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

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 $\{\mathrm{\perp },\wedge ,\mathrm{\neg }\}$ and $\{\mathrm{\perp },\wedge ,\to \}$ 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.