🤖 AI Summary
This work addresses the limitations of traditional institutional approaches, which rely on signature morphisms to handle variables indirectly, thereby necessitating cumbersome side conditions in logical formalizations. Within the framework of D-institutions, the paper introduces functor categories for the first time to directly model variable structures, defines a category for predicate logic, and formalizes the construction of complex sentences as functorial operations. Building on this foundation, the authors develop a corresponding proof system and establish its completeness. This approach substantially simplifies the axiomatic presentation of variable-rich logical systems, enhancing both formal rigor and notational conciseness.
📝 Abstract
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.