Quantum Set Theory: Transfer Principle and De Morgan’s Laws

Por • 20 feb, 2020 • Sección: Opinion

Masanao Ozawa

In quantum logic introduced by Birkhoff and von Neumann, De Morgan’s Laws play an important role for 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 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 the assignment of the truth value for 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, respectively. 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.

arXiv:2002.06692v1 [quant-ph]

Quantum Physics (quant-ph); Logic (math.LO)

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