Apartness relations between propositions

We classify all apartness relations definable in propositional logics extending intuitionistic logic using Heyting algebra semantics. We show that every Heyting algebra which contains a non‐trivial apartness term satisfies the weak law of excluded middle, and every Heyting algebra which contains a t...

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Bibliographic Details
Published inMathematical logic quarterly Vol. 70; no. 4; pp. 414 - 428
Main Author Kocsis, Zoltan A.
Format Journal Article
LanguageEnglish
Published Berlin Wiley Subscription Services, Inc 01.11.2024
Subjects
Boolean algebra
Semantics
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Summary:We classify all apartness relations definable in propositional logics extending intuitionistic logic using Heyting algebra semantics. We show that every Heyting algebra which contains a non‐trivial apartness term satisfies the weak law of excluded middle, and every Heyting algebra which contains a tight apartness term is in fact a Boolean algebra. This answers a question of Rijke regarding the correct notion of apartness for propositions, and yields a short classification of apartness terms that can occur in a Heyting algebra. We also show that Martin‐Löf Type Theory is not able to construct non‐trivial apartness relations between propositions.
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ISSN:0942-5616
1521-3870
DOI:10.1002/malq.202300055