Variables are a crucial element in logic and are also addressed in institution theory, an effort to axiomatize logic. In institution theory, we typically use extensions (signature morphisms) obtained from variables instead of introducing variables directly. While this approach appears simple at first glance because it does not introduce new structures, it often requires numerous conditions to describe variable structures, which can actually complicate the discussion. In this paper, we propose introducing variable structures directly by utilizing a generalization of category of functors. We define a category of predicate logics and formulate the introduction of compound sentences as a functor. We also introduce a proof system and prove a completeness theorem.

翻译：变量是逻辑学中的关键要素，在旨在公理化逻辑的制度理论中亦得到关注。制度理论通常采用由变量导出的扩充（签名词态）而非直接引入变量。这种方法虽因不引入新结构而看似简洁，但描述变量结构往往需要大量条件，反而使讨论复杂化。本文提出通过利用函子范畴的推广形式直接引入变量结构：定义谓词逻辑范畴，将复合语句的引入形式化为一个函子，同时建立证明系统并证明完备性定理。