Labelled proof theory has been famously successful for modal logics by mimicking their relational semantics within deductive systems. Simpson in particular designed a framework to study a variety of intuitionistic modal logics integrating a binary relation symbol in the syntax. In this paper, we present a labelled sequent system for intuitionistic modal logics such that there is not only one but two relation symbols appearing in sequents: one for the accessibility relation associated with the Kripke semantics for normal modal logics and one for the pre-order relation associated with the Kripke semantics for intuitionistic logic. This puts our system in close correspondence with the standard birelational Kripke semantics for intuitionistic modal logics. As a consequence, it can be extended with arbitrary intuitionistic Scott–Lemmon axioms. We show soundness and completeness, together with an internal cut elimination proof, encompassing a wider array of intuitionistic modal logics than any existing labelled system.

中文翻译：

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用于直觉模态逻辑的完全标记的证明系统

标记证明理论通过在演绎系统中模仿它们的关系语义，在模态逻辑方面取得了著名的成功。辛普森特别设计了一个框架来研究在句法中集成二元关系符号的各种直觉模态逻辑。在本文中，我们提出了一个用于直觉模态逻辑的标记序列系统，使得序列中不仅出现一个而是两个关系符号：一个用于与正常模态逻辑的 Kripke 语义相关的可访问性关系，一个用于预序与直觉逻辑的 Kripke 语义相关的关系。这使我们的系统与直觉模态逻辑的标准双关系 Kripke 语义密切对应。因此，它可以用任意直观的 Scott-Lemmon 公理进行扩展。