Neighborhood and algebraic models for predicate modal logics with $ω$-rules
Yoshihito Tanaka
TL;DR
The study develops neighborhood and algebraic semantics for predicate modal logics with $ω$-rules, including non-normal cases, and proves model-existence and completeness results under constant-domain semantics. A central achievement is a model existence theorem yielding neighborhood models with constant domains for logics that admit countably many axiom schemata and an $ω$-rule, without requiring the Barcan formula; this underpins completeness results for both normal and non-normal logics. The authors establish the completeness of a predicate extension of $GL$ with respect to neighborhood frames and show that a predicate common-knowledge logic is Kripke incomplete yet neighborhood complete, highlighting semantic distinctions driven by $ω$-rules. Overall, the work extends prior results by Tanaka, Arló-Costa, Pacuit, and related studies, connecting canonical algebras, Q-filter constructions, and ω-rule axiomatizations to obtain robust constant-domain semantics for predicate modal logics.
Abstract
This paper investigates neighborhood and algebraic models for predicate modal logics with $ω$-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 $ω$-rules were obtained by Tanaka, while similar results for non-normal modal logics without $ω$-rules were presented by Arló-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.