S5-Style Non-Standard Modalities in a Hypersequent Framework

Yaroslav Petrukhin

doi:10.12775/LLP.2021.020

Authors

Yaroslav Petrukhin Department of Logic, University of Lódź https://orcid.org/0000-0002-7731-1339

DOI:

https://doi.org/10.12775/LLP.2021.020

Keywords

hypersequent calculus, cut elimination, proof theory, modal logic, contingency logic, essence logic, accident logic

Abstract

The aim of the paper is to present some non-standard modalities (such as non-contingency, contingency, essence and accident) based on S5-models in a framework of cut-free hypersequent calculi. We also study negated modalities, i.e. negated necessity and negated possibility, which produce paraconsistent and paracomplete negations respectively. As a basis for our calculi, we use Restall's cut-free hypersequent calculus for S5. We modify its rules for the above-mentioned modalities and prove strong soundness and completeness theorems by a Hintikka-style argument. As a consequence, we obtain a cut admissibility theorem. Finally, we present a constructive syntactic proof of cut elimination theorem.

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