Logical limits of abstract argumentation frameworks

Abstract

Dung’s (1995) argumentation framework takes as input two abstract entities: a set of arguments and a binary relation encoding attacks between these arguments. It returns acceptable sets of arguments, called extensions, w.r.t. a given semantics. While the abstract nature of this setting is seen as a great advantage, it induces a big gap with the application that it is used to. This raises some questions about the compatibility of the setting with a logical formalism (i.e., whether it is possible to instantiate it properly from a logical knowledge base), and about the significance of the various semantics in the application context. In this paper we tackle the above questions. We first propose to fill in the previous gap by extending Dung’s (1995) framework. The idea is to consider all the ingredients involved in an argumentation process. We start with the notion of an abstract monotonic logic which consists of a language (defining the formulas) and a consequence operator. We show how to build, in a systematic way, arguments from a knowledge base formalised in such a logic. We then recall some basic postulates that any instantiation should satisfy. We study how to choose an attack relation so that the instantiation satisfies the postulates. We show that symmetric attack relations are generally not suitable. However, we identify at least one ‘appropriate’ attack relation. Next, we investigate under stable, semi-stable, preferred, grounded and ideal semantics the outputs of logic-based instantiations that satisfy the postulates. For each semantics, we delimit the number of extensions an argumentation system may have, characterise the extensions in terms of subsets of the knowledge base, and finally characterise the set of conclusions that are drawn from the knowledge base. The study reveals that stable, semi-stable and preferred semantics either lead to counter-intuitive results or provide no added value w.r.t. naive semantics. Besides, naive semantics either leads to arbitrary results or generalises the coherence-based approach initially developed by Rescher and Manor (1970). Ideal and grounded semantics either coincide and generalise the free consequence relation developed by Benferhat, Dubois, and Prade (1997), or return arbitrary results. Consequently, Dung’s (1995) framework seems problematic when applied over deductive logical formalisms.