Semantics based on weak admissibility were recently introduced to overcome a problem with self-defeating arguments that has not been solved for more than 25 years. The recursive definition of weak admissibility mainly relies on the notion of a reduct regarding a set E which only contains arguments which are neither in E, nor attacked by E. At first glance the reduct seems to be tailored for the weaker versions of Dung-style semantics only. In this paper we show that standard Dung semantics can be naturally reformulated using the reduct revealing that this concept is already implicit. We further identify a new abstract principle for semantics, so-called modularization describing how to obtain further extensions given an initial one. Its importance for the study of abstract argumentation semantics is shown by its ability to alternatively characterize classical and non-classical semantics.