Undecidability and non-axiomatizability of modal many-valued logics

TL;DR

This paper investigates the decidability and axiomatizability of certain modal many-valued logics based on Kripke frames over residuated lattices, revealing many are undecidable and not recursively axiomatizable, unlike classical modal logic.

Contribution

It demonstrates the undecidability and non-axiomatizability of broad classes of modal many-valued logics, including those based on Lukasiewicz and Product algebras, and addresses open questions in the field.

Findings

Many modal many-valued logics are undecidable.

Global modal Lukasiewicz and Product logics are not recursively axiomatizable.

The open question on the equivalence of global and local modal logics remains unresolved.

Abstract

In this work we study the decidability of a class of global modal logics arising from Kripke frames evaluated over certain residuated lattices, known in the literature as modal many-valued logics. We exhibit a large family of these modal logics which are undecidable, in contrast with classical modal logic and propositional logics defined over the same classes of algebras. This family includes the global modal logics arising from Kripke frames evaluated over the standard Lukasiewicz and Product algebras. We later refine the previous result, and prove that global modal Lukasiewicz and Product logics are not even recursively axiomatizable. We conclude by solving negatively the open question of whether each global modal logic coincides with its local modal logic closed under the unrestricted necessitation rule.