Frame definability in finitely-valued modal logics

TL;DR

This paper investigates frame definability in finitely-valued modal logics, showing they do not extend classical modal logic's definability and establishing their equivalence in class-defining power, with implications for complexity and the Goldblatt--Thomason theorem.

Contribution

It proves finitely-valued modal logics do not define more frame classes than classical logic and simplifies the proof of a key theorem, extending classical results to these logics.

Findings

Finitely-valued modal logics do not define more classes of frames than classical modal logic.

A large family of finitely-valued modal logics define exactly the same classes of frames as classical modal logic.

The results allow determination of the computational complexity of many finitely-valued modal logics.

Abstract

In this paper we study frame definability in finitely-valued modal logics and establish two main results via suitable translations: (1) in finitely-valued modal logics one cannot define more classes of frames than are already definable in classical modal logic (cf.~\citep[Thm.~8]{tho}), and (2) a large family of finitely-valued modal logics define exactly the same classes of frames as classical modal logic (including modal logics based on finite Heyting and \MV-algebras, or even \BL-algebras). In this way one may observe, for example, that the celebrated Goldblatt--Thomason theorem applies immediately to these logics. In particular, we obtain the central result from~\citep{te} with a much simpler proof and answer one of the open questions left in that paper. Moreover, the proposed translations allow us to determine the computational complexity of a big class of finitely-valued modal logics.