Gödel-McKinsey-Tarski and Blok-Esakia for Heyting-Lewis Implication

De Groot J, Litak TM, Pattinson D (2021)

Publication Type: Conference contribution

Publication year: 2021

Journal

Proceedings - Symposium on Logic in Computer Science Institute of Electrical and Electronics Engineers

Publisher: Institute of Electrical and Electronics Engineers Inc.

Book Volume: 2021-June

Conference Proceedings Title: Proceedings - Symposium on Logic in Computer Science

Event location: Virtual

ISBN: 9781665448956

DOI: 10.1109/LICS52264.2021.9470508

Abstract

Heyting-Lewis Logic is the extension of intuitionistic propositional logic with a strict implication connective that satisfies the constructive counterparts of axioms for strict implication provable in classical modal logics. Variants of this logic are surprisingly widespread: they appear as Curry-Howard correspondents of (simple type theory extended with) Haskell-style arrows, in preservativity logic of Heyting arithmetic, in the proof theory of guarded (co)recursion, and in the generalization of intuitionistic epistemic logic.Heyting-Lewis Logic can be interpreted in intuitionistic Kripke frames extended with a binary relation to account for strict implication. We use this semantics to define descriptive frames (generalisations of Esakia spaces), and establish a categorical duality between the algebraic interpretation and the frame semantics. We then adapt a transformation by Wolter and Zakharyaschev to translate Heyting-Lewis Logic to classical modal logic with two unary operators. This allows us to prove a Blok-Esakia theorem that we then use to obtain both known and new canonicity and correspondence theorems, and the finite model property and decidability for a large family of Heyting-Lewis logics.

Authors with CRIS profile

Tadeusz Michal Litak Lehrstuhl für Informatik 8 (Theoretische Informatik)

Involved external institutions

Australian National University (ANU) AU Australia (AU)

How to cite

APA:

De Groot, J., Litak, T.M., & Pattinson, D. (2021). Gödel-McKinsey-Tarski and Blok-Esakia for Heyting-Lewis Implication. In Proceedings - Symposium on Logic in Computer Science. Virtual: Institute of Electrical and Electronics Engineers Inc..

MLA:

De Groot, Jim, Tadeusz Michal Litak, and Dirk Pattinson. "Gödel-McKinsey-Tarski and Blok-Esakia for Heyting-Lewis Implication." Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2021, Virtual Institute of Electrical and Electronics Engineers Inc., 2021.

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