Craig interpolation theorem fails in bi-intuitionistic predicate logic

  • In this article we show that bi-intuitionistic predicate logic lacks the Craig Interpolation Property. We proceed by adapting the counterexample given by Mints, Olkhovikov and Urquhart for intuitionistic predicate logic with constant domains [13]. More precisely, we show that there is a valid implication \(\phi \rightarrow \psi\) with no interpolant. Importantly, this result does not contradict the unfortunately named "Craig interpolation" theorem established by Rauszer in [24] since that article is about the property more correctly named "deductive interpolation" (see Galatos, Jipsen, Kowalski and Ono’s use of this term in [5]) for global consequence. Given that the deduction theorem fails for bi-intuitionistic logic with global consequence, the two formulations of the property are not equivalent.

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Author:Grigory OlkhovikovORCiDGND, Guillermo BadiaORCiDGND
Parent Title (English):The review of symbolic logic
Publisher:Cambridge University Press
Place of publication:Cambridge
Document Type:Article
Date of Publication (online):2024/01/18
Date of first Publication:2022/08/12
Publishing Institution:Ruhr-Universität Bochum, Universitätsbibliothek
Tag:Craig Interpolation Theorem; bi-asimulation; bi-intuitionistic predicate logic
First Page:1
Last Page:23
Institutes/Facilities:Institut für Philosophie I
open_access (DINI-Set):open_access
faculties:Fakultät für Philosophie und Erziehungswissenschaft
Licence (English):License LogoCreative Commons - CC BY 4.0 - Attribution 4.0 International