This paper contends that Stoic logic (i.e. Stoic analysis) deserves more attention from contemporary logicians. It sets out how, compared with contemporary propositional calculi, Stoic analysis is closest to methods of backward proof search for Gentzen-inspired substructural sequent logics, as they have been developed in logic programming and structural proof theory, and produces its proof search calculus in tree form. It shows how multiple similarities to Gentzen sequent systems combine with intriguing dissimilarities that may enrich contemporary discussion. Much of Stoic logic appears surprisingly modern: a recursively formulated syntax with some truth-functional propositional operators; analogues to cut rules, axiom schemata and Gentzen’s negation-introduction rules; an implicit variable-sharing principle and deliberate rejection of Thinning and avoidance of paradoxes of implication. These latter features mark the system out as a relevance logic, where the absence of duals for its left and right introduction rules puts it in the vicinity of McCall’s connexive logic. Methodologically, the choice of meticulously formulated meta-logical rules in lieu of axiom and inference schemata absorbs some structural rules and results in an economical, precise and elegant system that values decidability over completeness.

中文翻译：

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斯多葛序列逻辑和证明理论

本文认为斯多葛逻辑（即斯多葛分析）值得当代逻辑学​​家更多的关注。它阐述了与当代命题演算相比，斯多葛分析如何最接近于受 Gentzen 启发的子结构序列逻辑的后向证明搜索方法，因为它们是在逻辑规划和结构证明理论中发展起来的，并在树中生成其证明搜索演算形式。它展示了与 Gentzen 序列系统的多重相似性如何与可能丰富当代讨论的有趣差异相结合。斯多葛逻辑的大部分看起来都出奇地现代：一种递归公式化的语法，带有一些真值泛函命题运算符；类似于切割规则、公理模式和根岑的否定引入规则；隐含的变量共享原则和故意拒绝细化和避免蕴涵悖论。后面的这些特征将系统标记为相关逻辑，其中左右引入规则的对偶的缺失使其接近 McCall 的连接逻辑。在方法论上，选择精心制定的元逻辑规则代替公理和推理模式吸收了一些结构规则，并产生了一个经济、精确和优雅的系统，该系统重视可判定性而不是完整性。