Beyond Maxi-Consistent Argumentation Operators

The question whether Dung's abstract argumentation theory can be instantiated with classical propositional logic has drawn a considerable amount of attention among scientists in recent years. It was shown by Cayrol in 1995 that if direct undercut is used, then stable extensions of an argumentation system correspond exactly to maximal (for set inclusion) consistent subsets of the knowledge base from which the argumentation system was constructed. Until now, no other correspondences were found between the extensions of an argumentation framework and its knowledge base (except if preferences are also given at the input of the system). This paper's contribution is twofold. First, we identify four intuitive conditions describing a class of attack relations which return extensions corresponding exactly to the maximal (for set inclusion) consistent subsets of the knowledge base. Second, we show that if we relax those conditions, it is possible to instantiate Dung's abstract argumentation theory with classical propositional logic and obtain a meaningful result which does not correspond to the maximal consistent subsets of the knowledge base used for constructing arguments. Indeed, we define a whole class of instantiations that return different results. Furthermore, we show that these instantiations are sound in the sense that they satisfy the postulates from argumentation literature (e.g. consistency, closure). In order to illustrate our results, we present one particular instantiation from this class, which is based on cardinalities of minimal inconsistent sets a formula belongs to.