An Inductive Construction for Many-Valued Coalgebraic Modal Logic

TL;DR

This paper introduces a new inductive framework for many-valued coalgebraic modal logic, enabling soundness, completeness, and finite model properties without restrictive language assumptions.

Contribution

It generalizes existing stratification methods to many-valued settings and employs an induction principle for model construction, relaxing previous language restrictions.

Findings

Proves soundness and completeness of the logic.

Establishes finite model property.

Lifts restrictions on language expressiveness.

Abstract

In this paper, we present an abstract framework of many-valued modal logic with the interpretation of atomic propositions and modal operators as predicate lifting over coalgebras for an endofunctor on the category of sets. It generalizes Pattinson's stratification method for colagebraic modal logic to the many-valued setting. In contrast to standard techniques of canonical model construction and filtration, this method employs an induction principle to prove the soundness, completeness, and finite model property of the logics. As a consequence, we can lift a restriction on the previous approach~ \cite{Lin2022} that requires the underlying language must have the expressive power to internalize the meta-level truth valuation operations.