On abstract F-systems. A graph-theoretic model for paradoxes involving a falsity predicate and its application to argumentation frameworks

Abstract: F-systems are digraphs that enable to model sentences that predicate the falsity of other sentences. Paradoxes like the Liar and Yablo's can be analyzed with that tool to find graph-theoretic patterns. In this paper we present the F-systems model abstracting from all the features of the language in which the represented sentences are expressed. All that is assumed is the existence of sentences and the binary relation '... affirms the falsity of ...' among them. The possible existence of non-referential sentences is also considered. To model the sets of all the sentences that can jointly be valued as true we introduce the notion of conglomerate, the existence of which guarantees the absence of paradox. Conglomerates also enable to characterize referential contradictions, i.e. sentences that can only be false under a classical valuation due to the interactions with other sentences in the model. A Kripke's style fixed point characterization of groundedness is offered and fixed points which are complete (meaning that every sentence is deemed either true or false) and consistent (meaning that no sentence is deemed true and false) are put in correspondence with conglomerates. Furthermore, argumentation frameworks are special cases of F-systems. We show the relation between local conglomerates and admissible sets of arguments and argue about the usefulness of the concept for argumentation theory.