Redefining Strong Completeness via Abstract Model Theory

This paper explores a novel approach to the concept of strong completeness within the expansive framework of abstract model theory. Traditionally, strong completeness relates to a logic's ability to derive all semantic consequences from a set of premises, often tied to properties like compactness and the L"owenheim-Skolem theorems. However, classical definitions are often bound to specific logical systems like first-order logic. We propose a redefinition of strong completeness that transcends these limitations, grounding it within the structural properties and universal constructions inherent to abstract model theory. By leveraging concepts such as abstract elementary classes, categoricity, and amalgamation properties, we develop a framework where strong completeness is characterized not by the mere existence of a proof system, but by the robust interplay between syntax and semantics across a broad spectrum of logics. This redefinition aims to provide a more general and foundational understanding of completeness, applicable to logics beyond first-order and encompassing various model-theoretic behaviors. We demonstrate how this abstract perspective unifies disparate notions of completeness and opens new avenues for investigating the foundational properties of diverse logical systems.