论文标题
量子集理论：转移原则和de Morgan的定律
Quantum Set Theory: Transfer Principle and De Morgan's Laws
论文作者
论文摘要
在伯克霍夫（Birkhoff）和冯·诺伊曼（von Neumann）提出的《量子逻辑》中，德·摩根（De Morgan）的定律在量子力学中观察命题的投影价值真实价值分配中起着重要作用。 Takeuti的量子集理论将此分配扩展到量子集宇宙上的所有集合理论陈述。但是，Takeuti的量子集理论存在问题，因为De Morgan的定律不存在通用和存在的界数量词。在这里，我们通过为满足De Morgan定律的有限量化符引入新的真实价值分配来解决这个问题。为了证明新的分配是合理的，我们证明了转移原则，表明对每个有限的ZFC定理的真实价值的分配具有由换向器确定的下限，即公式中常数的投射值值的换向程度。我们研究了最一般的真理价值分配，并为他们获得满足转移原则的必要条件，以满足De Morgan的法律并满足这两者。对于具有多个可确定的逻辑操作的分配类别，我们确切地确定了满足转移原则的36个任务，恰好满足转移原则和De Morgan定律的6个任务。
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.
