Several enrichments have been proposed for Dung abstract argumentation framework. These enrichments modify the standard setting by adding extra properties, thus changing the way arguments (and possibly other elements) are selected. The use of logical tools to encode and capture the original framework and some of its enrichments, have already been proposed. We extend these works by proposing a generic logical encoding which aims at being as modular as possible and which enables, by its genericity, not only to encompass the existing encodings, but also to define encodings for enriched frameworks that have not yet been studied through this scope.