逻辑的关系伴随
Relational Companions of Logics
AI总结：
本文研究了Hilbert式逻辑的变量包含伴随与受限规则伴随重合的充分必要条件，并借鉴泛逻辑学，提出了逻辑结构的关系伴随这一更一般的框架。
摘要：
逻辑的变量包含伴随最近被多位作者深入研究。这些伴随大致分为两类：左变量包含伴随和右变量包含伴随。近期的一篇论文引入了另一种由Hilbert式表述（Hilbert-style logics）诱导的逻辑伴随。该文证明了Hilbert式逻辑的受限规则伴随与其左变量包含伴随重合的充分条件，但必要条件尚不明确。本文分为两部分。在第一部分中，我们给出了Hilbert式逻辑的左变量包含伴随与受限规则伴随重合的充分必要条件。在本文的其余部分，我们认识到用于定义逻辑 $\langle\mathcal{L},\vdash\rangle$ 的变量包含伴随的变量包含限制，实际上是 $\mathcal{P}(\mathcal{L})$ 到 $\mathcal{L}$ 的关系。这引出了逻辑结构的关系伴随这一更一般的概念，这是一个我们借鉴自泛逻辑领域的框架。最后，我们表明，即使是Hilbert式逻辑及其受限规则伴随，也可以纳入本文讨论的逻辑结构及其关系伴随的一般概念的框架之中。
英文摘要：
The variable inclusion companions of logics have lately been thoroughly studied by multiple authors. There are broadly two types of these companions: the left and the right variable inclusion companions. Another type of companions of logics induced by Hilbert-style presentations (Hilbert-style logics) were introduced in a recent paper. A sufficient condition for the restricted rules companion of a Hilbert-style logic to coincide with its left variable inclusion companion was proved there, while a necessary condition remained elusive. The present article has two parts. In the first part, we give a necessary and sufficient condition for the left variable inclusion and the restricted rules companions of a Hilbert-style logic to coincide. In the rest of the paper, we recognize that the variable inclusion restrictions used to define variable inclusion companions of a logic $\langle\mathcal{L},\vdash\rangle$ are relations from $\mathcal{P}(\mathcal{L})$ to $\mathcal{L}$. This leads to a more general idea of a relational companion of a logical structure, a framework that we borrow from the field of universal logic. We end by showing that even Hilbert-style logics and the restricted rules companions of these can be brought under the umbrella of the general notions of logical structures and their relational companions that are discussed here.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
