Abstract
\({{{\mathcal {F}}}}\)-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.
Similar content being viewed by others
Notes
A well-known dissenting opinion on the non-circularity of Yablo’s paradox is that of Priest (1997).
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.
This notation is taken from Walicki (2008).
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.
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.
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.
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 \(y\in A\).
Posed by an anonymous reviewer.
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.
A double path is a graph consisting of two non-trivial paths, both with common origin and end.
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.
References
Bench-Capon, T. J. M. (2003). Persuasion in practical argument using value-based argumentation frameworks. Journal of Logic and Computation, 13(3), 429–448. https://doi.org/10.1093/logcom/13.3.429
Beringer, T., & Schindler, T. (2017). A graph-theoretic analysis of the semantic paradoxes. The Bulletin of Symbolic Logic, 23(4), 442–492. https://doi.org/10.1017/bsl.2017.37
Bolander, T. (2002). Self-reference and logic. Phi News, 1, 9–44.
Caminada, M. (2006). On the Issue of reinstatement in argumentation. In Logics in artificial intelligence, 10th European conference, JELIA 2006, Liverpool, UK, September 13-15, 2006, Proceedings (pp. 111–123). https://doi.org/10.1007/11853886-11.
Cook, R. T. (2004). Patterns of paradox. Journal of Symbolic Logic, 69(3), 767–774. https://doi.org/10.2178/jsl/1096901765
Cook, R. T. (2014). The Yablo paradox: An essay on circularity. Oxford: Oxford University Press.
Cook, R. T. (2020). An intensional theory of truth: An informal report. The Philosophical Forum, 51(2), 115–126. https://doi.org/10.1111/phil.12253.
Dung, P. M. (1995). On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games. Artificial Intelligence, 77(2), 321–357.
Dyrkolbotn, S. (2012). Argumentation, paradox and kernels in directed graphs. PhD Thesis, University of Bergen, Norway.
Fitting, M. (1986). Notes on the mathematical aspects of Kripke’s theory of truth. Notre Dame Journal of Formal Logic, 27(1), 75–88. https://doi.org/10.1305/ndjfl/1093636525
Galeana-Sanchez, H., & Neumann-Lara, V. (1984). On kernels and semikernels of digraphs. Discrete Mathematics, 48(1), 67–76.
Kripke, S. (1975). Outline of a theory of truth. Journal of Philosophy, 72(19), 690–716. https://doi.org/10.2307/2024634
Omori, H., & Szmuc, D. (2017). Conjunction and disjunction in infectious logics. In A. Baltag, J. Seligman, & T. Yamada (Eds.), Logic, rationality, and interaction (LORI 2017, Sapporo, Japan) (pp. 268–283). Berlin: Springer.
Priest, G. (1997). Yablo’s paradox. Analysis, 57(4), 236–242.
Rabern, L., Rabern, B., & Macauley, M. (2013). Dangerous reference graphs and semantic paradoxes. Journal of Philosophical Logic, 42(5), 727–765. https://doi.org/10.1007/s10992-012-9246-2
Szmuc, D. E. (2016). Defining LFIs and LFUs in extensions of infectious logics. Journal of Applied Non-Classical Logics, 26(4), 286–314. https://doi.org/10.1080/11663081.2017.1290488.
Tourville, N., & Cook, R. T. (2018). Embracing intensionality: Paradoxicality and semi-truth operators in fixed point models. Logic Journal of the IGPL, 28(5), 747–770. https://doi.org/10.1093/jigpal/jzy058.
Walicki, M. (2008). Reference, paradoxes and truth. Synthese, 171(1), 195. https://doi.org/10.1007/s11229-008-9392-9
Yablo, S. (1993). Paradox without self-reference. Analysis, 53(4), 251–252. https://doi.org/10.2307/3328245.
Acknowledgements
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
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). https://doi.org/10.1007/s10849-023-09394-1
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10849-023-09394-1