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Modal and Intuitionistic Variants of Extended Belnap–Dunn Logic with Classical Negation

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Abstract

In this study, we introduce Gentzen-type sequent calculi BDm and BDi for a modal extension and an intuitionistic modification, respectively, of De and Omori’s extended Belnap–Dunn logic BD+ with classical negation. We prove theorems for syntactically and semantically embedding BDm and BDi into Gentzen-type sequent calculi S4 and LJ for normal modal logic and intuitionistic logic, respectively. The cut-elimination, decidability, and completeness theorems for BDm and BDi are obtained using these embedding theorems. Moreover, we prove the Glivenko theorem for embedding BD+ into BDi and the McKinsey–Tarski theorem for embedding BDi into BDm.

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Notes

  1. Negation of this kind, which appears in Belnap–Dunn logic, is also referred to as De Morgan negation.

  2. The name “LK” is from Gentzen’s original sequent calculus LK for first-order classical logic (Gentzen 1969). However, in this paper, the name LK is used for a non-essential modification of the propositional fragment of the original LK.

  3. The name “LJ” is from Gentzen’s original sequent calculus LJ for first-order intuitionistic logic (Gentzen 1969). However, in this paper, the name LJ is used for a non-essential modification of the propositional fragment of the original LJ.

  4. Other issues on paraconsistent negation, see (Wansing and Odintsov 2016; Kamide 2017) and the references therein. For example, an issue on the double negation elimination axiom \({\sim }{\sim }\alpha \leftrightarrow \alpha \) in paraconsistent logics was discussed in (Kamide 2017).

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Acknowledgements

We would like to thank the anonymous referees for their valuable comments and suggestions. This work was supported by JSPS KAKENHI Grant Numbers JP18K11171 and JP16KK0007.

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  1. Department of Information and Electronic Engineering, Teikyo University, Toyosatodai 1-1, Utsunomiya, Tochigi, 320-8551, Japan

    Norihiro Kamide

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Kamide, N. Modal and Intuitionistic Variants of Extended Belnap–Dunn Logic with Classical Negation. J of Log Lang and Inf 30, 491–531 (2021). https://doi.org/10.1007/s10849-021-09330-1

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