An extension of J\'{o}nsson-Tarski representation and model existence in predicate non-normal modal logics

TL;DR

This paper extends the Jönsson-Tarski representation theorem to include non-normal modal algebras using neighborhood frames and Q-filters, establishing model existence and completeness for predicate and infinitary modal logics.

Contribution

It introduces a generalized representation theorem for non-normal modal algebras and proves model existence and completeness for predicate and infinitary modal logics.

Findings

Extended Jönsson-Tarski theorem for non-normal modal algebras

Model existence for predicate modal logics on neighborhood frames

Completeness results for infinitary modal logics

Abstract

In this paper, we give an extension of the J\'{o}nsson-Tarski representation theorem for both normal and non-normal modal algebras so that it preserves countably many infinitary meets and joins. To extend the J\'{o}nsson-Tarski representation to non-normal modal algebras we consider neighborhood frames instead of Kripke frames just as Do\v{s}en's duality theorem for modal algebras, and to deal with infinite meets and joins, we make use of Q-filters instead of prime filters. Then, we show that every predicate modal logic, whether it is normal or non-normal, has a model defined on a neighborhood frame with constant domains, and give completeness theorem for some predicate modal logics. We also show the same results for infinitary modal logics.