We investigate a recent proposal for modal hypersequent calculi. The interpretation of relational hypersequents incorporates an accessibility relation along the hypersequent. These systems give the same interpretation of hypersequents as Lellman’s linear nested sequents, but were developed independently by Restall for S5 and extended to other normal modal logics by Parisi. The resulting systems obey Došen’s principle: the modal rules are the same across different modal logics. Different modal systems only differ in the presence or absence of external structural rules. With the exception of S5, the systems are modular in the sense that different structural rules capture different properties of the accessibility relation. We provide the first direct semantical cut-free completeness proofs for K, T, and D, and show how this method fails in the case of B and S4.

中文翻译：

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模态逻辑 K、T 和 D 的模块化超序列计算的无截断完整性

我们研究了最近关于模态超后续结石的提议。关系超序列的解释包含了沿超序列的可访问性关系。这些系统对超序列的解释与 Lellman 的线性嵌套序列相同，但由 Restall 为 S5 独立开发，并由 Parisi 扩展到其他正常模态逻辑。由此产生的系统遵循 Došen 的原则：模态规则在不同的模态逻辑中是相同的。不同的模态系统仅在是否存在外部结构规则方面有所不同。除了 S5，系统是模块化的，因为不同的结构规则捕获了可访问性关系的不同属性。我们为 K、T 和 D 提供了第一个直接的语义无切割完整性证明，