Journal of Philosophical Logic

, Volume 47, Issue 2, pp 301–324 | Cite as

Proof Theory of Paraconsistent Quantum Logic

Article

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.

Keywords

Paraconsistent logic Quantum logic Sequent calculus Cut-elimination theorem 

Notes

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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Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Teikyo University, Faculty of Science and Engineering, Department of Information and Electronic EngineeringUtsunomiyaJapan

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