Skip to main content

Natural Deduction for Quantum Logic

Abstract

This paper presents a natural deduction system for orthomodular quantum logic. The system is shown to be provably equivalent to Nishimura’s quantum sequent calculus. Through the Curry–Howard isomorphism, quantum \(\lambda \)-calculus is also introduced for which strong normalization property is established.

This is a preview of subscription content, access via your institution.

Notes

  1. We are using slightly different symbols and names than those used in [13].

References

  1. Chajda, I., Halaš, R.: An implication in orthologic. Int. J. Theor. Phys. 44, 735–744 (2005)

    MathSciNet  Article  Google Scholar 

  2. Chajda, I.: The axioms for implication in orthologic. Czechoslov. Math. J. 58, 15–21 (2008)

    MathSciNet  Article  Google Scholar 

  3. Cutland, N.J., Gibbins, P.F.: A regular sequent calculus for quantum logic in which \(\wedge \) and \(\vee \) are dual. Logique Anal. (N.S.) 25, 221–248 (1982)

    MathSciNet  MATH  Google Scholar 

  4. Dalla Chiara, M.L., Giuntini, R.: Quantum logics. In: Gabbay, D.M., Guenthner, F. (eds.), Handbook of Philosophical Logic, vol. 6, Springer, pp. 129–228 (2002)

  5. Delmas-Rigoutsos, Y.: A double deduction system for quantum logic based on natural deduction. J. Philos. Log. 26, 57–67 (1997)

    MathSciNet  Article  Google Scholar 

  6. Engesser, K., Gabbay, D., Lehmann, D.: Nonmonotonicity and holicity in quantum logic. In: Handbook of Quantum Logic and Quantum Structures: Quantum Logic, Engesser, K., Gabbay, D., Lehmann, D. (eds.), Elsevier, pp. 587–623 (2009)

  7. Faggian, C., Sambin, G.: From basic logic to quantum logics with cut-elimination. Int. J. Theor. Phys. 37, 31–37 (1998)

    MathSciNet  Article  Google Scholar 

  8. Girard, J.Y., Taylor, P., Lafont, Y.: Proofs and Types. Cambridge University Press (1989)

  9. Hardegree, G.M.: The conditional in quantum logic. Synthese 29, 63–80 (1974)

    Article  Google Scholar 

  10. Harding, J.: The source of the orthomodular law. In: Engesser, K., Gabbay, D., Lehmann, D. (eds.), Handbook of Quantum Logic and Quantum Structures: Quantum Structures, Elsevier, pp. 555–586 (2007)

  11. Herman, L., Marsden, E.L., Piziak, R.: Implication connectives in orthomodular lattices. Notre Dame J. Formal Log. 16, 305–328 (1975)

    MathSciNet  Article  Google Scholar 

  12. Malinowski, J.: The deduction theorem for quantum logic: some negative results. J. Symb. Log. 55, 615–625 (1990)

    MathSciNet  Article  Google Scholar 

  13. Nishimura, H.: Sequential method in quantum logic. J. Symb. Log. 45, 339–352 (1980)

    MathSciNet  Article  Google Scholar 

  14. Nishimura, H.: Gentzen methods in quantum logic. In: Engesser, K., Gabbay, D., Lehmann, D. (eds.), Handbook of Quantum Logic and Quantum Structures: Quantum Logic, Elsevier, pp. 227–260 (2009)

  15. Pavičić, M.: Minimal quantum logic with merged implications. Int. J. Theor. Phys. 26, 845–852 (1987)

    MathSciNet  Article  Google Scholar 

  16. Pavičić, M., Megill, N.D.: Binary orthologic with modus ponens is either orthomodular or distributive. Helv. Phys. Acta 71, 610–628 (1998)

    MathSciNet  MATH  Google Scholar 

  17. Restall, G.: Normal proofs, cut free derivations and structural rules. Stud. Logica 102, 1143–1166 (2014)

    MathSciNet  Article  Google Scholar 

  18. Roman, L., Zuazua, R.E.: Quantum implication. Int. J. Theor. Phys. 38, 793–797 (1999)

    MathSciNet  Article  Google Scholar 

  19. Sambin, G., Battilotti, G., Faggian, C.: Basic logic: reflection, symmetry, visibility. J. Symb. Log. 65, 979–1013 (2000)

    MathSciNet  Article  Google Scholar 

  20. Selinger, P., Valiron, B.: A lambda calculus for quantum computation with classical control. In: Urzyczyn, P. (eds.), Lecture Notes in Computer Science, vol. 3461, Springer, pp. 227–260 (2005)

  21. van Tonder, A.: A lambda calculus for quantum computation. SIAM J. Comput. 33, 1109–1135 (2004)

    MathSciNet  Article  Google Scholar 

  22. Ying, M.: A theory of computation based on quantum logic (I). Theoret. Comput. Sci. 344, 134–207 (2005)

    MathSciNet  Article  Google Scholar 

  23. Younes, Y., Schmitt, I.: On quantum implication. Quantum Mach. Intell. 1, 53–63 (2019)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to K. Tokuo.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Tokuo, K. Natural Deduction for Quantum Logic. Log. Univers. (2022). https://doi.org/10.1007/s11787-022-00307-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11787-022-00307-7

Keywords

  • Quantum logic
  • Natural deduction
  • \(\lambda \)-Calculus
  • Curry–Howard isomorphism
  • Normalization

Mathematics Subject Classification

  • Primary 03G12
  • Secondary 03B60
  • 03F03
  • 68N18
  • 81P10