Neighborhood and algebraic models for predicate modal logics with $\omega$-rules

TL;DR

This paper explores neighborhood and algebraic models for predicate modal logics with $$-rules, establishing conditions for completeness and applying results to specific modal logic extensions.

Contribution

It extends existing results by providing new conditions for neighborhood models with constant domains in predicate modal logics with $$-rules.

Findings

Predicate extension of GL is sound and complete with neighborhood frames with constant domains.

Predicate common knowledge logic is Kripke incomplete but neighborhood complete.

Established sufficient conditions for neighborhood models in non-normal and normal modal logics with $$-rules.

Abstract

This paper investigates neighborhood and algebraic models for predicate modal logics with $\omega$-rules, including non-normal cases. We establish sufficient conditions under which such logics have neighborhood models with constant domains and satisfy the completeness theorem with respect to neighborhood frames with constant domains. Related results for normal modal logics with $\omega$-rules were obtained by Tanaka, while similar results for non-normal modal logics without $\omega$-rules were presented by Arl\'{o}-Costa and Pacuit and by Tanaka. The results presented here extend these works. As applications, we prove that a predicate extension of GL is sound and complete with respect to a class of neighborhood frames with constant domains, and that a predicate common knowledge logic is Kripke incomplete but neighborhood complete.