Labelled Sequents for Inquisitive First-Order Modal Logic

2026-06-30

AI Summary

This work addresses the absence of a formal proof system suitable for interrogative first-order modal logic—a framework designed to capture modal dependencies such as global supervenience. The paper constructs a labeled sequent calculus that extends the framework previously introduced by Litak and Sano for a weaker variant of the logic, thereby providing the first complete proof system for this stronger logic. Integrating techniques from both modal and interrogative semantics, the proposed calculus supports rigorous formal derivations and is shown to be strongly complete. Moreover, it satisfies essential structural properties, including rule invertibility and admissibility of the cut rule, ensuring its robustness and proof-theoretic adequacy.

Abstract

In recent work, an inquisitive first-order modal logic has been proposed to reason about relations of modal dependence, including the notion of global supervenience (functional dependence among the extensions of predicates relative to a space of possibilities). At present, no proof system exists for this logic. We provide a complete labelled sequent calculus, extending a calculus developed by Litak and Sano for a weak version of inquisitive first-order logic. We prove strong completeness for the calculus and show that it enjoys desirable structural properties, including the invertibility of its rules and the admissibility of cut.

Problem

• inquisitive logic
• first-order modal logic
• proof system
• modal dependence
• global supervenience

Innovation

• labelled sequent calculus
• inquisitive logic
• first-order modal logic
• strong completeness
• cut admissibility