A Characterization Result for Non-Distributive Logics

TL;DR

This paper extends van Benthem's characterization to non-distributive logics by reducing them to sorted residuated modal logics, simplifying their analysis and establishing a correspondence with first-order logic fragments.

Contribution

It introduces a reduction method of non-distributive logics to sorted residuated modal logics, enabling easier characterization and analysis of these logics.

Findings

Reduction of non-distributive logics to sorted modal logics demonstrated

Van Benthem characterization extended to non-distributive logics

Simplified proof of correspondence with first-order logic fragments

Abstract

Recent published work has addressed the Shalqvist correspondence problem for non-distributive logics. The natural question that arises is to identify the fragment of first-order logic that corresponds to logics without distribution, lifting van Benthem's characterization result for modal logic to this new setting. Carrying out this project is the contribution of the present article. The article is intended as a demonstration and application of a project of reduction of non-distributive logics to (sorted) residuated modal logics. The reduction is an application of recent representation results by this author for normal lattice expansions and a generalization of a canonical and fully abstract translation of the language of substructural logics into the language of their companion sorted, residuated modal logics. The reduction of non-distributive logics to sorted modal logics makes the proof of a van Benthem characterization of non-distributive logics nearly effortless, by adapting and reusing existing results, demonstrating the usefulness and suitability of this approach in studying logics that may lack distribution.