Skip to main content
SpringerLink
Log in

An axiom system for orthomodular quantum logic

  • Published:
Studia Logica Aims and scope Submit manuscript

Abstract

Logical matrices for orthomodular logic are introduced. The underlying algebraic structures are orthomodular lattices, where the conditional connective is the Sasaki arrow. An axiomatic calculusOMC is proposed for the orthomodular-valid formulas.OMC is based on two primitive connectives — the conditional, and the falsity constant. Of the five axiom schemata and two rules, only one pertains to the falsity constant. Soundness is routine. Completeness is demonstrated using standard algebraic techniques. The Lindenbaum-Tarski algebra ofOMC is constructed, and it is shown to be an orthomodular lattice whose unit element is the equivalence class of theses ofOMC.

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

Access this article

Log in via an institution

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. J. C. Abbott,Orthoimplication algebras,Studia, Logica 35 (1976), pp. 173–77.

    Google Scholar 

  2. A. R. Anderson andN. D. Belnap,Entailment vol. I, Princeton University Press, Princeton 1975.

    Google Scholar 

  3. G. Birkhoff,Lattice Theory, American Mathematical Society Colloquium Publications, Providence 1967.

    Google Scholar 

  4. G. Birkhoff andJ. Von Neumann,The logic of quantum mechanics Annals of Mathematics 37 (1936), pp. 823–43.

    Google Scholar 

  5. T. S. Blyth andM. F. Janowitz,Residuation Theory, Pergamon Press, Oxford 1972.

    Google Scholar 

  6. I. D. Clark,An axiomatization of quantum logic,The Journal of Symbolic Logic 38 (1973), pp. 389–92.

    Google Scholar 

  7. J. M. Dunn,The algebra of R,Entailment, vol. I, Princeton University Press, Princeton 1975, pp. 349–71.

    Google Scholar 

  8. P. D. Finch,Sasaki projections on orthocomplemented posets,Bulletin of the Australian Mathematical Society 1 (1969), pp. 319–24.

    Google Scholar 

  9. —,Quantum logic as an implication algebra, ibid., 2 (1970), pp. 101–6.

    Google Scholar 

  10. R. I. Goldblatt,Semantic analysis of orthologic,Journal of Philosophical Logic 3 (1974), pp. 19–35.

    Google Scholar 

  11. R. J. Greechie andS. P. Gudder,Quantum logics, in C.A. Hooker (ed.),Coniemporary Research in the Foundations and Philosophy of Quantum Theory, D. Reidel, Dordrecht 1973.

    Google Scholar 

  12. I. Hacking,Why orthomodular quantum logic is logic, unpublished manuscript.

  13. G. M. Hardegree,The conditional in quantum logic,Synthese 29 (1974), pp. 63–80, reprinted in P. A. Suppes (ed.)Logic and Probability in Quantum Mechanics D. Reidel, Dordrecht 1976, pp. 55–72.

    Google Scholar 

  14. —,Quasi-implicative lattices and the logic of quantum mechanics,Zeitschrift für Naturforschung 30a (1975), pp. 1347–60.

    Google Scholar 

  15. —,Stalnaker conditionals and quantum logic,Journal of Philosophical Logic 4 (1975), pp. 399–421.

    Google Scholar 

  16. -,Material implication in orthomodular (and Boolean) lattices,Notre Dame Journal of Formal Logic in press.

  17. -,Quasi-implication algebras, part I:Elementary Theory, part II:Structure Theory,Algebra Universalis, in press.

  18. —,The conditional in abstract and concrete quantum logic, in C. A. Hooker (ed.)The Logico-Algebraic Approach to Quantum Mechanics, vol. 2, D. Reidel, Dordrecht 1979, pp. 49–108.

    Google Scholar 

  19. -,Axiomatizing the class of orthomodular quantum logically valid arguments, to appear.

  20. L. Herman, E. Marsden andR. Piziak,Implication connectives in orthomodular lattices,Notre Dame Journal of Formal Logic 16 (1975), pp. 305–28.

    Google Scholar 

  21. L. Herman andR. Piziak,Modal propositional logic on an orthomodular basis,The Journal of Symbolic Logic 39 (1974), pp. 478–88.

    Google Scholar 

  22. S. S. Holland,The current interest in orthomodular lattices, in J. C. Abbott (ed.)Trends in Lattice Theory, Van Nostrand Reinhold, New York, 1970.

    Google Scholar 

  23. G. Kalmbach,Orthomodular logic,Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 20 (1974), pp. 395–406.

    Google Scholar 

  24. R. J. Kimble,Ortho-implication algebras,Notices of the American Mathematical Society 16 (1969), p. 772.

    Google Scholar 

  25. J. Kotas,An axiom system, for the modular logic,Studia Logica 21 (1967), pp. 17–38.

    Google Scholar 

  26. H. Kunsemüller,Zur axiomatik der quantenlogik,Philosophia Naturalis 8 (1964), pp. 363–76.

    Google Scholar 

  27. P. Mittelstaedt,On the interpretation of the lattice of subspaces of Hubert Space as a prepositional calculus,Zeitschrift für Naturforschung 27a (1972), pp, 1358–62.

    Google Scholar 

  28. P. Mittelstaedt andE. W. Stachow,Operational foundation of quantum logic.Foundations of Physics 4 (1974), pp. 355–65.

    Google Scholar 

  29. R. Piziak,Orthomodular lattices as implication algebras,Journal of Philosophical Logic 3 (1974), pp. 413–18.

    Google Scholar 

  30. U. Sasaki,Orthocomplemented lattices satisfying the exchange axiom,Journal of Science of the Hiroshima University 17a (1954), pp. 293–302.

    Google Scholar 

  31. B. C. van Fraassen,Formal Semantics and Logic, Macmillan, New York 1971.

    Google Scholar 

  32. J. Von Neumann,The Mathematical Foundations of Quantum Mechanics Tr. R. T. Beyer, Princeton University Press, Princeton 1955.

    Google Scholar 

  33. J. J. Zeman,Quantum logic with implication,Notre Dame Journal of Formal Logic 20 (1979), pp. 723–728.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Philosophy, University of Massachusetts, USA

    Gary M. Hardegree

Authors
  1. Gary M. Hardegree
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This research was supported by National Science Foundation Grant Number SOC76-82527.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hardegree, G.M. An axiom system for orthomodular quantum logic. Stud Logica 40, 1–12 (1981). https://doi.org/10.1007/BF01837551

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01837551

Keywords

Search

Navigation