Advertisement

Quantum Logics

  • Maria Luisa Dalla Chiara
  • Roberto Giuntini
Chapter
Part of the Handbook of Philosophical Logic book series (HALO, volume 6)

Abstract

The official birth of quantum logic is represented by a famous article of Birkhoff and von Neumann “The logic of quantum mechanics” [Birkhoff and von Neumann, 1936]. At the very beginning of their paper, Birkhoff and von Neumann observe:

One of the aspects of quantum theory which has attracted the most general attention, is the novelty of the logical notions which it presupposes .... The object of the present paper is to discover what logical structures one may hope to find in physical theories which, like quantum mechanics, do not conform to classical logic.

Keywords

Classical Logic Effect Algebra Quantum Logic Accessibility Relation Intuitionistic Logic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. [Battilotti, 1998]
    G. Battilotti. Embedding classical logic into basic orthologic with a primitive modality, Logic Journal of the IGPL, 6, 383–402, 1998.CrossRefGoogle Scholar
  2. [Battilotti and Sambin, 1999]
    G. Battilotti and G. Sambin. Basic logic and the cube of its extensions, in A. Cantini, E. Casari, and P. Minari (eds), Logic and Foundations of Mathematics, pp. 165–186. Kluwer, Dordrecht, 1999.Google Scholar
  3. [Bell and Slomson, 1969]
    J. L. Bell and A. B. Slomson. Models and Ultraproducts: An Introduction, North-Holland, Amsterdam, 1969.Google Scholar
  4. [Beltrametti and Cassinelli, 1981]
    E. Beltrametti and G. Cassinelli. The Logic of Quantum Mechanics, Vol. 15 of Encyclopedia of Mathematics and its Applications, Addison-Wesley, Reading, 1981.Google Scholar
  5. [Birkhoff, 1995]
    G. Birkhoff. Lattice Theory, Vol. 25 of Colloquium Publications, 38 edn, American Mathematical Society, Providence, 1995.Google Scholar
  6. [Birkhoff and von Neumann, 1936]
    G. Birkhoff and J. von Neumann. The logic of quantum mechanics, Annals of Mathematics 37, 823–843, 1936.Google Scholar
  7. [Cattaneo and Laudisa, 1994]
    G. Cattaneo and F. Laudisa. Axiomatic unsharp quantum mechanics, Foundations of Physics 24, 631–684, 1984.CrossRefGoogle Scholar
  8. [Cattaneo and Nisticò, 1986]
    G. Cattaneo and G. Nisticò. Brouwer—Zadeh posets and three-valued Lukasiewicz posets, Fuzzy Sets and Systems 33, 165–190, 1986.Google Scholar
  9. [Chang, 1957]
    C. C. Chang. Algebraic analysis of many valued logics, Transactions of the American Mathematical Society 88, 74–80. 1957.Google Scholar
  10. [Chang, 1958]
    C. C. Chang. A new proof of the completeness of Lukasiewicz axioms, Transactions of the American Mathematical Society 93, 467–490, 1958.CrossRefGoogle Scholar
  11. [Dalla Chiara, 1981]
    M. L. Dalla Chiara. Some metalogical pathologies of quantum logic, in E. Beltrametti and B. V. Fraassen (eds), Current Issues in Quantum Logic, Vol. 8 of Ettore Majorana International Science Series, Plenum, New York, pp. 147–159, 1981.CrossRefGoogle Scholar
  12. [Cutland and Gibbins, 1982]
    N. Cutland and P. Gibbins. A regular sequent calculus for quantum logic in which A and V are dual, Logique et Analyse–Nouvelle Serie - 25 (45), 221–248, 1982.Google Scholar
  13. [da Costa et al., 1992]
    N. C. A. da Costa, S. French and D. Krause. The Schrödinger problem, in M. Bibtol and O. Darrigol (eds), Erwin Schrödinger: Philosophy and the Birth of Quantum Mechanics, Editions Frontières, pp. 445–460, 1992.Google Scholar
  14. [Dalla Chiara and Giuntini, 1994]
    M. L. Dalla Chiara and R. Giuntini. Unsharp quantum logics, Foundations of Physics 24, 1161–1177, 1994.CrossRefGoogle Scholar
  15. [Dalla Chiara and Giuntini, 1995]
    M. L. Dalla Chiara and R. Giuntini. The logics of orthoalgebras, Studia Logica 55, 3–22, 1995.CrossRefGoogle Scholar
  16. [Dalla Chiara and Toraldo di Francia, 1993]
    M. L. Dalla Chiara and G. Toraldo di Francia. Individuals, kinds and names in physics, in G. Corsi, M. L. Dalla Chiara and G. Ghirardi (eds), Bridging the Gap: Philosophy, Mathematics, and Physics, Kluwer Academic Publisher, Dordrecht, pp. 261–283, 1993.CrossRefGoogle Scholar
  17. [Davies, 1976]
    E. B. Davies.Quantum Theory of Open SystemsAcademic, New York, 1976 Google Scholar
  18. [Dishkant, 1972]
    H. Dishkant. Semantics of the minimal logic of quantum mechanics, Studia Logica 30, 17–29, 1972.Google Scholar
  19. [Dummett, 1976]
    M. Dummett. Introduction to quantum logic, unpublished, 1976.Google Scholar
  20. [Dvurecenskij and Pulmannov£, 1994]
    A. Dvurecenskij and S. Pulmannovâ. D-test spaces and difference poset, Reports on Mathematical Physics 34, 151–170, 1994.CrossRefGoogle Scholar
  21. [Faggian, 1997]
    C. Faggian. Classical proofs via basic logic, in Proceedings of CSL ‘87, pp. 203–219. Lectures Notes in Computer Science 1414,Springer, Berlin, 1997.Google Scholar
  22. [Faggian and Sambin, 1997]
    C Faggian and G. Sambin. From basic logic to quantum logics with cut elimination, International Journal of Theoretical Physics 12, 1997.Google Scholar
  23. [Finch, 1970]
    P. D. Finch. Quantum logic as an implication algebra, Bulletin of the Australian Mathematical Society 2, 101–106, 1970.CrossRefGoogle Scholar
  24. [Foulis and Bennett, 1994]
    D. J. Foulis and M. K. Bennett. Effect algebras and unsharp quantum logics, Foundations of Physics 24, 1325–1346, 1994.CrossRefGoogle Scholar
  25. [Gibbins, 1985]
    P. Gibbins. A user-friendly quantum logic, Logique-et-Analyse.- Nouvelle-Serie 28, 353–362, 1985.Google Scholar
  26. [Gibbins, 1987]
    P. Gibbins. Particles and Paradoxes - The Limits of Quantum Logic, Cambridge University Press, Cambridge, 1987.CrossRefGoogle Scholar
  27. [Girard, 1987]
    J. Y. Girard. Liner logic, Theoretical Computer Science 50, 1–102, 1987.CrossRefGoogle Scholar
  28. [Giuntini, 1991]
    R. Giuntini. A semantical investigation on Brouwer-Zadeh logic, Journal of Philosophical Logic 20, 411–433, 1991.CrossRefGoogle Scholar
  29. [Giuntini, 1992]
    R. Giuntini. Brouwer-Zadeh logic, decidability and bimodal systems, Studia Logica 51, 97–112, 1992.CrossRefGoogle Scholar
  30. [Giuntini, 1993]
    R. Giuntini. Three-valued Brouwer-Zadeh logic, International Journal of Theoretical Physics 32, 1875–1887, 1993.CrossRefGoogle Scholar
  31. [Giuntini, 1995]
    R. Giuntini. Quasilinear QMV algebras, International Journal of Theoretical Physics 34, 1397–1407, 1995.CrossRefGoogle Scholar
  32. [Giuntini, 1996]
    R. Giuntini. Quantum MV algebras, Studia Logica 56, 393–417, 1996.CrossRefGoogle Scholar
  33. [Goldblatt, 1984]
    R. H. Goldblatt. Orthomodularity is not elementary, Journal of Symbolic Logic 49, 401–404, 1984.CrossRefGoogle Scholar
  34. [Goldblatt, 1974]
    R. Goldblatt. Semantics analysis of orthologic, Journal of Philosophical Logic 3, 19–35, 1974.CrossRefGoogle Scholar
  35. [Greechie, 1981]
    R. J. Greechie. A non-standard quantum logic with a strong set of states, in E. G. Beltrametti and B. C. van Fraassen (eds), Current Issues in Quantum Logic, Vol. 8 of Ettore Majorana International Science Series, Plenum, New York, pp. 375–380, 1981.CrossRefGoogle Scholar
  36. [Greechie and Gudder, n.d.]
    R. J. Greechie and S. P. Gudder. Effect algebra counterexamples, preprint, n. d.Google Scholar
  37. [Gudder, 1995]
    S. P. Gudder. Total extensions of effect algebras, Foundations of Physics Letters 8, 243–252, 1995.CrossRefGoogle Scholar
  38. [Hardegree, 1975]
    G. H. Hardegree. Stalnaker conditionals and quantum logic, Journal of Philosophical Logic 4, 399–421, 1975.CrossRefGoogle Scholar
  39. [Hardegree, 1976]
    G. M. Hardegree. The conditional in quantum logic, in P. Suppes (ed.), Logic and Probability in Quantum Mechanics, Reidel, Dordrecht, pp. 55–72, 1976.Google Scholar
  40. [Kalmbach, 1983]
    G. Kalmbach. Orthomodular Lattices, Academic Press, New York, 1983.Google Scholar
  41. [Keller, 1980]
    H. A. Keller. Ein nichtklassischer Hilbertscher Raum, Mathematische Zeitschrift 172, 41–49, 1980.CrossRefGoogle Scholar
  42. [Kôpka and Chovanec, 1994]
    F. Kôpka and F. Chovanec. D-posets, Mathematica Slovaca 44, 21–34, 1994.Google Scholar
  43. [Kraus, 1983]
    K. Kraus. States, Effects and Operations, Vol. 190 of Lecture Notes in Physics, Springer, Berlin, 1983.Google Scholar
  44. [Ludwig, 1983]
    G. Ludwig. Foundations of Quantum Mechanics, Vol. 1, Springer, Berlin, 1983.CrossRefGoogle Scholar
  45. [Mackey, 1957]
    G. Mackey. The Mathematical Foundations of Quantum Mechanics, Benjamin, New York, 1957.Google Scholar
  46. [Mangani, 1973]
    P. Mangani. Su certe algebre connesse con logiche a più valori, Bollettino Unione Matematica Italiana 8, 68–78, 1973.Google Scholar
  47. [Minari, 1987]
    P. Minari. On the algebraic and Kripkean logical consequence relation for orthomodular quantum logic, Reports on Mathematical Logic 21, 47–54, 1987.Google Scholar
  48. [Mittelstaedt, 1972]
    P. Mittelstaedt. On the interpretation of the lattice of subspaces of Hilbert space as a propositional calculus, Zeitschrift für Naturforschung 27a, 1358–1362, 1972.Google Scholar
  49. [Morash, 1973]
    R. P. Morash. Angle bisection and orthoautomorphisms in Hilbert lattices, Canadian Journal of Mathematics 25, 261–272, 1973.Google Scholar
  50. [Nishimura, 1980]
    H. Nishimura. Sequential method in quantum logic, Journal of Symbolic Logic 45, 339–352, 1980.CrossRefGoogle Scholar
  51. [Nishimura, 1994]
    H. Nishimura. Proof theory for minimal quantum logic I and II, In-ternational Journal of Theoretical Physics 33, 102–113, 1427–1443, 1994.Google Scholar
  52. [Pratt, 1993]
    V. Pratt. Linear logic for generalized quantum mechanics. In Proceeings of the Worksho on Physics and Computation, pp. 166–180, IEEE, 1993.Google Scholar
  53. [Ptâk and Pulmannovâ, 1991]
    P. Ptak and S. Pulmannovâ. Orthomodular Structures as Quantum Logics, number 44 in Fundamental Theories of Physics, Kluwer, Dordrecht, 1991.Google Scholar
  54. [Putnam, 1969]
    H. Putnam. Is logic empirical?, Vol. 5 of Boston Studies in the Philosophy of Science, Reidel, Dordrecht, pp. 216–241, 1969.Google Scholar
  55. [Sambin, 1996]
    G. Sambin. A new elementary method to represent every complete Boolean algebra, in A. Ursini, and P. Aglianò (eds), Logic and Algebra, Marcel Dekker, New York, pp. 655–665, 1996.Google Scholar
  56. [Sambin et al., 1998]
    G. Sambin, G. Battilotti and C. Faggian. Basic logic: reflection, symmetry, visibility, The Journal of Symbolic Logic,to appear.Google Scholar
  57. [Solèr, 1995]
    M. P. Solèr. Characterization of Hilbert space by orthomodular spaces, Communications in Algebra, 23, 219–243, 1995.CrossRefGoogle Scholar
  58. [Takeuti, 1981]
    G. Takeuti. Quantum set theory, in E. G. Beltrametti and B. C. van Fraassen (eds), Current Issues in Quantum Logic, Vol. 8 of Ettore Majorana International Science Series, Plenum, New York, pp. 303–322, 1981.CrossRefGoogle Scholar
  59. [Tamura, 1988]
    S. Tamura. A Gentzen formulation without the cut rule for ortholattices, Kobe Journal of Mathematics 5, 133–150, 1988.Google Scholar
  60. [Varadarajan, 1985]
    V. S. Varadarajan. Geometry of Quantum Theory, 2 edn, Springer, Berlin, 1985.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2002

Authors and Affiliations

  • Maria Luisa Dalla Chiara
    • 1
  • Roberto Giuntini
    • 2
  1. 1.Università FirenzeItaly
  2. 2.Università CagliariItaly

Personalised recommendations