Grelling's revenge

Abstract

The heterological paradox of Kurt Grelling is often mentioned but seldom critically examined in the history of symbolic logic.' The paradox is that heterologicality, the property of not being truly predicated of itself, is itself heterological if and only if it is not heterological. The paradox is sometimes thought to make a compelling argument for the need to stratify logical syntax into a hierarchy of types, as Russell and Whitehead argue (1927), in order to avoid formal syntactical inconsistency in similar paradoxes.2 I maintain in what follows that type theory, although it appears tailor-made to solve the problem, does not actually forestall Grelling's paradox, but that the paradox remains derivable in spite of rigorously enforced type restrictions. The paradox for that reason is even more urgently addressed in order to avoid antinomy in anything like a classical inferential framework. Accordingly, I conclude by proposing an alternative, non-type-theoretical solution to Grelling's paradox that preserves logical integrity in a bivalent semantics.