On the Contingency of Logic in Possible World Semantics

Abstract 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 ( $$\mathsf{{CPC}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>CPC</mml:mi> </mml:math> ) and intuitionistic propositional logic ( $$\mathsf{{IPC}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>IPC</mml:mi> </mml:math> ) in the possible world semantics. We define the class of mixed models $$\mathcal {M}\mathcal {M}(\mathsf{{CPC}}, \mathsf{{IPC}})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mi>M</mml:mi> <mml:mo>(</mml:mo> <mml:mi>CPC</mml:mi> <mml:mo>,</mml:mo> <mml:mi>IPC</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , together with a subclass of concrete mixed models ( $${\mathcal {CMM}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>CMM</mml:mi> </mml:math> ), and establish their semantic properties. Our main result shows that the set of formulas valid in $$\mathcal {M}\mathcal {M}(\mathsf{{CPC}}, \mathsf{{IPC}})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mi>M</mml:mi> <mml:mo>(</mml:mo> <mml:mi>CPC</mml:mi> <mml:mo>,</mml:mo> <mml:mi>IPC</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> corresponds exactly to the intuitionistic modal logic $$\mathsf{{iK}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>iK</mml:mi> </mml:math> extended with the Box Excluded Middle axiom ( $$\mathsf{{iK}+\mathsf{{bem}}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>iK</mml:mi> <mml:mo>+</mml:mo> <mml:mi>bem</mml:mi> </mml:mrow> </mml:math> ). To demonstrate this, we prove soundness and completeness results linking $$\mathcal {M}\mathcal {M}(\mathsf{{CPC}}, \mathsf{{IPC}})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mi>M</mml:mi> <mml:mo>(</mml:mo> <mml:mi>CPC</mml:mi> <mml:mo>,</mml:mo> <mml:mi>IPC</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and $${\mathcal {CMM}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>CMM</mml:mi> </mml:math> , and birelational models for $$\mathsf{{iK}+\mathsf{{bem}}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>iK</mml:mi> <mml:mo>+</mml:mo> <mml:mi>bem</mml:mi> </mml:mrow> </mml:math> .