Maximality of bi-intuitionistic propositional logic

TL;DR

This paper characterizes propositional bi-intuitionistic logic as the maximal logic satisfying certain properties, extending Lindström's theorem to a more complex, backward-looking modal setting.

Contribution

It provides a Lindström-style characterization of bi-intuitionistic propositional logic, incorporating new complexities due to the backward-looking modality.

Findings

Bi-intuitionistic logic is maximal with respect to certain properties.

The characterization extends previous work on intuitionistic logic.

The logic satisfies compactness, Tarski union property, and preservation under bi-asimulations.

Abstract

In the style of Lindstr\"om's theorem for classical first-order logic, this article characterizes propositional bi-intuitionistic logic as the maximal (with respect to expressive power) abstract logic satisfying a certain form of compactness, the Tarski union property and preservation under bi-asimulations. Since bi-intuitionistic logic introduces new complexities in the intuitionistic setting by adding the analogue of a backwards looking modality, the present paper constitutes a non-trivial modification of previous work done by the authors for intuitionistic logic in: G. Badia and G. Olkhovikov. A Lindstr\"om theorem for intuitionistic propositional logic. Notre Dame Journal of Formal Logic, 61 (1): 11-30 (2020).