ABSTRACT
The purpose of this chapter is to vindicate the axiom of reducibility not in the sense of adducing fresh grounds for believing it, but in the sense of demolishing once and for all an argument which has repeatedly been used against it (by Ramsey, Chwistek and Copi in particular). The argument runs as follows. To avoid the semantical paradoxes, Russell complicated simple type theory by introducing orders; this led to ramified type theory. However, it proved impossible to construct classical mathematics in this framework, in part because of a difficulty connected with mathematical induction; Russell therefore introduced the axiom of reducibility, according to which every propositional function was materially equivalent to a predicative propositional function, i.e. one whose order was greater by 1 than the order of its arguments. But (so runs the argument) this renders the ramified theory indistinguishable from the simple one, apart from a vacuous multiplication of subscripts; consequently, since the simple theory did not eliminate the semantical paradoxes, the ramification has failed of its purpose. This is as though a man had built a wall around himself to keep out his enemies and then, finding himself in need of a door, had knocked out part of the wall and let them all in again. Or so the critics claim. We shall show that they are wrong.
