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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2503.16651 |
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| _version_ | 1866912864772554752 |
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| author | Chen, Zhicheng |
| author_facet | Chen, Zhicheng |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_16651 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fundamental Propositional Logic with Strict Implication Chen, Zhicheng Logic 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. |
| title | Fundamental Propositional Logic with Strict Implication |
| topic | Logic |
| url | https://arxiv.org/abs/2503.16651 |