On a widespread naturalist metasemantic view, the meanings of logical and mathematical terms are determined, and can only be determined, by the way we use logical and mathematical language – in particular, by the basic logical and mathematical principles we’re disposed to accept. But it’s mysterious how this can be so: well-known results in model theory entail that any sufficiently powerful first-order theory is non-categorical, in which case, it seems, no such theory can pick out any one interpretation of our mathematical language. A standard response is to appeal to second-order mathematical theories, for which categoricity results – for instance, Dedekind’s categoricity theorem for second-order PA and Zermelo’s quasi-categoricity theorem for ZFC – are available. These results, though, require the full interpretation of second-order logic. So the problem has only been pushed back, since the principles of second-order logic are themselves non-categorical: those principles are compatible with restricted interpretations of the second-order quantifiers on which Dedekind’s and Zermelo’s results are no longer available.
In fact, an analogous problem arises even for the principles of propositional and first-order logic – as Carnap showed, those principles are compatible with non-standard interpretations of our logical vocabulary and so don’t seem able to pick out the intended interpretations. Pending a solution to these problems, there’s no hope for a naturalist metasemantics.
We provide a novel, unified solution to these problems, thus demonstrating the tenability of a naturalist metasemantics for logical and mathematical language. We start by offering a new solution to Carnap’s problem for propositional and first-order logic, one that doesn’t require revising the standard proof-theoretic framework. Our explanation of how the first-order quantifiers in particular get their intended interpretation is that we’re disposed to reason in accordance with the quantifier rules in an open-ended way, in which case the interpretation of the quantifiers must be permutation-invariant and so, by a theorem recently proved by Bonnay & Westerståhl, must be the standard interpretation. This result is particularly significant because it generalizes to the second-order case: the open-endedness of the second-order quantifier rules again guarantees permutation invariance, and we show, by generalizing Bonnay & Westerståhl’s theorem, that this permutation invariance guarantees the full interpretation of second-order logic. This in turn makes available categoricity results for our second-order mathematical theories.
The metasemantic picture we develop also – we hope – yields a novel, largely syntactic criterion for logicality, a moderate form of pluralism, and an attractive epistemology of the a priori.