We extend to natural deduction the approach of Linear Nested Sequents and of 2-Sequents. Formulas are decorated with a spatial coordinate, which allows a formulation of formal systems in the original spirit of natural deduction: only one introduction and one elimination rule per connective, no additional (structural) rule, no explicit reference to the accessibility relation of the intended Kripke models. We give systems for the normal modal logics from K to S4. For the intuitionistic versions of the systems, we define proof reduction, and prove proof normalization, thus obtaining a syntactical proof of consistency. For logics K and K4 we use existence predicates (à la Scott) for formulating sound deduction rules.

中文翻译：

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从 2 序列和线性嵌套序列到正常模态逻辑的自然演绎

我们将线性嵌套序列和 2 序列的方法扩展到自然演绎。公式用空间坐标装饰，允许以自然演绎的原始精神对形式系统进行公式化：每个连接词只有一个介绍和一个消除规则，没有附加（结构）规则，没有明确引用预期的可访问性关系克里普克模型。我们给出了从 K 到 S4 的正常模态逻辑系统。对于系统的直觉版本，我们定义了证明缩减，并证明了证明规范化，从而获得了一致性的句法证明。对于逻辑 K 和 K4，我们使用存在谓词（à la Scott）来制定合理的演绎规则。