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).