A Van Benthem Characterization Result for Distribution-Free Logics

Abstract

This article contributes to recent results in the model theory of distribution-free logics (which include a Goldblatt-Thomason theorem and a development of their Sahlqvist theory) by lifting van Benthem’s characterization result for modal logic to the more general setting of the logics of normal lattice expansions. Our proof approach makes use of a fully abstract translation of the language of the logics of interest into the language of sorted residuated modal logic, building on an analogous translation of substructural logics recently published by the author. 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 makes the proof of a van Benthem characterization of non-distributive logics possible, by adapting, reusing and generalizing existing results.