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Foundations of Physics

, Volume 30, Issue 9, pp 1503–1524 | Cite as

Łukasiewicz Operations in Fuzzy Set and Many-Valued Representations of Quantum Logics

  • Jarosław Pykacz
Article

Abstract

It, is shown that Birkhoff –von Neumann quantum logic (i.e., an orthomodular lattice or poset) possessing an ordering set of probability measures S can be isomorphically represented as a family of fuzzy subsets of S or, equivalently, as a family of propositional functions with arguments ranging over S and belonging to the domain of infinite-valued Łukasiewicz logic. This representation endows BvN quantum logic with a new pair of partially defined binary operations, different from the order-theoretic ones: Łukasiewicz intersection and union of fuzzy sets in the first case and Łukasiewicz conjunction and disjunction in the second. Relations between old and new operations are studied and it is shown that although they coincide whenever new operations are defined, they are not identical in general. The hypothesis that quantum-logical conjunction and disjunction should be represented by Łukasiewicz operations, not by order-theoretic join and meet is formulated and some of its possible consequences are considered.

Keywords

Probability Measure Binary Operation Quantum Logic Fuzzy Subset Orthomodular Lattice 
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.

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Copyright information

© Plenum Publishing Corporation 2000

Authors and Affiliations

  • Jarosław Pykacz
    • 1
  1. 1.Instytut MatematykiUniwersytet GdańskiGdańskPoland;

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