Categoricity by convention

On a widespread naturalist view, the meanings of mathematical terms are determined, and can only be determined, by the way we use mathematical language in particular, by the basic mathematical principles were disposed to accept. But its mysterious how this can be so, since, as is well known, minimally strong first-order theories, i.e. theories whose quantifiers range over objects, are non- categorical and so are compatible with countless non- isomorphic interpretations. As for second-order theories, i.e. theories whose quantifiers range over sets of objects: though they typically enjoy categoricity results (i.e. they can be shown to have only one kind of interpretation) for instance, Dedekinds categoricity theorem for second-order PA and Zermelos quasi- categoricity theorem for second-order ZFC these results require full second-order logic. So appealing to these results seems only to push the problem 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 Dedekinds and Zermelos results are no longer available. In this project, we develop a naturalist-friendly, non-revisionary solution to an analogous but seemingly more basic problem Carnaps categoricity problem for propositional and first-order logic and show that our solution generalizes, giving us full second-order logic and thereby securing the categoricity or quasi- categoricity of second-order mathematical theories. Briefly, the first-order quantifiers have their intended interpretation, we claim, because were disposed to follow the quantifier rules in an open- ended way. As we show, given this open-endedness, the interpretation of the quantifiers must be permutation-invariant and so, by a theorem recently proved by Bonnay & Westerstahl, must be the standard interpretation. Analogously for the second- order case: we prove, by generalizing Bonnay & Westerstahls theorem, that the permutation invariance of the interpretation of the second-order quantifiers, guaranteed once again by the open-endedness of our inferential dispositions, suffices to yield full second-order logic. Our view yields a novel, largely syntactic criterion for logicality, a moderate form of pluralism, and an attractive epistemology of the a priori. Or so we hope to be able to show.