The main object of this article is to give two novel proofs of the admissibility of Ackermann’s rule (γ) for the propositional relevant logic R. The results are established as corollaries of cut elimination for systems of tableaux for R. Cut elimination, in turn, is established both nonconstructively (as a corollary of completeness) and constructively (using Gentzen-like methods). The extensibility of the techniques is demonstrated by showing that (γ) is admissible for RQ* (R with constant domain quantifiers). The status of the admissibility of (γ) for RQ* was, to the best of the author’s knowledge, an open problem. Further extensions of these results will be explored in the sequel(s).

中文翻译：

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切割和伽玛 I：命题和常数域 R

本文的主要目的是对阿克曼规则的可接受性给出两个新颖的证明（γ) 为命题相关逻辑R. 结果被确定为画面系统的削减消除的推论R. 反过来，削减消除是非建设性的（作为完整性的推论）和建设性的（使用类似 Gentzen 的方法）。这些技术的可扩展性通过证明（γ) 可接纳为请求*(R具有常数域量词）。(γ） 为了请求*据作者所知，这是一个悬而未决的问题。这些结果的进一步扩展将在续集中进行探索。