Abstract
We develop a Gentzen-style proof theory for super-Belnap logics (extensions of the four-valued Dunn–Belnap logic), expanding on an approach initiated by Pynko. We show that just like substructural logics may be understood proof-theoretically as logics which relax the structural rules of classical logic but keep its logical rules as well as the rules of Identity and Cut, super-Belnap logics may be seen as logics which relax Identity and Cut but keep the logical rules as well as the structural rules of classical logic. A generalization of the cut elimination theorem for classical propositional logic is then proved and used to establish interpolation for various super-Belnap logics. In particular, we obtain an alternative syntactic proof of a refinement of the Craig interpolation theorem for classical propositional logic discovered recently by Milne.
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Special Issue: 40 years of FDE.
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Přenosil, A. Cut Elimination, Identity Elimination, and Interpolation in Super-Belnap Logics. Stud Logica 105, 1255–1289 (2017). https://doi.org/10.1007/s11225-017-9746-8
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DOI: https://doi.org/10.1007/s11225-017-9746-8