量子集合论：转移原理和德摩根定律
Quantum Set Theory: Transfer Principle and De Morgan's Laws
摘要：
在量子逻辑中，由Birkhoff和von Neumann引入的德摩根定律在量子力学中观测命题的投影值真值赋值中扮演着重要的角色。Takeuti的量子集合论将这个分配扩展到所有关于量子集合论的集合论陈述。然而，Takeuti的量子集合论有一个问题，即德摩根定律在泛有界量词和存在有界量词之间不成立。在这里，我们解决这个问题，通过引入一个新的真值分配满足德摩根定律的有界量词。为了证明新的分配，我们证明了转移原理，表明这种分配的真值的每一个有界ZFC定理有一个下界确定的交换子，投影值的交换度，常数的公式。我们研究了最一般的真值赋值类，并得到了它们满足传递原理、满足德摩根定律以及满足这两个条件的充分必要条件。对于多项式可定义的逻辑运算的分配类，我们确定正好36个分配满足转移原则和正好6个分配满足转移原则和德摩根定律。
英文摘要：
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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