Proof Theory of Paraconsistent Quantum Logic
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.
KeywordsParaconsistent logic Quantum logic Sequent calculus Cut-elimination theorem
We would like to thank anonymous referee for his or her valuable comments and information on the papers  and . 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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