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On \({{{\mathcal {F}}}}\)-Systems: A Graph-Theoretic Model for Paradoxes Involving a Falsity Predicate and Its Application to Argumentation Frameworks

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\({{{\mathcal {F}}}}\)-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 ‘\(\ldots \) affirms the falsity of\(\ldots \)’ 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 cases of \(\mathcal{F}\)-systems. We showed the relation between local conglomerates and admissible sets of arguments and argued about the usefulness of the concept for the argumentation theory.

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  1. A well-known dissenting opinion on the non-circularity of Yablo’s paradox is that of Priest (1997).

  2. Though we include sinks in \(\mathcal{F}\)-systems, the systems of Beringer and Schindler (2017) and Walicki (2008) are still more general to the extent that referential sentences may attribute truth to other sentences as well. This is also the case with Rabern et al.’s (2013) main proposal beyond \({{{\mathcal {F}}}}\)-systems.

  3. We borrowed the term ‘labelling’ from Caminada (2006). In Beringer and Schindler (2017), the term ‘decoration’ is used instead.

  4. This notation is taken from Walicki (2008).

  5. We can certainly think of other kinds of sentences that could be expressed. For example, if we have \(F=\{(x, y), (y, z)\}\), then we can realize that the model is expressing that x affirms that z is true. However, note that this interpretation does not depend strictly on the model, but on the intended semantics (like the one we will see next). Indeed, we can think of some infectious semantics (Omori & Szmuc, 2017; Szmuc, 2016) under which z is undetermined, y is false, and x is true, meaning that x is true because it affirms that z is either true or undetermined.

  6. The conditions for the assignment of \(\texttt{T}\) and \(\texttt{F}\) to non-sink nodes are comparable to those of Cook’s (2004) acceptable assignments. We add the label \(\texttt{U}\) following the general lines of Caminada’s (2006) labelling semantics, which has the spirit of the strong Kleene three-valued logic. On request of a reviewer, we should say that the weak Kleene logic –i.e., that in which all the connectives receive the undetermined value if any component is undetermined– is not useful here to the aim of characterizing paradoxes according to the intuitions expressed in the following definitions.

  7. Classical labellings play the role here of acceptable colorings on serial digraphs, as defined by Cook (2014). \(\texttt{T}\) and \(\texttt{F}\) correspond to colors turquoise and fuchsia, respectively.

  8. Kernels differ from conglomerates only in the absorption property, which says that \((S{\setminus } A)\subseteq \overleftarrow{F}(A)\). Therefore, we have essentially the notion of kernel used in Cook’s sink-free system, modulo the fact that sinks can be placed on the outside. We are informally saying here that A absorbs x with the meaning that \((x, y)\in F\) for some \(y\in A\).

  9. In Sect. 6, we will generalize these notions for every \({{{\mathcal {F}}}}\)-system (not only the non-paradoxical ones) (Definition 12).

  10. Posed by an anonymous reviewer.

  11. Cook (2020) gave the canonical example of the tautology-teller (‘This sentence is either true or false’) as semi-true, but that is clearly not expressible in the present framework.

  12. A double path is a graph consisting of two non-trivial paths, both with common origin and end.

  13. In terms of argumentation, x is controversial w.r.t. y if and only if x indirectly attacks y (odd-length path) and indirectly defends y (even-length path). In terms of sentences, we would be tempted to say that x indirectly affirms the falsity of y and indirectly affirms the truth of y, but this is not necessarily the case, since the possible existence of shortcuts between both paths could give rise to different interpretations.


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I want to thank the following people and institutions: three anonymous reviewers, for major criticisms that helped me improve the present work; Enrique Hernández-Manfredini and the Department of Mathematics of the Universidad de Aveiro (Portugal), for inviting me for a research visit in September 2019, during which the main ideas of this paper were presented; and Eduardo Barrio and everyone from the BA-Logic Group (Buenos Aires, Argentina), for a careful reading and fruitful discussion of the first draft of this paper during the Work In Progress Seminar in August 2020.

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Correspondence to Gustavo Bodanza.

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This work was partially supported by the National Agency for Scientific and Technological Promotions (ANPCYT) (Grant PICT 2017-1702), and Universidad Nacional del Sur (Grant PGI 24/I265), Argentina.

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Bodanza, G. On \({{{\mathcal {F}}}}\)-Systems: A Graph-Theoretic Model for Paradoxes Involving a Falsity Predicate and Its Application to Argumentation Frameworks. J of Log Lang and Inf 32, 373–393 (2023).

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