“ Now, if a man had an unlimited income, it is an immediate inference that, however low income-tax might be, he would have to pay annually to the Exchequer of his nation a sum equal in value to his whole income. Further, if his income was derived from a capital invested at a finite rate of interest (as is usual) , the annual payments of income-tax would each be equal in value to the man’s whole capital. If, then, the man with an unlimited income chose to be discontented, he would be sure of a sympathetic audience among philosophers and business acquaintances ”

“ 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. ”

“ 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.” ”

“ When the clown makes a bad joke for the first time, we keep our countenance, but at the fifth repetition we smile, and at the tenth we roar with laughter, because we begin to feel him a mere automaton. ”

“ The most noteworthy reformation of recent years in logic is the discovery and development by Mr. Bertrand Russell of the fact that the paradoxes—of Burali-Forti, Russell, König, Richard, and others—which have appeared of late years in the mathematical theory of aggregates and have just been referred to, are of an entirely logical nature, and that their avoidance requires us to take account of a principle which has been hitherto unrecognized, and which renders invalid several well-known arguments in refutation of scepticism, agnosticism, and the statement of a man that he asserts nothing. ”

“ By observing a very simple formula they will know infallibly what is funny and what is not; if they sometimes surprise their unlearned friends, they have only to mention their authority in order to silence doubt. ‘The attitudes, gestures and movements of the human body,’ says M. Bergson, ‘are laughable in exact proportion as that body reminds us of a mere machine.’ When an elderly gentleman slips on a piece of orange-peel and falls, we laugh, because his body follows the laws of dynamics instead of a human purpose. ”

“ 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. ”

“ 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. ”

“ Similarly the imaginary sceptic who asserts that he knows nothing and is refuted by being asked if he knows that he knows nothing, has asserted nonsense, and has been fallaciously refuted by an argument which involves a vicious-circle fallacy. In order that the sceptic’s assertion may become significant it is necessary to place some limitation upon the things of which he is asserting his ignorance; the proposition that he is ignorant of every member of this collection must not itself be one of the collection. Hence any significant scepticism is not open to the above form of refutation. ”

“ 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” ”

“ 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.” ”

“ Now, I suggest the extremely unlikely possibility that Tristram Shandy, by reflection on his own life and literary labours, was led to the correct course of accepting the paradox which resulted from this reflection and rejecting the Achilles. Thus, he concluded that an infinite whole may be similar (or, in Cantor’s terminology, “equivalent”) to a proper part of itself, and hence, by a confusion of similarity with identity (or equivalence with equality) which he shares with some subsequent philosophers, [86] that a whole may be equal to a proper part of itself. ”

“ 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. ”

“ Fortunately the discovery of logic by some mathematicians has removed this danger. Still to many mathematicians logic is still unknown, and to them—to Professor A. Schoenflies for example—modern mathematics, owing to its alliance with logic, appears to be sinking into scholasticism. It is true that the word “scholasticism” is not used by Professor Schoenflies in any intentionally precise signification, but merely as a vague epithet of disapproval, as the word “socialism” is used by the ordinary philistine, and this would certainly serve as a sufficient excuse. ”

“ 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. ”

“ 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. ”

“ People who refer to “the Oxford Movement” imply that Oxford only moved once; and those quaint people who say that “A is quite the gentleman” imply both the doubtful proposition that there is only one gentleman in the world, and the indubitably false proposition that he is that man. ”

“ 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 Tweedle [Pg 14] dum and Tweedledee, and even practical idealists like the White Knight, explicitly accept laws like the law of identity and the excluded middle. ”

“ 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.” ”

“ In the same way, ω may be said, without contradiction, to transcend all the finite integers. But if some one else, C, should say: “There are some things which no human being can ever know anything about,” he is talking nonsense. [91] And in the same way if we succeeded in imagining a number which transcends all numbers, we have succeeded in imagining the absurdity of a number which transcends itself.
All the paradoxes of logic (or “the theory of aggregates”) [Pg 72] are analogous to the difficulty arising from a man’s statement: “I am lying.” ”

“ Arguments which are condemned by the vicious-circle principle will be called ‘vicious-circle fallacies.’ Such arguments, in certain circumstances, may lead to contradictions, but it often happens that the conclusions to which they lead are in fact true, though the arguments are fallacious. Take, for example, the law of excluded middle in the form ‘all propositions are true or false.’ If from this law we argue that, because the law of excluded middle is a proposition, [Pg 80] therefore the law of excluded middle is true or false, we incur a vicious-circle fallacy. ”

“ 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. ”

“ A little reflection would have shown the man that he was not taking up a paradoxical attitude. A “paradoxical attitude” is of course the assertion of one or more propositions of which the truth cannot be perceived by a philosopher—and particularly an idealist—and can be perceived by a logician and occasionally, but not always, by a man of common-sense. ”

“ Mathematicians usually try to found mathematics on two principles: [78] one is the principle of confusion between the sign and the thing signified (they call this principle the foundation-stone of the formal theory) , and the other is the Principle of the Identity of Discernibles (which they call the principle of the permanence of equivalent forms) . ”

“ To conceive a body all black and yet white would only be to conceive two different sensations as produced in us simultaneously by the same object—a conception familiar to our experience—and we should probably be as well able to conceive a round square as a hard square, or a heavy square, if it were not that in our uniform experience, at the instant when a thing begins to be round, it ceases to be square, so that the beginning of the one impression is inseparably associated with the departure or cessation of the other. ”

“ 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. ”

“ 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. ”

“ 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.” ”

“ 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. ”

“ 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. ”

“ If he decided, by his theory, that it was a joke, he would solemnly say: “There is that joke.” If this narration is accepted as a joke, since it cannot be reduced to one of the proto-Aryan jokes under pain of leading us to commit a vicious-circle fallacy, we must conclude that there is at least one joke which is not proto-Aryan; and, in fact, is of a higher type. There is no great difficulty in forming a hierarchy of jokes of various types. Thus a joke of the fourth type (or order) is as follows: A joke of the first order was told to a Scotchman, who, as we would expect, was unable to see it. ”