In quantum logic, introduced by Birkhoff and von Neumann, De Morgans Laws play an important role in the projection-valued truth value assignment of observational propositions in quantum mechanics. Takeutis quantum set theory extends this assignment to all the set-theoretical statements on the universe of quantum sets. However, Takeutis quantum set theory has a problem in that De Morgans 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 Morgans 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 Morgans 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 Morgans Laws.
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