We translate the three-valued quantum logic into modal logic, and prove 3-equivalence between the valuation of the three-valued logic and a kind of Kripke model in regard to this translation. To prove 3-equivalence, we introduce an observable-dependent logic, which is a fragment of the many-valued quantum logic. Compared to the Birkhoff and von Neumann’s quantum logic, some notions about observables, the completeness relation for example, in quantum mechanics can be utilized if the observable-dependent logic is employed.

中文翻译：

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从三值量子逻辑到模态逻辑的转换

我们将三值量子逻辑转换为模态逻辑，并证明了三值逻辑的取值与关于这种转换的一种Kripke模型之间的3-等价关系。为了证明 3-等价，我们引入了一个可观察依赖逻辑，它是多值量子逻辑的一个片段。与伯克霍夫和冯诺依曼的量子逻辑相比，如果采用与可观察相关的逻辑，则可以利用一些关于可观察的概念，例如量子力学中的完整性关系。