Tuesday, November 27, 2007

6.1233 A world can be conceived in which the axiom of reducibility does not hold. But it is clear that logic has nothing to do with the question whether our world really is thus or not.

What grounds did Russell have for calling the Axiom of Reducibility a principle of logic? The axiom in question says that any higher-order property or proposition can be reduced to an equivalent first-order one. It means that the same class is determined by two propositional functions that are equivalent.[1] Every propositional function is thus logically equivalent to a predicative function. The axiom is introduced because without it the ramified theory of types makes certain mathematical proofs impossible. This kind of thing belongs to logic, in Russell’s view, because it is necessary in order to overcome logical, not just mathematical, paradoxes, e.g. about the class of classes not members of themselves as well as paradoxes about infinity. Ramsey and Wittgenstein showed that the axiom was not necessary.

[1] See p. 299 of the Cambridge Companion to Russell.

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