The evaluation of argument strength lies at the core of any argumentation system. Numerous semantics have been proposed for this purpose, along with a variety of principles (or axioms) that such semantics are expected to satisfy. Most existing semantics in the literature have been analyzed and compared in light of these principles. While this body of work marks a significant step toward establishing the theoretical foundations of argumentation semantics, it remains incomplete. In particular, characterizations of entire classes of semantics that uniquely satisfy specific subsets of axioms are still lacking, leaving open questions on the kind of semantics that can still be defined and their added values. This paper addresses this gap by establishing representation theorems that explicitly relate subsets of principles to corresponding classes of semantics. These semantics are defined through two mathematical functions: an impact function and an aggregation operator, each satisfying specific structural properties. We demonstrate how these principles offer a uniform and concise explanatory framework for the identified semantics. Finally, we show that classical extension-based semantics do not belong to these classes.