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.

CHAPTER II