Yablo’s Paradox, The Liar, and Referential Contradictions from a Graph Theory Point of View

Аннотация

F-systems are useful digraphs to model sentences that predicate the falsity of other sentences.   Paradoxes like the Liar and the one of Yablo  can be analyzed with that tool to     find graph-theoretic patterns. In this paper we studied this general model consisting of a set of sentences and the binary relation ‘  affirms the falsity of  ’ among them. The possible existence of non-referential sentences was also considered.  To  model the sets of all the sentences that   can jointly be valued as true we introduced the notion of conglomerate, the existence of which guarantees the absence of paradox. Conglomerates also enabled us 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-style fixed-point characterization of groundedness was offered, and complete (meaning that every sentence is deemed either true or false) and consistent (meaning that no sentence is deemed true and false) fixed points were put  in correspondence with conglomerates. Furthermore, argumentation frameworks are special arguments and argued about the usefulness of the concept for the argumentation theory.