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| Main Authors: | , |
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| 格式: | Recurso digital |
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| 出版: |
Zenodo
2025
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| 在線閱讀: | https://doi.org/10.5281/zenodo.17834321 |
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書本目錄:
- This paper introduces a novel, unified framework for understanding intuitionistic, modal, and paraconsistent logics through the lens of sheaf theory. By conceptualizing "logical states" as varying sections over a topological space, we demonstrate how the core principles and semantic structures of these diverse logical systems can be elegantly captured and interrelated within a single mathematical paradigm. We define sheaves of logical states, where the stalks represent local interpretations or truth values, and the restriction maps embody the consistency requirements or epistemic access across open sets. For intuitionistic logic, this framework naturally models the Heyting algebra structure arising from open sets of a topological space. For modal logic, Kripke frames are shown to be representable as specific types of topological spaces over which sheaves of propositions are constructed, with accessibility relations encoded in the topology or the sheaf's structure. Crucially, the framework extends to paraconsistent logics by allowing for non-classical truth assignments in the stalks that can tolerate contradictions locally, while global consistency can be maintained or analyzed. This unification not only provides a deeper categorical and topological understanding of these logics but also opens new avenues for exploring their foundational properties, computational implementations, and applications in areas such as knowledge representation, distributed systems, and formal semantics. The paper details the formal constructions, provides examples for each logical system, and discusses the implications of this topological approach for future research in non-classical logics.