标题： 多值模态逻辑 显示英文标题

标题： Many-Valued Modal Logic

摘要： 我们将模态逻辑和多值逻辑的概念以一种普遍和全面的方式结合起来。 具体来说，给定任何有限的线性有序真值集合以及由真值表定义的命题联结词集合，我们定义了多值最小正规模态逻辑，该逻辑以类似Gentzen的序列演算形式呈现，并证明了它相对于多值Kripke模型的合理性与强完备性。 该逻辑独立处理必然性和可能性，即它们不是彼此定义的，因此它们之间的对偶性在证明系统本身中得到体现。 我们还证明了该逻辑的有限模型性质（这意味着强可判定性），并考虑了其一些扩展。 此外，我们证明了只有一种方式可以定义否定，使得必然性和可能性之间的德摩根对偶性成立。 另外，我们将多值直觉主义逻辑嵌入到我们多值模态逻辑的一个扩展中。 显示英文摘要

摘要： We combine the concepts of modal logics and many-valued logics in a general and comprehensive way. Namely, given any finite linearly ordered set of truth values and any set of propositional connectives defined by truth tables, we define the many-valued minimal normal modal logic, presented as a Gentzen-like sequent calculus, and prove its soundness and strong completeness with respect to many-valued Kripke models. The logic treats necessitation and possibility independently, i.e., they are not defined by each other, so that the duality between them is reflected in the proof system itself. We also prove the finite model property (that implies strong decidability) of this logic and consider some of its extensions. Moreover, we show that there is exactly one way to define negation such that De Morgan's duality between necessitation and possibility holds. In addition, we embed many-valued intuitionistic logic into one of the extensions of our many-valued modal logic.