Russellian Type Theory and Semantical Paradoxes

Abstract

We compare Grelling’s paradox with Russell’s paradox about propositions, in order to illuminate their different natures. We criticize Church’s reconstruction of Grelling’s paradox, according to which it would be an inten- sional antinomy arising within the simple type theory and suitably solved within the minified type theory. We claim, on the contrary, that this paradox is not genuinely intensional and that ramification is of no help for its resolution. We argue that the paradox rests on a highly problematic assumption, which can be rejected within the simple theory itself, since its alleged evidence has nothing to do with the lack of ramification. In contrast, we show that a reconstruction, similar to the one proposed by Church for Grelling’s paradox, is appropriate for Russell’s paradox about propositions, which turns out to be really intensional and capable of an adequate solultion by means of ramification.