Quantum Set Theory: Transfer Principle and De Morgan's Laws
- URL: http://arxiv.org/abs/2002.06692v4
- Date: Sat, 13 Feb 2021 02:08:53 GMT
- Title: Quantum Set Theory: Transfer Principle and De Morgan's Laws
- Authors: Masanao Ozawa
- Abstract summary: Takeuti's quantum set theory has a problem in that De Morgan's Laws do not hold between universal and existential bounded quantifiers.
We introduce a new truth value assignment for bounded quantifiers that satisfies De Morgan's Laws.
- Score: 0.0
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: 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.
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