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Proof Theory of Paraconsistent Quantum Logic

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Abstract

Paraconsistent quantum logic, a hybrid of minimal quantum logic and paraconsistent four-valued logic, is introduced as Gentzen-type sequent calculi, and the cut-elimination theorems for these calculi are proved. This logic is shown to be decidable through the use of these calculi. A first-order extension of this logic is also shown to be decidable. The relationship between minimal quantum logic and paraconsistent four-valued logic is clarified, and a survey of existing Gentzen-type sequent calculi for these logics and their close relatives is addressed.

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Notes

  1. The denomination “orthologic” is now much more widespread.

  2. The information on the paper [17] was given by an anonymous referee

  3. The information on the paper [23] was given by an anonymous referee

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Acknowledgments

We would like to thank anonymous referee for his or her valuable comments and information on the papers [17] and [23]. We would also like to thank Prof. Mitio Takano for his helpful comments on an early version of this paper. This work was supported by JSPS KAKENHI Grant (C) JP26330263.

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

    Norihiro Kamide

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  1. Norihiro Kamide
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Correspondence to Norihiro Kamide.

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Kamide, N. Proof Theory of Paraconsistent Quantum Logic. J Philos Logic 47, 301–324 (2018). https://doi.org/10.1007/s10992-017-9428-z

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