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In 1981, Takeuti introduced set theory based on quantum logic by constructinga model analogous to Boolean-valued models for Boolean logic. He defined thequantum logical truth value for every sentence of set theory. He showed thatequality axioms do not hold, while axioms of ZFC set theory hold ifappropriately modified with the notion of commutators. Here, we consider theproblem in Takeuti's quantum set theory that De Morgan's laws do not hold forbounded quantifiers. We construct a counter-example to De Morgan's laws forbounded quantifiers in Takeuti's quantum set theory. We redefine the truthvalue for the membership relation and bounded existential quantification toensure that De Morgan's laws hold. Then, we show that the truth value of everytheorem of ZFC set theory is lower bounded by the commutator of constantstherein as quantum transfer principle.

Reforming Takeuti's Quantum Set Theory to Satisfy De Morgan's Laws