统一守恒为一般演算的翻译
Unifying Conservation as Translation for General Calculi
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中文总结 AI 辅助
提出一个不依赖结构性质的抽象框架，统一处理逻辑演算间的守恒与翻译定理，涵盖否定翻译、极小性定理及Kuroda定理的表征。
中文摘要 AI 辅助
我们提出了一个关于逻辑演算之间守恒定理和翻译定理的抽象框架。与以往的方法不同，我们的框架不要求底层推论关系满足诸如切消之类的结构性质，从而特别允许无切演算。此外，我们研究任意演算之间的翻译，而不仅仅是从稳定扩张到原始演算的翻译。翻译在矢列层面通过作用于前件和后件的函数对来表述，该框架针对多后件演算开发，同时涵盖单后件情形。这产生了对经典结果的统一处理，包括否定翻译、极小性定理、Orevkov守恒类，以及满足Kuroda双重否定定理的最小逻辑的表征。
英文摘要
We present an abstract framework for conservation and translation theorems between logical calculi. Unlike previous approaches, our setting does not require the underlying consequence relations to satisfy structural properties such as cut, allowing in particular for cut-free calculi. Moreover, we study translations between arbitrary calculi rather than only from a stable extension to the original calculus. Translations are formulated at the level of sequents by pairs of functions acting on antecedents and succedents, and the framework is developed for multi-succedent calculi while subsuming the single-succedent case. This yields a uniform treatment of classical results including negative translations, minimality theorems, Orevkov's conservation classes, and a characterisation of the least logic satisfying Kuroda's double negation theorem.
发表机构
- Università degli Studi di Verona（维罗纳大学）
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