Advertisement

Studia Logica

, Volume 101, Issue 1, pp 193–217 | Cite as

Systems of Quantum Logic

  • Satoko Titani
  • Heiji Kodera
  • Hiroshi Aoyama
Article
  • 170 Downloads

Abstract

Logical implications are closely related to modal operators. Lattice-valued logic LL and quantum logic QL were formulated in Titani S (1999) Lattice Valued Set Theory. Arch Math Logic 38:395–421, Titani S (2009) A Completeness Theorem of Quantum Set Theory. In: Engesser K, Gabbay DM, Lehmann D (eds) Handbook of Quantum Logic and Quantum Structures: Quantum Logic. Elsevier Science Ltd., pp. 661–702, by introducing the basic implication → which represents the lattice order. In this paper, we fomulate a predicate orthologic provided with the basic implication, which corresponds to complete ortholattices, and then formulate a quantum logic which is equivalent to QL, by using a modal operator instead of the basic implication.

Keywords

Logic Modality Quantum theory Set theory 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Birkhoff G., von Neumann J.: The logic of quantum mechanics. Annals of Mathematics 37, 823–843 (1936)CrossRefGoogle Scholar
  2. 2.
    McNeille H.: Partially ordered sets. Transactions of the American Mathematical Society 42, 416–460 (1937)CrossRefGoogle Scholar
  3. 3.
    Piron, C., Foundations of Quantum Physics, W.A. Benjamin, Inc., Massachusetts, 1976.Google Scholar
  4. 4.
    Takano M.: Strong completeness of lattice valued logic. Archive for Mathematical Logic 41, 497–505 (2002)CrossRefGoogle Scholar
  5. 5.
    Takeuti, G., Two Applications of Logic to Mathematics, Iwanami and Princeton University Press, Tokyo and Princeton, 1978.Google Scholar
  6. 6.
    Takeuti, G., Quantum set theory, in E. Beltrametti and B. C. van Frassen (eds.), Current Issues in Quantum Logic, pp. 303–322, Plenum, New York, 1981.Google Scholar
  7. 7.
    Titani S.: Lattice valued set theory. Archive for Mathematical Logic 38, 395–421 (1999)CrossRefGoogle Scholar
  8. 8.
    Titani, S., A completeness theorem of quantum set theory, in K. Engesser, D. M. Gabbay, and D. Lehmann (eds.), Handbook of Quantum Logic and Quantum Structures: Quantum Logic, pp. 661–702, Elsevier Science Ltd., 2009.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Chubu UniversityKasugaiJapan
  2. 2.Aichi University of EducationKariyaJapan
  3. 3.Faculty of HumanitiesTokaigakuen UniversityNagoyaJapan

Personalised recommendations