What Is Carnap’s Problem and How Can We Solve It?

On a plausible approach to logical metasemantics, our dispositions to treat a logical expression’s I- and E-rules as valid determine its contribution to the truth conditions of sentences in which it appears. Carnap’s so-called Categoricity Problem is that the rules in question don’t seem to fix a unique interpretation of our logical vocabulary: there appear to be deviant interpretations of both the connectives and the quantifiers that are compatible with the validity of their rules. And although standard responses are available to Carnap’s problem as it applies to propositional logic (by appeal, e.g., to bilateral rules or to a local notion of validity), the case of the quantifiers is more difficult. Here we develop a more precise account of how Carnap-style arguments work than has ever before been given, one that makes clear why certain such arguments succeed while others fail. In so doing, we demonstrate that despite recent criticisms, the account of the categoricity of the quantifiers we offered in previous work isn’t threatened by any of the alleged deviant interpretations that have been discussed in the literature: each of these either is incompatible with the validity of the quantifier rules or else results in an illegitimate Carnap-style argument.