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In 1981, Takeuti introduced quantum set theory by constructing a model of set theory based on quantum logic represented by the lattice of closed linear subspaces of a Hilbert space in a manner analogous to Boolean-valued models of set theory, and showed that appropriate counterparts of the axioms of Zermelo–Fraenkel set theory with the axiom of choice (ZFC) hold in the model. In this paper, we aim at unifying Takeuti’s model with Boolean-valued models by constructing models based on general complete orthomodular lattices, and generalizing the transfer principle in Boolean-valued models, which asserts that every theorem in ZFC set theory holds in the models, to a general form holding in every orthomodular-valued model. One of the central problems in this program is the well-known arbitrariness in choosing a binary operation for implication. To clarify what properties are required to obtain the generalized transfer principle, we introduce a class of binary operations extending the implication on Boolean logic, called generalized implications, including even nonpolynomially definable operations. We study the properties of those operations in detail and show that all of them admit the generalized transfer principle. Moreover, we determine all the polynomially definable operations for which the generalized transfer principle holds. This result allows us to abandon the Sasaki arrow originally assumed for Takeuti’s model and leads to a much more flexible approach to quantum set theory.

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ArakiH. (2000). Mathematical Theory of Quantum Fields. Oxford: Oxford University Press.
BellJ. L. (2005). Set Theory: Boolean-Valued Models and Independence Proofs (third edition). Oxford: Oxford University Press.
BerberianS. K. (1972). Baer *-Rings. Berlin: Springer.
BirkhoffG. & von NeumannJ. (1936). The logic of quantum mechanics. Annals of Mathematics, 37, 823843.
BrunsG. & KalmbachG. (1973). Some remarks on free orthomodular lattices. In SchmidtJ., editor. Proceedings of the Lattice Theory Conference, Houston, TX, pp. 397408.
ChevalierG. (1989). Commutators and decompositions of orthomodular lattices. Order, 6, 181194.
CohenP. J. (1963). The independence of the continuum hypothesis I. Proceedings of the National Academy of Sciences of the United States of America, 50, 11431148.
CohenP. J. (1966). Set Theory and the Continuum Hypothesis. New York: Benjamin.
DiracP. A. M. (1958). The Principles of Quantum Mechanics (fourth edition). Oxford: Oxford University Press.
FourmanM. P. & ScottD. S. (1979). Sheaves and logic. In FourmanM. P., MulveyC. J., and ScottD. S., editors. Applications of Sheaves. Lecture Notes in Mathematics, Vol. 753. Berlin: Springer, pp. 302401.
GeorgacarakosG. N. (1979). Orthomodularity and relevance. Journal of Philosophical Logic, 8, 415432.
GraysonR. J. (1979). Heyting-valued models for intuitionistic set theory. In FourmanM. P., MulveyC. J., and ScottD. S., editors. Applications of Sheaves. Lecture Notes in Mathematics, Vol. 753. Berlin: Springer, pp. 402414.
HardegreeG. M. (1981). Material implication in orthomodular (and Boolean) lattices. Notre Dame Journal of Formal Logic, 22, 163182.
HermanL., MarsdenE. L., & PiziakR. (1975). Implication connectives in orthomodular lattices. Notre Dame Journal of Formal Logic, 16, 305328.
HusimiK. (1937). Studies on the foundation of quantum mechanics I. Proceedings of the Physico-Mathematical Society of Japan, 19, 766778.
JohnstoneP. T. (1977). Topos Theory. London: Academic.
KalmbachG. (1983). Orthomodular Lattices. London: Academic.
KotasJ. (1967). An axiom system for the modular logic. Studia Logica, 21, 1738.
MarsdenE. L. (1970). The commutator and solvability in a generalized orthomodular lattice. Pacific Journal of Mathematics, 33, 357361.
OzawaM. (2005). Perfect correlations between noncommuting observables. Physics Letters A, 335, 1119.
OzawaM. (2006). Quantum perfect correlations. Annals of Physics, 321, 744769.
OzawaM. (2007). Transfer principle in quantum set theory. Journal of Symbolic Logic, 72, 625648.
OzawaM. (2016). Quantum set theory extending the standard probabilistic interpretation of quantum theory. New Generation Computing, 34, 125152.
PulmannováS. (1985). Commutators in orthomodular lattices. Demonstratio Mathematica, 18, 187208.
SasakiU. (1954). Orthocomplemented lattices satisfying the exchange axiom. Journal of Science of the Hiroshima University: Series A, 17, 293302.
ScottD. & SolovayR. (1967). Boolean-valued models for set theory. Unpublished manuscript for Proceedings of AMS Summer Institute on Set Theory. Los Angeles: University of California, 1967.
TakeutiG. (1981). Quantum set theory. In BeltramettiE. G. and van FraassenB. C., editors. Current Issues in Quantum Logic. New York: Plenum, pp. 303322.
TakeutiG. & ZaringW. M. (1973). Axiomatic Set Theory. New York: Springer.
TitaniS. (1999). A lattice-valued set theory. Archive for Mathematical Logic, 38, 395421.
TitaniS. & KozawaH. (2003). Quantum set theory. International Journal of Theoretical Physics, 42, 25752602.
UrquhartA. (1983). Review. Journal of Symbolic Logic, 48, 206208.
von NeumannJ. (1955). Mathematical Foundations of Quantum Mechanics. Princeton, NJ: Princeton University Press. [Originally published: Mathematische Grundlagen der Quantenmechanik (Springer, Berlin, 1932)].
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The Review of Symbolic Logic
  • ISSN: 1755-0203
  • EISSN: 1755-0211
  • URL: /core/journals/review-of-symbolic-logic
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