Studia Logica

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

Systems of Quantum Logic

  • Satoko Titani
  • Heiji Kodera
  • Hiroshi Aoyama


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.


Logic Modality Quantum theory Set theory 


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© 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

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