We consider Dung’s argumentation framework, in which an argument system consists of a set of arguments and a binary relation between arguments representing the notion of a conflict. The semantics given by Dung define (with respect to each argument system) acceptable sets of arguments called extensions. For his so-called stable semantics, Dung also gives an alternative definition in terms of an equation that a set satisfies if and only if that set is a stable extension. However, neither the original definition nor the equation reflect the fact that the stable semantics (similarly to all of Dung’s semantics) rely upon the notion of an admissible set. Moreover, none of Dung’s other semantics have been characterized by such an equation. Our first goal is to provide such characterizations for the other semantics: We capture Dung’s semantics by means of equations that a set satisfies if and only if it is an extension under the semantics at hand. Not only do we give such equations, but we also take care of providing them as a unified characterization expressing the common grounds of Dung’s semantics. Beyond Dung’s semantics, we are interested in semantics (within Dung’s argumentation framework) relying upon the notion of an admissible set. Our second goal is to show that many of those semantics are captured like Dung’s, using the same unified characterization.