This paper and its sequel “look under the hood” of the usual sorts of proof-theoretic systems for certain well-known intuitionistic and classical propositional modal logics. Section 1 is preliminary. Of most importance: a marked formula will be the result of prefixing a formula in a propositional modal language with a step-marker, for this paper either 0 or 1 . Think of 1 as indicating the taking of “one step away from 0 . ” Deductions will be constructed using marked formulas. Section 2 presents the model-theoretic concepts, based on those in [ 7 ], that guide the rest of this paper. Section 3 presents Natural Deduction systems IK and CK, formalizations of intuitionistic and classical one-step versions of K. In these systems, occurrences of step-markers allow deductions to display deductive structure that is covered over in familiar “no step” proof-theoretic systems for such logics. □ $\square $ and ♢ are governed by Introduction and Elimination rules; the familiar K rule and Necessitation are derived (i.e. admissible) rules. CK will be the result of adding the 0 -version of the Rule of Excluded Middle to the rules which generate IK. Note: IK is the result of merely dropping that rule from those generating CK, without addition of further rules or axioms (as was needed in [ 7 ]). These proof-theoretic systems yield intuitionistic and classical consequence relations by the obvious definition. Section 4 provides some examples of what can be deduced in IK. Section 5 defines some proof-theoretic concepts that are used in Section 6 to prove the soundness of the consequence relation for IK (relative to the class of models defined in Section 2.) Section 7 proves its completeness (relative to that class). Section 8 extends these results to the consequence relation for CK. (Looking ahead: Part 2 will investigate one-step proof-theoretic systems formalizing intuitionistic and classical one-step versions of some familiar logics stronger than K.)

中文翻译：

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一步模态逻辑，直觉和经典，第 1 部分

本文及其续集“深入了解”某些众所周知的直觉主义和经典命题模态逻辑的常用证明理论系统。第 1 节是初步的。最重要的是：标记的公式将是在命题模态语言中为公式加上步进标记的结果，对于本文，为 0 或 1 。将 1 视为表示“离 0 一步之遥”。”扣除将使用标记的公式构建。第 2 节介绍了基于 [7] 中的模型理论概念，这些概念指导了本文的其余部分。第 3 节介绍自然演绎系统 IK 和 CK，K 的直观和经典一步版本的形式化。在这些系统中，步进标记的出现允许演绎显示演绎结构，该结构覆盖在此类逻辑的熟悉的“无步骤”证明理论系统中。□ $\square $ 和♢ 受引入和消除规则的约束；熟悉的 K 规则和 Necessitation 是派生的（即可接受的）规则。CK 将是将排除中间规则的 0 版本添加到生成 IK 的规则中的结果。注意：IK 只是从那些生成 CK 的规则中删除该规则的结果，没有添加进一步的规则或公理（如 [7] 中所需要的那样）。这些证明理论系统通过明显的定义产生直觉的和经典的结果关系。第 4 节提供了一些可以在 IK 中推导出的示例。第 5 节定义了一些在第 6 节中使用的证明理论概念，以证明 IK 的后果关系的合理性（相对于第 2 节中定义的模型类）。第 7 节证明其完整性（相对于该类）。第 8 节将这些结果扩展到 CK 的后果关系。（展望未来：第 2 部分将研究一步证明理论系统，这些系统将一些比 K 更强大的熟悉逻辑的直观和经典一步版本形式化。）