Non-Reflexive Nonsense: Proof Theory of Paracomplete Weak Kleene Logic

Pub Date : 2024-03-18 DOI:10.1007/s11225-023-10086-x

引用次数: 0

Abstract

Our aim is to provide a sequent calculus whose external consequence relation coincides with the three-valued paracomplete logic ‘of nonsense’ introduced by Dmitry Bochvar and, independently, presented as the weak Kleene logic \(\textbf{K}_{\textbf{3}}^{\textbf{w}}\) by Stephen C. Kleene. The main features of this calculus are (i) that it is non-reflexive, i.e., Identity is not included as an explicit rule (although a restricted form of it with premises is derivable); (ii) that it includes rules where no variable-inclusion conditions are attached; and (iii) that it is hybrid, insofar as it includes both left and right operational introduction as well as elimination rules.

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非反身胡言乱语：准完全弱克莱因逻辑的证明理论

摘要我们的目的是提供一种时序微积分，它的外部结果关系与德米特里-波赫瓦尔（Dmitry Bochvar）提出的三值准完全逻辑 "无稽之谈"（of nonsense）相吻合，并由斯蒂芬-克莱因（Stephen C. Kleene）作为弱克莱因逻辑（\textbf{K}_{\textbf{3}}^{textbf{w}}/\）独立提出。Kleene.这种微积分的主要特点是：(i) 它是非反身的，即 "同一性"（Identity）不作为显式规则（尽管它的一种带前提的限制形式是可推导的）；(ii) 它包括不附加变量包含条件的规则；(iii) 它是混合的，因为它既包括左运算引入规则，也包括右运算引入规则，还包括消元规则。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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