Journal of Philosophical Logic

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

Proof Theory of Paraconsistent Quantum Logic

  • Norihiro Kamide


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.


Paraconsistent 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 [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.


  1. 1.
    Almukdad, A., & Nelson, D. (1984). Constructible falsity and inexact predicates. Journal of Symbolic Logic, 49(1), 231–233.Google Scholar
  2. 2.
    Anderson, A.R., Belnap, N.D., & et al. (1975). Entailment: the logic of relevance and necessity, Vol. 1, Princeton University Press.Google Scholar
  3. 3.
    Aoyama, H. (2003). On a weak system of sequent calculus. Journal of Logical Philosophy, 3, 29–37.Google Scholar
  4. 4.
    Aoyama, H. (2004). LK, LJ, dual intuitionistic logic, and quantum logic. Notre Dame Journal of Formal Logic, 45(4), 193–213.CrossRefGoogle Scholar
  5. 5.
    Belnap, N.D. (1977). Modern uses of multiple-valued logic. In Epstein, G., & Dunn, J.M. (Eds.), Useful four-valued logic, A (pp. 7–37). Dordrecht: Reidel.Google Scholar
  6. 6.
    Birkhoff, G., & von Neumann, J. (1936). The logic of quantum mechanics. Annals of Mathematics, 37, 823–843.CrossRefGoogle Scholar
  7. 7.
    Cockett, J.R., & Seely, R.A.G. (2001). Finite sum-product logic. Theory and Applications of Categories, 8(5), 63–99.Google Scholar
  8. 8.
    Cutland, N.J., & Gibbins, P.F. (1982). A regular sequent calculus for quantum logic in which ∧ and ∨ are dual. Logique et Analyse, 99, 221–248.Google Scholar
  9. 9.
    Dalla Chiara, M.L., & Giuntini, R. (1989). Paraconsistent quantum logics. Foundations of Physics, 19(7), 891–904.CrossRefGoogle Scholar
  10. 10.
    Dunn, J.M. (1976). Intuitive semantics for first-degree entailment and ‘coupled trees’. Philosophical Studies, 29(3), 149–168.CrossRefGoogle Scholar
  11. 11.
    Dunn, J.M. (2000). Partiality and its dual. Studia Logica, 65, 5–40.CrossRefGoogle Scholar
  12. 12.
    Faggian, C., & Sambin, G. (1998). From basic logic to quantum logics with cut-elimination. International Journal of Theoretical Physics, 37(1), 31–37.CrossRefGoogle Scholar
  13. 13.
    Font, J.N. (1997). Belnap’s four-valued logic and de Morgan lattices. Logic Journal of the IGPL, 5(3), 413–440.CrossRefGoogle Scholar
  14. 14.
    Goldblatt, R. (1974). Semantic analysis of orthologic. Journal of Philosophical Logic, 3(1-2), 19–35.CrossRefGoogle Scholar
  15. 15.
    Kamide, N., & Wansing, H. (2015). Proof theory of N4-related paraconsistent logics, Studies in Logic 54. College Publications.Google Scholar
  16. 16.
    Mey, D. (1989). A predicate calculus with control of derivations, Proceedings of the 3rd Workshop on Computer Science Logic, Lecture Notes in Computer Science, 440, 254–266.CrossRefGoogle Scholar
  17. 17.
    Mönting, J.S. (1981). Cut elimination and word problems for varieties of lattices. Algebra Universalis, 12, 290–321.CrossRefGoogle Scholar
  18. 18.
    Nelson, D. (1949). Constructible falsity. Journal of Symbolic Logic, 14, 16–26.CrossRefGoogle Scholar
  19. 19.
    Nishimura, H. (1980). Sequential method in quantum logic. Journal of Symbolic Logic, 45, 339–352.CrossRefGoogle Scholar
  20. 20.
    Nishimura, H. (1994). Proof theory for minimal quantum logic I. International Journal of Theoretical Physics, 33(1), 103–113.CrossRefGoogle Scholar
  21. 21.
    Nishimura, H. (1994). Proof theory for minimal quantum logic II. International Journal of Theoretical Physics, 33(7), 1427–1443.CrossRefGoogle Scholar
  22. 22.
    Pynko, A.P. (1995). Characterizing Belnap’s logic via de Morgan’s laws. Mathematical Logic Quarterly, 41, 442–454.CrossRefGoogle Scholar
  23. 23.
    Restall, G., & Paoli, F. (2005). The geometry of nondistributive logics. Journal of Symbolic Logic, 70(4), 1108–1126.CrossRefGoogle Scholar
  24. 24.
    Sambin, G., Battilotti, C., & Faggian, C. (2000). Basic logic: reflection, symmetry, visibility. Journal of Symbolic Logic, 65(3), 979–1013.CrossRefGoogle Scholar
  25. 25.
    Takano, M. (1995). Proof theory for minimal quantum logic: a remark. International Journal of Theoretical Physics, 34(4), 649–654.CrossRefGoogle Scholar
  26. 26.
    Tamura, S. (1988). A Gentzen formulation without the cut rule for ortholattices. Kobe Journal of Mathematics, 5, 133–15-.Google Scholar

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© 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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