Distribution-Free Normal Modal Logics

TL;DR

This paper develops a semantic framework for distribution-free normal modal logics, establishing foundational results like canonicity and completeness, and extends the approach to various modal axioms and intuitionistic modal logic.

Contribution

It introduces a uniform relational semantics for distribution-free normal modal logics, including canonicity and completeness proofs, and offers a new semantic perspective on intuitionistic modal logic.

Findings

Proved canonicity and completeness for minimal distribution-free normal modal logic.

Extended semantic results to logics with D, T, B, S4, S5 axioms.

Provided a new semantic approach to intuitionistic modal logic.

Abstract

This article initiates the semantic study of distribution-free normal modal logic systems, laying the semantic foundations and anticipating further research in the area. The article explores roughly the same area, though taking a different approach, with a recent article by Bezhanishvili, de Groot, Dmitrieva and Morachini, who studied a distribution-free version of Dunn's Positive Modal Logic (PML). Unlike PML, we consider logics that may drop distribution and which are equipped with both an implication connective and modal operators. We adopt a uniform relational semantics approach, relying on recent results on representation and duality for normal lattice expansions. We prove canonicity and completeness in the relational semantics of the minimal distribution-free normal modal logic, assuming just the K-axiom, as well as of its axiomatic extensions obtained by adding any of the D, T, B, S4 or S5 axioms. Adding distribution can be easily accommodated and, as a side result, we also obtain a new semantic treatment of Intuitionistic Modal Logic.