Paraconsistent quantum logics


Paraconsistent quantum logics are weak forms of quantum logic, where the noncontradiction and the excluded-middle laws are violated. These logics find interesting applications in the operational approach to quantum mechanics. In this paper, we present an axiomatization, a Kripke-style, and an algebraic semantical characterization for two forms of paraconsistent quantum logic. Further developments are contained in Giuntini and Greuling's paper in this issue.

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  1. 1.

    A. R. Anderson and N.D. Belnap, Jr.,Entailment: The Logic of Relevance and Necessity (Princeton University Press, Princeton, 1975).

    Google Scholar 

  2. 2.

    A. Avron, “The semantics and proof theory of linear logic” (Edinburgh, 1987).

  3. 3.

    G. Birkhoff,Lattice Theory (American Mathematical Society, Prividence, 1966).

    Google Scholar 

  4. 4.

    A. Cantini,Properties and Operations, revised and translated edition ofProprieta' e Operazioni (Bibliopolis, Napoli, 1983), in preparation.

    Google Scholar 

  5. 5.

    G. Cattaneo, C. Garola, and G. Nistico', “Preparation-effect versus question-proposition structures,” to be published inInt. J. Theoret. Phys.

  6. 6.

    A. Church, “A formulation of the logic of sense and denotation,” inStructure, Method, and Meaning (The liberal Art Press, 1951).

  7. 7.

    N. Da Costa, “The philosophical import of paraconsistent logic,”J. Non-Class. Logics 1, 1–19 (1982).

    Google Scholar 

  8. 8.

    M. L. Dalla Chiara, “Quantum logic,” inHandbook of Philosophical Logic, III, D. Gabbay and F. Guenthner, eds. (Reidel, Dordrecht, 1986).

    Google Scholar 

  9. 9.

    M. L. Dalla Chiara, “An approach to intensional semantics,”Synthese 73, 479–496 (1987).

    Google Scholar 

  10. 10.

    M. L. Dalla Chiara and G. Toraldo di Francia, “Individuals, kinds, and names in physics,”Versus 40, 29–50 (1985).

    Google Scholar 

  11. 11.

    J. M. Dunn, “Relevance logic and entailment,” inHandbook of Philosophical Logic, III, D. Gabbay and F. Guenthner, eds. (Reidel, Dordrecht, 1986).

    Google Scholar 

  12. 12.

    S. Feferman, “Intensionality in mathematics,”J. Philos. Logic. 14, 41–55 (1985).

    Google Scholar 

  13. 13.

    K. Fine, “Some connections between elementary and modal logic,” inProceedings of the Third Scandinavian Logic Symposium, S. Kanger, ed. (North-Holland, Amsterdam, 1975).

    Google Scholar 

  14. 14.

    J. Y. Girard, “Linear logic,”Theor. Comput. Sci. 50, 1–102 (1987).

    Google Scholar 

  15. 15.

    R. Giuntini, “Brouwer-Zadeh logic and the operational approach to quantum mechanics,” preprint.

  16. 16.

    R. Giuntini, “Brouwer-Zadeh logic,” submitted toStud. Logica.

  17. 17.

    R. Giuntini and H. Greuling, “Toward a formal language for unsharp properties,Found. Phys. 19, 931 (1989).

    Google Scholar 

  18. 18.

    R. Goldblatt, “Semantic analysis of orthologic,”Philos. Logic 3, 19–35 (1974).

    Google Scholar 

  19. 19.

    K. Kraus,States, Effects, and Operations (Springer, Berlin, New York, 1983).

    Google Scholar 

  20. 20.

    G. Ludwig,Foundations of Quantum Mechanics, I (Springer, Berlin, New York, 1983).

    Google Scholar 

  21. 21.

    P. Mittelstaedt,Quantum Logic (Reidel, Dordrecht, 1978).

    Google Scholar 

  22. 22.

    R. Routley and R. K. Meyer, “The semantics of entailment, I,” inTruth, Syntax, and Semantics, H. Leblanc, ed. (North-Holland, Amsterdam, 1973).

    Google Scholar 

  23. 23.

    A. Tarski and J. Lukasiewicz, “Investigations into the sentential calculus,” inLogic, Semantics, Metamathematics (Oxford University Press, Oxford, 1956).

    Google Scholar 

  24. 24.

    A. Urquhart, “Semantics for relevant logics,”J. Symb. Logic 37, 159–169 (1972).

    Google Scholar 

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Chiara, M.L.D., Giuntini, R. Paraconsistent quantum logics. Found Phys 19, 891–904 (1989).

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  • Quantum Mechanic
  • Weak Form
  • Operational Approach
  • Interesting Application
  • Quantum Logic