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Proof Theory of Paraconsistent Weak Kleene Logic

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Abstract

Paraconsistent Weak Kleene Logic (PWK) is the 3-valued propositional logic defined on the weak Kleene tables and with two designated values. Most of the existing proof systems for PWK are characterised by the presence of linguistic restrictions on some of their rules. This feature can be seen as a shortcoming. We provide a cut-free calculus (a hybrid between a natural deduction calculus and a sequent calculus) for PWK that is devoid of such provisos. Moreover, we introduce a Priest-style tableaux calculus for PWK.

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Acknowledgements

A preliminary version of this paper was presented at the First Bilateral Workshop UNICA-UNAM (Cagliari, February 2019). Thanks are due to the audience of this talk for their insightful suggestions. We warmly thank Thomas Ferguson, Graham Priest and Damian Szmuc for the very useful feedback, and two anonymous reviewers for their detailed and extremely pertinent comments. F. Paoli gratefully acknowledges the support of the Horizon 2020 program of the European Commission: SYSMICS project, number: 689176, MSCA-RISE-2015 and of MIUR: Project “Theory and applications of resource sensitive logics”, PRIN 2017, Prot. 20173WKCM5. Both authors express their gratitude for the support of Fondazione di Sardegna within the project “Science and its Logics: The Representation’s Dilemma”, Cagliari, CUP: F72 F16 003 220 002, and the Regione Autonoma della Sardegna within the project: “Le proprietà d’ordine in matematica e fisica”, CUP: F72 F16 002 920 002.

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Correspondence to Francesco Paoli.

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Paoli, F., Pra Baldi, M. Proof Theory of Paraconsistent Weak Kleene Logic. Stud Logica 108, 779–802 (2020). https://doi.org/10.1007/s11225-019-09876-z

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