Are the open-ended rules for negation categorical?

Abstract

Vann McGee has recently argued that Belnap’s criteria constrain the formal rules of classical natural deduction to uniquely determine the semantic values of the propositional logical connectives and quantifiers if the rules are taken to be open-ended, i.e., if they are truth-preserving within any mathematically possible extension of the original language. The main assumption of his argument is that for any class of models there is a mathematically possible language in which there is a sentence true in just those models. I show that this assumption does not hold for the class of models of classical propositional logic. In particular, I show that the existence of non-normal models for negation undermines McGee’s argument.