Colin Zwanziger (Institute of Philosophy): Inquisitive logic from the perspective of category theory
Abstract It was observed in the propositional case by Holliday (2020) that the language of (intuitionistic) inquisitive logic can be identified with (intuitionistic) logic, together with a geometric modality in the sense of Goldblatt (1981), also known as a lax modality. From the inquisitive perspective, the modality is understood as extracting the presupposition behind a question. Holliday's algebraic semantics interprets the modality as a nucleus on a Heyting algebra.
Categorical semantics provides a nice general approach to extending algebraic semantics from propositional logic to quantified logic, and this approach remains natural in the inquisitive case. To interpret higher-order intuitionistic inquisitive logic, we replace Heyting algebras by toposes and nuclei by Cartesian reflectors. Several more concrete approaches to semantics of quantified inquisitive logic, including the most standard possible-worlds semantics, can be subsumed under the notion of Cartesian reflector on a topos.
