A van Benthem Theorem for Atomic and Molecular Logics

TL;DR

This paper generalizes van Benthem's modal characterization theorem to atomic and molecular logics, defining bisimulation notions from truth conditions and applying the results to various logical systems.

Contribution

It introduces a framework for bisimulation in atomic and molecular logics and extends the van Benthem theorem to these logics, including case studies.

Findings

Bisimulation notions can be derived from truth conditions.

A generalized van Benthem theorem holds for molecular logics.

Application to modal, Lambek calculus, and intuitionistic logic.

Abstract

After recalling the definitions of atomic and molecular logics, we show how notions of bisimulation can be automatically defined from the truth conditions of the connectives of any of these logics. Then, we prove a generalization of van Benthem modal characterization theorem for molecular logics. Our molecular connectives should be uniform and contain all conjunctions and disjunctions. We use modal logic, the Lambek calculus and modal intuitionistic logic as case study and compare in particular our work with Olkhovikov's work.