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]

CHAPTER XXII