In a previous publication, we introduced an abstract logic via an abstract notion of quantifier. Drawing upon concepts from categorical logic, this abstract logic interprets formulas from context as subobjects in a specific category, e.g., Cartesian, regular, or coherent categories, Grothendieck, or elementary toposes. We proposed an entailment system formulated as a sequent calculus which we proved complete. Building on this foundation, our current work explores model theory within abstract logic. More precisely, we generalize one of the most important and powerful classical model theory methods, namely the ultraproduct method, and show its fundamental theorem, i.e., Los's theorem. The result is shown as independently as possible of a given quantifier.

翻译：在先前的研究中，我们通过抽象的量化概念引入了一种抽象逻辑。借鉴范畴逻辑的思想，该抽象逻辑将上下文中的公式解释为特定范畴（如笛卡尔范畴、正则范畴、相干范畴、格罗滕迪克拓扑斯或初等拓扑斯）中的子对象。我们提出了一种以相继式演算形式表述的蕴涵系统，并证明了其完备性。基于此基础，本研究进一步探讨抽象逻辑中的模型论。具体而言，我们推广了经典模型论中最重要且强大的方法之一——超积方法，并证明了其基本定理（即洛斯定理）。该结果的证明过程尽可能独立于特定的量化子。