We present natural deduction systems and associated modal lambda calculi for the necessity fragments of the normal modal logics K, T, K4, GL and S4. These systems are in the dual-context style: they feature two distinct zones of assumptions, one of which can be thought as modal, and the other as intuitionistic. We show that these calculi have their roots in in sequent calculi. We then investigate their metatheory, equip them with a confluent and strongly normalizing notion of reduction, and show that they coincide with the usual Hilbert systems up to provability. Finally, we investigate a categorical semantics which interprets the modality as a product-preserving functor.

中文翻译：

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模态逻辑的双上下文演算

我们提出了正常模态逻辑 K、T、K4、GL 和 S4 的必要性片段的自然演绎系统和相关的模态 lambda 演算。这些系统采用双上下文风格：它们具有两个不同的假设区域，其中一个可以被认为是模态的，另一个是直觉的。我们表明，这些微积分的根源在于后续的微积分。然后我们研究他们的元理论，为他们配备一个融合的和强烈规范化的归约概念，并表明他们与通常的希尔伯特系统一致，直到可证明。最后，我们研究了一种将模态解释为产品保留函子的分类语义。