THE PHILOSOPHY OF
MR. B*RTR*ND R*SS*LL

WITH AN APPENDIX OF LEADING
PASSAGES FROM CERTAIN OTHER WORKS

EDITED BY
PHILIP E. B. JOURDAIN

LONDON: GEORGE ALLEN & UNWIN LTD.
RUSKIN HOUSE 40 MUSEUM STREET, W.C. 1
CHICAGO: THE OPEN COURT PUBLISHING CO.

First published in 1918

(All rights reserved)

EDITOR’S NOTE

When Mr. B*rtr*nd R*ss*ll, following the advice of Mr. W*ll**m J*m*s, again “got into touch with reality” and in July 1911 was torn to pieces by Anti-Suffragists, many of whom were political opponents of Mr. R*ss*ll and held strong views on the Necessity of Protection of Trade and person, a manuscript which was almost ready for the press was fortunately saved from the flames on the occasion when a body of eager champions of the Sacredness of Personal Property burnt the late Mr. R*ss*ll’s house. This manuscript, together with some further fragments found in the late Mr. R*ss*ll’s own interleaved copy of his Prayer-Book of Free Man’s Worship, which was fortunately rescued with a few of the great author’s other belongings, was first given to the world in the Monist for October 1911 and January 1916, and has here been arranged and completed by some other hitherto undecipherable manuscripts. The title of the above-mentioned Prayer-Book, it may perhaps be mentioned, was apparently suggested to Mr. R*ss*ll by that of the Essay on “The Free Man’s Worship” in the Philosophical Essays (London, 1910, pp. 59-70[1]) of Mr. R*ss*ll’s distinguished contemporary, Mr. Bertrand Russell, from whom much of Mr. R*ss*ll’s philosophy was derived. And, indeed, the influence of Mr. Russell extended even beyond philosophical views to arrangement and literary style. The method of arrangement of the present work seems to have been borrowed from Mr. Russell’s Philosophy of Leibniz of 1900; in the selection of subjects dealt with, Mr. R*ss*ll seems to have been guided by Mr. Russell’s Principles of Mathematics of 1903; while Mr. R*ss*ll’s literary style fortunately reminds us more of Mr. Russell’s later clear and charming subtleties than his earlier brilliant and no less subtle obscurities. But, on the other hand, some important points of Mr. Russell’s doctrine, which first appeared in books published after Mr. R*ss*ll’s death, were anticipated in Mr. R*ss*ll’s notes, and these anticipations, so interesting for future historians of philosophy, have been provided by the editor with references to the later works of Mr. Russell. All editorial notes are enclosed in square brackets, to indicate that they were not written by the late Mr. R*ss*ll.

At the present time we have come to take a calm view of the question so much debated seven years ago as to the legitimacy of logical arguments in political discussions. No longer, fortunately, can that intense feeling be roused which then found expression in the famous cry, “Justice—right or wrong,” and which played such a large part in the politics of that time. Thus it will not be out of place in this unimpassioned record of some of the truths and errors in the world to refer briefly to Mr. R*ss*ll’s short and stormy career. Before he was torn to pieces, he had been forbidden to lecture on philosophy or mathematics by some well-intentioned advocates of freedom in speech who thought that the cause of freedom might be endangered by allowing Mr. R*ss*ll to speak freely on points of logic, on the grounds, apparently, that logic is both harmful and unnecessary and might be applied to politics unless strong measures were taken for its suppression. On much the same grounds, his liberty was taken from him by those who remarked that, if necessary, they would die in defence of the sacred principle of liberty; and it was in prison that the greater part of the present work was written. Shortly after his liberation, which, like all actions of public bodies, was brought about by the combined honour and interests of those in authority, occurred his lamentable death to which we have referred above.

Mr. R*ss*ll maintained that the chief use of “implication” in politics is to draw conclusions, which are thought to be true, and which are consequently false, from identical propositions, and we can see these views expressed in Chapters III and XIX of the present work. These chapters were apparently written before the Government, in the spring of 1910, arrived at the famous secret decision that only “certain implications” are permitted in discussion. Naturally the secret decision gave rise to much speculation among logicians as to which kinds of implication were barred, and Mr. R*ss*ll and Mr. Bertrand Russell had many arguments on the subject, which naturally could not be published at the time. However, after Mr. R*ss*ll’s death, successive prosecutions which were made by the Government at last made it quite clear that the opinion held by Mr. R*ss*ll was the correct one. There had been numerous prosecutions of people who, from true but not identical premisses, had deduced true conclusions, so that the possible legitimate forms of “implication” were reduced. Further, the other doubtful cases were cleared up in course of time by the prosecution of (1) members of the Aristotelian Society for deducing true conclusions from false premisses; (2) members of the Mind Association for deducing false conclusions from false premisses; and also by the attempted prosecution of an eminent lady for deducing true conclusions from identities. Fortunately this lady was able to defend herself successfully by pleading that one eminent philosopher believed them to be true—which, of course, means that the conclusions are false. Thus appeared the true nature of legitimate political arguments.

[1] [This Essay is also reprinted in Mr. Russell’s Mysticism and Logic, London and New York, 1918, pp. 46-57.—Ed.]

“Even a joke should have some meaning....”

(The Red Queen, T. L. G., p. 105).

CONTENTS

ABBREVIATIONS

CHAPTER I

THE INDEFINABLES OF LOGIC

The view that the fundamental principles of logic consist solely of the law of identity was held by Leibniz,[2] Drobisch, Uberweg,[3] and Tweedledee. Tweedledee, it may be remembered,[4] remarked that certain identities “are” logic. Now, there is some doubt as to whether he, like Jevons,[5] understood “are” to mean what mathematicians mean by “=,” or, like Schröder[6] and most logicians, to have the same meaning as the relation of subsumption. The first alternative alone would justify our contention; and we may, I think, conclude from an opposition to authority that may have been indicated by Tweedledee’s frequent use of the word “contrariwise” that he did not follow the majority of logicians, but held, like Jevons,[7] the mistaken[8] view that the quantification of the predicate is relevant to symbolic logic.

It may be mentioned, by the way, that it is probable that Humpty-Dumpty’s “is” is the “is” of identity. In fact, it is not unlikely that Humpty-Dumpty was a Hegelian; for, although his ability for clear explanation may seem to militate against this, yet his inability to understand mathematics,[9] together with his synthesis of a cravat and a belt, which usually serve different purposes,[10] and his proclivity towards riddles seem to make out a good case for those who hold that he was in fact a Hegelian. Indeed, riddles are very closely allied to puns, and it was upon a pun, consisting of the confusion of the “is” of predication with the “is” of identity—so that, for example, “Socrates” was identified with “mortal” and more generally the particular with the universal—that Hegel’s system of philosophy was founded.[11] But the question of Humpty-Dumpty’s philosophical opinions must be left for final verification to the historians of philosophy: here I am only concerned with an a priori logical construction of what his views might have been if they formed a consistent whole.[12]

If the principle of identity were indeed the sole principle of logic, the principles of logic could hardly be said to be, as in fact they are, a body of propositions whose consistency it is impossible to prove.[13] This characteristic is important and one of the marks of the greatest possible security. For example, while a great achievement of late years has been to prove the consistency of the principles of arithmetic, a science which is unreservedly accepted except by some empiricists,[14] it can be proved formally that one foundation of arithmetic is shattered.[15] It is true that, quite lately, it has been shown that this conclusion may be avoided, and, by a re-moulding of logic, we can draw instead the paradoxical conclusion that the opinions held by common-sense for so many years are, in part, justified. But it is quite certain that, with the principles of logic, no such proof of consistency, and no such paradoxical result of further investigations is to be feared.

Still, this re-moulding has had the result of bringing logic into a fuller agreement with common-sense than might be expected. There were only two alternatives: if we chose principles in accordance with common-sense, we arrived at conclusions which shocked common-sense; by starting with paradoxical principles, we arrived at ordinary conclusions. Like the White Knight, we have dyed our whiskers an unusual colour and then hidden them.[16]

The quaint name of “Laws of Thought,” which is often applied to the principles of Logic, has given rise to confusion in two ways: in the first place, the “Laws,” unlike other laws, cannot be broken, even in thought; and, in the second place, people think that the “Laws” have something to do with holding for the operations of their minds, just as laws of nature hold for events in the world around us.[17] But that the laws are not psychological laws follows from the facts that a thing may be true even if nobody believes it, and something else may be false if everybody believes it. Such, it may be remarked, is usually the case.

Perhaps the most frequent instance of the assumption that the laws of logic are mental is the treatment of an identity as if its validity were an affair of our permission. Some people suggest to others that they should “let bygones be bygones.” Another important piece of evidence that the truth of propositions has nothing to do with mind is given by the phrase “it is morally certain that such-and-such a proposition is true.” Now, in the first place, morality, curiously enough, seems to be closely associated with mental acts: we have professorships and lectureships of, and examinations in, “mental and moral philosophy.” In the second place, it is plain that a “morally certain” proposition is a highly doubtful one. Thus it is as vain to expect any information about our minds from a study of the “Laws of Thought” as it would be to expect a description of a certain social event from Miss E. E. C. Jones’s book An Introduction to General Logic.

Fortunately, the principles or laws of Logic are not a matter of philosophical discussion. Idealists like Tweedledum and Tweedledee, and even practical idealists like the White Knight, explicitly accept laws like the law of identity and the excluded middle.[18] In fact, throughout all logic and mathematics, the existence of the human or any other mind is totally irrelevant; mental processes are studied by means of logic, but the subject-matter of logic does not presuppose mental processes, and would be equally true if there were no mental processes. It is true that, in that case, we should not know logic; but our knowledge must not be confounded with the truths which we know.[19] An apple is not confused with the eating of it except by savages, idealists, and people who are too hungry to think.

[2] Russell, Ph. L., pp. 17, 19, 207-8.

[3] Schröder, A. d. L., i. p. 4.

[4] See [Appendix A]. This Appendix also illustrates the importance attached to the Principle of Identity by the Professor and Bruno.

[5] S. o. S., pp. 9-15.

[6] A. d. L., i. p. 132.

[7] Cf., besides the reference in the last note but one, E. L. L., pp. 183, 191. “Contrariwise,” it may be remarked, is not a term used in traditional logic.

[8] S. L., 1881, pp. 173-5, 324-5; 1894, pp. 194-6.

[9] Cf. [Appendix C], and William Robertson Smith, “Hegel and the Metaphysics of the Fluxional Calculus,” Trans. Roy. Soc., Edinb., vol. xxv., 1869, pp. 491-511.

[10] See [Appendix B].

[11] [This is a remarkable anticipation of the note on pp. 39-40 of Mr. Russell’s book, published about three years after the death of Mr. R*ss*ll, and entitled Our Knowledge of the External World as a Field for Scientific Method in Philosophy, Chicago and London, 1914.—Ed.]

[12] Cf. Ph. L., pp. v.-vi. 3.

[13] Cf. Pieri, R. M. M., March 1906, p. 199.

[14] As a type of these, Humpty-Dumpty, with his inability to admit anything not empirically given and his lack of comprehension of pure mathematics, may be taken (see [Appendix C]). In his (correct) thesis that definitions are nominal, too, Humpty-Dumpty reminds one of J. S. Mill (see [Appendix D]).

[15] See Frege, Gg., ii. p. 253.

[16] See [Appendix E].

[17] See Frege, Gg., i. p. 15.

[18] See the above references and also [Appendix F].

[19] Cf. B. Russell, H. J., July 1904, p. 812.

CHAPTER II

OBJECTIVE VALIDITY OF THE “LAWS OF THOUGHT”

I once inquired of a maid-servant whether her mistress was at home. She replied, in a doubtful fashion, that she thought that her mistress was in unless she was out. I concluded that the maid was uncertain as to the objective validity of the law of excluded middle, and remarked that to her mistress. But since I used the phrase “laws of thought,” the mistress perhaps supposed that a “law of thought” has something to do with thinking and seemed to imagine that I wished to impute to the maid some moral defect of an unimportant nature. Thus she remonstrated with me in an amused way, since she probably imagined that I meant to find fault with the maid’s capacity for thinking.

CHAPTER III

IDENTITY

In the first chapter we have noticed the opinion that identities are fundamental to all logic. We will now consider some other views of the value of identities.

Identities are frequently used in common life by people who seem to imagine that they can draw important conclusions respecting conduct or matters of fact from them. I have heard of a man who gained the double reputation of being a philosopher and a fatalist by the repeated enunciation of the identity “Whatever will be, will be”; and the Italian equivalent of this makes up an appreciable part of one of Mr. Robert Hichens’ novels. Further, the identity “Life is Life” has not only been often accepted as an explanation for a particular way of living but has even been considered by an authoress who calls herself “Zack” to be an appropriate title for a novel; while “Business is Business” is frequently thought to provide an excuse for dishonesty in trading, for which purpose it is plainly inadequate.

Another example is given by a poem of Mr. Kipling, where he seems to assert that “East is East” and “West is West” imply that “never the twain shall meet.” The conclusion, now, is false; for, since the world is round—as geography books still maintain by arguments which strike every intelligent child as invalid[20]—what is called the “West” does, in fact, merge into the “East.” Even if we are to take the statement metaphorically, it is still untrue, as the Japanese nation has shown.

The law-courts are often rightly blamed for being strenuous opponents of the spread of modern logic: the frequent misuse of and, or, the, and provided that in them is notorious. But the fault seems partly to lie in the uncomplicated nature of the logical problems which are dealt with in them. Thus it is no uncommon thing for somebody to appear there who is unable to establish his own identity, or for A to assert that B was “not himself” when he made a will leaving his money to C.

The chief use of identities is in implication. Since, in logic, we so understand implication[21] that any true proposition implies and is implied by any other true proposition; if one is convinced of the truth of the proposition Q, it is advisable to choose one or more identities P, whose truth is undoubted, and say that P implies Q. Thus, Mr. Austen Chamberlain, according to The Times of March 27, 1909, professed to deduce the conclusion that it is not right that women should have votes from the premisses that “man is man” and “woman is woman.” This method requires that one should have made up one’s mind about the conclusion before discovering the premisses—by what, no doubt, Jevons would call an “inverse or inductive method.” Thus the method is of use only in speeches and in giving good advice.

Mr. Austen Chamberlain afterwards rather destroyed one’s belief in the truth of his premisses by putting limits to the validity of the principle of identity. In the course of the Debate on the Budget of 1909, he maintained, against Mr. Lloyd George, that a joke was a joke except when it was an untruth: Mr. Lloyd George, apparently, being of the plausible opinion that a joke is a joke under all circumstances.

[20] The argument about the hull of a ship disappearing first is not convincing, since it would equally well prove that the surface of the earth was, for example, corrugated on a large scale. If the common-sense of the reader were supposed to dismiss the possibility of water clinging to such corrugations, it might equally be supposed to dismiss the possibility of water clinging to a spherical earth. Traditional geography books, no doubt, gave rise to the opinions held by Lady Blount and the Zetetic Society.

[21] The subject of Implication will be further considered in Chapter XIX.

CHAPTER IV

IDENTITY OF CLASSES

I once heard of a meritorious lady who was extremely conventional; on the slender grounds of carefully acquired habits of preferring the word “woman” to the word “lady” and of going to the post-office without a hat, imagined that she was unconventional and altogether a remarkable person; and who once remarked with great satisfaction that she was a “very queer person,” and that nothing shocked her “except, of course, bad form.”

Thus, she asserted that all the things which shocked her were actions in bad form; and she would undoubtedly agree, though she did not actually state it, that all the things which were done in bad form would shock her. Consequently she asserted that the class of things which shocked her was the class of actions in bad form. Consequently the statement of this lady that some or all of the actions done in bad form shocked her is an identical proposition of the form “nothing shocks me, except, of course, the things which do, in fact, shock me”; and this statement the lady certainly did not intend to make.

This excellent lady, had she but known it, was logically justified in making any statement whatever about her unconventionality. For the class of her unconventional actions was the null class. Thus she might logically have made inconsistent statements about this class of actions. As a matter of fact she did make inconsistent statements, but unfortunately she justified them by stating that, “It is the privilege of woman to be inconsistent.” She was one of those persons who say things like that.

CHAPTER V

ETHICAL APPLICATIONS OF THE LAW OF IDENTITY

It may be remembered that Mr. Podsnap remarked, with sadness tempered by satisfaction, that he regretted to say that “Foreign nations do as they do do.” Besides aiding the comforting expression of moral disapproval, the law of identity has yet another useful purpose in practical ethics: It serves the welcome purpose of providing an excuse for infractions of the moral law. There was once a man who treated his wife badly, was unfaithful to her, was dishonest in business, and was not particular in his use of language; and yet his life on earth was described in the lines:

This man maintained a wife’s a wife,
Men are as they are made,
Business is business, life is life;
And called a spade a spade.

One of the objects of Dr. G. E. Moore’s Principia Ethica[22] was to argue that the word “good” means simply good, and not pleasant or anything else. Appropriately enough, this book bore on its title-page the quotation from the preface to the Sermons, published in 1726, of Bishop Joseph Butler, the author of the Analogy: “Everything is what it is and not another thing.”

But another famous Butler—Samuel Butler, the author of Hudibras—went farther than this, and maintained that identities were the highest attainment of metaphysics itself. At the beginning of the first Canto of Hudibras, in the description of Hudibras himself, Butler wrote:

He knew what’s what, and that’s as high
As metaphysic wit can fly.

I once conducted what I imagined to be an æsthetic investigation for the purpose of discovery, by the continual use of the word “Why?”[23] the grounds upon which certain people choose to put milk into a tea-cup before the tea. I was surprised to discover that it was an ethical, and not an æsthetic problem; for I soon elicited the fact that it was done because it was “right.” A continuance of my patient questioning elicited further evidence of the fundamental character of the principle of identity in ethics; for it was right, I learned, because “right is right.”

It appears that some people unconsciously think that the principle of identity is the foundation, in certain religions, of the reasons which can be alleged for moral conduct, and are surprised when this fact is pointed out to them. The late Sir Leslie Stephen, when travelling by railway, fell into conversation with an officer of the Salvation Army, who tried hard to convert him. Failing in this laudable endeavour, the Salvationist at last remarked: “But if you aren’t saved, you can’t go to heaven!” “That, my friend,” replied Stephen, “is an identical proposition.”

[22] Cambridge, 1903.

[23] Cf. P. E., p. 2.

CHAPTER VI

THE LAW OF CONTRADICTION IN MODERN LOGIC

Considering the important place assigned by philosophers and logicians to the law of contradiction, the remark will naturally be resented by many of the older schools of philosophy, and especially by Kantians, that “in spite of its fame we have found few occasions for its use.”[24] Also in modern times, Benedetto Croce, an opponent of both traditional logic and mathematical logic, began the preface of the book of 1908 on Logic[25] by saying that that volume “is and is not” a certain memoir of his which had been published in 1905.

[24] Pa. Ma., p. 116.

[25] [English translation of the third Italian edition by Douglas Ainslie, under the title: Logic as the Science of the Pure Concept, London 1917.—Ed.]

CHAPTER VII

SYMBOLISM AND MEANING

When people write down any statement such as “The curfew tolls the knell of parting day,”[26] which we will call “C” for shortness, what they mean is not “C” but the meaning of “C”; and not “the meaning of ‘C’” but the meaning of “the meaning of ‘C’.” And so on, ad infinitum. Thus, in writing or in speech, we always fail to state the meaning of any proposition whatever. Sometimes, indeed, we succeed in conveying it; but there is danger in too great a disregard of statement and preoccupation with conveyance of meaning. Thus many mathematicians have been so anxious to convey to us a perfectly distinct and unmetaphysical concept of number that they have stripped away from it everything that they considered unessential (like its logical nature) and have finally delivered it to us as a mere sign. By the labours of Helmholtz, Kronecker, Heine, Stolz, Thomae, Pringsheim, and Schubert, many people were persuaded that, when they said “‘2’ is a number” they were speaking the truth, and hold that “Paris” is a town containing the letter “P.” When Frege pointed out[27] this difficulty he was almost universally denounced in Germany as “spitzfindig.” In fact, Germans seem to have been influenced perhaps by that great contemner of “Spitzfindigkeit,” Kant, to reject the White Knight’s[28] distinctions between words and their denotations and to regard subtlety with disfavour to such a degree that their only mathematical logician except Frege, namely Schröder—the least subtle of mortals, by the way—seems to have been filled with such fear of being thought subtle, that he made his books so prolix that nobody has read them.

Another term which, as we shall see when discussing the paradoxes of logic, mathematicians are accustomed to apply to thought which is more exact than any to which they are accustomed is “scholastic.”[29] By this, I suppose, they mean that the pursuits of certain acute people of the Middle Ages are unimportant in contrast with the great achievements of modern thought, as exemplified by a method of making plausible guesses known as induction,[30] the bicycle, and the gramophone—all of them instruments of doubtful merit.

[26] Cf. Md, N. S., vol. xiv., October 1905, p. 486.

[27] In Z. S., for example.

[28] See [Appendix G].

[29] Cf. Chapter XXXVII below.

[30] Cf. P. M., p. 11, note.

CHAPTER VIII

NOMINALISM

De Morgan[31] said that, “if all mankind had spoken one language, we cannot doubt that there would have been a powerful, perhaps universal, school of philosophers who would have believed in the inherent connexion between names and things; who would have taken the sound man to be the mode of agitating the air which is essentially communicative of the ideas of reason, cookery, bipedality, etc.... ‘The French,’ said the sailor, ‘call a cabbage a shoe; the fools! Why can’t they call it a cabbage, when they must know it is one?’”

One of the chief differences between logicians and men of letters is that the latter mean many different things by one word, whereas the former do not—at least nowadays. Most mathematicians belong to the class of men of letters.

I once had a manservant who told me on a certain occasion that he “never thought a word about it.” I was doubtful whether to class him with such eminent mathematicians as are mentioned in the last chapter, or as a supporter of Max Müller’s theory of the identity of thought and language. However, since the man was very untruthful, and he told me that he meant what he said and said what he meant,[32] the conclusion is probably correct that he really believed that the meanings of his words were not the words themselves. Thus I think it most probable that my manservant had been a mathematician but had escaped by the aid of logic.

As regards his remark that he meant what he said and said what he meant, he plainly wished to pride himself on certain virtues which he did not possess, and was not indifferent to applause, which, however, was never evoked. The virtues, if so they be, and the applause were withheld for other reasons than that the above statements are either nonsensical or false. Suppose that “I say what I mean” expresses a truth. What I say (or write) is always a symbol—words (or marks); and what I mean by the symbol is the meaning of the symbol and not the symbol itself. So the remark cannot express a truth, any more than the name “Wellington” won the battle of Waterloo.

[31] F. L., pp. 246-7.

[32] The Hatter (see [Appendix H]) pointed out that there is a difference between these two assertions. Thus, he clearly showed that he was a nominalist, and philosophically opposed to the March Hare who had recommended Alice to say what she meant.

CHAPTER IX

AMBIGUITY AND SYMBOLIC LOGIC

The universal use of some system of Symbolic Logic would not only enable everybody easily to deal with exceedingly complicated arguments, but would prevent ambiguous arguments. In denying the indispensability of Symbolic Logic in the former state of things, Keynes[33] is probably alone, against the need strongly felt by Alice when speaking to the Duchess,[34] and most modern logicians. It may be noticed that the Duchess is more consistent than Keynes, for Keynes really uses the signs for logical multiplication and addition of Boole and Venn under the different shapes of the words “and” and “or.”

As regards ambiguity, a translation of Hymns Ancient and Modern into, say, Peanesque, would prevent the puzzle of childhood as to whether the “his” in

And Satan trembles when he sees
The weakest saint upon his knees

refers to the saint’s knees or Satan’s.

[33] In his Fm. L.

[34] See [Appendix I].

CHAPTER X

LOGICAL ADDITION AND THE UTILITY OF SYMBOLISM

Frequently ordinary language contains subtle psychological implications which cannot be translated into symbolic logic except at great length. Thus if a man (say Mr. Jones) wishes to speak collectively of himself and his wife, the order of mentioning the terms in the class considered and the names applied to these terms are, logically speaking, irrelevant. And yet more or less definite information is given about Mr. Jones according as he talks to his friends of:

In case (1) one is probably correct in placing Mr. Jones among the clergy or the small professional men who make up the bulk of the middle-class; in case (2) one would conclude that Mr. Jones belonged to the lower middle-class; the form (3) would be used by Mr. Jones if he were a member of the upper, upper middle, or lower class; while form (4) is only used by retired shopkeepers of the lower middle-class, of which a male member usually combines belief in the supremacy of man with belief in the dignity of his wife as well as himself. A further complication is introduced if a wife is referred to as “the wife.”[35] Cases (2) and (3) then each give rise to one more case. Cases (1) and (4) do not, since nobody has hitherto referred to his wife as “the Mrs. Jones”—at least without a qualifying adjective before the “Mrs.”

On the other hand, certain descriptive phrases and certain propositions can be expressed more shortly and more accurately by means of symbolic logic. Let us consider the proposition “No man marries his deceased wife’s sister.” If we assume, as a first approximation, that all marriages are fertile and that all children are legitimate, then, with only four primitive ideas: the relation of parent to child (P) and the three classes of males, females, and dead people, we can define “wife” (a female who has the relation formed by taking the relative product of P and P̌[36] to a male), “sister,” “deceased wife,” and “deceased wife’s sister” in terms of these ideas and of the fundamental notions of logic. Then the proposition “No man marries his deceased wife’s sister” can be expressed unambiguously by about twenty-nine simple signs on paper, whereas, in words, the unasserted statement consists of no less than thirty-four letters. Although, legally speaking, we should have to adopt somewhat different definitions and possibly increase the complications of our proposition, it must be remembered that, on the other hand, we always reduce the number of symbols in any proposition by increasing the number of definitions in the preliminaries to it.

But the utility of symbolic logic should not be estimated by the brevity with which propositions may sometimes be expressed by its means. Logical simplicity, in fact, can very often only be obtained by apparently complicated statements. For example, the logical interpretation of “The father of Charles II was executed” is, “It is not always false of x that x begat Charles II, and that x was executed and that ‘if y begat Charles II, y is identical with x’ is always true of y.”[37] From the point of view of logic, we may say that the apparently simple is most often very complicated, and, even if it is not so, symbolism will make it seem so,[38] and thus draw attention to what might otherwise easily be overlooked.

[35] Cf. Chapter XXIV below.

[36] C. S. Peirce’s notation for the relation “converse of P.”

[37] Russell, Md., N. S., vol. xiv., October 1905, p. 482.

[38] Russell, International Monthly, vol. iv., 1901, pp. 85-6; cf. M., vol. xxii., 1912, p. 153. [This essay is reprinted in Mysticism and Logic, London and New York, 1918, pp. 74-96.—Ed.]

CHAPTER XI

CRITICISM

Those people who think that it is more godlike to seem to turn water into wine than to seem to turn wine into water surprise me. I cannot imagine an intolerable critic. It seems to me that, if A resents B’s criticism in trying to put his (A’s) discovery in the right or wrong place, A acts as if he thought he had some private property in truth. The White Queen seems to have shared the popular misconception as to the nature of criticism.[39]

[39] See [Appendix J].

CHAPTER XII

HISTORICAL CRITICISM

From a problem in Diophantus’s Arithmetic about the price of some wine it would seem that the wine was of poor quality, and Paul Tannery has suggested that the prices mentioned for such a wine are higher than were usual until after the end of the second century. He therefore rejected the view which was formerly held that Diophantus lived in that century.[40]

The same method applied to a problem given by the ancient Hindu algebraist Brahmagupta, who lived in the seventh century after Christ, might result in placing Brahmagupta in prehistoric times. This is the problem:[41] “Two apes lived at the top of a cliff of height h, whose base was distant mh from a neighbouring village. One descended the cliff and walked to the village, the other flew up a height x and then flew in a straight line to the village. The distance traversed by each was the same. Find x.”

[40] W. W. Rouse Ball, A Short Account of the History of Mathematics, 4th edition, London, 1908, p. 109.

[41] Ibid., pp. 148-9.

CHAPTER XIII

IS THE MIND IN THE HEAD?

The contrary opinion has been maintained by idealists and a certain election agent with whom I once had to deal and who remarked that something slipped his mind and then went out of his head altogether. At some period, then, a remembrance was in his head and out of his mind; his mind was not, then, wholly within his head. Also, one is sometimes assured that with certain people “out of sight is out of mind.” What is in their minds is therefore in sight, and cannot therefore be inside their heads.

CHAPTER XIV

THE PRAGMATIST THEORY OF TRUTH

The pragmatist theory that “truth” is a belief which works well sometimes conflicts with common-sense and not with logic. It is commonly supposed that it is always better to be sometimes right than to be never right. But this is by no means true. For example, consider the case of a watch which has stopped; it is exactly right twice every day. A watch, on the other hand, which is always five minutes slow is never exactly right. And yet there can be no question but that a belief in the accuracy of the watch which was never right would, on the whole, produce better results than such a belief in the one which had altogether stopped. The pragmatist would, then, conclude that the watch which was always inaccurate gave truer results than the one which was sometimes accurate. In this conclusion the pragmatist would seem to be correct, and this is an instance of how the false premisses of pragmatism may give rise to true conclusions.

From the text written above the church clock in a certain English village, “Be ye ready, for ye know not the time,” it would be concluded that the clock never stopped for a period as long as twelve hours. For the text is rather a vague symbolical expression of a propositional function which is asserted to be true at all instants. The proposition that a presumably not illiterate and credulous observer of the clock at any definite instant does not know the time implies, then, that the clock is always wrong. Now, if the clock stopped for twelve hours, it would be absolutely right at least once. It must be right twice if it were right at the first instant it stopped or the last instant at which it went;[42] but the second possibility is excluded by hypothesis, and the occurrence of the first possibility—or of the analogous possibility of the stopped clock being right three times in twenty-four hours—does not affect the present question. Hence the clock can never stop for twelve hours.

The pragmatist’s criterion of truth appears to be far more difficult to apply than the Bellman’s,[43] that what he said three times is true, and to give results just as insecure.

[42] Both cases cannot occur; the question is similar to that arising in the discussion of the mortality of Socrates (see Chapter XXII).

[43] See [Appendix K].

CHAPTER XV

ASSERTION

The subject of the present chapter must not be confused with the assertion of ordinary life. Commonly, an unasserted proposition is synonymous with a probably false statement, while an asserted proposition is synonymous with one that is certainly false. But in logic we apply assertion also to true propositions, and, as Lewis Carroll showed in his version of “What the Tortoise said to Achilles,”[44] usually pass over unconsciously an infinite series of implications in so doing. If p and q are propositions, p is true, and p implies q, then, at first sight, one would think that one might assert q. But, from (A) p is true, and (B) p implies q, we must, in order to deduce (Z) q is true, accept the hypothetical: (C) If A and B are true, Z must be true. And then, in order to deduce Z from A, B, and C, we must accept another hypothetical: (D) If A, B, and C are true, Z must be true; and so on ad infinitum. Thus, in deducing Z, we pass over an infinite series of hypotheticals which increase in complexity. Thus we need a new principle to be able to assert q.

Frege was the first logician sharply to distinguish between an asserted proposition, like “A is greater than B,” and one which is merely considered, like “A’s being greater than B,” although an analogous distinction had been made in our common discourse on certain psychological grounds, for long previously. In fact, soon after the invention of speech, the necessity of distinguishing between a considered proposition and an asserted one became evident, on account of the state of things referred to at the beginning of this chapter.

[44] Md. N. S., vol. iv., 1895, pp. 278-80. Cf. Russell, P. M., p. 35.

CHAPTER XVI

THE COMMUTATIVE LAW

Often the meaning of a sentence tacitly implies that the commutative law does not hold. We are all familiar with the passage in which Macaulay pointed out that, by using the commutative law because of exigencies of metre, Robert Montgomery unintentionally made Creation tremble at the Atheist’s nod instead of the Almighty’s. This use of the commutative law by writers of verse renders it doubtful whether, in the hymn-line:

The humble poor believe,

we are to understand a statement about the humble poor, or a doubtful maxim as to the attitude of our minds to statements made by the humble poor.

The non-commutativity of English titles offers difficulties to some novelists and Americans who refer to Mary Lady So-and-So as Lady Mary So-and-So, and vice versa.

CHAPTER XVII

UNIVERSAL AND PARTICULAR PROPOSITIONS

People who are cynical as to the morality of the English are often unpleasantly surprised to learn that “All trespassers will be prosecuted” does not necessarily imply that “some trespassers will be prosecuted.” The view that universal propositions are non-existential is now generally held: Bradley and Venn seem to have been the first to hold this, while older logicians, such as De Morgan,[45] considered universal propositions to be existential, like particular ones.

If the Gnat[46] had been content to affirm his proposition about the means of subsistence of Bread-and-Butter flies, in consequence of their lack of which such flies always die, without pointing out such an insect and thereby proving that the class of them is not null, Alice’s doubt as to the existence of the class in question, even if it were proved to be well founded, would not have affected the validity of the proposition.

This brings us to a great convenience in treating universal propositions as non-existential: we can maintain that all x’s are y’s at the same time as that no x’s are y’s, if only x is the null-class. Thus, when Mr. MacColl[47] objected to other symbolic logicians that their premisses imply that all Centaurs are flower-pots, they could reply that their premisses also imply the more usual view that Centaurs are not flower-pots.

[45] Cf., e.g., F. L., p. 4.

[46] See [Appendix L].

[47] Cf., e.g., Md., N. S., vol. xiv., July, 1905, pp. 399-400.

CHAPTER XVIII

DENIAL OF GENERALITY AND GENERALITY OF DENIAL

The conclusion of a certain song[48] about a young man who poisoned his sweetheart with sheep’s-head broth, and was frightened to death by a voice exclaiming:

“Where’s that young maid
What you did poison with my head?”

at his bedside, gives rise to difficulties which are readily solved by a symbolism that brings into relief the principle that the denial of a universal and non-existential proposition is a particular and existential one. The conclusion of the song is:

Now all young men, both high and low,
Take warning by this dismal go!
For if he’d never done nobody no wrong,
He might have been here to have heard this song.

It is an obvious error, say Whitehead and Russell,[49] though one easy to commit, to assume that the cases: (1) all the propositions of a certain class are true; and (2) no proposition of the class is true; are each other’s contradictories. However, in the modification[50] of Frege’s symbolism which was used by Russell

while the contradictory of (1) is:

not (x). x.

The last line but one of the above verse may, then, be written:

(t). not (x). not not ϕ(x, t),

where “ϕ(x, t)” denotes the unasserted propositional function “the doing wrong to the person x at the instant t.” By means of the principle of double negation we can at once simplify the above expression into:

(t). not (x). ϕ(x, t);

which can be thus read: “If at every instant of his life there was at least one person x to whom he did no wrong (at that instant).” It is difficult to imagine any one so sunk in iniquity that he would not satisfy this hypothesis. We are forced, then, unless our imagination for evil is to be distrusted, to conclude that any one might have been there to have heard that song. Now this conclusion is plainly false, possibly on physical grounds, and certainly on æsthetic grounds. It may be added, by the way, that it is quite possible that De Morgan was mistaken in his interpretation of the above proposition owing to the fact that he was unacquainted with Frege’s work. In fact, if he had not noticed the fact that any two of the “not’s” cannot be cancelled against one another he would have concluded that the interpretation was: “If he had never done any wrong to anybody.”

According as the symbol for “not” comes before the (x) or between the (x) and the ϕ, we have an expression of what Frege called respectively the denial of generality, and the generality of denial. The denial of the generality of a denial is the form of all existential propositions, while the assertion of or denial of generality is the general form of all non-existential or universal propositions.

[48] To which De Morgan drew attention in a letter; see (Mrs.) S. E. De Morgan, Memoir of Augustus De Morgan, London, 1882, p. 324.

[49] Pa. Ma., p. 16.

[50] However, here, for the printer’s convenience, we depart from Mr. Russell’s usage so far as to write “not” for a curly minus sign.

CHAPTER XIX

IMPLICATION

A good illustration of the fact that what is called “implication” in logic is such that a false proposition implies any other proposition, true or false, is given by Lewis Carroll’s puzzle of the three barbers.[51]

Allen, Brown, and Carr keep a barber’s shop together; so that one of them must be in during working hours. Allen has lately had an illness of such a nature that, if Allen is out, Brown must be accompanying him. Further, if Carr is out, then, if Allen is out, Brown must be in for obvious business reasons. The problem is, may Carr ever go out?

Putting p for “Carr is out,” q for “Allen is out” and r for “Brown is out,” we have:

Lewis Carroll supposed that “q implies r” and “q implies not-r” are inconsistent, and hence that p must be false. But these propositions are not inconsistent, and are, in fact, both true if q is false. The contradictory of “q implies r” is “q does not imply r” which is not a consequence of “q implies not-r.” It seems to be true theoretically that, if Mr. X is a Christian, he is not an Atheist, but we cannot conclude from this alone that his being a Christian does not imply that he is an Atheist, unless we assume that the class of Christians is not null. Thus, if p is true, q is false; or, if Carr is out, Allen is in. The odd part of this conclusion is that it is the one which common-sense would have drawn in that particular case.

A distinguished philosopher (M) once thought that the logical use of the word “implication”—any false proposition being said to “imply” any proposition true or false—is absurd, on the grounds that it is ridiculous to suppose that the proposition “2 and 2 make 5” implies the proposition “M is the Pope.” This is a most unfortunate instance, because it so happens that the false proposition that 2 and 2 make 5 can rigorously be proved to imply that M, or anybody else other than the Pope, is the Pope. For if 2 and 2 make 5, since they also make 4, we would conclude that 5 is equal to 4. Consequently, subtracting 3 from both sides, we conclude that 2 would be equal to 1. But if this were true, since M and the Pope are two, they would be one, and obviously then M would be the Pope.

The principle that the false implies the true has very important applications in political arguments. In fact, it is hard to find a single principle of politics of which false propositions are not the main support.

If p and q are two propositions, and p implies q; then, if, and only if, q and p are both false or both true, we also have: q implies p. The most important applications of this invertibility were made by the late Samuel Butler[52] and Mr. G. B. Shaw. A political application may be made as follows: In a country where only those with middling-sized incomes are taxed, conservative and bourgeois politicians would still maintain that the proposition “the rich are taxed” implies the proposition “the poor are taxed,” and this implication, which is true because both premiss and conclusion are false, would be quite unnecessarily supported by many false practical arguments. It is equally true that “the poor are taxed” implies that “the rich are taxed.” And this can be proved, in certain cases, on other grounds. For the taxation of the poor would imply, ultimately, that the poor could not afford to pay a little more for the necessities of life than, in strict justice, they ought; and this would mean the cessation of one of the chief means of production of individual wealth.

We also see why a valuable means for the discovery of truth is given by the inversion of platitudinous implications. It may happen that another platitude is the result of inversion; but it is the fate of any true remark, especially if it is easy to remember by reason of a paradoxical form, to become a platitude in course of time. There are rare cases of a platitude remaining unrepeated for so long that, by a converse process, it has become paradoxical. Such, for example, is Plato’s remark that a lie is less important than an error in thought.

Of late years, a method of disguising platitudes as paradoxes has been too extensively used by Mr. G. K. Chesterton. The method is as follows. Take any proposition p which holds of an entity a; choose p so that it seems plausible that p also holds of at least two other entities b and c; call a, b, c, and any others for which p holds or seems to hold, the class A, and p the “A-ness” or “A-ity” of A; let d be an entity for which p does not hold; and put d among the A’s when you think that nobody is looking. Then state your paradox: “Some A’s do not have A-ness.” By further manipulation you can get the proposition “No A’s have A-ness.” But it is possible to make a very successful coup if A is the null-class, which has the advantage that manipulation is unnecessary. Thus, Mr. Chesterton, in his Orthodoxy put A for the class of doubters who doubt the possibility of logic, and proved that such agnostics refuted themselves—a conclusion which seems to have pleased many clergymen.

In this way, Mr. Chesterton has been enabled readily to write many books and to maintain, on almost every page, such theses as that simplicity is not simple, heterodoxy is not heterodox, poets are not poetical, and so on; thereby building up the gigantic platitude that Mr. Chesterton is Chestertonian.

In the chapter on Identity we have illustrated the use of a case of the principle that any proposition implies any true proposition. This important principle may be called the principle of the irrelevant premiss;[53] and is of great service in oratory, because it does not matter what the premiss is, true or false. There is a principle of the irrelevant conclusion, but, except in law-courts, interruptions of meetings, and family life, this is seldom used, partly because of the limitation involved in the logical impossibility for the conclusion to be false if the premiss be true, but chiefly because the conclusion is more important than the premiss, being usually a matter of prejudice.

Certain modern logicians, such as Frege, have found it necessary so to extend the meaning of implication of q by p that it holds when p is not a proposition at all. Hitherto, politicians, finding that either identical or false propositions are sufficient for their needs, have made no use of this principle; but it is obvious that their stock of arguments would be vastly increased thereby.

Logical implication is often an enemy of dignity and eloquence. De Morgan[54] relates “a tradition of a Cambridge professor who was once asked in a mathematical discussion, ‘I suppose you will admit that the whole is greater than its part?’ and who answered, ‘Not I, until I see what use you are going to make of it.’” And the care displayed by cautious mathematicians like Poincaré, Schoenflies, Borel, Hobson, and Baire in abstaining from pushing their arguments to their logical conclusions is probably founded on the unconscious—but no less well-grounded—fear of appearing ridiculous if they dealt with such extreme cases as “the series of all ordinal numbers.”[55] They are, probably, as unconscious of implication as Gibbon, when he remarked that he always had a copy of Horace in his pocket, and often in his hand, was of the necessary implication of these propositions that his hand was sometimes in his pocket.

[51] Md., N. S., vol. iii., 1894, pp. 436-8. Cf. the discussions by W. E. Johnson (ibid., p. 583) and Russell (P. M., p. 18, note, and Md., N. S., vol. xiv., 1905, pp. 400-1).

[52] The inhabitants of “Erewhon” punished invalids more severely than criminals. In modern times, one frequently hears the statement that crime is a disease; and if so, it is surely false that criminals ought to be punished.

[53] Irrelevant in a popular sense; one would not say, speaking loosely, that the fact that Brutus killed Cæsar implies that the sea is salt; and yet this conclusion is implied both by the above premiss, and the premiss that Cæsar killed Brutus. Cf. on such questions Venn, S. L., 2nd ed., pp. 240-4.

[54] F. L., p. 264.

[55] Cf. Chapters XXIX and XXXVII.

CHAPTER XX

DIGNITY

We have seen, at the end of the preceding chapter, that logical implication is often an enemy of dignity. The subject of dignity is not usually considered in treatises on logic, but, as we have remarked, many mathematicians implicitly or explicitly seem to fear either that the dignity of mathematics will be impaired if she follows out conclusions logically, or that only an act of faith can save us from the belief that, if we followed out conclusions logically, we should find out something alarming about the past, present, or future of mathematics.

Thus it seems necessary to inquire rather more closely into the nature of dignity, with a view to the discovery of whether it is, as is commonly supposed, a merit in life and logic.

The chief use of dignity is to veil ignorance. Thus, it is well known that the most dignified people, as a rule, are schoolmasters, and schoolmasters are usually so occupied with teaching that they have no time to learn anything. And because dignity is used to hide ignorance, it is plain that impudence is not always the opposite of dignity, but that dignity is sometimes impudence. Dignity is said to inspire respect; and this may be in part why respect for others is an error of judgment and self-respect is ridiculous.

Self-respect is, of course, self-esteem. William James has remarked that self-esteem depends, not simply upon our success, but upon the ratio of our success to our pretensions, and can therefore be increased by diminishing our pretensions. Thus if a man is successful, but only then, can he be both ambitious and dignified. James also implies that happiness increases with self-esteem. Likeness of thought with one’s friends, then, does not make one happy, for otherwise a man who esteemed himself little would be indeed happy. Also if a man is unhappy he could not, from our premisses, by the principles of the syllogism and of contraposition, be dignified—a conclusion which should be fatal to many novelists’ heroes.

A reflection on pessimism to which this discussion gives rise is the following: It would appear that a man’s self-esteem would be increased by a conviction of the unworthiness of his neighbours. A man, therefore, who thinks that the world and all its inhabitants, except himself, are very bad, should be extremely happy. In fact, the effects would hardly be distinguishable from those of optimism. And optimism, as everybody knows, is a state of mind induced by stupidity.

CHAPTER XXI

THE SYNTHETIC NATURE OF DEDUCTION

Doubt has often been expressed as to whether a syllogism can add to our knowledge in any way. John Stuart Mill and Henri Poincaré, in particular, held the opinion that the conclusion of a syllogism is an “analytic” judgment in the sense of Kant, and therefore could be obtained by the mere dissection of the premisses. Any one, then, who maintains that mathematics is founded solely on logical principles would appear to maintain that mathematics, in the last instance, reduces to a huge tautology.

Mill, in Chapter III of Book II of his System of Logic, said that “it must be granted that in every syllogism, considered as an argument to prove the conclusion, there is a petitio principii. When we say

it is unanswerably urged by the adversaries of the syllogistic theory, that the proposition, Socrates is mortal, is presupposed in the more general assumption, All men are mortal; that we cannot be assured of the mortality of all men unless we are already certain of the mortality of every individual man; that if it be still doubtful whether Socrates, or any other individual we choose to name, be mortal or not, the same degree of uncertainty must hang over the assertion, All men are mortal; that the general principle, instead of being given as evidence of the particular case, cannot itself be taken for true without exception until every shadow of doubt which could affect any case comprised with it is dispelled by evidence aliunde; and then what remains for the syllogism to prove? That, in short, no reasoning from general to particular can, as such, prove anything, since from a general principle we cannot infer any particulars but those which the principle itself assumes as known. This doctrine appears to me irrefragable....”

But it is not difficult to see that in certain cases at least deduction gives us new knowledge.[56] If we already know that two and two always make four, and that Asquith and Lloyd George are two and so are the German Emperor and the Crown Prince, we can deduce that Asquith and Lloyd George and the German Emperor and the Crown Prince are four. This is new knowledge, not contained in our premisses, because the general proposition, “two and two are four,” never told us there were such people as Asquith and Lloyd George and the German Emperor and the Crown Prince, and the particular premisses did not tell us that there were four of them, whereas the particular proposition deduced does tell us both these things. But the newness of the knowledge is much less certain if we take the stock instance of deduction that is always given in books on logic, namely “All men are mortal; Socrates is a man, therefore Socrates is mortal.” In this case what we really know beyond reasonable doubt is that certain men, A, B, C, were mortal, since, in fact, they have died. If Socrates is one of these men, it is foolish to go the roundabout way through “all men are mortal” to arrive at the conclusion that probably Socrates is mortal. If Socrates is not one of the men on whom our induction is based, we shall still do better to argue straight from our A, B, C, to Socrates, than to go round by the general proposition, “all men are mortal.” For the probability that Socrates is mortal is greater, on our data, than the probability that all men are mortal. This is obvious, because if all men are mortal, so is Socrates; but if Socrates is mortal, it does not follow that all men are mortal. Hence we shall reach the conclusion that Socrates is mortal, with a greater approach to certainty if we make our argument purely inductive than if we go by way of “all men are mortal” and then use deduction.

Many years ago there appeared, principally owing to the initiative of Dr. F. C. S. Schiller of Oxford, a comic number of Mind. The idea was extraordinarily good, not so the execution. A German friend of Dr. Schiller was puzzled by the appearance of the advertisements, which were doubtfully humorous. However, by a syllogistic process, he acquired information which was new and useful to him, and thus incidentally refuted Mill. Presumably he started from the title of the magazine (Mind!), for a mark of exclamation seems nearly always in German to be a sign of an intended joke (including of course the mark after the politeness expressed in the first sentence of a private letter or a public address). There would be, then, the following syllogism:

Thus the syllogism may be almost as powerful an agent in the detection of humour as M. Bergson’s criterion, to be described in a future chapter.[57]

[56] [The following passage is almost word for word the same as a passage on pp. 123-5 of Mr. Russell’s Problems of Philosophy, first published in 1912, a year after Mr. R*ss*ll’s death. It is easy hastily to conclude that Mr. Russell was indebted to Mr. R*ss*ll to a greater degree than is usually supposed. But an examination of the internal evidence leads us to another conclusion. The two texts, it will be found, differ only in the names of the German Emperor, the Crown Prince and the other personages being replaced, in the book of 1912, by those of Messrs. Brown, Jones, Smith, and Robinson. Now, Mr. Russell, in a new edition of his Problems issued near the beginning of the European war and before the Russian revolution, substituted “the Emperor of Russia” for “the Emperor of China” of the first edition. Hence it seems quite likely that Mr. Russell, who has always shown a tendency to substitute existents for nonentities, wrote Mr. R*ss*ll’s notes.—Ed.]

[57] [See Chapter XLII.—Ed.]

CHAPTER XXII

THE MORTALITY OF SOCRATES

The mortality of Socrates is so often asserted in books on logic that it may be as well briefly to consider what it means. The phrase “Socrates is mortal” may be thus defined: “There is at least one instant t such that t has not to Socrates the one-many relation R which is the converse of the relation ‘exists at,’ and all instants following t have not the relation R to Socrates, and there is at least one instant t´ such that neither t´ nor any instant preceding t´ has the relation R to Socrates.”

This definition has many merits. In the first place, no assumption is made that Socrates ever lived at all. In the second place, no assumption is made that the instants of time form a continuous series. In the third place, no assumption is made as to whether Socrates had a first or last moment of his existence. If time be indeed a continuous series, then we can easily deduce[58] that there must have been either a first moment of his non-existence or a last one of his existence, but not both; just as there seems to be either a greatest weight that a man can lift or a least weight that he cannot lift, but not both.[59] This may be set forth as follows: for the present we will not concern ourselves with evidence for or against human immortality; I will merely try to present some logical questions which persistently arise whenever we think of eternal life. One of the greatest merits of modern logic is that it has allowed us to give precision to such problems, while definitely abandoning any pretensions of solving them; and I will now apply the logico-analytical method to one of the problems of our knowledge of the eternal world.[60]

We will start from the generally accepted proposition that all men are mortal. Clearly, if we could know each individual man, and know that he was mortal, that would not enable us to know that all men are mortal, unless we knew, in addition, that those were all the men there are. But we need not here assume any such knowledge of general propositions; and, though most of us will admit that the proposition in question has great intrinsic plausibility, it is not strictly necessary for our present purpose to assume anything more than the still more probable proposition “Socrates is mortal.” This last proposition, quite apart from the fact that we have a large amount of historical evidence for its truth, has been repeated so often in books on logic that it has taken on the respectable air of a platitude while preserving the character of an exceedingly probable truth. The truth also results from the fact that it is used as the conclusion of a syllogism. For it is a well-known fact that syllogisms can only be regarded as forming part of a sound education if the conclusions are obviously true. The use of a syllogism of the form “All cats are ducks and all ducks are mice, therefore all cats are mice,” would introduce grave doubts into the University of Oxford as to whether logic could any longer be considered as a valuable mental training for what are amusingly called the “learned professions.”

If, then, we divide all the instants of time, whether past, present, or future, into two series—those instants at which Socrates was alive, and those instants at which he was not alive—and leave out of consideration, for the sake of greater simplicity, all those instants before he lived, we see at once, by the simple application of Dedekind’s Axiom, that, if Socrates entered into eternal life after his death, there must have been either a last moment of his earthly life or a first moment of his eternal life, but not both.

Logic alone can give us no information as to which of these cases actually occurred, and we are thrown back on to a discussion of empirical evidence. It is no unusual thing to read of people who thought “that every moment would be their last.” In this case it is quite obvious that they consequently thought that eternity would have no beginning.

Now here we must consider two things: (1) It is plainly unsafe to conclude from what people think will happen to what will happen; (2) even if we could so conclude, it would be unsafe to deduce that there was a last moment in the life of Socrates: we could only make the guess plausible, as we should be using the inductive method.

There are two other pieces of evidence that there is a last moment of any earthly existence, which we may now briefly consider. That this was so was held by Carlo Michaelstaedter; but since he apparently only believed this because he wanted, by attributing a supposed ethical value to that moment, to give support to his theory of suicide, we ought not to give great weight to this evidence. Secondly, Thomas Hobbes objected to the principle “that a quantity may grow less and less eternally, so as at last to be equal to another quantity; or, which is all one, that there is a last in eternity” as “void of sense.” Now, the principle meant is true, so that, although the other proposition mentioned by Hobbes does not follow logically from the first, there is some evidence that this other is true. In fact, that Hobbes thought that such-and-such a proposition followed from another proposition which he wrongly believed to be false, is far better evidence for the truth of such-and-such a proposition than any we have for the truth of most of our most cherished beliefs.

Thirdly, Leibniz, in a dialogue[61] written on his journey of 1676 to visit Spinoza, raised the question whether the moment at which a man dies may be regarded as both the last moment at which he is alive and the first at which he is dead, as it must be by Aristotle’s theory of continuity. Agreement with this view violates the law of contradiction; denial of it implies that two moments can be immediately adjacent. By the denial, then, we are led to regard space and time as made up of indivisible points and moments, and thus, since we can draw one and only one parallel from any point in the diagonal of a square to a given side, the diagonal will contain the same (infinite) number of points as that side, and will therefore be equal to it. In this Leibniz repeated an argument used by the ancient Arabs, Roger Bacon, and William of Occam. This Leibniz considered to be a proof that a line cannot be an aggregate of points. Indeed, their number would be “the number of all numbers” of the greatest possible integer, which is not.

It does not seem, further, that any light is thrown on the logical question of human mortality or immortality by legal decisions. It would appear that one can, legally speaking, be alive for any period less than twenty-four hours after one is dead and be dead for any period less than twenty-four hours before one’s death. At least, according to Salkeld, i. 44, it was “adjudged that if one be born the first of February at eleven at night, and the last of January in the twenty-first year of his age, at one of the clock in the morning, he makes his will of lands, and dies, it is a good will, for he was then of age.” In Sir Robert Howard’s case (ibid., ii. 625) it was held by Chief Justice Holt that “if A be born on the third day of September; and on the second day of September twenty-one years afterwards he make his will, this is a good will; for the law will make no fraction of a day, and by consequence he was of age.” But it is hardly necessary to remark that in this way the problem with which we are concerned is merely shifted and not solved. For the question as to whether there is or is not a last moment of a man’s life is not answered by the decision that he dies legally twenty-four hours before or after he dies in the usual sense of the word, and the problem arises as to whether there is or is not a last moment of his legal age.[62]

So assuming that there was a last moment of Socrates’s earthly life, and consequently no first moment of his eternal life, we see, further, that, unless the possibility of infinite numbers is granted, it would be quite possible for us logically to doubt the possibility of an eternal life for Socrates on the same grounds as those which led Zeno to assert that motion was impossible and that Achilles could never overtake the Tortoise. If, on the other hand, it be admitted that eternity, at least in the case of Socrates, had a beginning, these same arguments of Zeno would lead any one who denies the possibility of infinite number to conclude that Socrates, like the worm, can never die. Thus is it quite plain that the difficulties about immortality which meet us at the very outset of our inquiry can partly be solved only by the help of the theory of infinite numbers and partly, it would seem, not at all.

There is another difficulty about immortality which is quite distinct from this and is analogous to another argument of Zeno. If, indeed, all the instants of time be divided, as before, into the two series of instants at which Socrates was alive and instants at which he was not alive, it follows at once that no instant of time is not accounted for. At none of these instants, however, does Socrates die; obviously he cannot die either when he is alive or when he is dead. Thus it would appear that Socrates never died, and that we ought to re-define the term “mortal” to mean “a human being who is alive at some moments and dead at some.” Consequently we must avoid the very tempting conclusion that, because Socrates never died, he was therefore immortal.

It is very important carefully to distinguish between the two arguments I have just set forth. The second argument proves quite rigidly that Socrates and, indeed, anybody else, never dies, whether there is or is not a last moment of his life on earth. The first argument proves that, if there is a first moment of Socrates’s eternal life, his life on earth never ends. But we have seen that we cannot conclude that this unending life proves that he never is or will be in a state of eternity.