Logical argumentation is a well-known approach to modelling nonmonotonic reasoning with conflicting information. In this paper we provide a proof-theoretic study of properties of logical argumentation frameworks. Given some desiderata in terms of rationality postulates, we consider the conditions that an argumentation framework should fulfill for the desiderata to hold. The rationality behind this approach is to assist designers to ``plug-in'' pre-defined formalisms according to actual needs. This work extends related research on the subject in several ways: more postulates are characterized, a more abstract notion of arguments is considered, and it is shown how the nature of the attack rules (subset attacks versus direct attacks) affects the properties of the whole setting.