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 $\neg$, $\wedge$, $\vee$. 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 ${\perp, \wedge, \vee, \rightarrow}$. Soundness and completeness theorems are established for both systems.

💡 Research Summary

The paper extends Wesley Holliday’s “fundamental propositional logic” (FPL) – a logic that captures the inferential behavior of ¬, ∧, and ∨ solely through their introduction and elimination rules – by adding the strict implication connective (→). In the original FPL, the semantic consequence relation is the same over two classes of Kripke‑style frames: pseudo‑reflexive pseudo‑symmetric 2⊥‑models (denoted D₁) and reflexive pseudo‑symmetric 2⊥‑models (denoted D₂). The authors show that once → is present, these two consequence relations diverge: there are formulas (e.g., {p, p→q}⊭₁ q) that are not derivable in D₁ but are valid in D₂. This divergence stems from the fact that reflexivity guarantees the usual transitivity property of strict implication, which pseudo‑reflexivity alone does not.

To address the split, the paper provides two separate Hilbert‑style axiomatizations, ⊢₁ for D₁ and ⊢₂ for D₂, both formulated in the language {⊥, ∧, ∨, →} (negation is defined as α→⊥). The basic rule set includes the usual structural rules (cut, monotonicity), conjunction and disjunction introduction/elimination, and three rules governing →: →‑introduction (α⊢β ⇒ ⊢α→β), →‑distribution over ∧, and →‑transitivity. On top of these, the authors add frame‑specific rules:

• (Abs) α ∧ ¬α ⊢ ⊥, which corresponds to strong pseudo‑reflexivity;
• (¬¬I) α ⊢ ¬¬α, which corresponds to weak pseudo‑symmetry;
• (Refl1) and (Refl2), which encode the reflexivity condition in a way that works uniformly for both systems.

The paper proves that (Abs) is sound exactly on frames that are strongly pseudo‑reflexive, while (¬¬I) is sound exactly on weakly pseudo‑symmetric frames. Both (Refl1) and (Refl2) are shown to be sound on any reflexive frame. Consequently, ⊢₁ (basic rules + (Abs) + (Refl1, Refl2)) is sound and complete for D₁, and ⊢₂ (basic rules + (¬¬I) + (Refl1, Refl2)) is sound and complete for D₂.

The completeness proofs follow a canonical‑model construction. Worlds are pairs of sets of formulas (Γ⁺, Γ⁻) satisfying closure conditions derived from the proof system. The authors define a Galois connection between the operators ⊥F = 2F∅ and 2F, and use the closure operator 2F⊥F to interpret ∧, ∨, and → on frames. By showing that every maximally consistent set yields a world in a model belonging to the appropriate class (D₁ or D₂), they establish the usual equivalence Γ⊢ₖϕ ⇔ Γ⊨ₖϕ for k = 1,2.

An appendix supplies the completeness proof for the minimal 2⊥‑logic ⊢K, which underlies both ⊢₁ and ⊢₂. The authors also discuss the role of successor‑serial frames, noting that on such frames the defined negation coincides with the standard strict negation.

Overall, the paper demonstrates that strict implication introduces a subtle but crucial distinction between pseudo‑reflexive and reflexive frames, necessitating separate axiomatizations. This result deepens our understanding of how implication interacts with non‑classical lattice‑based semantics and provides a solid proof‑theoretic foundation for future work on quantum‑style logics that incorporate a strict conditional.