New Generation Computing

, Volume 34, Issue 1–2, pp 125–152 | Cite as

Quantum Set Theory Extending the Standard Probabilistic Interpretation of Quantum Theory

  • Masanao OzawaEmail author


The notion of equality between two observables will play many important roles in foundations of quantum theory. However, the standard probabilistic interpretation based on the conventional Born formula does not give the probability of equality between two arbitrary observables, since the Born formula gives the probability distribution only for a commuting family of observables. In this paper, quantum set theory developed by Takeuti and the present author is used to systematically extend the standard probabilistic interpretation of quantum theory to define the probability of equality between two arbitrary observables in an arbitrary state. We apply this new interpretation to quantum measurement theory, and establish a logical basis for the difference between simultaneous measurability and simultaneous determinateness.


Quantum Logic Quantum Set Theory Quantum Theory Quantum Measurements von Neumann Algebras 


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© Ohmsha and Springer Japan 2016

Authors and Affiliations

  1. 1.Nagoya UniversityChikusa-ku, NagoyaJapan

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