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CUT-FREE COMPLETENESS FOR MODULAR HYPERSEQUENT CALCULI FOR MODAL LOGICS K, T, AND D

Part of: Proof theory and constructive mathematics General logic

Published online by Cambridge University Press:  21 July 2020

SAMARA BURNS  and
RICHARD ZACH
SAMARA BURNS
Affiliation:
DEPARTMENT OF PHILOSOPHY COLUMBIA UNIVERSITY 1150 AMSTERDAM AVENUE NEW YORK, NY10027, USAE-mail: [email protected]
RICHARD ZACH
Affiliation:
UNIVERSITY OF CALGARY DEPARTMENT OF PHILOSOPHY 2500 UNIVERSITY DRIVE NW CALGARY, ABT2N 1N4, CANADAE-mail: [email protected]: https://richardzach.org/
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Abstract

We investigate a recent proposal for modal hypersequent calculi. The interpretation of relational hypersequents incorporates an accessibility relation along the hypersequent. These systems give the same interpretation of hypersequents as Lellman’s linear nested sequents, but were developed independently by Restall for S5 and extended to other normal modal logics by Parisi. The resulting systems obey Došen’s principle: the modal rules are the same across different modal logics. Different modal systems only differ in the presence or absence of external structural rules. With the exception of S5, the systems are modular in the sense that different structural rules capture different properties of the accessibility relation. We provide the first direct semantical cut-free completeness proofs for K, T, and D, and show how this method fails in the case of B and S4.

Keywords

sequent calculushypersequentslinear nested sequentsmodal logiccompleteness

MSC classification

Primary: 03B45: Modal logic (including the logic of norms)
Secondary: 03F05: Cut-elimination and normal-form theorems
Type
Research Article
Information
The Review of Symbolic Logic , Volume 14 , Issue 4 , December 2021 , pp. 910 - 929
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Copyright
© Association for Symbolic Logic, 2020

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