Fundamental Propositional Logic with Strict Implication

Fundamental logic was introduced by Wesley Holliday (2023) to unify intuitionistic logic and quantum logic from a proof-theoretic perspective, capturing the logic determined solely by the introduction and elimination rules of connectives $\mathrm{\neg }$, $\wedge$, $\vee$. This paper incorporates strict implication – standard in intuitionistic logic and a significant candidate for quantum logic – into the framework of fundamental propositional logic. We demonstrate that, unlike the original language, the presence of strict implication causes the semantic consequence relations over pseudo-reflexive pseudo-symmetric frames and reflexive pseudo-symmetric frames to diverge. Consequently, we provide separate axiomatizations for these two logics in the language $\{\perp ,\wedge ,\vee ,\to \}$. Soundness and completeness theorems are established for both systems.