逻辑学的关系伴侣

Sankha S. Basu, Sayantan Roy
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摘要

最近,多位学者对逻辑的变包含同伴进行了深入研究。这些伴随体大致分为两类:左变量包含伴随体和右变量包含伴随体。最近的一篇论文介绍了由希尔伯特式陈述(希尔伯特式逻辑)诱导的另一种逻辑的同伴。该文证明了希尔伯特式逻辑的受限规则同伴与其左变量包含同伴重合的充分条件,而一个必要条件仍然难以捉摸。本文分为两部分。在第一部分中,我们给出了希尔伯特式逻辑的左变量包含和受限规则同伴重合的必要条件和充分条件。在本文的其余部分,我们认识到用于定义逻辑$langle\mathcal{L},\vdash\rangle$的变量包含限制是从$mathcal{P}(\mathcal{L})$到$\mathcal{L}$的关系。这就引出了逻辑结构的关系伴生的更一般的概念,我们从普遍逻辑领域借用了这一框架。最后,我们将证明,即使是希尔伯特式逻辑及其受限规则同伴,也可以纳入本文所讨论的逻辑结构及其关系同伴的一般概念之中。
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Relational Companions of Logics
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.
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