Introduction to Mathematical Philosophy

by Bertrand Russell

. So long as only extensional functions of functions are involved, this gives us in practice results which would otherwise have required the impossible notion of "all

-functions." One particular region where this is vital is mathematical induction.

The axiom of reducibility involves all that is really essential in the theory of classes. It is therefore worth while to ask whether there is any reason to suppose it true.

This axiom, like the multiplicative axiom and the axiom of infinity, is necessary for certain results, but not for the bare existence of deductive reasoning. The theory of deduction, as explained in Chapter XIV., and the laws for propositions involving "all" and "some," are of the very texture of mathematical reasoning: without them, or something like them, we should not merely not obtain the same results, but we should not obtain any results at all. We cannot use them as hypotheses, and deduce hypothetical consequences, for they are rules of deduction as well as premisses. They must be absolutely true, or else what we deduce according to them does not even follow from the premisses. On the other hand, the axiom of reducibility, like our two previous mathematical axioms, could perfectly well be stated as an hypothesis whenever it is used, instead of being assumed to be actually true. We can deduce its consequences hypothetically; we can also deduce the consequences of supposing it false. It is therefore only convenient, not necessary. And in view of the complication of the theory of types, and of the uncertainty of all except its most general principles, it is impossible as yet to say whether there may not be some way of dispensing with the axiom of reducibility altogether. However, assuming the correctness of the theory outlined above, what can we say as to the truth or falsehood of the axiom?

The axiom, we may observe, is a generalised form of Leibniz's identity of indiscernibles. Leibniz assumed, as a logical principle, that two different subjects must differ as to predicates. Now predicates are only some among what we called "predicative functions," which will include also relations to given terms, and various properties not to be reckoned as predicates. Thus Leibniz's assumption is a much stricter and narrower one than ours. (Not, of course, according to his logic, which regarded all propositions as reducible to the subject-predicate form.) But there is no good reason for believing his form, so far as I can see. There might quite well, as a matter of abstract logical possibility, be two things which had exactly the same predicates, in the narrow sense in which we have been using the word "predicate." How does our axiom look when we pass beyond predicates in this narrow sense? In the actual world there seems no way of doubting its empirical truth as regards particulars, owing to spatio-temporal differentiation: no two particulars have exactly the same spatial and temporal relations to all other particulars. But this is, as it were, an accident, a fact about the world in which we happen to find ourselves. Pure logic, and pure mathematics (which is the same thing), aims at being true, in Leibnizian phraseology, in all possible worlds, not only in this higgledy-piggledy job-lot of a world in which chance has imprisoned us. There is a certain lordliness which the logician should preserve: he must not condescend to derive arguments from the things he sees about him.

Viewed from this strictly logical point of view, I do not see any reason to believe that the axiom of reducibility is logically necessary, which is what would be meant by saying that it is true in all possible worlds. The admission of this axiom into a system of logic is therefore a defect, even if the axiom is empirically true. It is for this reason that the theory of classes cannot be regarded as being as complete as the theory of descriptions. There is need of further work on the theory of types, in the hope of arriving at a doctrine of classes which does not require such a dubious assumption. But it is reasonable to regard the theory outlined in the present chapter as right in its main lines, i.e. in its reduction of propositions nominally about classes to propositions about their defining functions. The avoidance of classes as entities by this method must, it would seem, be sound in principle, however the detail may still require adjustment. It is because this seems indubitable that we have included the theory of classes, in spite of our desire to exclude, as far as possible, whatever seemed open to serious doubt.

The theory of classes, as above outlined, reduces itself to one axiom and one definition. For the sake of definiteness, we will here repeat them. The axiom is:

There is a type