Strong Completeness of Coalgebraic Modal Logics

TL;DR

This paper develops a generic coalgebraic framework for constructing canonical models in modal logic, ensuring strong completeness across various non-normal modal logics, including probabilistic and graded modalities.

Contribution

It introduces a universal canonical model construction method in coalgebraic modal logic, enabling new and known strong completeness results for diverse modal systems.

Findings

Proves strong completeness of graded modal logic with finite multiplicities.

Establishes strong completeness of the modal logic of exact probabilities.

Reconstructs canonical model theorems for existing modal logics.

Abstract

Canonical models are of central importance in modal logic, in particular as they witness strong completeness and hence compactness. While the canonical model construction is well understood for Kripke semantics, non-normal modal logics often present subtle difficulties - up to the point that canonical models may fail to exist, as is the case e.g. in most probabilistic logics. Here, we present a generic canonical model construction in the semantic framework of coalgebraic modal logic, which pinpoints coherence conditions between syntax and semantics of modal logics that guarantee strong completeness. We apply this method to reconstruct canonical model theorems that are either known or folklore, and moreover instantiate our method to obtain new strong completeness results. In particular, we prove strong completeness of graded modal logic with finite multiplicities, and of the modal logic of exact probabilities.