Year: 2025
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Author: Chen, Zhicheng
Language: English
Topic: Advanced Algebra and Logic
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4417150412
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