Dag Westerståhl: The meaning of ⎕ and other logical constants

We study the extent to which logical consequence relations in a language L determine the meaning of the logical constants of L. For example: Do the laws of classical logical consequence in first-order logic fix the meaning of \(\forall\)? Carnap asked this question in 1943 about the connectives in classical propositional logic (and implicitly answered it). In [1], we extended this result to first-order logic, and to the framework of possible worlds semantics. In this talk, which is about work in progress, I consider the same question for some intensional logics. First, there are two simple observations on the interpretation of the connectives in intuitionistic propositional logic, in the setting of Kripke semantics. Then I focus on modal logic. Roughly, what range of interpretations of \(\Box\) do various (normal) modal logics allow? Some results and some open issues will be presented. This is joint work with Denis Bonnay.

[1] D. Bonnay and D. Westerståhl, ‘Compositionality solves Carnap’s Problem’, Erkenntnis 80 (4), 2015 (online first).