On the Minimum Many-Valued Modal Logic over a Finite Residuated Lattice

TL;DR

This paper explores the foundational aspects of many-valued modal logics over finite residuated lattices, providing axiomatizations for different classes of Kripke frames and extending existing axiomatizations with canonical constants.

Contribution

It introduces axiomatizations of many-valued modal logics over finite residuated lattices for various frame classes, including expansions with canonical constants and specific cases like finite MV chains.

Findings

Axiomatizations for modal logics over residuated lattices are established.

Extensions with canonical constants are developed for the lattice-based modal logic.

Axiomatizations for finite MV chains without canonical constants are provided.

Abstract

This paper deals with many-valued modal logics, based only on the necessity operator, over a residuated lattice. We focus on three basic classes, according to the accessibility relation, of Kripke frames: the full class of frames evaluated in the residuated lattice (and so defining the minimum modal logic), the ones evaluated in the idempotent elements and the ones evaluated in 0 and 1. We show how to expand an axiomatization, with canonical constants in the language, of a finite residuated lattice into one of the modal logic, for each one of the three basic classes of Kripke frames, over the very lattice. And we also give axiomatizations for the case of a finite MV chain but this time without canonical constants.