On the Contingency of Logic in Possible World Semantics
Fuente:
arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | |
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| _version_ | 1866915563489460224 |
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| author | van der Giessen, Iris Joosten, Joost J. Mayaux, Paul Arroyo, Vicent Navarro |
| author_facet | van der Giessen, Iris Joosten, Joost J. Mayaux, Paul Arroyo, Vicent Navarro |
| contents | This paper investigates the contingency of logic within the framework of possible world semantics. Possible world semantics captures the meaning of necessitation, i.e., a statement is necessarily true if it holds in all possible worlds. Standard Kripkean semantics assumes that all possible worlds are governed by one single logic. We relax this assumption and introduce mixed models, in which different worlds may obey different logical systems. The paper provides a first case study where we mix classical propositional logic ($\mathsf{CPC}$) and intuitionistic propositional logic ($\mathsf{IPC}$) in the possible world semantics. We define the class of mixed models $\mathcal{M}\mathcal{M}(\mathsf{CPC}, \mathsf{IPC})$, together with a subclass of concrete mixed models ($\mathcal{CMM}$), and establish their semantic properties. Our main result shows that the set of formulas valid in $\mathcal{M}\mathcal{M}(\mathsf{CPC}, \mathsf{IPC})$ corresponds exactly to the intuitionistic modal logic $\mathsf{iK}$ extended with the Box Excluded Middle axiom ($\mathsf{iK} + \mathsf{bem}$). To demonstrate this, we prove soundness and completeness results linking $\mathcal{M}\mathcal{M}(\mathsf{CPC}, \mathsf{IPC})$ and $\mathcal{CMM}$, and birelational models for $\mathsf{iK} + \mathsf{bem}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_17302 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Contingency of Logic in Possible World Semantics van der Giessen, Iris Joosten, Joost J. Mayaux, Paul Arroyo, Vicent Navarro Logic Primary 03B45, Secondary 03B05, 03B20 This paper investigates the contingency of logic within the framework of possible world semantics. Possible world semantics captures the meaning of necessitation, i.e., a statement is necessarily true if it holds in all possible worlds. Standard Kripkean semantics assumes that all possible worlds are governed by one single logic. We relax this assumption and introduce mixed models, in which different worlds may obey different logical systems. The paper provides a first case study where we mix classical propositional logic ($\mathsf{CPC}$) and intuitionistic propositional logic ($\mathsf{IPC}$) in the possible world semantics. We define the class of mixed models $\mathcal{M}\mathcal{M}(\mathsf{CPC}, \mathsf{IPC})$, together with a subclass of concrete mixed models ($\mathcal{CMM}$), and establish their semantic properties. Our main result shows that the set of formulas valid in $\mathcal{M}\mathcal{M}(\mathsf{CPC}, \mathsf{IPC})$ corresponds exactly to the intuitionistic modal logic $\mathsf{iK}$ extended with the Box Excluded Middle axiom ($\mathsf{iK} + \mathsf{bem}$). To demonstrate this, we prove soundness and completeness results linking $\mathcal{M}\mathcal{M}(\mathsf{CPC}, \mathsf{IPC})$ and $\mathcal{CMM}$, and birelational models for $\mathsf{iK} + \mathsf{bem}$. |
| title | On the Contingency of Logic in Possible World Semantics |
| topic | Logic Primary 03B45, Secondary 03B05, 03B20 |
| url | https://arxiv.org/abs/2510.17302 |