A unified relational semantics for BPL, IPL and OL -- axiomatization without disjunction

TL;DR

This paper introduces a unified relational semantics for several propositional logics, axiomatizes them with various frame conditions, and provides translations into modal logics, enhancing understanding of their foundational relationships.

Contribution

It presents a novel unified semantics for BPL, IPL, and OL, along with axiomatizations and modal translations, bridging different logical systems.

Findings

Unified semantics for BPL, IPL, and OL established.

Axiomatizations for various frame conditions provided.

Translations into modal logics developed.

Abstract

In this paper, we propose a relational semantics of propositional language, which unifies the relational semantics of intuitionistic logic, Visser's Basic Propositional Logic and orthologic. Working in language $\{\bot,\land,\neg\}$ and $\{\bot,\land,\to\}$ respectively, we axiomatize this basic logic as well as stronger ones corresponding to different combinations of frame conditions: reflexivity, symmetry, and transitivity. We also provide translations from these propositional logics into modal logics.