Unification of Modal Logic via Topological Categories

TL;DR

This paper introduces a unifying topological categorical framework that links various semantic interpretations and syntactic extensions of modal logic, enhancing conceptual understanding and providing a uniform interpretative approach.

Contribution

It establishes an exact correspondence between syntactic extensions of modal logic and semantic structures in topological categories, unifying diverse modal semantics.

Findings

Unified treatment of modal logic semantics

Deeper conceptual understanding of modal structures

Framework applicable to various modal extensions

Abstract

In this paper we provide a unifying description of different types of semantics of modal logic found in the literature via the framework of topological categories. In the style of categorical logic, we establish an exact correspondence between various syntactic extensions of modal logic on one hand, including modal dependence, group agent structures, and logical dynamic, and semantic structures in topological categories on the other hand. This framework provides us a uniform treatment of interpreting these syntactic extensions in all different types of semantics of modal logic, and it deepens our conceptual understanding of the abstract structure of modal logic.