ABSTRACT
Hw = (3(f>)(wR(cf>x) a ~] <f>w) In particular, H'H' = (3<f>XH'R(<f>x) a n p H 9). Now, we might wish to argue, by the axiom of reducibility, there is a predicative function (in the sense of Principia) Fsuch that Fw = Hw. Therefore, since ‘H'RH , we get F'H 9 = 1 H'H', and therewith H ‘H 9 == 1 H ‘H'. Hence, it would seem that the axiom of reduci bility reintroduces the semantic contradictions. Now the theory of types does not exclude either the substitution of ‘7 /’ for w or the substitution of H for F, but simply remains silent on such discourses. This is an indication that if we wish to give a unified treatment of mathematics and semantics, we have to introduce explicitly semantic concepts and rules governing their use.
