Many-valued coalgebraic logic over semi-primal varieties

TL;DR

This paper develops a systematic method to extend classical coalgebraic logics to many-valued logics over semi-primal algebras, preserving key properties like completeness and expressivity, with applications to modal logic.

Contribution

It introduces a novel lifting technique for endofunctors from Boolean algebras to semi-primal algebras, enabling the construction of many-valued coalgebraic logics.

Findings

Lifting endofunctors preserves completeness and expressivity.

Axiomatizations of classical logics can be extended to many-valued logics.

Method applied successfully to classical modal logic.

Abstract

We study many-valued coalgebraic logics with semi-primal algebras of truth-degrees. We provide a systematic way to lift endofunctors defined on the variety of Boolean algebras to endofunctors on the variety generated by a semi-primal algebra. We show that this can be extended to a technique to lift classical coalgebraic logics to many-valued ones, and that (one-step) completeness and expressivity are preserved under this lifting. For specific classes of endofunctors, we also describe how to obtain an axiomatization of the lifted many-valued logic directly from an axiomatization of the original classical one. In particular, we apply all of these techniques to classical modal logic.