Modal meet-implication logic

TL;DR

This paper introduces a modal meet-implication logic extending propositional intuitionistic logic with meet-preserving and monotone modalities, providing semantics, duality, and completeness results, and exploring its expressivity within dialgebraic frameworks.

Contribution

It develops a new modal logic framework for meet-implication fragments, including semantics, duality, and completeness, and extends it with monotone modalities within dialgebraic logic.

Findings

Semantic characterization via semilattices

Completeness and duality results established

Extension with monotone modality proven complete

Abstract

We extend the meet-implication fragment of propositional intuitionistic logic with a meet-preserving modality. We give semantics based on semilattices and a duality result with a suitable notion of descriptive frame. As a consequence we obtain completeness and identify a common (modal) fragment of a large class of modal intuitionistic logics. We recognise this logic as a dialgebraic logic, and as a consequence obtain expressivity-somewhere-else. Within the dialgebraic framework, we then investigate the extension of the meet-implication fragment of propositional intuitionistic logic with a monotone modality and prove completeness and expressivity-somewhere-else for it.