S5-Style Non-Standard Modalities in a Hypersequent Framework
DOI:
https://doi.org/10.12775/LLP.2021.020Keywords
hypersequent calculus, cut elimination, proof theory, modal logic, contingency logic, essence logic, accident logicAbstract
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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