<p>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 (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathsf{{CPC}} $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="sans-serif">CPC</mi> </math></EquationSource> </InlineEquation>) and intuitionistic propositional logic (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathsf{{IPC}} $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="sans-serif">IPC</mi> </math></EquationSource> </InlineEquation>) in the possible world semantics. We define the class of mixed models <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {M}\mathcal {M}(\mathsf{{CPC}}, \mathsf{{IPC}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">M</mi> <mi mathvariant="script">M</mi> <mo stretchy="false">(</mo> <mi mathvariant="sans-serif">CPC</mi> <mo>,</mo> <mi mathvariant="sans-serif">IPC</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, together with a subclass of concrete mixed models (<InlineEquation ID="IEq4"> <EquationSource Format="TEX">${\mathcal {CMM}} $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">CMM</mi> </math></EquationSource> </InlineEquation>), and establish their semantic properties. Our main result shows that the set of formulas valid in <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {M}\mathcal {M}(\mathsf{{CPC}}, \mathsf{{IPC}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">M</mi> <mi mathvariant="script">M</mi> <mo stretchy="false">(</mo> <mi mathvariant="sans-serif">CPC</mi> <mo>,</mo> <mi mathvariant="sans-serif">IPC</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> corresponds exactly to the intuitionistic modal logic <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathsf{{iK}} $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="sans-serif">iK</mi> </math></EquationSource> </InlineEquation> extended with the Box Excluded Middle axiom (<InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathsf{{iK}+\mathsf{{bem}}} $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="sans-serif">iK</mi> <mo>+</mo> <mi mathvariant="sans-serif">bem</mi> </mrow> </math></EquationSource> </InlineEquation>). To demonstrate this, we prove soundness and completeness results linking <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathcal {M}\mathcal {M}(\mathsf{{CPC}}, \mathsf{{IPC}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">M</mi> <mi mathvariant="script">M</mi> <mo stretchy="false">(</mo> <mi mathvariant="sans-serif">CPC</mi> <mo>,</mo> <mi mathvariant="sans-serif">IPC</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">${\mathcal {CMM}} $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">CMM</mi> </math></EquationSource> </InlineEquation>, and birelational models for <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\mathsf{{iK}+\mathsf{{bem}}} $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="sans-serif">iK</mi> <mo>+</mo> <mi mathvariant="sans-serif">bem</mi> </mrow> </math></EquationSource> </InlineEquation>.</p>

On the Contingency of Logic in Possible World Semantics

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2025年11月8日

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Iris van der Giessen,
Joost J. Joosten,
Paul Mayaux,
Vicent Navarro Arroyo

摘要

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}} $ $CPC$ ) and intuitionistic propositional logic ( $\mathsf{{IPC}} $ $IPC$ ) in the possible world semantics. We define the class of mixed models $\mathcal {M}\mathcal {M}(\mathsf{{CPC}}, \mathsf{{IPC}})$ $M M (CPC, IPC)$ , together with a subclass of concrete mixed models ( ${\mathcal {CMM}} $ $CMM$ ), and establish their semantic properties. Our main result shows that the set of formulas valid in $\mathcal {M}\mathcal {M}(\mathsf{{CPC}}, \mathsf{{IPC}})$ $M M (CPC, IPC)$ corresponds exactly to the intuitionistic modal logic $\mathsf{{iK}} $ $iK$ extended with the Box Excluded Middle axiom ( $\mathsf{{iK}+\mathsf{{bem}}} $ $iK + bem$ ). To demonstrate this, we prove soundness and completeness results linking $\mathcal {M}\mathcal {M}(\mathsf{{CPC}}, \mathsf{{IPC}})$ $M M (CPC, IPC)$ and ${\mathcal {CMM}} $ $CMM$ , and birelational models for $\mathsf{{iK}+\mathsf{{bem}}} $ $iK + bem$ .