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MONADIC INTUITIONISTIC AND MODAL LOGICS ADMITTING PROVABILITY INTERPRETATIONS

Part of: General logic Proof theory and constructive mathematics

Published online by Cambridge University Press:  02 December 2021

GURAM BEZHANISHVILI ,
KRISTINA BRANTLEY  and
JULIA ILIN
GURAM BEZHANISHVILI
Affiliation:
DEPARTMENT OF MATHEMATICAL SCIENCES NEW MEXICO STATE UNIVERSITY LAS CRUCES, NM 88003, USA E-mail: guram@nmsu.edu
KRISTINA BRANTLEY*
Affiliation:
DEPARTMENT OF MATHEMATICAL SCIENCES NEW MEXICO STATE UNIVERSITY LAS CRUCES, NM 88003, USA E-mail: guram@nmsu.edu
JULIA ILIN
Affiliation:
INSTITUTE OF LOGIC, LANGUAGE AND COMPUTATION UNIVERSITY OF AMSTERDAM P.O.BOX 94242, 1090 GE AMSTERDAM, THE NETHERLANDS
*
E-mail: kleifest@gmail.com
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Abstract

The Gödel translation provides an embedding of the intuitionistic logic $\mathsf {IPC}$ into the modal logic $\mathsf {Grz}$ , which then embeds into the modal logic $\mathsf {GL}$ via the splitting translation. Combined with Solovay’s theorem that $\mathsf {GL}$ is the modal logic of the provability predicate of Peano Arithmetic $\mathsf {PA}$ , both $\mathsf {IPC}$ and $\mathsf {Grz}$ admit provability interpretations. When attempting to ‘lift’ these results to the monadic extensions $\mathsf {MIPC}$ , $\mathsf {MGrz}$ , and $\mathsf {MGL}$ of these logics, the same techniques no longer work. Following a conjecture made by Esakia, we add an appropriate version of Casari’s formula to these monadic extensions (denoted by a ‘+’), obtaining that the Gödel translation embeds $\mathsf {M^{+}IPC}$ into $\mathsf {M^{+}Grz}$ and the splitting translation embeds $\mathsf {M^{+}Grz}$ into $\mathsf {MGL}$ . As proven by Japaridze, Solovay’s result extends to the monadic system $\mathsf {MGL}$ , which leads us to a provability interpretation of both $\mathsf {M^{+}IPC}$ and $\mathsf {M^{+}Grz}$ .

Keywords

Intuitionistic logicmodal logicGödel translationprovability interpretationpredicate logicsmonadic logicsfinite model property

MSC classification

Primary: 03B45: Modal logic (including the logic of norms)
Secondary: 03F45: Provability logics and related algebras (e.g., diagonalizable algebras) 03B55: Intermediate logics
Type
Article
Information
The Journal of Symbolic Logic , Volume 88 , Issue 1 , March 2023 , pp. 427 - 467
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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