of argument. Another example is as follows : The Divine Law commands us to honour kings ; Louis XIV. is a king ; 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 of 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 arguments, 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 properly to the science of language. XVIII.] SYLLOGISMS. 159 I may here explain that the syllogism and the sorites 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 the metals are part of the elements; but in regard to in- tension, that is to say the qualities impHed 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 of the great class Exogens; but in intension the cha- racter of Exogen is part of the character of Ranuncu- laceas, 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 : Exte7isive Syllogism. All Ranunculaces are Exogens ; The Anemone is one of the Ranunculaceas ; Therefore the Anemone is an Exogen. Intensive Syllogism. All the qualities of Ranunculacese are qualities of Anemone ; All the qualities of Exogen are quaUties of Ranun- culaceas ; Therefore all the qualities of Exogen are qualities of Anemone. Any sorites can be similarly represented either in ex- tension 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, 7th ed. Edinburgh. i6o OF CONDITIONAL [less. LESSON XIX. OF CONDITIONAL ARGUMENTS. It will be remembered that when treating of propositions we divided them into two distinct kinds, Categorical Pro- positions, and Conditional Propositions. The former kind alone has hitherto been considered, and we must now proceed to describe Conditional propositions and the ar- guments which may be composed of them. Logicians have commonly described Conditional pro- positions as composed of two or 7nore Categofical pro- positiofis united by a conjunction. This union may happen in two ways, giving rise to two very different species of conditionals, which we shall call Hypothetical Propositions and Disjunctive Propositions. The way in which the several kinds of propositions are related will be seen in the following diagram : {v^ategoncai. ^_.^.,^ Hypothetical Disjunctive. A conditional proposition may be further described as one which makes a statement under a certain con- dition or qualification restricting its application. In the hypothetical form this condition is introduced by the conjunction if, or some other word equivalent to it. Thus— " If iron is impure, it is brittle " is a hypothetical proposition consisting of two distinct categorical propositions, the first of which, " Iron is im- pure," is called the Antecedent; the second, "It is brittle," ^ XIX.] ARGUMENTS. i6i the Consequent. In this case " impurity " is the condition or qualification which limits the application of the pre- dicate brittle to iron. It was asserted by Home Tooke in his celebrated work The Diversio7is of Picrley^ that all conjunctions are the remains or corrupted forms of verbs. This is certainly true in the case of the hypothetical con- junction ; for the word if in old English is written gif or gyf and is undoubtedly derived from the verb to give. We may actually substitute at present any verb of similar meaning, as for instance — g7'a?it, allow^ suppose. Thus we may say — " Grant that iron is impure, and it is brittle." " Supposing that iron is impure, it is brittle." The hypothetical proposition might be employed in arguments of various form, but only two of these are of sufficient importance to receive special names. The hy- pothetical syllogism consists of two premises, called the major and minor, as in the case of the ordinary syllo- gism. The major premise is h}^othetical in form ; the minor premise is categorical, and according as it is af- firmative or negative the argument is said to be a Construc- tive or a Destructive hypothetical syllogism. Thus the form. If ^ is ^, C is D\ ^ But ^ is ^; Therefore C is Z>, is a constructive hypothetical syllogism. It must be carefully observed that the minor premise afifirms the antecedent of the major premise, whence the argument is said to be of the modus poneiis^ or mood which posits or affirms. It is probably one of the most familiar and common kinds of argument. The form, \i A \s B, C \% D; But C is not D ; Therefore A is not B- II i62 OF CONDITIONAL [less. represents the corresponding Destructive hypothetical syllogism, 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. Xhdl either the antecedent imist be affirmedy or the conseqiient denied. If either part of this rule be broken, a serious fallacy will be committed. Thus the apparent argument, If A is B, C is n ; But C is D; Therefore A is B, is really a fallacy which we may call sXv^ fallacy of affirm- ing the consegue7it, 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 E is F^ or G is //", 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 useful 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 him 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 the ante- cede fit y as in the form — If ^ is ^, C is Z> ; But A is not B ; Therefore C is not D, XIX.] ARGUMENTS. 163 The error may be explained in the same way ; for as "-4 being j5" is not stated to be the only condition of C being i9, 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 deser\^e 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 doiial 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 minor term ; thus in the case, II— 2 i64 OF CONDITIONAL [less. ^ If iron is impure it is brittle ; But it is impure ; ^ Therefore it is brittle, we have to substitute for the indefinite pronoun //, the iron in question, and we obtain a correct categorical syl- logism in the mood Barbara : Impure iron is brittle ; The iron in question is impure iron ; Therefore the iron in question is brittle. ^ Sometimes the reduction requires a more extensive change of language. For instance, If the barometer is falling, bad weather is coming ; But the barometer is falling ; Therefore bad weather is coming, may be represented in the following form : The circumstances of the barometer faUing are the cir- cumstances of bad weather coming ; But these are the circumstances of the barometer fall- ing; Therefore these are the circumstances of bad weather coming. As an instance of the Destructive Hypothetical syl- logism we may take : If Aristotle is right, slavery is a proper form of society; But slavery is not a proper form of society; Therefore Aristotle is not right. This becomes as a categorical : The case of Aristotle being right is the case of slavery being a proper form of society; But this is not the case ; Therefore this is not the case of Aristotle being right. If not reducible by any other form of expression, hypo- theticals can always be reduced by the use of the words case of. XIX.] ARGUMENTS. 165 It will now be easily made apparent that the fallacy of affirming the consequent is really a breach of the 3rd rule of the syllogism, leading to an undistributed middle term. Our example may be as before : If a man is avaricious he wiU refuse money ; But he does refuse money ; Therefore he is avaricious. This becomes as a categorical syllogism, All avaricious men refuse money ; But this man refuses money ; Therefore this man is avaricious. This is the mood AAA in the second figure ; and the middle term, refusing money, is undistributed in both premises, so that the argument is entirely fallacious. Again, the fallacy of denying the antecedent is equiva- lent to the illicit process of the major. Our former example (p. 163) may thus be represented: "A science which furnishes the mind with a multitude of useful facts deserves cultivation ; but Logic is not such a science ; therefore Logic does not deserve cultivation." This apparent syllogism is of the mood AEE in the first figure, which breaks the fourth rule of the syllogism, because the major term, deserving cultivation, is dis- tributed in the negative conclusion, but not in the affirma- tive major premise. We now pass to the consideration of the disjunctive proposition, which instead of a single predicate has several alternatives united by the disjunctive conjunction or, any one of which may be affirmed of the subject. "A member of the House of Commons is either a representa- tive of a county, or of a borough, or of a University," is an instance of such a proposition, containing three alterna- tives ; but there may be any number of alternatives from two upwards. 166 OF CONDITIONAL [less. The disjunctive syllogism consists of a disjunctive major premise with a categori-cal proposition, either af- firmative or negative, forming the minor premise. Thus arise two moods, of which the affirmative mood is called by the Latin words modus po7iendo tollens (the mood which by affirming denies), and may be thus stated : A is either B or C, Butyi is^; Therefore A is not C. This form of argument proceeds on the supposition that if one alternative of a disjunctive proposition be held true, the others cannot also be true. Thus " the time of year must be either spring, summer, autumn or winter," and if it be spring it cannot be summer, autumn or winter ; and so on. But it has been objected by Whately, Han- sel, Mill, as well as many earlier logicians, that this does not always hold true. Thus if we say that " a good book is valued either for the usefulness of its contents or the excellence of its style," it does not by any means follow because the contents of a book are useful that its style is not excellent. We generally choose alternatives which are inconsistent with each other; but this is not logically necessary. The other form of disjunctive syllogism, called the modus tollendopone7ts (the mood which by denying affirms), is always of necessity cogent, and is as follows : A is either B or C^ But ^ is not ^; Therefore A is C. Thus if we suppose a book to be valued only for the usefulness of its contents or the excellence of its style, it follows that if a book be valued but not for the former reason it must be for the latter; and vice versa. If the time of year be not spring, it must be summer, autumn or XIX.] ARGUMENTS. 167 winter; if it be not autumn nor winter, it must be either spring or summer; and so on. In short if any alternatives be denied, the rest remain to be affirmed as before. It will be noticed that the disjunctive syllogism is governed by totally different rules from the ordinary categorical syllogism, since a negative premise gives an affirmative conclusion in the former, and a negative conclusion in the latter. There yet remains a form of argument called the Dilemma, because it consists in assuming two alternatives, usually called the horns of the dilemma, and yet proves something in either case (Greek 5t- two ; Xrjfifia, assump- tion). Mr Mansel defines this argument as " a syllogism, having a conditional major premise with more than one antecedent, and a disjunctive minor." There are at least three forms in which it may be stated. The first form is called the Simple Constructive Dilemma : If A is B,C\s D\ and \i E \s F, C \s D \ But either A is B, or E is F; Therefore C is D. Thus "if a science furnishes useful facts, it is worthy of being cultivated; and if the study of it exercises the reasoning powers, it is worthy of being cultivated ; but either a science furnishes useful facts, or its study exercises the reasoning powers ; therefore it is worthy of being cultivated." The second form of dilemma is the Complex Con- structive Dilemma, which is as follows : If^ is^, C'ls D] and if^isi^, 6^ is H ; But either ^ is ^, or ^ is i^; Therefore either C is D, or G is H. It is called complex because the conclusion is in the disjunctive form. As an instance we may take the argu- i68 OF CONDITIONAL [less. ment, "If a statesman who sees his former opinions to be wrong does not alter his course he is guilty of deceit ; and if he does alter his course he is open to a charge of inconsistency ; but either he does not alter his course or he does ; therefore he is either guilty of deceit, or he is open to a charge of inconsistency." In this case as in the greater number of dilemmas the terms A^ B, C, D, &c. are not all different. The Destructive Dilemma is always complex, because it could otherwise be resolved into two unconnected de- structive hypothetical syllogisms. It is in the following form: If ^ is ^, C is Z>; and if ^ is F, G is H; But either C is not D, or G is not H; Therefore either A is not B, or E is not F. For instance, " If this man were wise, he would not speak irreverently of Scripture in jest; and if he were good, he would not do so in earnest ; but he does it either in jest or earnest; therefore he is either not wise, or not good*." Dilemmatic arguments are however more often fal- lacious than not, because it is seldom possible to find instances where two alternatives exhaust all the possible cases, unless indeed one of them be the simple negative of the other m accordance with the law of excluded mid- dle (p. 119). Thus if we were to argue that "if a pupil is fond of learning he needs no stimulus, and that if he dis- likes learning no stimulus will be of any avail, but as he is either fond of learning or dislikes it, a stimulus is either needless or of no avail," we evidently assume improperly the disjunctive minor premise. Fondness and dislike are not the only two possible alternatives, for there may be ♦ Whately. h ^""* •iv XIX.] ARGUMENTS. 169 some who are neither fond of learning nor dishke it, and to these a stimulus in the shape of rewards may be de- sirable. Almost anything can be proved if we are allowed thus to pick out two of the possible alternatives which are in our favour^ and argue from these alone. A dilemma can often be retorted by producing as cogent a dilemma to the contrary effect. Thus an Athe- nian mother, according to Aristotle, addressed her son in the following words : " Do not enter into public business ; for if you say what is just, men will hate you ; and if you say what is unjust, the Gods will hate you." To which Aristotle suggests the following retort : " I ought to enter into public affairs ; for if I say what is just, the Gods will love me ; and if I say what is unjust, men will love me." Mansel's Aldrich, App. Note I, on the Hypothetical Syllogism. LESSON XX. LOGICAL FALLACIES. In order to acquire a satisfactory knowledge of the rules of correct thinking, it is essential that we should become acquainted with the most common kinds of fallacy ; that is to say, the modes in which, by neglecting the rules of logic, we often fall into erroneous reasoning. In previous lessons we have considered, as it were, how to find the right road ; it is our task here to ascertain the turnings at which we are most liable to take the wrong road. In describing the fallacies I shall follow the order and adopt the mode of classification which has been usual for the last 2000 years and more, since in fact the great I70 LOGICAL FALLACIES. [less. teacher Aristotle first explained the fallacies. According to this mode of arrangement fallacies are divided into two principal groups, containing the logical and the material fallacies. 1. The logical fallacies are those which occur in the mere form of the statement ; or as it is said in the old Latin expressions, in diciwne, or 2?i voce. It is supposed accordingly that fallacies of this kind can be discovered without a knowledge of the subject-matter with which the argument is concerned. 2. The material fallacies, on the contrary, arise out- side of the mere verbal statement, or as it is said,, extra dicttonejft; they are concerned consequently with the sub- ject of the argument, or in re (in the matter), and cannot be detected and set right but by those acquainted with the subject. The first group of logical fallacies may be further di- vided into t\ve pU7'ely logical and the semi-logical, and we may include in the former class the distinct breaches of the syllogistic rules which have already been described. Thus we may enumerate as Purely Logical Fallacies : 1. Fallacy of four terms {Qiiaternio Ter7ninortiiri) — Breach of Rule i ; 2. Fallacy of undistributed middle — Breach of Rule 3 ; 3. Fallacy of ilHcit process, of the major or minor term — Breach of Rule 4 ; 4. Fallacy of negative premises — Breach of Rule 5 ; as well as breaches of the 6th rule, to which no distinct name has been given. Breaches of the 7th and cSth rules may be resolved into the preceding (p. 151), but they may also be described as in p. 135. The other part of the class of logical fallacies contains Semi-logical fallacies, which are six in number, as follows ; -XX.] LOGICAL FALLACIES. 171 1. Fallacy of Equivocation. 2. Fallacy of Amphibology. 3. Fallacy of Composition. 4. Fallacy of Division. 5. Fallacy of Accent. 6. Fallacy of Figure of Speech. * *These I shall describe and illustrate in succession. Equivocation consists in the same term being used in two distinct senses ; any of the three terms of the syl- logism may be subject to this fallacy, but it is usually the middle term which is used in one sense in one premise ' and in another sense in the other. In this case it is often called the fallacy of ambiguoics middle, and when we dis- tinguish the two meanings by using other suitable modes of expression it becomes apparent that the supposed syl- logism contains four terms. The fallacy of equivocation -.may accordingly be considered a disguised fallacy of four terms. Thus if a person were to argue that " all criminal actions ought to be punished by law; prosecutions for theft are criminal actions ; therefore prosecutions for ► theft ought to be punished by law," it is quite apparent that the term "criminal action" means totally different ' things in the two premises, and that there is no true middle term at all. Often, however, the ambiguity is of *"a subtle and difficult character, so that different opinions may be held concerning it. Thus we might argue : " He who harms another should be punished. He who communicates 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. Many 172 LOGICAL FALLACIES. [LESS,- difficult legal questions are of this nature, as for in- . 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 of nuisance which the law would interfere to prevent. Or again ; Interference with another man's business is illegal; Underselling interferes with another man's business; Therefore underselling 4s 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 y 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 Henr>'-, 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 4 as we add the 3 after or before multiplying. In the careless construction of sentences it is often impossible to ^xx.] 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 distribii lively, that is, of each and any separately, and then we may in the con- clusion infer the same of the whole piit 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 ' ^wo 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 veiy likely to judge erroneously, the jury as a whole are also very 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 is supposed that all trades at once might be benefited simi- 174 LOGICAL FALLACIES. [less.^ 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 the 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 v/ord 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 JiimP The italics indicate that the word him v/as suppHed by the translators of the authorized version^^ but it may suggest a very difterent 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 xx.] LOGICAL FALLACIES. 175 as emphasis is thrown upon one word or another. Thus the sentence "The study of Logic is not supposed to communicate a knowledge of many useful facts," may be 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, 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 w^hich 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-fiavoicred^ 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 walks the whole day; therefore he tramples on the day." Here an adverbial phrase is converted into a noun object. LESSON XXI. MATERIAL FALLACIES. The Material fallacies are next to be considered; and their importance is very great, although it is not easy to illustrate them by brief examples. There are altogether 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 sequitur. 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 rule. This latter fallacy is usu- ally described by the Latin phrase a dicto seamdurn quid ad dictum simpliciter, meaning " from a statement under 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 froDi one special case to another special case. • Formal Logic^ Chapter XIII. LESS. XXL] 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 re- 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 of the sabstance 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 punished." 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 shewn to have been done under exceptional cir- cumstances, as by a duly qualified surgeon acting for the good 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 the 12 178 MATERIAL FALLACIES. [less. case of a surgeon would be to infer from one special case to another special case. It is undoubtedly true that to give to beggars promotes mendicancy and causes evil ; but if we interpret this to mean that assistance is never to be given to those who solicit it, we fall into the converse fallacy of accident, inferring of all who solicit alms what is only true of those who sohcit 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 and 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 Igiioratio Elenchi^ 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 provec^. 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 plaintiff's attorney." Whoever thus uses what is known as argu7nentum ad hominem^ 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. If a man is accused of a crime it is no answer to say that. XXI.] MATERIAL FALLACIES. ijg the prosecutor is as bad. If a great change in the law is proposed in Parliament, it is an Irrelevant Conclusion to 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 argujnentum ad poptihim is another form of 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. Petitio Principii is a familiar name, and the nature of the fallacy it denotes is precisely expressed in the phrase begging the guestiojt. Another apt name for the fallacy is circuhis in probanda, 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, ^is C, A isB, therefore A is C, the conclusion is proved by being deduced from two propositions, neither of which is identical with it; but if the tnith of one of these premises itself depends upon the following syllogism, CisB, A is C, tlierefore A is i?, 12—2 i8o MATERIAL FALLACIES. [less. 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 ivhej^e it is at one moment and the place whei'e 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 that the doctrine is hei'esy, 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 a XXI.] MATERIAL FALLACIES, i8i 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 Q,uestio7i-begging Epithets. In like manner we beg the question when we oppose any change by saying that it is U7i-E7iglish. The Fallacy of the Consequent is better understood by the familiar phrase 7io?i seqjutiir. 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 7io7t causa pro caiisd. 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 /^j/ hoc 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 l82 MATERIAL FALLACIES, [less. xxi. 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 are due to that misgovernment. Lastly there is the somewhat trivial Fallacy of Many 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 off 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 their 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 in a single answer." Read Prof, de Morgan's excellent and amusing Chapter on Fallacies, Formal Logic, Ch. Xlll. Whatel/s remarks on Fallacies, Elements of Logic ^ Book III., are often very original and 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 ^vriter Euclid, so Logic has been taught nearly as Aristotle taught it about the year 335 B.C. But within the last few years teachers hav'e 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 imiprovements from the very basis of the science. This reform in Logic is called by the somewhat mys- terious name of the quantification of tlie predicate, but the reader who has found no insuperable difficulty in the preceding lessons need not fear one here. To quan- tify the predicate is simply to state whether the whole or the part only of the predicate agrees with or diff'ersfrojn the stcbject. In this proposition, " All metals are elements," 1 84 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 soine elements," the httle 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 are singular and identical with each other, and both are accordingly universal. As soon as we allow the quantity of the predicate to be 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," XXII.] GF THE PREDICATE. 185 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." The particular negative proposition " Some men are not (any) trustworthy persons " 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 E 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 nohcy is 1 86 THE QUANTIFICATION [less. not honesty," and also " what is not honesty is not the best poHcy." From this proposition in fact we can draw two contrapositives ; but the reader will carefully remember that from the ordinary unquantified proposition A we can only draw one contrapositive (see p. 84). Thus if "metals are elements" we must not say that "what are not metals are not elements." But if we quantify the predicate thus, "All metals are sotne elements," we may infer that " what are not metals are not soine 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 ^ is all Y, 1 I Some X is some Y. I Affirmative A All X is some Y. j propositions, Y Some X is all K J E NoJiTis (any) K j « Some X is not some Y. (, Negative Tj No ^ is some Y. j propositio}is, O Some X is no K The letters X and Y are used to stand for any subject and 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 refer- XXII.] OF THE PREDICATE. 1S7 ring to each of the eight propositions, and are very suitably chosen. The doubly universal affirmative pro- position is called U \ the simple converse of A is called Y; the Greek letter y\ {Eta, e) is applied to the proposi- tion obtained by changing the universal predicate of E into a particular predicate ; and the Greek « {Omega, 0) 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, 11 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 Outliiie of a New Systein of Logic, published in 1827, several years previous to the earliest of the logical publications of Sir W. Hamilton. But Mr Bentham considered that some of the propositions are hardly to be distinguished from others; as Y from A, of which it is the simple converse; or T] 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 'oy Hamilton and Thomson. When the whole eight propositions are admitted into use we are obhged 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 Thotight, Sir W. Hamilton also devised a curious system of notation for exhibiting all the moods of the syllogism in a clear manner. He always employed the letter 3/ to denote the middle term of the syllogism, and the two letters C and r (the Greek capital letter Gamma) for the two terms appearing in the conclusion. The copula of the THE QUANTIFICATION [less. Table of Moods of the Syllogis7n. First Figure. Second Fig. Third Figure. Affirm. Neg. Affirm. Neg. Affirm. Neg. 1 UUU EUE UEE UUU EUE UEE UUU EUE UEE ii AYI 77Y CO AOco YYI OYco YOco AAI 7; Aco At; CO iii AAA YAA OAt; AYA t;Y,; AOt; iv YYY OYO YOO AYY 77YO AOO YAY OAO Yt;0 V All A CO CO YII OIo, Ycoco All t; I CO A CO CO vi lYI CO Yco lOco lYI toYco lOco lAI coAco I t; 00 vii UYY EYO UOO^ UYY EYO UOO UAY EAO Ut;0 viii AUA ^U^ AE;; YUA OUj; YE,; AUA t;Ut; AEt; ix UAA EAE UAA EAE U^t; UYA EYE UOt; X YUY ouo YEE ' AUY 7;UO AEE I YUY OUO YEE xi UII EIO Ucoo) 1 UII EIO ; U CO 0) UII EIO U CO a» xii lUI CO U 0) IE, lUI coUco i IE,; 1 lUI CO U CO IE,; proposition was indicated by a line thickened towards the subject ; thus C i^i AT means that " Cisi^." To indicate the quantity of the terms Hamilton inserted a .. XXII.] OF THE PREDICATE, 1S9 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 affii'mative propositions. C : m Mil , ill All Cs are some J/'s (A) C : mm : M All C's are aU ATs (U) C , smm^^^- — , 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 : JIM,,, p '.M No C is any Af (E) C , i^BBBB^M— — : il/ Some C is not any J/ (0) C , H^iBB^v— ^ , M Some C is not some M («) Any syllogism can be represented by placing Af the middle term in the centre and connecting it on each side with the other terms. The copula representing the con- . elusion can then be placed below ; Barbara is expressed as follows — C, ■■■■ :M, The negative mood Celarent is similarly- + Cesare in the second figure is thus represented— C: is3Es»- ,M: j,,,, :r Sir W. Hamilton also proposed a new law or supreme xanon of the syllogism by which the vaHdity of all forms I90 THE QUANTIFICATION [LESS. of the syllogism might be tested. This was stated in the 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 woi'se I'elation, 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 Syllabtcs of a new Systejn 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 together 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 XXII.] OF THE PREDICATE. 191 two majorities added together over the number of the meeting must have voted in favour of each resolution. This kmd 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 which give a distinct idea of the fraction or number of the subject involved in the assertion. T. Spencer Baynes, Essay on the new Analytic of Logical Forms; Edinburgh, 1850. Prof. Bowen's Treatise on Logic or the Laws of Pure Thotight, Cambridge, U. S. 1866 (Trubner 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; and 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 Thought stated in Lesson XIV., and the reader who may have thought them mere useless truisms will perhaps be sur- 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 h^on be the thing, and cojnbustibility 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 any 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 xxiiL] 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 this process of twofold division, which is really the pro- cess of Dichotomy mentioned in p. 107, the development of a term, because it enables us always to develope the utmost 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 next point is to ascertain how many do or may exist. The Law 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 four 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 jnust be struck out. There remains, then, only one alternative 13 194 BOOLE'S SYSTEM OF LOGIC. [less. which is capable of existing if the premises be true, and as 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 be 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-element, 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 m.etal 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- ment is not a metal and is not iron- As another example of this kind of deductive process I will take a case of the Disjunctive Syllogism, in the ne- gative mood, as follows : — A fungus is either plant or animal, . A fungus is not an animal ; Therefore it is a plar.t. XXIII.] BOOLE'S SYSTEM OF LOGIC. 195 Now if we develope all the possible ways in which fungus, plant and animal can be combined together, we obtain for the term fungus — (i) Fungus, plant, animal. (2) Fungus, plant, not-animal. (3) Fungus, not-plant, animal. (4) Fungus, not-plant, not-animal. Of these however the 4th cannot exist because by the prem.ise a fungus must be a plant, or if not a plant an animal. The ist and 3rd again cannot exist because the minor premise informs us that a fungus is not an animal. There remains then only the second combination, Fungus, plant, not-animal, from which we learn the syllogistic conclusion that " a fungus is a plant." The chief excellence of this mode of deduction consists in the fact that it is not restricted to any definite series of forms like the syllogism, but is applicable, without any additional rules, to all kinds of propositions or problems which can be conceived and stated. There may be any number of premises, and they may contain any number of terms ; all we have to do to obtain any possible inference is to develope the term required into all its alternatives and then to examine how many of these agree with the pre- mises. What remain after this examination necessarily form the description of the term. The only inconvenience of the method is that, as the number of terms increases, the number of alternatives to be examined increases very rapidly, and it soon becomes tedious to write them all out. This work may be abbreviated if we substitute single letters to stand for the terms, somewhat as in algebra; thus we may take^, B, C, D, &c., to stand for the affirm- ative terms, and a^ b, c, d, &c., for the corresponding nega- tive ones. Let us take as a first example the premises — 13—2 196 BOOLE'S SYSTEM OF LOGIC. [LESS Organic substance is either vegetable or animal. Vegetable substance consists mainly of carbon, hydrogen, and nitrogen. ^ Animal substance consists mainly of carbon, hydrogen, and nitrogen. It would take a long time to write out all the combi- nations of the four terms occurring in the above ; but if we substitute letters as follows — A = organic substance, ^ = vegetable substance, C= animal substance, Z> = consisting mainly of carbon, hydrogen, and nitrogen, we can readily represent all the combinations which can belong to the term A. (i) ABCD (2) ABCd (3) ABcD (4) ABcd Now the premises amount to the statements, that A must be either B or C, B must be D^ C must be D. The combinations (7) and (8) are inconsistent with the ftrst premise ; the combinations (2) and (4) with the second premise; and (6) is inconsistent with the third premise. There remain only, ABCD ABcD AbCD. Whence we learn at once that "organic substance {A) always consists mainly of carbon, hydrogen and nitrogeii," AbCD (5) AbCd (6) AbcD (7) Abed (8) XXIII.] BOOLE'S SYSTEM OF LOGIC. 197 because it always occurs in connexion ^vith D. The reader may perhaps notice that the term A BCD impHes that or- . ganic substance may be both vegetable {B) and animal (C), If the first premise be interpreted as meaning that this is not possible, of course this combination should also be struck out. It is an unsettled point whether the alter- natives of a disjunctive proposition can coexist or not (see p. 166), but I much prefer the opinion that they can; and as a matter of fact it is quite likely that there exist very simple kinds of living beings, which cannot be "distinctly asserted to be vegetable only or animal only, but partake of the nature of each. As a more complicated problem to shew the powers of this system, let us consider the premises which were treated by Dr Boole in his Laws of Thought^ p. 125, as follows : " Similar figures consist of all whose corresponding angles are equal, and whose corresponding sides are proportional. Triangles whose corresponding angles are equal have their corresponding sides proportional ; and vice versa. Triangles whose corresponding sides are proportional have their corresponding angles equal " Now if we take our symbol letters as follows : A = similar figure, ^ = triangle, C= having corresponding angles equal, Z> = having corresponding sides proportional, the premises will be seen to amount to the statements that A is identical with CD, and that BC is identical with BD; ,in other words, all A's ought to be CUs, CD's ought to 198 BOOLE'S SYSTEM OF LOGIC. [less. be A\ all BCs ought to be BU^ and all Biys ought to be BCs. The possible combinations in which the letters may be united are 16 in number and are shewn in the following table : ABCD aBCD ABCd aBCd ABcD aBcD ABcd aBcd AbCD abCD AbCd abCd AbcD abcD A bed abed Comparing each of these combinations with the premises we see that ABCd, ABeD, A Bed, and others, are to be struck out because every A is also to be CD. The com- binations aBCD and abCD are struck out because every CD should also be A. Again, aBCd is inconsistent with the condition that ever)' BC is also to be BD\ and ii the reader carefully follows out the same process of ex- amination, there will remain only six combinations, which agree with all the premises, thus — ABCD aBed AbCD abCd abcD abed From these combinations we can draw any description we like of the classes of things agreeing with the premises. The class A or similar figures is represented by only two combinations or alternatives ; the negative class a or dissimilar figures, by four combinations, whence we may draw the following conclusion: "Dissimilar figures con sist of all triangles which have not their corresponding angles equal, and sides proportional (aBed), and of al} XXIII.] BOOLE'S SYSTEM OF LOGIC. 199 figures, not being triangles, which have either their angles equal and sides not proportional iabCd), or their cor- responding sides proportional and angles not equal {abcD), or neither their corresponding angles equal nor corresponding sides proportional {abed)." In performing this method of inference it is soon seen to proceed in a very simple mechanical manner, and the only inconvenience is the large number of alternatives or 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 16 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, b, C, c, 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 is 200 BOOLE S SYSTEM OF LOGIC. [LESS. represented by a key in the middle of the series ; the full stop by one to the extreme right, while there are two other keys which serve for the disjunctive conjunction ' 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 334 OBSERVATION [LESS. may almost be said to experiment instead of merely ob- serving when they simultaneously employ instruments as 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 better 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, the pro- * p. 77- XXVII.] AND EXPERIMENT. 235 gress of knowledge has been slow, uncertain and irregu- lar ; while in such as admit of experiment, and in which mankind have ag^reed 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 with 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 or 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 of 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, occupations 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 discrhninate accurately betiveeji what we really do observe and what we only infer from the facts observed. So long as we only record and describe what our senses * See Lockyer's Elemeiitary Lessons in Astronomy, N05. XLVI, XLVII. 236 OBSERVATION [less. have actually witnessed, we cannot commit an error ; but the moment we presume or infer anything we are liable to 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 of 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 far as could be observed. It is by judgment only I can assert he was my brother, and that judgment may possi- bly be wrong. Nothing is more important in observation and experi- xxvil] and EXPERTMENT. 237 ment than to be uniniluenced by any prejudice or theoiy 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 exists 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 year 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 have 238 OBSERVATION, &^c. [LESS. no value. We must have some opinion or 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 little 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 conclusions have been reahzed"*^." The student is strongly recommended to read Sir J. Herschel's Preliminary Discourse on the Study of Natural Philosophy (Lardner's Cabinet Cyclo- pcEdia), especially Part ii. Chaps. 4 to 7, concerning Observation, Experiment, and the Inductive Pro- cesses generally. * Modern Ctdhtre, edited by Yoiimans, p. ■2I2. [Macmillan and Co.] XXVIII.] METHODS OF INDUCTTON. 239 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. I nduction consists in infer - ring from particulars to generals, or detecting a^ gene ral truth_ among its particular occur rences^ But in physical science the truths to be discovered generally relate to the connection of cause and effect, and we usually call them laws of causation or natural laws. By the Cause 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- I 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 trigger, which is only the last apparent cause or occasion 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^- 240 METHODS OF INDUCTION. [less. ture, but the possibility also of the escape of the vapour when it acquires a certain pressure. The freezing of 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 a s soon as t hese circumstances b€€om£ Joiown^ we__ have a law or uniforraity^ofna ture of grea ter^^less gene- rality. In this and the following Lessons I shall often have to 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 indispensable antecedejit is however idefttical with a cause, being that 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 cf)aLv6fX€vov, 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- XXVIII.] METHODS OF INDUCTION. 241 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 pheno7)ienon is 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- of-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 have iridescent colours like those of Mother-of-pearl. Hence it is evident that the form of the surface, which is the only invariable antecedent or condition requisite for the production of the colours, must be their cause. 16 242 METHODS OF INDUCTION. [less. The method of agreement is subject to a serious difficulty, called by Mr Mill the Plurality of Causes, 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 if 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 m 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 circumstances remaining the same, is the cause of the phenomenon in those circumstances. ( Thus we can clearly prove that friction is ofie 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- ceiver of an air-pump, as Hawksbee first did in 1705, and XXVIII.] METHODS OF INDUCTION. 343 then observing that when the receiver is full of air we 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 if 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 air, from which the oxygen has been withdrawn, it soon becomes suffocated. This is essentially the great method of experiment, and its utihty mainly depends upon the precaution of only varying one ciraunstance at a time, all other circum- stances being niai7itained just as they were. This is 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 till 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- 244 METHODS OF INDUCTION, [LESS. 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 experiment. 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 other 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 at a time. Thus we cannot, generally speaking, try the XXVIII.] METHODS OF INDUCTION. 245 qualities of the same substance in the sohd 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 wemight 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, calfed- double refraction, also belongs to many other crystals ; and we might at once prove it to be due to crystaUine 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 cr)^stalline 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 ]\Ir Mill's TMrd 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 in- ductive 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 be variously combined, and let «, b, c, d, e, &c., be effects following from them. If then we can collect the following sets of antecedents and effects — Antecedents. Consequents. ABC abc ADE ade AFG afg AHK ahk we may apply the method of agreement, and little doubt will remain that A^ the sole invariable cintecedent, is the cause 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. The joint method of agreement and difference is similarly represented by — XXVIII.] METHODS OF INDUCTION. 247 Antecedents. ABC Consequents. ode ADE adc AFG AHK "PQ RS pq rs TV tv XV xy Here the presence of A is followed as in the simple method of agreement by a ; and the absence of ^, in circumstances differing from the previous ones, is followed by the ab- sence of a. Hence there is a very high probability that A is the cause of a. But it will easily be seen that A is not the only circumstance in which the two sets of in- stances differ, otherwise to any pair we might apply the simple method of difference. But the presence oi A is a circumstance in which one set invariably, or uniformly, or always, differs, from the other set. This joint method is thus a substitute for the simpler method of difference in cases where that cannot be properly brought into action. Herschel's Discourse^ part II. chap. 6, p. 144. Mill's System of Logic ^ book III. chaps. 8 and 9. LESSON XXIX. METHODS OF QUANTITATIVE INDUCTION. The methods of Induction described in the last Lesson related merely to the happening or not happening of the event, the cause of which was sought. Thus we learnt that friction was one cause of heat by observing that two 248 METHODS OF [less. solid bodies, even two pieces of ice, rubbed together, pro- duced heat, but that when they were not rubbed there was no such production of heat. This, however, is a very elementary sort of experiment ; and in the progress of an investigation we always require to measure the exact quantity of an effect, if it be capable of being more or less, and connecting it with the quantity of the cause. There is in fact a natural course of progress through which we proceed in every such inquiry, as may be stated in the following series of questions. 1. Does the antecedent invariably produce an effect? 2. In what direction is that effect.? ^ 3. How much is that effect in proportion to the cause? 4. Is it uniformly in that proportion? 5. If not, according to what law does it vary? » Take for instance the effect of heat in altering the dimensions of bodies. The first question is, whether the heating of a solid body, say a bar of iron, alters its length ; the simple method of difference enables us to answer that it does. The next inquiry shows that almost all sub- stances are lengthened or increased in dimensions by heat, but that a very few, such as india rubber, and water below 4'o8° Cent., are decreased. We next ascertain the proportion of the change to each degree of temperature, which is called the coefficient of expansion. Thus iron expands 0*0000122 of its own length for every i" Centi- grade between o^ and 100". Still more minute inquiry shows, however, that the expansion is not uniformly proportional to temperature; most metals expand more and more rapidly the hotter they are, but the details of the subject need not be con- sidered here. The fixed stars, again, have often been mentioned in these Lessons, but the reader is probably aware that they are not really fixed. Taking any particular star, the XXIX.] QUANTITATIVE INDUCTION. 249 astronomer has really to answer the several five questions stated below. Firstly. Does the star move ? 2ndly. In what direction does it move.^ 3rdly. How much does it move in a^^ear or a century.^ 4thly. Does it move uniformly.? 5thly. If not, according to what law does the motion vary in direction and rapidity .? Every science and every question in science is first a matter of fact only, then a matter of quantity, and by degrees becomes more and more precisely quantitative. Thirty years ago most of the phenomena of electricity and electro-magnetism were known merely as facts ; now they can be for the most part exactly measured and calculated. As soon as phenomena can thus be measured we can apply a further Method of Induction of a very im- portant character. It is the Method of Difference indeed applied under far more favourable circumstances, where every degree and quantity of a phenomenon gives us a new experiment and proof of connection between cause and effect. It may be called the Method of Concomitant Variations, and is thus stated by Mr Mill, in what he entitles the Fifth Canon of Induction : "Whatever phenomenon varies in any manner when- ever another phenomenon varies in some particular man- ner, is either a cause or an effect of that phenomenon, or is connected with it through some fact of causation." Sir John Herschel's statement of the same method is as follows : — " Increase or diminution of the effect, with the increased or diminished intensity of the cause, in cases which admit of increase and diminution," to which he adds, " Reversal of the effect with that of the cause." The illustrations of this method are infinitely nu- merous. Thus Mr Joule, of Manchester, conclusively proved that friction is a cause of heat by expending exact 250 METHODS OF [less. quantities of force in rubbing one substance against another, and showed that the heat produced was exactly greater or less in proportion as the force was greater or less. We can apply the method to many cases which had previously been treated by the simple method of dif- ference ; thus instead of striking a bell in a complete vacuum we can strike it with a very little air in the receiver of the air-pump, and we then hear a very faint sound, which increases or decreases ever)' time we in- crease or decrease the density of the air. This experi- ment conclusively satisfies any person that air is the cause of the transmission of sound. It is this method which often enables us to detect the material connection which exists between two bodies. For a long time it had been doubtful whether the red flames seen in total eclipses of the sun belonged to the sun or the moon ; but during the last eclipse of the sun it was noticed that the flames moved with the sun, and were gradually covered and uncovered by the moon at successive instants of the eclipse. No one could doubt thenceforth that they belonged to the sun. Whenever, again, phenomena go through Periodic Changes, alternately increasing and decreasing, we should seek for other phenomena which go through changes in exactly the same periods, and there will probably be a connection of cause and effect. It is thus that the tides are proved to be due to the attraction of the moon and sun, because the periods of high and low, spring and neap tides, succeed each other in intervals corresponding to the apparent revolutions of those bodies round the earth. The fact that the moon revolves upon its own axis in exactly the same period that it revolves round the earth, so that for unknown ages past the same side of the moon has always been turned towards the earth, is a most perfect case of concomitant variations, conclusively prov- XXIX.] QUANTITATIVE INDUCTION. 251 ing that the earth's attraction governs the motions of the moon on its own axis. The most extraordinary case of variations howevei consists in the connection which has of late years been ^ shown to exist between the Aurora Boreahs, magnetic storms, and the spots on the sun. It has only in the last 30 or 40 years become known that the magnetic compass needle is subject at intervals to very slight but curious movements ; and that at the same time there are usually natural currents of electricity produced in tele- * graph-wires so as to interfere with the transmission of mes- sages. These disturbances are known as magnetic storms, * and are often observed to occur when a fine display of the Northern or Southern Lights is taking place in some part of the earth. Observations during many years have shown that these storms come to their worst at the end of every eleven years, the maxnnum taking place about the present year 1870, and then diminish in intensity until the next period of eleven years has passed. Close obser- vations of the sun durmg 30 or 40 years have shown that the size and number of the dark spots, which are gigantic J. storms going on upon the sun's surface, increase and decrease exactly at the same periods of time as the mag- netic storms upon the earth's surface. No one can doubt, then, that these strange phenomena are connected to- gether, though the mode of the connection is quite un- known. It is now believed that the planets Jupiter, Saturn, Venus and Mars, are the real causes of the dis- turbances ; for Balfour Stewart and Warren de la Rue ~'have shown that an exact correspondence exists between the motions of these planets and the periods of the sun- spots. This is a most remarkable and extensive case of concomitant variations. We have now to consider a method of Induction which must be employed when several causes act at once 252 METHODS OF [LiiSS. and their effects are all blended together, producing a joint effect of the same kind as the separate effects. If in one experiment friction, combustion, compression and electric action are all going on at once, each of these causes will produce quantities of heat which will be added together, and it will be difficult or impossible to say how much is due to each cause separately. We may call this a case of the homogeneous intermixture of effects, the name indicating that the joint effect is of the same kind as the separate effects. It is distinguished by Mr Mill from cases of the heterogeneous, or, as he says, the hetero- pathic intermixture of effects, where the joint effect is totally different in kind from the separate effects. Thus if we bend a bow too much it breaks instead of bending further ; if we warm ice it soon ceases to rise in tempera- ture and melts ; if we warm water it rises in temperature homogeneously for a time but then suddenly ceases, and an effect of a totally different kind, the production of vapour, or possibly an explosion, follows. Now when the joint effect is of a heterogeneous kind the method of difference is sufficient to ascertam the cause of its occurrence. Whether a bow or a spring will break with a given weight may easily be tried, and whether water will boil at a given temperature in any given state of the barometer may also be easily ascertained. But in the homogeneous intermixture of effects we have a more complicated task. There are several causes each pro- ducing a part of the effect, and we want to know how much is due to each. In this case we must employ a further Inductive Method, called by Mr Mill the Method of Residues, and thus stated in his Fourth Canon : — "Subduct from any phenomenon such part as is known by previous inductions to be the effect of certain antece- dents, and the residue of the phenomenon is the effect of the remaining antecedents." XXIX.] QUANTITATIVE INDUCTION. 253 If we know that the joint effect a, b, c is due to the causes A, B, and C, and can prove that a is due to A and b to Bf it follows that c must be due to C. There cannot be a simpler case of this than ascertaining the exact ^ weight of any commodity in a cart by weighing the cart and load, and then subtracting the tare or weight of the cart alone, which had been previously ascertained. We can thus too ascertain how much of the spring tides is lue to the attraction of the sun, provided we have pre- viously determined the height of the tide due to the moon, which will be about the average height of the tides during the whole lunar month. Then subtracting the moon's ' tide the remainder is the sun's tide. Newton employed this method in a beautiful experi- ment to determine the elasticity of substances by allow- ing balls made of the substances to swing against each other, and then observing how far they rebounded com- jDared with their original fall. But the loss of motion is due partly to imperfect elasticity and partly to the resist- ance of the air. He determined the amount of the latter effect in the simplest manner by allowing the balls to ■* swing without striking each other, and observing how much each vibration was less than the last. In this way he was enabled easily to calculate the quantity that must be subtracted for the resistance of the air. It is this method that we employ in making allowance for the errors or necessary corrections in observations. Few thermometers are quite correct ; but if we put a ther- mometer into melting snow, which has exactly the tem- perature of o" Centigrade, or 32" Fahr., we can observe exactly how much below or above the true point the mercury stands, and this will indicate how much we ^ught to add or subtract from readings of the thermometer to make them correct. The height of the barometer is affected by several causes besides the variation of ihe 254 METHODS OF [less. pressure of the air. It is decreased by the capillary repulsion between the glass tube and the mercury ; it is increased by the expansion of the mercury by heat, if the temperature be above 32^* Fahr. ; and it may be increased or decreased by any error in the length of the measure employed to determine the height. In an accurate obser- vation all these effects are calculated and allowed for in the final result. In chemical analysis this method is constantly em- ployed to determine the proportional weight of substances which combine together. Thus the composition of water is ascertained by taking a known weight of oxide of copper, passing hydrogen over it in a heated, tube, and condensing the water produced in a tube containing sul- phuric acid. If we subtract the original weight of the condensing tube from its final weight we learn how much water is produced ; the quantity of oxygen in it is found by subtracting the final weight of the oxide of copper from its original weight. If we then subtract the weight of the oxygen from that of the water we learn the weight of the hydrogen, which we have combined with the oxygen. When the experiment is very carefully performed, as de- scribed in Ur Roscoe's Lessons in Elementary Chemistry^ (p. 38), we find that SS'Sg parts by weight of oxygen unite with in I parts of hydrogen to form 100 parts of water. In all sciences which allow of measurement of quan- tities this method is employed, but more especially in astronomy, the most exact of all the sciences. Almost all the causes and effects in astronomy have been found out as residual phenomena, that is, by calculating the effects of all known attractions upon a planet or satellite, and then observing how far it is from the place thus predicted. When this was very carefully done in the case of Uranus, it was still found that the planet was sometimes before and sometimes behind its true place. This residual effect XXIX.] QUANTITATIVE INDUCTION. 255 pointed to the existence of some cause of attraction not then known, but which was in consequence soon dis- covered in the shape of the planet Neptune. The motions of several comets have in this way been calculated, but it is observed that they fail to return at the expected time. There is a discrepancy which points to the existence of some obstructive power in the space passed through, the nature of which is not yet understood. Mill's System of Logic, Book III. Chap. 10, Of the Plurality of Causes ; and of the Intermixtjire of Effects. LESSON XXX. EMPIRICAL AND DEDUCTIVE METHODS. We have hitherto treated of Deduction and Induction as if 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 from 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 / 256 EMPIRICAL AND DEDUCTIVE [less. 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 indue- \ tion from certain observations or experiments, and has no other guarantee for its truth, it is said to be an empirical 1 law. As Mr Mill says, "Scientific inquirers give the name of Empirical Laws to uniformities which observation or 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 efinetpla, 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 quahties 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 usetul may be empirical knowledge, it is yet of slight im.portance 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 \\ ill 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 tlian 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 KUmentary C/ie?nistr)'. 17 258 EMPIRICAL AND DEDUCTIVE [LESb. 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 laborator)'^, 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 last. 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 Priucipia, 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 Combined 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. 3. Veriiication. XXX.] METHODS. 259 The tirst process consists in such a rough and simple appeal to experience as may give us a glimpse of the laws which operate, without being sufficient to establish their 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 intermediate use 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 resembhng those of the rainbow. He adopted the theory that white hght 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 de , 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- tance, 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 be varied in another way by carrying the mercurial barome- 17—2 26o 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 lower on the mountain than below, because only the higher parts of the atmosphere pressed upon the mountain. The success of the experiment com- pletely verihed the original hypothesis. The progress of the experimental sciences mainly depends upon the mode in which one experiment thus leads to others, and dis- ; closes new facts, which would in all probability have never * 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 of certain empirical laws concerning the motions of the pla- XXX.] AfETHOnS. 261 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 lime required by the planet to traverse it. Newton was 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 ha 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 16) that Newton, hearing of some more exact data from which he could calculate the distance of the moon. 262 EMPIRICAL AND DEDUCTIVE [less. was able to explain the discrepancy. His theon' 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 w^aters 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 \vhich 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 w^ould be so only by accident. The theory is never proved com- pletely true, and never can be ; but the more accurately the results of 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 of Kepler and the few slight facts which worked on New- XXX.] METHODS. 263 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 other great theories have followed nearly the same course. The undulator}^ 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 of 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 modem 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 observ^a- tion and induction must ever be the ground of all certain knowledge of 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 264 EXPLANATION, TENDENCY, [less. comprehension of their laws. He who frames a theory 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- gence in comparing them with undoubted facts, and the greatest candour in confessing the ninety-nine mistakes he has made in reachins;^ 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 to 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 some part of the world and subsequently hear that a neighbouring volcano has broken out, we say that the earthquake is thus partially explained. The eruption shows that there were great forces operating beneath the earth's syurface, and the earthquake is obviously an effect of such causes. The scratches which maybe plainly seen upon the surface of rocks in certain parts of Wales and Cumberland, are explained by the former existence of gla- ciers in those mountains; the scratches exactly harmonize xxxi.l HYPOTHESIS, THEORY, AND FACT. 265 with the effects of <]:laciers now existing in Switzerland, Greenland, and elsewhere. These may be considered ex- planatlons of fact by fact. A fact may also be explained by a general law of , 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 differeni cases. Thus the cracking of 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 moves 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 the separate effects, and is of the same kind with them. Our last example of the trade-winds really comes under this case, for we find that there is one law or tendency which causes \vinds to blow from the arctic regions towards the equator, and a second tendency which causes then to blow 266 EXPLANATTON, TENDENCY, [less. from east to west. These tendencies are combined to- gether, and cause the trade-winds to blow from the North- East in the northern hemisphere, and from the South-East in the southern hemisphere. The law according to which the 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 water in bringing warmth or carrying it 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, but the upward motion we have impressed upon it disguises the 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 to proceed onward in a straight line. The laws of motion and gravity are such that this balance must always be preserved ; if the moon by any cause were brought nearer to the earth its tendency to fly off would be increased, and would exceed the effect of gravity until it had regained XXXI.] HYPOTHESIS, THEORY, AND FACT. 267 its proper distance. A tendency then is a cause which .may or may not be coiotteracted. In the second case of explanation an efifect is shown to be due, not to the supposed cause directly, but to an Intermediate effect of that cause. Instead o{ A being the < cause of C, it is found that A is the cause of ^, and Bt\\Q cause 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 it 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- har oxide of vanadium, lately discovered by Dr Roscoe. 4lt would be impossible to understand the effect at all un- less we knew that it is probably due to active oxygen or 268 P:XPLANATI0N, TENDENCY, [less. 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 ex- planation is where one law is shown to be a case of a more general law. As was explained in Lesson XX i v. we naturally discover the less general first, and gradually penetrate to the more simple but profound secrets of nature. 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 Eliza- beth ; common frictional electricity was carefuUy 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 of Copenhagen gave in 1819 the first link between them, by pointing out that an electric current may cause move- ments in a compass- needle. Ampere and Faraday worked * Watts' Dictionary oj Chemistry^ Vol. I. p. 601. Kxxi.] HYPOTHESIS, THEORY, AXD 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 v^hich are being gradually rendered more certain and accurate, but the 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 in connection with 270 EXPLANATIOA', TENDENCY, [less the subject we are discussing, and its meaning must be - considered. It is derived from the Greek words Jtto, iuider, and dearis, plcicing, and is therefore exactly synony- mous with the Latin word supposition a placing under, whence our common word supposition. It appears to ' mean in science the imagining of some thing, force or cause, which underlies the phenomena we are examining, and is the agent in their production without admitting of direct observation. In making an 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 dorniitive 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 law XXXI.] HYPOTHESIS, THEORY, AND FACT. 271 that the heat increases as we sink mines in any part of the earth's surface is stronger evidence. The intensely heated condition of the sun and other stars is strongly confirmatory as showing that other bodies do exist in the ' supposed condition of the earth's interior. The cool state of the earth's surface is perfectly consistent with its comparatively small size and the known facts and laws concerning the conduction and radiation of heat. And " the more we learn concerning the way in which the sun's heat is supplied by the fall of meteoric matter, the more it is probable that the earth may have been intensely heated like the sun at some former time, although for an immense period it has been growing slowly colder. A supposition coinciding with so many facts, laws, and other probable hypotheses, almost ceases to be hypothetical, and its high probability causes it to be regarded as a known fact. Provided it is consistent with the laws of thought there is nothing that we may not have to accept as a probable hypothesis, however difficult it may be to conceive and understand. The force of gravity is hypothetical in so far that we know it only by its effects upon the motions of bodies. Its decrease at a distance harmonizes exactly indeed with the way in which light, sound, electric or magnetic attractions, and in fact all influences which " emanate from a point and spread through space, decrease ; hence it is probable that the law of the inverse square is absolutely true. But in other respects gravity is strongly opposed to all our ideas. If sound could travel to the sun as rapidly as in the earth's atmosphere it would re- quire nearly fourteen years to reach its destination ; were the sun and earth united by a solid continuous bar of iron, . a strong pull at one end would not be felt at the other until nearly three years had passed. Light indeed comes from the sun in rather more than eicrht minutes ; but what 272 EXPLANATION, TENDENCY, [less. are we to think of the force of gravity, which appears to reach the sun in an instant — so short that no calculations have yet been able to detect any interval at all ? In fact there seems some reason to suppose that gravity is felt instantaneously throughout the immeasurable regions of space. The undulatory hypothesis of light presents features equally extraordinary and inconceivable. That light does consist of minute but excessively rapid vibrations of. something occupying space, is almost certain, because of tlie great harmony which this hypothesis introduces into the exceedingly various and complicated phenomena of light, and the explanation which it affords of the analog)' of light to sound. It is difficult indeed to imagine that anything can oscillate so rapidly as to strike the retina of the eye 831,479,000,000,000 in one second, as must be the case with violet light according to this hypothesis. But this is nothing to the difficulty of imagining space to be filled with solid ether of extreme rigidity and elasticity, ' but which nevertheless offers no appreciable resistance to the passage through it of ordinary matter, and does not itself possess any gravity*. It has been asserted indeed that the retardation in the return of comets is due to friction against this ether, and Mr Balfour Stewart be- lieves he has produced heat by friction of a metallic disc against the ether in a vacuum. Should these assertions . prove to be true we have new facts in harmony with the theory of light, which would thereby become less hypo- thetical than before. There is no difficulty now in perceiving the part which hypothesis plays in the deductive method of scientific investigation considered in the last lesson. The pre- liminary induction is replaced more or less completely by • See Sir John Herschel's Familiar Lectures, p. 315, &c XXXI.] HYPOTHESIS, THEORY, AND FACT. 273 imagining the existence of agents which we think adequate " to produce the known effects in question. If it is our object to explain the causes of ebbing and flowing wells, which occur in many parts of the world, we cannot J possibly proceed by first exploring the interior of the earth, until we can discover the source of a spring, and observe its circumstances. We are obliged to imagine cavities and channels of various forms, until we conceive ^ such an apparatus as will, in accordance with known laws of hydrostatics, occasion the irregular flowing of water in the way observed. If we can show that cavities of a particular form will produce that effect, and can think of ' no other mode in which it could be produced, the hypo- thesis becomes established as almost a certain fact. It is the same with any great hypothesis like that of the theory of light. We have no means of directly observing and measuring the qualities of the ether which is the medium of light. All we know about this ether at present is derived from the observed phenomena of light. Hence we are driven to invent something and endow it with qualities from which we may calculate, according to some , of the principles of mechanics, the effect to be expected ; and finding that these effects may be made to harmonize with those actually observed, we depend upon this coinci- dence to prove the existence of the ether. The truth of a hypothesis thus altogether depends upon subsequent verification and accordance with observed facts. To invent hypotheses which cannot thus be verified, or to invent them and then neglect the verification, leads to no result at all, or to fallacy. But when the verification is careful and complete no reproach can be brought against the employment of hypothesis. It becomes, perhaps, as certain as any other mode of investigation, and is at any rate indispensable. There was, in fact, little truth or reason in Newton's celebrated protest against the use of 18 274 EXPLANATION, TENDENCY, [less. hypothesis — "Hypotheses non fingo." The fact is that as his theon' of gravitation rested upon the greatest and most successful of hypotheses, so his views of the material nature of light and the causes of its peculiar phenomena involved a false hypothesis, which has long since been completely disproved. The word theory has constantly been used in the last few lessons, and deserves some examination. It comes from the Greek de-copia, meaning contemplation, reflection or speculation; but this gives us little clue to its modern use. In reality the word is highly ambiguous, being sometimes used as equivalent to hypothesis, at other times as equivalent to general law or truth. When people form theories concerning comets, the sun, the cause of earthquakes, &c., they imagine a great many things which may or may not exist ; such theories are really complicated hypotheses, and should be so called. In this sense there are two theories of electricity, one of which supposes the existence of a single fluid w^hich accumulates in some places and has then a tendency to discharge itself towards places where there is a deficiency, just as water always tends to find its level ; the other supposes the existence of two fluids which are commonly united, but when separated tend to rush back into union again. These so-called theories are really hypotheses, be- cause we have no independent evidence of the existence of any fluid, and it is now almost certain that there is no such thing. The atomic theory, again, is really a hypo- thesis suggested by Dalton to explain the remarkable laws which he detected in the proportions of chemical elements which combine together. It is a valid hypothesis in so far as it does really explain the fixedness of the quantities which combine; but it is purely h>'pothetical as regards the shapes, properties or absolute magnitudes of the atoms, because we have no facts which it can har- XXXI.] HYPOTHESIS, THEORY, AND FACT. 275 monise in these respects, and no apparent means of gaming them. In another and more proper sense theory is opposed to practice, just as the general is opposed to the particular. The theory of gravitation means all the more general laws of motion and attraction on which Newton founded his system of the Universe. We may know what those laws are without being able to determine the place of a planet or make any practical use of them ; the particular results must be calculated out by skilful astronomers before navigators, travellers or others can make practical use of them in the determination of the latitude or longitude. When we speak of the mathematical theory of sound, the lunar theory, the theory of the tides, the word is employed without any special reference to hypothesis, and is merely equivalent to general knowledge or science, implying the possession of a complete series of general and accurate , laws, but in no way distinguishing them from accurate knowledge in general. When a word is really used in an equivocal manner like theory, it is not desirable to attempt to give it an accurate definition which would be imagi- nary and artificial. The word fact is used very often in this as in most books, and demands a few remarks. It is derived from factum, the past participle of facere, to do, and would thus mean something which is done, an act, or deed ; but the meaning is evidently greatly extended by analogy. We usually oppose to each other fact and tlieory, but just as theory seems to have two ambiguous meanings, so I believe that fact is ambiguous. Sometimes it means wliat is certain and known by the evidence of the senses, as opposed to what is known only probably by hypothesis and inference; at other times it is contrasted to a general law, and is equivalent to a particular instance or case. A law of great generality may often be as certain and true, 18—2 276 CLASSIFICATION, [less. especially in mathematics, as the particular facts coming under it, so that the contrast must in this case be that between the general and particular. We often use the word too in common life, as merely equivalent to truth; thus we might say, " It is a fact that the primary laws of - thought are the foundation of reasoning." In short, as theory means ambiguously what is hypothetical, general, abstract or uncertain, so fact is equally ambiguous, and means confusedly what is intuitively known, particular, concrete or certain. Mill's Systein of Logic, Book in. Chapters 12, 13 and 14, Of Explanation, and Hypothesis. LESSON XXXII. CLASSIFICATION, AND ABSTRACTION. In an earlier Lesson, upon the subject of the Predicables, we considered the doctrine of classification as it was treated by logicians many centuries ago. The progress of science, however, during the last two centuries has caused great attention to be given to the true principles on which we can arrange a great multitude of diverse objects in order, and we have to consider what are the characteristics of a natural and perfect system of classifi- cation. It maybe said, indeed, that the subject we are treating is coextensive with the science of logic. All thought, all reasoning, so far as it deals with general names or general notions, may be said to consist in classification. Every common or general name is the name of a class, and every name of a class is a common name. "Metal" is the name xxxiL] AND ABSTRACTTON. 277 of one class of substances so often used in our syllogistic examples ; '' Element" of another class, of which the former class is part. Reasoning has been plausibly represented to consist in affirming of the parts of a class whatever may be affirmed of the whole. Every law of nature which we arrive at enables us to classify together a number of facts, and it would hardly be too much to define logic as the theory of classification. Here we deal, however, with that more conscious and distinct arrangement of objects or notions, which is espe- cially employed in the natural sciences, such as Botany, Zoology, Mineralogy and Palaeontology. The derivation of the word class is somewhat curious. In ancient Rome it was the practice to summon the whole people together at certain periods, and this cere- mony was known as a cldsis, from the Greek xXaa-ty, or kXtjo-is, derived from /caXe'o), to call together. Serv-ius Tullius is said to have divided the people into six orders, according to the amount of tribute they could pay, and these orders were not unnaturally called the classes of the people. Hence the name came by degrees to be applied to any organized body of people, such as an army ; thence it was transferred to a fleet of vessels as marshalled in a fixed order, and was finally extended by analogy to any collection of objects carefully arranged. When, however, we now speak of the lower or higher classes of the people It is curious that we ire restoring the word very nearly to its original meaning. Classification may perhaps be best defined as ^Ae ar- rangement of things, or our notions of them, according to their resemblances or identities. Every class should so be constituted as to contain objects exactly resembling each other in certain definite qualities, which are stated in the definition of the class. The more numerous and extensive the resemblances which are thus indicated by 278 CLASSIFICATION, [less. any system of classes, the more perfect and useful must that system be considered. Mr Mill thus describes his view of the meaning — "Classification is a contrivance for the best possible ordering of the ideas of objects in our minds ; for causing the ideas to accompany or succeed one another in such a way as shall give us the greatest command over our know- ledge already acquired, and lead most directly to the acquisition of more. The general problem of classifica- tion, in reference to these purposes, may be stated as follows : To provide that things shall be thought of in such groups, and those groups in such an order, as will best conduce to the remembrance, and to the ascertain- ment of their laws." A collection of objects may generally be classified in an indefinite number of ways. Any quality which is possess- ed by some and not by others may be taken as the first difference, and the groups thus distinguished may be sub- divided in succession by any other qualities taken at will. Thus a library of books might be arranged, (i) according to their size, (2) according to the language in which they are wi itten, (3) according to the alphabetic order of their authors' names, (4) according to their subjects ; and in various other ways. In large libraries and in catalogues such modes of arrangement ate adopted and variously combined. Each different arrangement presents some peculiar convenience, and that mode must be selected which best meets the especial purpose of the library or catalogue. The population of a kingdom, again, may be classified in an almost endless number of ways with regard to different purposes or sciences. The popu- lation of the United Kingdom may be divided according to their place of birth, as English, Welsh, Scotch, Irish, colonial-born, and aliens. The ethnographer would divide them into Anglo-Saxons, Cymri, Gaels, Picts, XXxil] and abstraction. 2^9 Scandinavians, &c. The statist arranges them accord- ing to age ; to condition, as married, unmarried, wdowed, &c. ; to state of body, as able, incapacitated, blind, im- becile. The political economist regards the innumerable trades which are carried on, and classifies them in a complex manner. The lawyer again treats every one as a minor, an adult, a feme sole, a feme couverte, a guardian, ward, trustee, felon, and so on. In the natural world, again, we may make various classifications. Plants may be arranged according to the country from which they are derived; the kind of place or habitat in which they flourish ; the time they live, as annual, biennial, perennial; their size, as herbs, shrubs, trees; their properties, as esculents, drugs, or poisons: all these are distinct from the classifications which the botanist devises to represent the natural affinities or relationships of plants. It is thus evident that in making a classification we have no one fixed method wliich can be ascertained by rule, but that an indefinite number of choices or alternatives are usually open to us. Logic cannot in such cases do much ; and it is really the work of the special sciences to investigate the character of the classification required. All that logic can do is to point out certain general requirements and principles. The first requisite of a good classification is, that it shall be appropriate to the purpose in liand ; that is to say, the points of resemblance selected to form the leading classes shall be those of importance to the practical use of the classification. All those things must be arranged together which require to be treated alike, and those things must be separated which require to be treated separately. Thus a lawyer has no need to classify per- sons according to the counties of England they were born in, because the law is the same independently of counties ; but so far as a Scotchman, a Manx man, or an alien, is 28o CLASSIFICATION, [less. under different laws from the English born man, we shall require to classify them apart. A gardener is quite right in classifying plants as annuals, biennials, perennials; as herbs, shrubs, trees ; as evergreen and deciduous ; or according to the soil, temperature and other circumstances which affect them, because these are points which must guide him in treating some differently from others. Another and, in a scientific point of view, the most important requisite of a good classification, is that it shall enable the greatest possible number of general assertions to be made. This is the criterion, as stated by Dr Whewell, which distinguishes ajiatural from an artificial system of classification, and we must carefully dwell upon its meaning. It will be apparent that a good classification is more than a mere orderly arrangement ; it involves a process of induction which will bring to light all the more general relations which exist between the things classified. An arrangement of books will generally be artificial ; the octavo volumes will not have any common character ex- cept being of an octavo size. An alphabetical arrange- ment of names again is exceedingly appropriate and con- venient to many purposes, but is artificial because it allows of few or no general assertions. We cannot make any general assertion whatever about persons because their names happen to begin with an A or a B, a P or a W. Even those who agree in bearing the name Sm.ith or Taylor or Robinson might be submitted to the inductive method of agreement without the discovery of any common circumstance which could be stated in a general proposition or law. It is true that if we investigated the antecedents of the Evanses and Joneses we should find them nearly all to be Welsh, and the Campbells to be Scotch, and those who bear a very peculiar name would often be found to descend from common ancestors. So far even an alphabetic arrangement embodies something xxxii.] AND ABSTRACTION. 281 that is natural in it, and enables general assertions to be made. Hardly any arrangement can be made, in fact, which will not indicate some vestiges of important rela- tions and resemblances ; but what we want is a system which will reveal all the most important general truths. For this purpose we must select as the ground of union those characters which carry with them most other characters. In Lesson xil. we considered the proprium as a quality which belongs to the whole of a class without forming part of the definition of the class. Now we ought to frame the definition of a class that it may con- tain as few characters as possible, but that as many other characters, properties, or propria, as possible, shall be attributable to the things contained in the class. Every one can see, for instance, that animals form one great group of beings, which have many characters in common, and that plants form another group. Animals have sen- sation, voluntary motion, consume carbonaceous food, and evolve carbonic acid, possess a stomach, and produce fat. Plants are devoid of sensation and voluntary motion, produce carbonaceous tissue, absorb carbonic acid, and evolve oxygen, possess no stomach, and produce starch. At one time it might have been thought that almost any of the characters named was a sufficient mark of the group to which a being belonged. Whatever had a stomach, was an animal ; whatever had not, was a plant ; whatever produced starch or evolved oxygen was called a plant ; whatever absorbed oxygen or produced fat was an animal. To the present day these statements remain generally true, so that we may make assertions in the form of the proposition U, that "all animals are all beings that evolve carbonic acid, and all plants are all beings that absorb carbonic acid." But in reality the exceptions are many, and increasing research makes it continually more apparent that there is no definite line to be drawn 282 CLA SSIFICA TION, [LEsa between animal and vegetable life. This, of course, is not a failure of logical science, but a fact of great sig- nificance concerning the things themselves. In a classification of plants we meet again with most deep and natural distinctions between the great classes called Exogens, Endogens, and Acrogens. The latter have no true sexual flowers and seeds, are formed almost wholly of cellular tissue, and have an epidermis without cuticular pores. The former two classes have much in common ; they have true flowers, woody tissue and cuticular pores, and hence may be united into one wider class, Vasculares. But exogens and endogens are also most strongly distinguished. Exogens have a stem or trunk consisting of distinct bark, pith, and wood in con- centric layers, leaves with reticular veins, seeds with two seed-leaves and a naked radicle ; generally speaking, too, the parts of the flower are some multiple of two or five in number. Endogens, on the contrary, have no distinct bark, pith, and wood, no concentric layers, leaves with parallel veins, seeds with one seed-leaf, and a radicle not naked ; they have, too, the parts of the flower generally a multiple of three in number. These are the very widest classes in what is called the natural system of botanical arrangement ; but similar principles are observed in all its minor classes. The continual efforts of botanists are directed to bringing the great multitudes of plants together in species, genera, orders, classes, and in various intermediate groups, so that the members of each group shall have the greatest number of points of mutual resemblance and the fewest points of resemblance to members of other groups. Thus is best fulfilled the great purpose of classification, which reduces multiplicity to unity, and enables us to infer of aU the other members of a class what we know of any one member, provided we distinguish properly between those XXXII.J AND ABSTRACTION. 283 qualities which are likely or are known to belong to the class, and those which are peculiar to the individual. It is a necessary condition of correct classification, as re- marked by Prof. Huxley, that the definition of a group shall hold exactly true of all members of the group, and not of the members of any other group. To carry out this condition in the natural sciences is, however, very difficult, because kinds of plants or animals are continually dis- covered which stand in an intermediate position between classes which would otherwise be well distinguished. Thus ferns much embarrass the fundamental division of plants, because though they have no true flowers, and in this and other respects agree with other acrogens, yet they have abundance of woody fibre, which would entitle them to rank with vasculares, the larger group of which exogens and endogens are the subdivisions. It may be remarked that the progress of chemistry is rapidly rendering it a science of classification ; and in fact the whole theory of chemical combination now depends on a correct grouping of elements and compounds. Dr Roscoe in his Lessons in Elementary Chemistry enu- merates no less than eleven classes of metals, each class having a number of properties in common. Thus the metals of the alkalies, namely. Potassium, Sodium, Caesium, Rubidium, Lithium, form a remarkably natural class. They are all soft, easily fusible, volatile at high tempera- tures ; they combine with great force with oxygen, decom- pose water at all temperatures, forming oxides which are very soluble in water, and become powerfully caustic and alkaline bodies from which water cannot be expelled by heat. Their carbonates are soluble in water, and each metal forms only one compound with chlorine. The metals of the alkaline earths, Calcium, Strontium, and Barium, also form a very natural class, distinguished by the fact that their carbonates are insoluble in pure 284 CLASSIFICATION, [lkss. water, but soluble in water containing carbonic acid in solution. The gold class contains the rare or valuable metals Gold, Platinum, Palladium, Rhodium, Ruthenium, Iridium, and Osmium, which are not acted on by nitric acid, and can only be dissolved by chlorine or the mixture of acids called aqua rcgia. The oxides can be reduced or deoxidised by simply heating them. Natural classifications give us the deepest resemblances and relations, and may lead us ultimately to a knowledge of the way in which the varieties of things are produced. They are, therefore, essential to a true science, and may almost be said to constitute the framework of the science. Yet it does not follow that they are appropriate for all purposes. When our purpose is merely to recognise the name of a chemical element, a plant or an animal, its character as defined in a natural system would give us little or no assistance. The chemist does not detect potassium by getting it into the state of metal, and trying whether it would decompose water. He merely observes which, among all the com^pounds of potassium, have the best marked and most peculiar characters ; thus a com- pound of potassium, platinum, and chlorine is most distinctive or characteristic of the metal, and is generally used as a means of recognising it ; but a fine violet colour which potash gives to the flame of a lamp was also used as an indication of its presence long before the spectroscope was introduced to analyse such colours. An artificial classification of the elements is thus ne- cessary to the detection of substances, and accordingly in any book on chemical analysis will be found arrange- ments of the elements according to characters of very minor importance, but which are selected on account of the ease and certainty with which they can be observed. In Botany, again, the natural system of classification is far from being well suited for determining the name of a xxxii.J AND ABSTRACTION, 285 plant, because the classes are often defined by the form of minute parts of the seed, the arrangement of the seed- vessel, and other parts which it is usually difficult or sometimes impossible to examine. Accordingly botanists usually arrange their genera and species in the order of the natural system, but contrive a sort of key or artificial arrangement, in which the most simple and apparent characters, often called characteristics, are employed for the discrimination of the plants. The best arrangement of this kind as regards British plants is to be found in Bentham's British Flora. In reality the celebrated Linna^an arrangement of plants was intended by its author to serve in this way. Linnaeus was too profound a philosopher to suppose that the numbers of stamens and pistils usually expressed the real relationships of plants. Many of his classes were really natural classes, but the stamens and pistils were selected as the general guide to the classes and orders, as being very plain and evident marks. Closely connected with the process of classification is that of abstraction. To abstract is to separate the qualities common to all individuals of a group from the peculiarities of each individual. The notion " triangle " is the result of abstraction in so far as we can reason concerning triangles, without any regard to the particular size or shape of any one triangle. All classification im- plies abstraction, for in framing and defining the class I must separate the common qualities from the peculiari- ties. When I abstract, too, I form a general conception, or one which, generally speaking, embraces many objects. If, indeed, the quality abstracted is a peculiar property of the class, or one which belongs to the whole and not to any other objects, I may not increase the extent of the notion, so that Mr Herbert Spencer is, perhaps, right in holding that we can abstract without generalizing. We 286 CLASSIFICATION, &c. [less. often use this word generalization, and the process may be defined as inferring of a whole class what we know only of a part. Whenever we regard the qualities of a thing as not confined to that thing only but as extended to other objects ; when, in fact, we consider a thing only as a member of a class, we are said to generalize. If, after studying the properties of the circle, we proceed to those of 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 of 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 facts. 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 these processes from the general process of natural classification which we have considered. \KxnL]OFA PHILOSOPHICAL LANGUAGE. 287 LESSON XXXIII. 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. I It is easily apparent, indeed, that language serves Y^hree 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. I 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 j notions. Some philosophers have held that it is impossible 10 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 288 REQUISITES OF A [less. reasoning. Though this is no doubt false (see p. 13), it must nevertheless be allowed that unless general ideas were fixed and represented by words, we could never attain to sustained thought such as we at present enjoy. The use of language in the second pui-pose is doubtless indispensable in a practical point of view, and reasoning may almost be considered identical with the correct use of words. When language is used solely to assist reason- ing there is no need that the meaning of each word should be fixed ; we might use names, as the letters x, y, z, a, b, 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 symboHcal 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 with sufficient accuracy. Hence an indispensable subsidiary of induction is the means of describing and recording our observations. Thus only can knowledge be accumulated, so that each observer shall start with the advantage of 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- XXXIII.] PHILOSOPHICAL LANGUAGE, 289 gate the requisite qualities of a philosophical language suitable to the needs of science. As an instnunent of record language must evidently possess 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 us 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 lis 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 names but from things which bear those names. If anyone were ignorant of the meaning of blue he could not be in- formed buw by reference to something that excited in him the sensation of blueness^ and had he been blind from birth he could not acquire any noiion of what blueness was. There are indeed a mmiber 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- 19 290 REQUISITES OF A (less. ence 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, &c. The shapes of leaves are described in Botany by such names as ovate, lanceolate, linear, pinnate, peltate, referring the mind respectively to an ^gg^ a lance, a line, a feather, and a shield. In recording dimensions it is equally im- possible to avoid comparison with the dimensions of other things. A yard or a foot has no meaning unless there be a definite standard yard or foot which fixes its meaning ; and the reader is probably aware that when the physical standard of a length is once completely lost it can never be recovered. The word is nothing unless we somewhere have the thing to which it corresponds. The first requisite of a philosopWcal language evident- ly is that "every general name must have a certain and knowable meaning." It need hardly be mentioned that singular or proper names, the names of distinct objects, must likewise be known; but as such names are merely marks imposed upon the things they do not need the same consideration. General names are a more difficult subject, because, as we have seen in Lesson v., they have a double meaning in denotation or extension, and connota- tion or intension. Of these two meanings the connotation is the one which must be fixed ; the other cannot as a general rule be limited and defined. Had the name planet been restricted to Jupiter, Saturn, Mars, Venus, and Mercury, the planets known before the invention of the telescope, we should have had to find a new name for those subsequently discovered, and should even then commit the fault of calling by different names those things which are closely similar. But if by planet we mean any xxxiii.] PHILOSOPHICAL LANGUAGE. 291 round body revolving round the sun in an orbit of slight ellipticity, it will include all such bodies as may be dis- covered from time to time, of which more than 100 are already known. Similarly locomotive engine is not merely the name of a number of engines now actually existing ; for if so a new name must be needed every week as some new engine is made or an old one destroyed. What is fixed in a general name is its connotation, or the qualities implied in the things bearing the name. We ought therefore as far as possible to define the meaning of every general name we use, not by naming the objects which it denotes, but the qualities, which it connotes. Having however considered the subject of definition in previous Lessons (XII. and Xlll.), we need only inquire here how far it is desirable to employ words which are in current use in preference to newly invented terms. The advantage of an old term is that it possesses force of meaning for all persons, and so far saves the necessity of learning the meaning of a strange technical expression. Every one knows what heat is, and the expression science of heat bears meaning to every person however unlearned. But there is this objection against old terms to be noted, that they are almost always subject to ambiguity; accord- ingly it will be found that the scientific man really uses the word heat differently from other persons. All things are more or less hot in science, whereas in common life we could never say that ice was hot or contained heat. In fact heat means ordinarily the excess of temperature above the ordinary mean, and the notion is purely relative to that of cold. We also apply the word analogously to sensations of taste, as when we say pepper is hot, or even to purely mental phenomena, as in a hot dispute, a hot temper, &c. If to avoid these ambiguities we invent a new term, Caloric^ we may give it any precision of meaning we like, but we raise one more obstacle to the 19—2 292 REQUISITES OF A [less study of science, because there is one more technical term to be learnt This difficulty is especially great in the science of political economy. We there deal with such familiar ideas as wealth, money, value, currency, capital, labour, exchange, but it is the very familiarity of the ideas which occasions the greatest difficulty, because different people attach different meanings to the words, and infinite logo- machy (Greek Xoyor, word ; /iax*?' battle), or disputes arising on merely verbal questions, is the result. Even if a writer carefully defines the meaning in which he uses ihose terms he cannot oblige other persons to bear the definitions in mind. The other alternative of inventing wholly new terms is out of the question, as it would un- doubtedly render a work intolerable to most readers. The only advice that can be given is to introduce a new term where it is likely to be readily accepted and to dis- place an old ambiguous term ; but otherwise to endeavour to remove the ambiguity of the old term by constantly keeping in view a precise definition of the intended meaning. A complete philosophical language will be composed of two distinct kinds of terms, which form respectively the descriptive terminology and the nomenclature of the science. A descriptive terminology, as pointed out by Dr WTiewell, must include all the terms required to describe exactly what has been observed concerning any object or phenomenon, in order that we may possess a permanent record of the obser\'^ation. For every quality, shape, circumstance, degree or quantity there must be an appro- priate name or mode of expression. Thus in recording the discovery of a new inineral we ought to be able to fix in words its exact crystalhne form, its colour, its degree of iiardness, its specific gravity, smell and taste if any, ^ xxxiii.] PHILOSOPHICAL LANGUAGE. 293 and many other qualities which may possess importance. Modern botany arose from the efforts of Linnasus to create a system of terms by which every part and character of a plant can be accurately described. The language of botany, as since improved, presents the most complete instance of a scientific terminology. Geology suffers much, as I apprehend, from the difficulty of find- ing accurate terms ; such names as trap, basalt, gneiss, granite, tuff, greenstone, trachyte, porphyry, lava, &c., are very vague, and there are no precise descriptive terms by which to define and distinguish them. Where a quality does not admit of degree or quantity it only requires a single name ; otherwise we must find some mode of exact measurement and expression. The invention of any in- strument for measuring a quality which has been before unmeasured is always an important step in science, and the construction of the thermometer by Fahrenheit and the pendulum clock by Huyghens were great eras in science. On the other hand, each science requires a nomen- clature or collection of names for the distinct objects or classes of objects treated in it. In mineralogy the names of separate minerals, such as hasmatite, topaz, amphibole, epidote, blende, polybasite, form the nomenclature ; in chemistry we have all the names of the elements, together with a vast apparatus of names for organic and other compounds, such as ethyl, acetyl, cyanogen, napthalin, benzol, &c. In astronomy the names of the planets, satellites, nebulas, constellations or individual stars, form a nomenclature of by no means a perfect or convenient kind ; and geology has similarly a nomenclature neces- sarily of an incomplete character, in the names of the successive formations, silurian, devonian, carboniferous, permian, triassic, eocene, miocene, pHocene, post-plio- cene, &c. It is evident that a nomenclature must possess names 294 REQUISITES OF A [less. of various degrees of generality, including individual objects if they need separate record, infimcc species if such there be, with wider classes, up to the summa g^enera, or widest notions embraced in the science. In astronomy we deal chiefly with the names of individual objects, and there is as yet but little scope for classi- fication. In such natural sciences as botany or zoology there is seldom or never any need of names for indi- viduals, as an indefinite multitude of individuals generally resemble each other very closely in a great number of properties, so as to constitute what has been called a natural kind. Mr Mill uses this term to denote " one of those classes which are distinguished from all others, not by one or a few definite properties, but by an unknown multitude of them ; the combination of properties on which the class is grounded being a mere index to an indefinite number of other distinctive attributes." According to Mr Mill's language he seems to include in a nomenclature only the names of supposed species ; for he says : — "A nomenclature maybe defined, the collec- tion of names of all kinds with which any branch of knowledge is conversant ; or more properly, of all the lowest kinds, or uifimcE species, those which may be sub- divided indeed, but not into kinds, and which generally accord with what in natural history are termed simply species." But the fact is that naturalists have now aban- doned the notion that the species is any definite form ; many species are divided already into subspecies and varieties, or even varieties of varieties; and according to the principles of Darwin's theory the subdivision might go on indefinitely. It is surely most reasonable to regard the natural kingdoms of vegetables and animals as ar- ranged in an indefinite series of classes and subclasses, and all the names attaching to any such classes belong to the nomenclature. NXXiTi.] PHILOSOPHICAL LANGUAGE. 295 Again, Mr Mill does not include in the nomenclature such general names as denote conceptions artificially formed in the course of induction and investigation. Ac- cordingly, besides a terminology suited for describing with precision the individual facts observed, there is a branch of language containing " a name for every com- mon property of any importance or interest, which we detect by comparing those facts : including (as the con- cretes corresponding to those abstract terms) names for the classes which we artificially construct in virtue of those properties, or as many of them, at least, as we have frequent occasion to predicate any thing of." As exam- ples of this class of names he mentions Circle, Limit, Momentum, Civilization, Delegation, Representation. While the nomenclature contains the names of natural classes, this third branch of language would apparently contain the names of artificial ideas or classes. But I feel great difficulty in giving a clear account of Mr Mill's views on this subject, and, as my object in these Lessons does not allow of the discussion of unsettled questions, I must conclude by referring the reader who desires to continue the subject, to the 4th and 6th chap- ters of the 4th Book of Mr Mill's System of Logic ^ which treat of the Requisites of a Philosophical Language, See Dr Whewell's " Aphorisms concerning the Lan- guage of Science," at the end of his Philosophy of the Inductive Sciences. Thomson's Outline of the Laws of Thought, con- tains most interesting remarks on the general nature and use of Language, §§ 17 — 31. QUESTIONS AND EXERCISES. Lesson I. — Introduction. 1. What are the meanings of a Law of Nature, and a Law of Thought ? 2. Explain the distinction between the Fonn of Thought, and the Matter of Thought. 3. In what sense may Logic be called the Science of Sciences ? 4. What is the derivation of the name Logic ? 5. How does a Science differ from an Art, and why is Logic more in the form of a Science than an Art ? 6. Can we say that Logic is a necessary aid in correct reasoning, when persons who have never studied logic reason correctly ? Lesson \\.— Three Parts of Logic. 1. Name the parts of which a syllogism is composed. 2. How far is it correct to say that Logic is concerned with language t 3. What are the three acts of mind considered in Logic? Which of them is more especially the subject of the Science ? 4. Can you state exactly what is meant by a general notion, idea, or conception ? 5. How do the Nominalists, Realists, and Concep- tualists differ in their opinions as to the nature of a general notion ? 5. What is the supposed fourth part of Logic ? QUESTIONS AND EXERCISES. 297 Lesson III. — Terms. 1. Define a name or term. 2. What is a categorematic term ? 3. Explain the distinction between a collective and a general term. 4. Distinguish the collective and distributive use of the word all in the following : — (i) Non omnis moriar {i.e. I shall not all die). (2) " All men find their own in all men's good, And all men join in noble brotherhood." Te7inyson. (3) Non omnia possumus omnes {i.e. we cannot all do all things). 5. Which of the following are abstract terms ? Act, ingratitude, home, hourly, homeliness, intro- duction, individuality, truth, true, trueness, yellow, yellowness, childhood, book, blue, in- tention, reason, rationality, reasonableness. 6. Define a negative term, and mention the mark by which you may recognise it. 7. Distinguish a privative from a negative term, and find some instances of privative terms. 8. Describe the logical characters of the following terms, with the precautions given at p. 26. Metropolis Consciousness Sect Book Lord Chancellor Nation Library Vegetable Kingdom Institution Great Britain Brilliance Light Csesar Weight Observation Void Sensation Tongue Gold Cffisar Air Prime Minister Csesarism Mentor Indigestibility Application Anarchy Manchester Individual Retribution Recollection Volume Solemnitv 2Q8 QUESTIONS AND EXERCISES. / Insignificant Language Understanding Brilliant Adornment Geology Independence Agreement Demeanour Heaviness Obliquity Resemblance Illustration Motionless Departure Section Henry VIII. Nestor Whiteness Formal Logic Alexander Lesson IV. — Ambiguity of Terms. T. Define univocal terms, and suggest some terms which are perfectly univocal. 2. What are the other names by which equivocal' terms are often called ? 3. Distinguish the three kinds of ambiguous terms, and find instances of each. 4. Distinguish the three causes by which the third and most important class of ambiguous terms have been produced. 5. Explain the ambiguity of any of the following terms, referring each to its proper cause, and * tracing out as far as possible the derivation of ^ each separate meaning from the original m^eaning. Bill Minister Subject Letter Table Clerk Object Star Term Order Earth Pole School Wood Law Reason Air BuU Sensation Bed Glass Volume Art Bowl Peer Scale Interest End Sense Feeling Paper Division Ball Kind Bolt Class Lesson V. — Twofold meaning of Terms. I, Distinguish very carefully the meanings in ex- tension and intension of the terms — Quadruped, railway, human being, engine, moun- tain, Member of Parliament. QUESTIONS AND EXERCISES. 299 2. Enumerate the synonyms or other nam.es used instead of extension and intension. 3. According to what law is the quantity of extension connected with the quantity of intension ? Show that the law holds true of the following series of terms — (i) Iron, metal, element, matter, substance. (2) Matter, organized matter, animal, man. (3) Ship, steamship, screw-steamship, iron screw- steamship, British iron screw steamship. (4) Book, printed book, dictionary, Latin dic- tionary. 4. Distinguish between the connotation and deno- tation of a term. 5. Select from the list of terms under Lesson in., Question 8 (p. 297), such terms as are non-con- notative according to Mr Mill's views. 6. Arrange the following terms in series as in ques- tion 3, placing each term of greater extension before a term of less extension. Point out which are the terms of greatest and least inten- sion in each series. Emperor Animal Planet Teacher Dissenter Mammalian Baptist Individual Matter Timber Jupiter Solicitor Person Ruler Quadruped Horse Organized substance Being Heavenly body Lawyer Napoleon III Christian Alexander Episcopalian Lesson VI. — Growth of Language, I. Trace out the generalization or specialization which has taken place in any of the following words: — 3<5o QUESTIONS AND EXERCISES. Kind, genus, class, species, order, rank, Augustus, president, speaker, Utopia, rock. Commons, doctor. 2. Point out metaphors derived from the notions of weight, straightness, rock, wind. 3. Distinguish as accurately as possible the meanings of the following synonyms : — Sickness, malady ; mud, mire ; confutation, refu- tation ; boundary, limit ; mind, intellect ; recol- lection, reminiscence ; procrastination, dilato- riness ; converse, reverse, obverse, inverse. 4- Form lists of all the words derived from any of the following roots :— (i) Tendere, to stretch, as in intention, attention, (2) Ponere, to place, as in position, supposition. (3) Genus, tribe or kind, as in genus, generation. (4) Munus, gift, as in remuneration, common (Latin, Commufiis). 1^5) Modus, shape or fashion, as in mood, moderate. (6) Scribere, to write, as in scribe, inscription, de- scribe. (7) Capere to take, as in deception, incipient. Lesson VII. — Leibnitz on Knoivledge. 1. What are the characters of perfect knowledge ? 2. Describe the character of the knowledge which we have of the following notions or objects : — A syllogism. Electricity. Motion. A triangle. Eternity. The weight of the eanh (5852 trillions of tonsl The colour of the sky. QUESTIONS AND EXERCISES. 301 3, Explain exactly what you mean by intuitive know- ledge. Lesson VIII. — Propositions. 1. Define a proposition, and name the parts of which it is composed. 2. How are propositions classified.-* 3. Name the four kinds of categorical propositions, and their symbols. 4. Under v/hich classes are singular and indefinite propositions placed ? 5. Enumerate the most usual signs of the quantity of a proposition. 6. What are modal propositions according to early logicians, and according to Thomson .'' 7. How far do logicians consider propositions with regard to their truth or falsity ? Lesson IX. — Opposition of Propositiojts. 1. State the quantity of the subject and predicate in each of the propositions A, E, I, 0. 2, Select out of the following propositions, pairs of contrary, contradictory, subaltern, and subcon- trary propositions : — (1) Some elements are known. (2) No elements are known. (3) All elements are known. (4) Not all elements are known. (5) Some elements are not known. (6't All elements are not known. 3. What propositions are true, false, or doubtful, (i) when A is false, (3) when I is false, (2) when E is false, (4) when is false? 4, Prove by means of the contradictory propositions 302 QUESTIONS AND EXERCISES. that subcontrary propositions cannot both be false. 5. Show by means of the subcontrary propositions that contrary propositions may both be false. 6. What quantity would you assign to each of the following propositions "i (i) Knowledge is power. (2) Nebulae are material bodies. (3) Light is the vibration of an ether. (4) Men are more to be trusted than we think. (5) The Chinese are industrious. 7. Why is it desirable in controversy to refute a state- ment by its contradictory and not by its contrary? Lesson X. — Conversion and Iininediate Infer etice. 1. Define inference and conversion. 2. What are converse and convertend propositions } 3. State the rules of valid conversion. 4. Name all the kinds of conversion. 5. By what process do we pass from each of the fol- lowing propositions to the next .? (i) No knowledge is useless. (2) No useless thing is knowledge. (3) All knowledge is not useless. (4) All knowledge is useful. (5) What is not useful is not knowledge. (6) What is useless is not knowledge. (7) No knowledge is useless. 6. Give the logical opposites of the following propo- sition, and the converse of its contradictor}' : — " He cannot become rich who will not labour." 7. Apply negative conception to the proposition " All men are fallible ;" then convert and show that the result is the contrapositive of the original QUESTIONS AND EXERCISES. 303 8. Classify the propositions subjoined into the four following groups: — a. Those which can be inferred from (i). d. Those from which (i) can be inferred. c. Those which do not contradict (i), but cannot be inferred from it. a. Those which contradict (i). (i) All just acts are expedient acts. (2) No expedient acts are unjust. (3) No just acts are inexpedient. (4) All inexpedient acts are unjust. (5) Some unjust acts are inexpedient. (6) No expedient acts are just. (7) Some inexpedient acts are unjust. (8) All expedient acts are just. (9) No inexpedient acts are just. (10) All unjust acts are inexpedient. (11) Some inexpedient acts are just acts. (12) Some expedient acts are just. (13) Some just acts are expedient. (14) Some unjust acts are expedient. Lessons VIII. IX. and X. — Examples of Propositiotis. The reader is desired to ascertain the logical character of each of the following propositions; he is to state of each 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 inferences from it by any proces's which may be applicable. (i) All birds are feathered. (2) No reptiles are feathered. (3) Fixed stars are self-luminous. 304 QUESTIONS AND EXERCISES. (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 kill. (25) I shall not all die. {No7i otmiis moriar.) (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 animals (30) Not many of the metals are brittle. (31) Many are the deserving men who are unfortunate. (32) Amalgams are alloys of mercur>\ (33) One kind of metal at least is Hquid. (34) Talents are often misused. (35) Some parallelograms have "their adjoinmg sides equal. (36) Britain is an island. (37) Romulus and Remus were twins. QUESTIONS AND EXERCISES. 305 (38) A man's a man. (39) Heaven is all mercy. (40) Every one is a good judge of his own interests. (41) All parallelograms have their opposite angles equal. (42) Familiarity breeds contempt. (43) No one is always happy. (44) Many a little makes a mickle. Lesson XI. — Logical Analysis of Sentmces. . How does the grammatical predicate differ from the logical predicate .? , Distinguish between a compound and a complex sentence ; and between coordinate and subordinate propositions. , Enumerate the grammatical expressions which may form (i) A subject. (4) An object. (2) An attribute. (5) An adverbial. (3) A predicate. , Examine the following sentences, ascertain which are compound or complex, and point out the co- ordinate or subordinate propositions, (i) Happy is the man that findeth wisdom, and the man that getteth understanding. (2) Heat, being motion, can be converted into me- chanical force. (3) Ceres, Pallas, Juno, and Vesta are minor planets, or asteroids. (4) Knowledge comes, but wisdom lingers. (5) Fortune often sells to the hasty what she gives to those who wait. (6) Thousands at His bidding speed. And post o'er land and ocean without rest ; They also serve who only stand and wait. 20 3o6 QUESTIONS AND EXERCISES. (7) Pride that dines on vanity, sups on contempt. (8) Nobody can be healthful without exercise, neither natural body, nor politic. (9) Nature is often hidden, sometimes overcome, seldom extinguished. (10} It is impossible to love and be wise. (11) Though gods they were, as men they died. (12) He that is not industrious envieth him that is. (13) Ye are my friends, if ye do whatsoever I command you. — John xv. 14. (14) The wisdom that is from above is first pure, then "" peaceable, gentle, and easy to be intreated, fiill of mercy, and good fruits, without par- tiality, and without hypocrisy. — James iii. 17. 5. Analyse in the form of a scheme or diagram any of the following sentences :— (1) The first aphorism of Bacon's AW?^;// Organum^ on p. 229. (2) Some judgments are merely explanatory of their subject, having for their predicate, a conception- which it fairly implies, to all who know and can define its nature. (3) There be none of the affections which have been noted to fascinate or bewitch, but love and envy : they both have vehement wishes ; they frame themselves readily into imaginations and suggestions ; and they come easily into the eye, especially upon the presence of the objects, which ai'e the points that conduce to fascination. if any such there be. Further examples for analysis must be sought in Dalgleish's Gra^nfnatical Analysis^ with Progressive Ex- ".rciscs. (Oliver and Boyd.) Edinburgh, r 866. Price 9^/ QUESTIONS AND EXERCISES. 307 Lesson XII.— 7>/^r Prcdicadles, etc. 1. Define each of the live predicables. 2. In what sense may we say that the genus is part of the species, and in what sense that the species is pan of the genus ? 3. Select from the terms in the 6th Question of Les- son v., p. 299, such as are genera, species, highest genera, or lowest species of other terms. 4. Explain the expressions sui generis, homogeneous, heterogeneous, summum genus, infima species, tree of Porphyry. 5. Name a property and accident of each of the follow- ing classes : — Circle, Planet, Bird, Member of Parliament, Ruminant Animal. 6. What are the rules of correct logical division. 7. The first name in each of the following series of terms is that of a class which you are to divide and subdivide so as to include all the subjoined minor classes in accordance with the laws of division. (3) Reasoning. Induction (Imperfece) Deduction Mediate luference Induction Hypothetical Syllogism Disjuncdve Syllogism (i) People. (2) Triangle. Laity Equiangular Aliens Isosceles Naturalized Right-angled Subjects Scalene Peers Obtuse-angled Natural-born Subjects Clergy Baronets Commons 8. Divide any of the following classes : — Governments, Sciences, Logical terms. Propositions. 9. Of what does a logical definition consist 1 20 — 2 3o8 QUESTIONS AND EXERCISES. 10. What are the rules of correct definition ? 1 1. What rules do the following definitions break ? (i) Life is the sum of the vital functions. (2) Genus is the material part of the species. (3) Illative conversion is that in which the truth of the converse can be inferred from that of the convertend. « (4) Mineral substances are those which have not been produced by the powers of vegetable or animal life. (5) An equilateral triangle is a triangle whose sides " and angles are respectively equal. (6) An acute-angled triangle is one which has an acute angle. Lesson XIII. — Pascal and Descartes on Method. (i) What is the use of nominal definitions? (2) How must we employ definitions in order to avoid confusion ? (3) How far can we be said to be free to use any name - for any object.? (4) What according to Pascal is the true method of avoiding error ? (5) How do we learn the meanings of words which cannot be defined .? (6) Give instances of words which can be clearly de- fined and of others which cannot ^^7) State the five rules of method given in the Port Royal Logic. (8) Explain Descartes' rules for the attainment of truth. Lesson XIV. — Laws of Tlwught. I, State the three Fundamental Laws of Thought, and apply them to the following notions : — QUESTIONS AND EXERCISES. 309 (i) Matter, organic, inorganic. (2) Undulations, polarized, non-polarized. (3) Figure, rectilinear, curvilinear. 2. Is it wrong to assert that animal cannot both be vertebrate and invertebrate, seeing that some animals are vertebrate and some are not .'* 3. Select from the following such terms as are nega- tives of the others, and such as are opposites : — Light, plenum, gain, heat, decrease, loss, darkness, cold, increase, vacuum. 4. How is Aristotle's dictum applicable to the follow- ing arguments? (i) Silver is a good conductor of electricity ; for such are all the metals. (2) Comets cannot be without weight ; for they are composed of matter, which is not without weight Lesson XV. — Syllogism: the Rules. 1. Distinguish mediate and immediate inference. 2. Define syllogism, and state with what it is synony- mous. 3. What are the six principal and two subordinate rules of the syllogism t 4. In the following syllogisms point out in succession the conclusion, the middle term, the major term, the minor term, the major premise and the minor premise, observing this precise order, (i) All men are fallible ; All kings are men ; Therefore all kings are fallible. (2) Platinum is a metal ; All metals combine with oxygen ; Therefore Platinum combines with oxygen. 3IO QUESTIONS AND EXERCISES. (3) Hottentots are capable of education ; for Hotten- tots are men, and all men are capable of edu- cation. 5. Explain carefully what is meant by non-distribution of the middle term. Lesson XVI. — The Moods and Figures of the Syllogism. r. Name the rules of the syllogism which are broken by any of the following moods, no regard being paid to figure : — AIA, EEI, TEA, lOI, IIA, AEI. 2. Write out all the 64 moods of the syllogism and strike out the 53 invalid ones. 3. Show in what figures the following premises give a vahd conclusion : — AA, AI, E A, OA. 4. In what figures are I E O and E 1 O valid ? 5. To what moods do the following valid syllogisms belong ? Arrange them in correct logical order. (i) Some Y's are Z's. (2) All Z's are Y's. No X's are Y's. No Y'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 premises : and state to what mood the syllogism belongs. (l) Some amphibious animals are mammalian. All mammalian animals are vertebrate. {2) All planets are heavenly bodies. No planets are self-luminous. (3) Mammalian animals are quadrupeds. No birds are quadrupeds. (4) Ruminant animals are not predacious The lion is predacious. QUESTIONS AND EXERCISES. 311 7. Invent examples to show that false premises may give true conclusions. 8. Supply premises to the following conclusions : — (i) 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 XVI I.— r^^ Syllogism; Reduction. r. State and explain the mnemonic lines Barbara, Celarent, &c. 2. 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 ngure : — 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 — EI, I A, OA, IE. 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 Impossibile. Lesson XVIII. — Irregular and Compound Syllogisms. 1. Describe the meaning of each of the terms — En- thymeme, Prosyllogism, Episyllogism, Epichei- rema. Sorites. 2. Make an example of a syllogism in which there are two prosyllogisms. 3. Construct a sorites of four premises and resolve it into distinct syllogisms. 4. What are the rules to which a sorites must conform? 312 QUESTIONS AND EXERCISES. 5. The 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 state it in correct logical form, and then if it be 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. Monarchs are men. Therefore monarchs are mortals. . 2. Personal deformity is an affliction of nature. Disgrace is not an affliction of nature. Therefore personal deformity is not disgrace. 3. Some statesmen are also authors ; for such are Mr Gladstone, Lord Derby, Lord Russell, and Sir G. C. Lewis. 4. This explosion must have been occasioned by gun- powder; for nothing else would have possessed sufficient force. 5. Every man should be moderate ; for excess will cause disease. 6. Blessed are the merciful; for they shall obtain mercy. 7. As almost all the organs of the body have a known use, the spleen must have some use. 8. Cogito, ergo sum. (I think, therefore I exist.) 9. Some speculative men are unworthy of trust ; for they are unwise, and no unwise man can be trusted. 10. 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 the rod. 12. Comets must consist of heavy matter; for other- wise they would not obey the law of gravitation. 13. 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. 16. 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 does not improve the reasoning powers. 20. Every candid man acknowledges merit in a rival ; every learned man does not do so; therefore every learned man is not candid. Lesson XIX. — Conditioftal Arguments. I. What are the kinds of conditional propositions, and by what signs can you recognise them? 314 QUESTIONS AND EXERCISES. 2. What are the rules of the hypothetical syllogism ? 3. To what categorical fallacies do breaches of these rules correspond? 4. Select from the following such as are valid argu- ments, and reduce them to the categorical form ; explain the fallacious reasoning in the others, (i) Rain has fallen if the ground is wet ; but the 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, rain 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 01 categorical form. 6. State the following in the form of a Disjunctive 01 Dilemmatic 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. 1. Classify fallacies. 2. Explain the following expressions : A dicto secundum quid ad dictum simpHciter ; igno- ratio elenchi ; argumentum ad hominem ; argu- mentum ad populum ; petitio principii ; circulus in probando ; non sequitur ; post hoc ergo propter hoc. QUESTIONS AND EXERCISES. 315 3. What is arguing in a circle; and what is a ques- tion-begging epithet? 4. What differences of meaning may be produced m the following sentence bv varying the accent? " 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. (i) He went to London and then to Brighton by the express train. (2) 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 /, 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. 21. Elementary substances alone are metals. Iron is a metal : therefore it is an elementarv substance 3i6 QUESTIONS AND EXERCISES. 22. No Athenians could have been Helots ; for all the Helots were slaves, and all Athenians were free men. 23. Aristotle must have been a man of extraordinary industry; for only such a man could have pro- duced his works. 24. Nothing is better than wisdom ; dry bread is better than nothing ; therefore dry bread is better than wisdom. 25. Pitt was not a great and useful minister; for though he would have been so had he carried out Adam Smith's doctrines of Free Trade, he did not carry out those doctrines. 26. Only the virtuous are truly noble; some who are called noble are not virtuous ; therefore some who are called noble are not truly noble. 27. Ireland is idle and therefore starves ; she starves, and therefore rebels. 28. No designing person ought to be trusted; en- gravers are by profession designers; therefore they ought not to be trusted. 29. 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. 30. Is a stone a body? Yes. Then is not an animal a body.? Yes. Are you an animal ? I think so. Ergo, you are a stone, being a body. — Lucian. 31. If ye were Abraham's children, ye would do the works of Abraham. — John viii. 39. 32. He that is of God heareth God's words : ye there- fore hear them not, because ye are not of God. — John viii. 47. S3. Mahomet was a wise lawgiver ; for he studied the character of his people. QUESTIONS AND EXERCISES. 317 34. Every one desires virtue, because every one desires happiness. 35. His imbecility of character might have been in- ferred from his proneness to favourites ; for all weak princes have this failing. — De Morgan. 36. He is brave who conquers his passions ; he who resists temptation conquers his passions; so .that he who resists temptation is brave. 37. 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 of antiquity. 38. Since all metals are elements, the most rare of all the metals must be the most rare of all the elements, 39. 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- ment. 42. All desires are not blameable ; all desires are liable to excess ; therefore some things liable to excess are not blameable. 43. 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. 46. Few towns in the United Kingdom have more than 3i« QUESTIONS AND EXERCISES. 3CXD,ooo inhabitants ; and as all such towns ought to be represented by three members in Parlia- ment, it is evident that few towns ought to have three representatives. 46. All the works of Shakspeare cannot be read in a day ; therefore the play of Hamlet, being one of the works of Shakspeare, cannot be read in a day. 47. In moral matters we cannot stand still ; therefore he who does not go forward is sure to fall behind. 48. 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. 49. 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 . 53. The holder of some shares m a lottery is sure to gain a prize ; and as I am the holder of some shares in a lotter)' I am sure to gain a prize. 54. 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 65. Over credulous persons ought never to be believed ; and as the Ancient Historians were in many instances over credulous they ought never to be believed. 56. Some mineral compounds are not decomposed by heat ; all organic substances are decomposed by heat; therefore no organic substances are mi- neral compounds. 57. 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. 60. All presuming men are contemptible; this man, therefore, is contemptible ; for he presumes to believe his opinions are correct. 61. 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. 62. 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 has improved in health ; therefore they are of value. 63. He who calls you a man speaks truly ; he who calls you a fool calls you a man ; therefore he who calls you a fool speaks truly. 64. Who is most hungry eats most; who eats least is most hungry ; therefore who eats least eats most. 65. What produces intoxication should be prohibited ; 320 QUESTIONS AND EXERCISES. the use of spirituous liquors causes intoxication ; therefore the use of spirituous Hquors should be prohibited. 66. What we eat grew in the fields ; loaves of bread are what we eat ; therefore loaves of bread grew in the fields. 67. If light consisted of material particles it would possess momentum ; it cannot therefore consist of material particles, for it does not possess momentum. 68. Everything is allowed by law which is morally right ; indulgence in pleasures is allowed by law ; therefore indulgence in pleasures is morally right. 69. All the trees in the park make a thick shade ; this is one of them, therefore this tree makes a thick shade, 70. All visible bodies shine by their own or by re- flected light. The moon does not shine by its own, therefore it shines by reflected light ; but the sun shines by its own light, therefore it cannot shine by reflected light. 71. Honesty deserves reward; and a negro is a fellow- creature ; therefore, an honest negro is a fellow- creature deserving of reward. 72. Nearly all the satellites revolve round their planets from west to east; the moon is a satellite; there- fore it revolves round its planet from west to east. 73. Italy is a Catholic country and abounds in beg- gars ; France is also a Catholic country, and therefore abounds in beggars. 74. Every law is either useless or it occasions hurt to 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 be abolished. QUESTIONS AND EXERCISES. 321 75. The end of a thing is its perfection ; death is the end of life ; therefore death is the perfection of hfe. 76. 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 ? 77. 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 of morality forbids me to receive the sum of money for assisting the person, 78. 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. 79. 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. 80. The stonemasons are benefitted 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. 81. 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 happiness requires the establishment of laws. 21 322 QUESTIONS AND EXERCISES, 82. He that can swim needs not despair to fly ; for to swim is to fly in a grosser fluid, and to fly is to swim in a subtler. 83. 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 Cassar; but they were obliged to go one way or the other ; therefore they were either 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. 86. 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. 88. 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 least safe guide. QUESTIONS AND EXERCISES. 323 89. 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 90. If men were prudent, they would act morally for their own good ; if benevolent, for the good of others. But many men will not act morally, either for their own good, or that of others ; such men, therefore, are not prudent or benevolent. 91. 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. 92. 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. 93. 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 obhged to perform them ; the obli- gation, therefore, to perform promises is essential to the same ends and in tne same degree. 96. 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. 21 — 2 324 QUESTIONS AND EXERCISES. 96. He who believes himself to be always in the right in his opinion, lays claim to infallibility ; you always believe yourself to be in the right in youi 4 opinion ; therefore you lay claim to infallibility. — Whately. -^ 97. 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. 98. 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. 99. 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. 100, The scarlet poppy belongs to the genus Papaver, of the natural order Papaveraceae ; which again is part of the subclass Thalamiflorse, belonging to the great class of Dicotyledons. Hence the scarlet poppy is one of the Dicotyledons. 101. Improbable events happen almost every day ; but what happens almost every day is a very pro- bable event ; therefore improbable events are very probable events. — Whately. Lesson Yy^ii.— Quantification of the Predicate. I- What does the quantification of the predicate mean? QUESTIONS AND EXERCISES. 325 2. Assign to each of the following propositions its proper symbol, and the symbol of its converse : (i) 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. 3. Draw all the contrapositive propositions and imme- diate inferences you can from the following pro- positions: — (i) 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 XXI IL — Boole's System of Logic. 1. Apply this system of inference to prove the syl- logisms on p. 141, in Cesare, and Camestres. 2. Show that if all A's are not ^'s, then no ^'s are A's ; and that if all yi's are all £'s, then all not A's are all not ^'s. 3. Develope the term substance^ as regards the terms vegetable^ a?iiinal, 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." 4. Test the validity of this argument : " Good always triumphs, and vice always fails ; therefore the victor cannot be wrong, nor the vanquished right." 326 QUESTIONS AND EXERCISES. 5. It is known of a certain class of things that — (i) Where the quality A is, B is not. (2) Where B is, and only where B is, C and D are. What can we infer from these premises of the class of things in which A is not pre- sent but C is present ? 6. If all A's are ^'s ; all ^'s are C's ; all C's are Us ; shew that all y^'s are Z>'s, and that all notZ>'s are not yi's. Lesson yiXW .—Method. 1. What is the supposed position of method accord- ing to former logical writers, and what are the rules of method .? 2. Explain the expressions nobis noiiora, and notiora naturcE. 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 a priori and a posterioi'i knowledge. 6. To which kind belongs our knowledge of the fol- lowing facts ? (i) The light of the stars takes a long time \o reach us. (2) Vaccination is a preservative against small-pox. (3) A meteor becomes heated in passing through the air. (4) There must be either some inhabitants or no inhabitants upon Jupiter. Lesson XXV. — Pet^fect hiduction. 1. Define and distinguish Deduction, Induction, and Traduction. QUESTIONS AND EXERCISES. 327 2. Find an instance of reasoning in Traduction. 3. Distinguish Perfect and Imperfect Induction. 4. How does Mr Mill define Induction, and what is his opinion of Perfect Induction? 5. What is the use of Perfect Induction? 6. Construct some instances of the inductive syllo- gism, and show that they may be thrown into a disjunctive form. Lesson XXVI. — Iftduction, Analogy and Example. 1. 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 XXVII. — Observation and Experwte7tt. 1. What is the false method of Science against which Bacon protested? 2. Explain the exact meaning of Bacon's assertions, that man is the Servant and Interpreter of Nature, and that Knowledge is Power. 3. How does experiment differ from obser/atiou? 328 QUESTIONS AND EXERCISES. 4. Classify the sciences according as they employ passive observation, experiment, or both. 5. Name the chief points in which experiment is superior to mere observation. 6. What is the principal precaution needful in obser- vation "i 7. Explain how it is possible to anticipate nature and yet establish all conclusions upon the results of experience. Lessons XXVIII. 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 : (i) 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 of Difference. 6. What can you infer from the following instances? Antecedents. Consequents. ABDE stqp BCD qsr BFG vqu ADE tsp BHK zqw ABFG .pquv ABE i>qt. QUESTIONS AND EXERCISES. 329 7. (i) Friction alters the temperature of the bodies rubbed together. (2) The sun is supposed to move through space. (31 A ray of hght passing into or out of a denser 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? 10. Supposing us to be unacquainted with the causes of the following phenomena, by what methods should we investigate each ? (i) 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 of Great Britain. Lesson YJ^^.— Empirical mid Deductive Methods. 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. 330 QUESTIONS AND EXERCISES. Lesson ^XXl.— Explanation, &c. 1. What do you mean by the explanation of a fact ? 2. State the three ways in which a law of nature may be explained, and suggest some additional in- stances of each case. 3. Define tendency. Do all causes consist only of tendencies, or can you find examples to the con- trary ? 4. Give a definition of hypothesis. How may a valid be distinguished from an invalid hypothesis ? 5. What place does hypothesis hold in the Deductive Method ? 6. Explain the ambiguities of the words theory and fact. Lesson XXXIL — Classification. 1. Define classification, and give the derivation of the word. 2. 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 abs- tracted ? Lesson XXXIII. — Requisites of a Philosophical Language. I. What are the three purposes for which we use language 1 QUESTIONS AND EXERCISES. 331 2. 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, Rosacese, Rose-like, Potassium, Alkaloid, Ruminant Animal, Ruminating, Ruby, Ruby-red. 5. What does Mr Mill mean by the expression Na- tural Kind ? INDEX. AND CONCISE VOCABULARY OF LOGICAL AND PHILOSOPHICAL TERMS, Abacus^ die logical, 199 Abscissio Infiniti (the cutting off of the infinite or negative part;, the process by which we determine the position of an object in a sj'stem of classes, by successive comparison and rejection ofthose classes to which it does not belong. Absolute terms, i.e. non-relative terms, 25 ; sometimes used as name of non-connotative terms, 41 Abstract terms, 20, 43 Abstraction^ 285 Accent^ fallacy of, 174 Accident, fallacy of, 176 ; the pre- dicable, 103 Accidental definition is a defi- nition which assigns the properties of a species, or the accidents of an iiuiividual ; it is more commonly called a Description. Acquired perceptions, 236 Added determinants, inference by, 86 Adequate knowledge, 56 A dicto secundum quid, &c., fallacy of, 176 Adjectives, 21 Adverbials, 93 Affirmative propositions, 63 Algebraic reasoning, 58, 219 Ambigruity of all, 20 ; oisome, 79 of many old terms, 291 ; of terms in Political Economy, 292 Ambiguous middle terra, 130, 171 AmpbibolOgy, fallacy of, 172 Ampliative propositions, 69 Analogue, a thing analogous to some other thing. Analysis, method of, 205 Analogy, the cause of ambiguity, 35) 50; reasoning by, 226 — 8 Analytics, [ja ' kvoXvuKo.,) the title ' given in the second century to por- tions of the Organon, or Logical Treatises of Aristotle ; they were distinguished as the Prior and Pes terior Analytics. Analytic syllogism, a syllogism in which the conclusion is placed first, the premises following as the reasons. See SyntJietic Syllogism; the distinction is unimportant. Antecedent, of a hypothetical pro- position, 160; of an event, »40 Anticipation of nature, 229 Antinomy [6.vri, against; v6[lo<;, law) , the opposition of one law or rule to another. Kant. A posteriori knowledge, 208 A priori knowledge, 208 Arbor Forpbyriana. see Tree of Porphyry. Argument, (Latin, urgiis, from dp-yos, clear, manifest, ) the process of reasoning, the shewing or proving that which is doubtful by that which is known. See In/ere?ice. The mid- dle term of a syllogism is sometimes called specially tJie argument. Argumentum a fortiori, an argument in which we prove that the case in question is more strong or probable than one already con- ceded to be sufficiently so. Argumentum ad hominem, 178 Argumentum ad judicium, an appeal to the common sense ofi mankind. INDEX. 333 Argumentum ad ig^oranti- ani; an argument founded on the ignorance of adversaries. Argtunentum ad populuxn^ 179 Argumentuin ad verecun- diani; an appeal to our respect for some great authority. Arguxnentum ex concesso, a proof derived from a proposition already conceded. Aristotle's Dicta^ 123 Art and Science, distinction of, 7 Artificial Classification, 284 Assertion^ {ad, to; sero, to join,) a statement or proposition, affirma- tive or negative. Association of ideas, [associo, to , accompany; sochis, a companion,) the natural connection existing in the mind between impressions which have previously coexisted, or which are similar. Any idea tends to bring into the mind its associated ideas, in accordance with the two great laws of association, the Law of Conti- guity, and the Law of Similarity. Assumption, [assumo, to take for granted,) any proposition taken as the basis of argument; in a special sense, the minor premise of a cate- gorical syllogism. Attribute^ [attribzw, to give or ascribe to,) a quality or circumstance which may be affirmed (or denied) of a thing; opposed to Substance^ which see. Attribute in grammar, 92 Attributive term, i. e. Connotative term, 41 Axioxn^ defininition of, 125 Baconian method, 255; Philoso- phy, 229 Barbara, Celarent, &c., 145 Begging the Question, 179 Belief, assent to a proposition, ad- mitting of any degree of strength, from the slightest probability to the fullest certainty ; see Probability. Bentliam, George, new .system of Logic, 187 Boole^ George, his system of Logic, 191 ; his Laws of Thought, 197 ; his logical works, 201 Canons of syllogism, 121 — 2; HamQ- ton's supreme Canon, 189 Ccinons of Mill's Inductive Methods, First, 240 ; Second, 242 ; Third, 245 ; Fourth, 252; Fifth, 249 Categorematic words, 18 Categorical propositions, 63 Categories, the sttuuna genera, or most extensive classes into which things can be distributed ; they are ten in number, as follows : OvcrCa, Substance ; Uoaov, Quan- tity ; notov, Quality ; IIpo? n, Re- lation ; TJoLeiv, Action ; Udaxfi-v, Passion, or suffering ; Uov, Place ; IIoTe, Time ; KeiTOai, Position : 'Exetv, Habit or condition. Ever>'thing which can be affirmed must come under one or other of these highest predicates, which were de- scribed in the first treatise of Aris- totle's Organou, called the Catego- ries. Cause, meaning of, 239 Aristotle distinguished four kinds of causes for the existence of a thing — I. The Material Cause, the sub- stance or matter composing it ; 2. The Formal Cause, the pattern, type or design, according to which it is shaped ; 3. The Efficient Cause, the force employed in shaping it ; 4. The Final Cause, the end, motive or purpose of the work. Cliance, ignorance of the causes which are in action ; see Probability. Character^ derivation of the word, 46 Cbaracteristics, 285 CirculUS in definiendo, no, 114 CirculUS in probando, 179 Clearness of knowledge, 54 Cognition, {cog^wsco, to know,) knowledge, or the action of mmd in acquiring knowledge. Colligation of Facts, Dr Whewell's expression for the mental union of facts by some suitable conception, see 2S6 Collective terms, 19 Combined or complete method of investigation, 258 Comparison, com, together ; par, equal or like,) the action of mind by which we judge whether two objects 334 INDEX. of thought are the same or different in certain points. See yudgment. Compatible terms are those which, though distinct, are not contradic- tory, and can therefore be affirmed of the same subject ; as " large " and " heavy ; " " bright-coloured " and "nauseous." Complex conception, inference by, 87 Complex sentence, 91 ; syllogism, 158 Composition of Causes, the principle which is exemplified in all cases in which the joint effect of several causes is identical with the sum of their separate effects, y. S. Mill. See pp. 252, 265 Composition, fallacy of, 173 Compound sentence, go Comprehension of terms, see In- tension. Computation, 127 Concept, that which is conceived, the result of the act of conception ; nearly synonymous with general no- tion, idea, thought. Conception {con, together ; capio, to take). An ambiguous term, mean- ing properly the action of mind in which it takes several things toge- ther, so as to form a general notion ; or again, in which it forms " a men- tal image of the several attributes given in any word or combination of words." Mansel. Conceptualists, 13 Conclusion of syllogism, 15, 127 ; weakened, 140 Concrete terms, 20 Conditional propositions, 62, 160 Confusion of words, ambiguity from, 31 Conjugate words, those which come from the same root or stock, as kfwwn, knowing, knowingly, know- ledge. Connotation of terms, 39, 41; ought to be exactly fixed, 290 Consciousness, the immediate knowledge which the mind has of its sensations and thoughts, and, in general, of all its present operations. Reid. Consectary = Corollary. Consequence, the connection be- tween antecedent and consequent ; but often used ambiguously for the latter. Consequent of a hypothetical pro-* position, 161 Consequent or effect of a cause, 240 Consequent, fallacy of the, 181 Conservation of energy, 263, 269 Consilience of Inductions, thb agreement of inductions derived from different and independent series of facts, as when we learn the mo- tion of the earth by entirely different modes of observation and reasoning. Wliewell. Consistency of propositions, 78 Consistent terms, see compatible terms. Contingent, [contingo, to touch,) that which may or may not happen ; opposed to the necessary and itn' possible. Contingent matter, 80 Continuity, Law of, the principle that nothing can pass from one ex- treme to another without passing through all the intermediate degrees; motion, for instance, cannot be instan- taneously produced or destroyed. Contradiction, Law of, 117, 193 Contradictory terms, 24, 119; propositions, 76 Contraposition, conversion by, 83, 186 Converse fallacy of accident, 176 Conversion of propositions, 82—85; with quantified predicate, 184 Convertend, 82 Coordinate propositions, 90 Copula, 16 Corollary, a proposition which fol- lows immediately from another which has been proved. Correction of observations, 253 Correlative terms, 25 Criterion [kpi.tt,p>.ov, from KpCvm, to judge), any fact, rule, knowledge, or means requisite to the formation of a judgment which shall decide a doubtful question. Cross division, 105 Data, (plural of datum, that which INDEX. ll"^ is given,) the facts or assertions from 1 which an inference is to be drawn. Deduction and Induction, 212 .Reductive or combined method, 258, 272 De factO; what actually or really ^ happens : opposed to de jure, what ought to happen by law or right. DeSnition^ the logical process, 109, 1 12 ; of logic, I ^Seg^ee^ terms expressing, 24; ques- tions of, 120 Demonstration, [demottstro, to point out,) strictly the pointing out the connection between premises and w- conclusion. The term 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 its contradictory, or shewing that the conclusion cannot be supposed untrue. Demonstrative Induction, 220 -De Morgan's logical discoveries and writings, 190 Denotation of terms, 39 'Depth of a notion, see hitension. Derivatives from the root sj>ec, ^ sight, 52 Descartes on Method, 116, 229 Description, see Accidental Deji- r nit ion. 1>escriptive terminology, 292 Destructive dilemma, 168; hypo- thetical syllogism, 162 — 4 Des3monymization of terms, 49 Determination, the distinguishing of parts of a genus by reunion of the genus and difference. Se.& Division. Development of a teim, 193 Diagrams, of sentences, 93 — 7 ; of ^ syllogisms, 129 — 133, 142; of pro- positions, 72 — 75 jDialectiC (StoAexTi»cij reKvri, the art of discourse, from SioAeyeaOai, to discourse). The original name of Logic, perhaps invented by Plato ; also used to denote the Logic of Probable Matter (Aristode), the right use of Reason and Language, the Science of Being ; it is thus a highly ambiguous term. Dichotomy, division by, 107, 193 Dicta de omni et nullo, 123 Difference, the predicable, 99 Differentiation of terms, 49 Dilemma, 167 Disbelief, the state of mind in which we are fully persuaded that some opinion is not true. y. S. Mill. It is equivalent to belief in the contra- dictory opinion or assertion, and is not to be confused vi\\.\^.Do2ibt, which see. Discourse, or reasoning, 15 Discovery, method of, 202 Disjunctive, propositions, 62, 160; syllogism, 166, 194 Distinct knowledge, 55 Distribution of terms, 19, 74—5, 82, 129 Division, logical, 105 ; metaphysical, 108; fallacy of, 174 Doubt, [diibito, 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, 174—5 Empiricism (e/xjreipia, experience), the doctrine of those who consider that all knowledge is derived merely from experience. Empirical Law, 256 Entli3rmeme, 153 Epicbeirema, 155 Episyllog^sm, 155 Equivocal terms, 29 Equivocation, 30 ; causes of, 31 ; fallacy of, 171 Essence, [essentia, from esse, to be,) " the very being of anything, where- by it is what it is." Locke. It is an ancient scholastic word, which can- not be really defined, and should be banished from use. Essential propositions, 68 Euler'S diagrams, 72 — 5, 1-29 — 133, Evidence, [e, and videre, to see,) literally the seeing of anything. The word now means any facts ap- prehended by the mind and made the grounds of knowledge and belief. 33^ INDEX. Examples, use of, 227 Exceptive propositions, 68 Excluded middle, law of, 117, 119, 19? Exclusive propositions,'68 Exhaustive division, 107, 192 Experience, 228 Experimentum CruciS, an ex- periment which decides betweentwo 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, 264; of laws, 265 Explicative propositions, 68 Exposita, a proposition given to be treated by some logical process. Extension and intension, 37, 208 Extensive Syllogfism, 159 Extremes of a proposition, are its ends or terms, the subject and predi- Fact, 275 Fallacy, purely logical, 170; semi- logical, 170 — 175 ; materia], 176^ 182 ; in hypothetical syllogism, 162 ; in dilemma, r68 False cause, 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, 4 Fundamentum divisionis,io5 Fundamentum relationis,the ground of relation, i.e. the series oC events or circumstances which es- tablish a relation between two cor- relative terms. Fundamental principles of syllo- gism, 121 Galenian, or 4th figure of the syl- logism, 145 General notions, 13 ; terms, 18 Generalization, 286 ; of names, ^^^ - Generic property, 102 Genus, gS : generalissimum, 100 Geometrical reasoning, 58, 218; Pascal on, 115 ^ Grammatical predica£&, 88 ; sen- tence, 89 Gravitation, theory of, 260 Hamilton, Sir W., Method of No- tation, 187 Herschel, Sir J., on active and passive observation, 234 Heterogeneous, loi ; intermi}**- ture of eff"ects, 252 Homogeneous, loi ; intermixture of effects, 252, 265 HomolOgue, whatever is kotnolo- gous. Homology, a special term for the,- analogy existing between parts or 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. n Homonymous terms, 30 H3rpotliesis, 269, 270 H3rpotlietical propositions, 62, 160 /■ syllogism, 161 — 2 Idea (i6ea, elSos, image), a term used ambiguously, but generally equiva- lent to thought, notion, concept. Defined by Locke as "Phantasm,^ notion, species, or v/hatever it is which the mind can be employed about in thinking." To have an idea/ of a thing is to think of that thing. Identity, law of, 117— 8 Idol (etSwAoi', e!6os, image), Bacon'^ figurative name for the sources of error ; he enumerated four kinds ; Idols of the Tribe, which affect al,' people ; Idols of the Cave, which are peculiar to an individual ; of the Forum, which arise in the inter-, course of men ; of the Theatre, which" proceed from the systems of philoso- phers. Ignoratio Elenchi, 178 1 Illation [illatum, past participle of in/ero, to bring in). See Inference Illative, that which can be inferrea. Illicit process, of the minor term, 131 ; of the major term, 132, 139 Immediate inference, 85—7 Imperfect figures of the syllo- gism, 145 Imperfect Induction, 213 « Impossible matter, 80 Inconsistent terms imply qualities which cannot coexist in the same thing. See co}7ipatible terms. INDEX. 337 Inconsistent propositions, 76 Indefinite propositions, 65 Indefinite or infinite term, is a ne- gative term which only marks an object by exclusion from a class. Indesignate propositions. See In- definite propositions. Indirect demonstration. See De- nt&nsiration. Indirect inference, method of, 192 Indirect reduction of the syllo- gism, 146, 148 — 9. Individual; what cannot be divided without losing its name and distinc- tive qualities, although generally capable of physical division or par- tition, which see. Induction, 212 Inductive syllogism, 211, 214 Inference, defined, Si ; immediate, 85 — 87 ; mediate, 126 In fiina species, 100 Innate ideas, see a priori truths,iQZ Inseparable accident, 103 Instances, use of, 227 Intension and extension of terms, 37, 99, 208 ; law of relation, 40 Intensive syllogism, 159 Intention, first and second, a dis- tinction between terms thus defined by Hobbes : — " Of the first inten- tion are the names of things, a man, stone, &c. ; of the second are the names of names, and speeches, as universal, particular , geJiits, species, sylU^gisrn, and the like." A term of the second intention expresses the mode in which the mind regards or classifies those of the first intention. Intermediate link, explanation by, 267 Intuitive knowledge, 57 Inversion of subject and predicate, 67 Irrelevant conclusion, fallacy of, 178 Judgment, 12 1 Language, the subject of logic, 10 I Language, requisites of philoso- phical, 290 ; three purposes of, 287 Laws of thought, I, 117 ; of nature. •23Q Leibnitz on Knowledge, 53 Lemma (Aa/u./3a'i'u>, to uke or as- sume), a proposition, a premise granted ; in geometry, a preliminary proposition. Limitation, conversion by, 82, 87 Logic, derivation of name, 6 Logical abacus, slate and machine, 199 Logomachy, 293 Lowest species, 100 IXEaclline; the logical, igg nSajor, term, 128 ; premise, 129 Many questions, fallacy of, 182 IVIaterial fallacies, 170, 176 SSatliematical induction, 220 nSatter of thought, 4 ; of proposi- tions, 80 IVCatter is defined by J. S. Mill as " the external cause to which we ascribe our sensations," or as Per- manent Possibility of Sensation. Mediate inference, 126 nSembra dividentia, the parts into which a class is divided ; the constituent species of a genus. Metaphor, 50 ^Metaphysical division, 108 IffetaphysiCS [ra. ixerd ra ^vaiKo), the works of Aristotle which fol- lowed or were studied after his Physics. First Philosophy, or the so-called science of things in theii own nature ; ontology or the science of Being. Method {fiedoBo^, fxera and bS6Ca., wis- dom), a false argument ; the name often implies that a false argument is consciously used for deception. Borites;, 156 Specialization of names, 45, 48 Species, in logic, 98 ; in natural history, loi Subaltern^ propositions, 77; genera and species, 100 Subaltemans, subaltern- ates, 77 Subcontrary Propositions, 77 Subject of a proposition, 62, 92 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 ; statu from stare, 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 Subsuniption (sitb, under ; sumo, to take or put), a name used by Sir W. Hamilton for the minor premise of a syllogism, because it brings or subsumes a special case under the rule expressed in the major premise or sumption. Subsumption of a law is Mr Mill's expression for the third mode of explaining a law by shewing it to be a particilar case of a more ge- neral law, a68 Sufficient Reason, Principle or Law of, 125 Sui generis, loi Summum genus, 100 Sumption {sumo, to take), Sir W. Hamilton's name for the major pre- mise of a syllogism. Supposition, 270 Syllogism, 10, 127; inductive, 211, 214 Symbolical knowledge, 57 Simcategorematic words, 18 S3mtlieBis, 205 Syntbetic syllogism, a syllo. gism in which the conclusion standj last ; see A nalytic syllogism. System, (o-u'cmjfxo, from oi'vianj^i to put together), a connected body at knowledge. Tacit premise, 153 TautOlogOUB propositions, 69 Tendency, 266 Terminology, 292 Terms, 10, 16, 17 Tertii adjacentis, 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 (^empia, contemplation), knowledge of principles, as opposed to practice ; ambiguously used, see . P- 274 Thesis (0€cri5, from ti^tj/xi, to place], an assertion or proposition which is put forth to be proved or supported by arguments. Thoughts on things, the object of logic, 10 Totum divisum, a class or notion which is divided into parts by 2 difference. Traduction, 212 Transfer of meaning of terras, 33 Tree of Porphyry, 103 Trilemma, an argument resem- bling a dilemma, but in which there arc three alternatives. Truisms, 69 Truth, conformity of our knowledge with the things known. Ultra-total distribution, iqi Uniformity of nature, 217 Universal propositions, ej, 66; affirmative, 71 ; negative, 7? Univocal terms, 29 Variations, method of, 249: pc» riodic, 250 Verb, 88 Weakened conclusion, 140 Worse relation (Hamilton), i^