Skip to main content
Log in

Quantum logic revisited

  • Published:
Foundations of Physics Aims and scope Submit manuscript

Abstract

An adequate conjunction-implication pair is given for complete orthomodular lattices. The resulting conjunction is noncommutative in nature. We use the well-known lattice of closed subspaces of a Hilbert space, to give physical meaning to the given lattice operation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. T. L. Bell, “A new approach to quantum logic,”J. Philos. Sci. 37, 83–95 (1986).

    Google Scholar 

  2. E. G. Beltrameti and G. Cassinelli,The Logic of Quantum Mechanics (Encyclopedia of Mathematics and Its Applications) (Addison-Wesley, Reading, Massachusetts, 1981).

    Google Scholar 

  3. G. Birkhoff and J. Von-Neumann, “The logic of quantum mechanics,”Ann. Math. 37, 823–843.

  4. T. A. Brody, “On quantum logic,”Found. Phys. 14, 409–430 (1984).

    Google Scholar 

  5. P. D. Finch, “Quantum logic as an implication algebra,”Bull. Aust. Math. Soc. 1, 101–106 (1970).

    Google Scholar 

  6. D. Foulis, “Baer *-semigroups,”Proc. Am. Math. Soc. 11, 648–654 (1960).

    Google Scholar 

  7. S. P. Gudder and R. H. Schelp, “Coordinatization of orthocomplemented and orthomodular posets,”Proc. Am. Math. Soc. 25, 229–237 (1970).

    Google Scholar 

  8. G. M. Hardegree, “Material implication in orthomodular (and Boolean) lattices,”Notre Dame J. Formal Logic 22, 163–181 (1981).

    Google Scholar 

  9. M. Jammer,The Philosophy of Quantum Mechanics (Wiley, New York, 1974).

    Google Scholar 

  10. J. M. Jauch,Foundations of Quantum Mechanics (Addison-Wesley, Reading, Massachusetts, 1973).

    Google Scholar 

  11. P. T. Johnstone,Stone Spaces (Cambridge University Press, Cambridge, 1982).

    Google Scholar 

  12. M. Maczynski, “Commutativity and generalized transition probability in quantum logic,”Current Issues in Quantum Logic, E. G. Beltrametti and Bas C. van Fraassen, eds. (Plenum, New York, 1981).

    Google Scholar 

  13. C. Mulvey, Communications at meetings in Oberwolfach, Paris, and Brighton, 1984.

  14. A. W. Naylor and G. R. Sell,Linear Operator Theory in Engineering and Science (Springer, New York, 1982).

    Google Scholar 

  15. C. Piron,Foundations of Quantum Physics (W. A. Benjamin, New York, 1976).

    Google Scholar 

  16. L. Román and B. Rumbos, “Remarks on material implication in orthomodular lattices,”Math. Rep. Acad. Sci. X, 279–284 (1988).

    Google Scholar 

  17. L. Román and B. Rumbos, “A characterization of nuclei in orthomodular and quantic lattices,” to appear inJ. Pure Appl. Algebra.

Download references

Author information

Authors and Affiliations

Authors

Additional information

To the memory of Thomas A. Brody.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Román, L., Rumbos, B. Quantum logic revisited. Found Phys 21, 727–734 (1991). https://doi.org/10.1007/BF00733278

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00733278

Keywords

Navigation