How to Argue for Anything: Enforcing Arbitrary Sets of Labellings using AFs

ABSTRACT. We contribute to the study of the meta-logical properties of argumentation frameworks, as introduced by Dung. Such frameworks are used in a variety of contexts in contemporary AI, not just as a means to model argumentative scenarios, but also as an abstract mathematical representation of non-monotonic reasoning patterns more generally. Importantly, Dung observed that by representing such patterns by directed graphs, various semantics for non-monotonic reasoning could be formulated in graph theoretic terms. In this paper we address so-called sceptical reasoning about argumentation frameworks, asking the following question: what are the propositional validities of abstract argumentation? We answer this by giving graph constructions demonstrating that the set of validities for a range of different argumentation semantics coincide and that it is axiomatized by three-valued Lukasiewicz logic.