Annals of Pure and Applied Logic 172 (4):102938 (2021)

In quantum logic, introduced by Birkhoff and von Neumann, De Morgan's Laws play an important role in the projection-valued truth value assignment of observational propositions in quantum mechanics. Takeuti's quantum set theory extends this assignment to all the set-theoretical statements on the universe of quantum sets. However, Takeuti's quantum set theory has a problem in that De Morgan's Laws do not hold between universal and existential bounded quantifiers. Here, we solve this problem by introducing a new truth value assignment for bounded quantifiers that satisfies De Morgan's Laws. To justify the new assignment, we prove the Transfer Principle, showing that this assignment of a truth value to every bounded ZFC theorem has a lower bound determined by the commutator, a projection-valued degree of commutativity, of constants in the formula. We study the most general class of truth value assignments and obtain necessary and sufficient conditions for them to satisfy the Transfer Principle, to satisfy De Morgan's Laws, and to satisfy both. For the class of assignments with polynomially definable logical operations, we determine exactly 36 assignments that satisfy the Transfer Principle and exactly 6 assignments that satisfy both the Transfer Principle and De Morgan's Laws.
Keywords Quantum set theory  Orthomodular-valued models  transfer principle  De Morgan's laws  Quantum logic  Orthomodular lattices  Commutator  Implication  Boolean-valued models  ZFC
Categories (categorize this paper)
DOI 10.1016/j.apal.2020.102938
Edit this record
Mark as duplicate
Export citation
Find it on Scholar
Request removal from index
Revision history

Download options

PhilArchive copy

Upload a copy of this paper     Check publisher's policy     Papers currently archived: 69,066
External links

Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
Through your library

References found in this work BETA

An Axiom System for the Modular Logic.Jerzy Kotas - 1967 - Studia Logica 21 (1):17 - 38.
Transfer Principle in Quantum Set Theory.Masanao Ozawa - 2007 - Journal of Symbolic Logic 72 (2):625 - 648.
Material Implication in Orthomodular (and Boolean) Lattices.Gary M. Hardegree - 1981 - Notre Dame Journal of Formal Logic 22 (2):163-182.
A Bridge Between Q-Worlds.Benjamin Eva, Masanao Ozawa & Andreas Doering - forthcoming - Review of Symbolic Logic:1-40.

View all 16 references / Add more references

Citations of this work BETA

No citations found.

Add more citations

Similar books and articles

Transfer Principle in Quantum Set Theory.Masanao Ozawa - 2007 - Journal of Symbolic Logic 72 (2):625 - 648.
Orthomodular-Valued Models for Quantum Set Theory.Masanao Ozawa - 2017 - Review of Symbolic Logic 10 (4):782-807.
Curie’s Principle and Spontaneous Symmetry Breaking.John Earman - 2004 - International Studies in the Philosophy of Science 18 (2 & 3):173 – 198.
The Principle Underlying Quantum Mechanics.Aage Bohr, Ben R. Mottelson & Ole Ulfbeck - 2004 - Foundations of Physics 34 (3):405-417.


Added to PP index

Total views
13 ( #765,780 of 2,498,773 )

Recent downloads (6 months)
3 ( #210,759 of 2,498,773 )

How can I increase my downloads?


My notes