Hostname: page-component-6587cd75c8-cpvbf Total loading time: 0 Render date: 2025-04-23T23:02:56.087Z Has data issue: false hasContentIssue false
  • English
  • Français

REFERENCE DIGRAPHS OF NON-SELF-REFERENTIAL PARADOXES

Part of: Graph theory Philosophical aspects of logic and foundations

Published online by Cambridge University Press:  30 September 2024

MING HSIUNG Open the ORCID record for MING HSIUNG [Opens in a new window]
MING HSIUNG*
Affiliation:
SCHOOL OF PHILOSOPHY AND SOCIAL DEVELOPMENT SOUTH CHINA NORMAL UNIVERSITY GUANGZHOU 510631 PEOPLE’S REPUBLIC OF CHINA
*
E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

All the known non-self-referential paradoxes share a reference pattern of Yablo’s paradox in that they all necessarily contain infinitely many sentences, each of which refers to infinitely many sentences. This raises a question: Does the reference pattern of Yablo’s paradox underlie all non-self-referential paradoxes, just as the reference pattern of the liar paradox underlies all finite paradoxes? In this regard, Rabern et al. [J Philos Logic 42(5): 727–765, 2013] prove that every dangerous acyclic digraph contains infinitely many points with an infinite out-degree. Building upon their work, this paper extends Rabern et al.’s result to the first-order arithmetic language with a primitive truth predicate, proving that all reference digraphs for non-self-referential paradoxes contain infinitely many sentences of infinite out-degree (called “social sentences”). We then strengthen this result in two respects. First, among these social sentences, infinitely many appear in one ray. Second, among these social sentences, infinitely many have infinitely many out-neighbors, none of which will eventually get to a sink. These observations provide helpful information towards the following conjecture proposed by Beringer and Schindler [Bull. of Symb. Logic 23(4): 442–492, 2017]: every dangerous acyclic digraph contains the Yablo digraph as a finitary minor.

Keywords

reference digraphself-referencesocial pointtruth predicateYablo’s paradox

MSC classification

Primary: 03A05: Philosophical and critical
Secondary: 05C20: Directed graphs (digraphs), tournaments
Type
Research Article
Information
The Review of Symbolic Logic , Volume 18 , Issue 1 , March 2025 , pp. 349 - 366
Check for updates
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Beringer, T., & Schindler, T. (2016). Reference graphs and semantic paradox. In Arazim, P., & Dančák, M. editors. Logica Yearbook 2015, London: College Publications, pp. 1–15.Google Scholar
Beringer, T., & Schindler, T. (2017). A graph-theoretic analysis of the semantic paradoxes. The Bulletin of Symbolic Logic, 23(4), 442–492.CrossRefGoogle Scholar
Boolos, G. (1993). Logic of Provability. Cambridge: Cambridge University Press.Google Scholar
Butler, J. M. (2018). An entirely non-self-referential Yabloesque paradox. Synthese, 195(11), 5007–5019.CrossRefGoogle Scholar
Cook, R. T. (2004). Patterns of paradox. Journal of Symbolic Logic, 69(3), 767–774.CrossRefGoogle Scholar
Cook, R. T. (2014). The Yablo Paradox: An Essay on Circularity. Oxford: Oxford University Press.CrossRefGoogle Scholar
Cutland, N. J. (1980). Computability: An Introduction to Recursive Function Theory. Cambridge: Cambridge University Press.Google Scholar
Dyrkolbotn, S., & Walicki, M. (2014). Propositional discourse logic. Synthese, 191(5), 863–899.CrossRefGoogle Scholar
Feferman, S. (1960). Arithmetization of metamathematics in a general setting. Fundamenta Mathematicae, 49(1), 35–92.Google Scholar
Gupta, A. (1982). Truth and paradox. Journal of Philosophical Logic, 11(1), 1–60.CrossRefGoogle Scholar
Halbach, V., Leitgeb, H., & Welch, P. (2003). Possible-worlds semantics for modal notions conceived as predicates. Journal of Philosophical Logic, 32(2), 179–223.Google Scholar
Herzberger, H. G. (1970). Paradoxes of grounding in semantics. The Journal of Philosophy, 67(6), 145–167.CrossRefGoogle Scholar
Herzberger, H. G. (1982). Notes on naive semantics. Journal of Philosophical Logic, 11(1), 61–102.CrossRefGoogle Scholar
Hsiung, M. (2020). What paradoxes depend on. Synthese, 197(2), 887–913.Google Scholar
Hsiung, M. (2021). Unwinding modal paradoxes on digraphs. Journal of Philosophical Logic, 50(2), 319–362.Google Scholar
Just, W., & Weese, M. (1997). Discovering modern set theory II: set-theoretic tools for every mathematician . Graduate Studies in Mathematics Volume 18. Providence, RI: American Mathematical Society.Google Scholar
Kripke, S. A. (1975). Outline of a theory of truth. Journal of Philosophy, 72(19), 690–712.CrossRefGoogle Scholar
Leitgeb, H. (2002). What is a self-referential sentence? Critical remarks on the alleged (non-)circularity of Yablo’s paradox. Logique et Analyse, 45(177-178), 3–14.Google Scholar
Leitgeb, H. (2005). What truth depends on. Journal of Philosophical Logic, 34(2), 155–192.CrossRefGoogle Scholar
McGee, V. (1985). How truthlike can a predicate be? A negative result. Journal of Philosophical Logic, 14(4), 399–410.Google Scholar
Priest, G. (1997). Yablo’s paradox. Analysis, 57(4), 236–242.CrossRefGoogle Scholar
Rabern, L., Rabern, B., & Macauley, M. (2013). Dangerous reference graphs and semantic paradoxes. Journal of Philosophical Logic, 42(5), 727–765.Google Scholar
Schlenker, P. (2007). The elimination of self-reference: generalized Yablo-series and the theory of truth. Journal of Philosophical Logic, 36(3), 251–307.Google Scholar
Von Neumann, J., & Morgenstern, O. (1944). Theory of Games and Economic Behavior. Princeton: Princeton University Press.Google Scholar
Walicki, M. (2017). Resolving infinitary paradoxes. Journal of Symbolic Logic, 82(2), 709–723.CrossRefGoogle Scholar
Walicki, M. (2019). Kernels of digraphs with finitely many ends. Discrete Mathematics, 342(2), 473–486.Google Scholar
Walicki, M., & Dyrkolbotn, S. (2017). Diagnosing paradoxes. Proceeding of TARK 2017.Google Scholar
Yablo, S. (1982). Grounding, dependence, and paradox. Journal of Philosophical Logic, 11(1), 117–137.Google Scholar
Yablo, S. (1985). Truth and reflection. Journal of Philosophical Logic, 14(3), 297–349.Google Scholar
Yablo, S. (1993). Paradox without self-reference. Analysis, 53(4), 251–252.CrossRefGoogle Scholar
Yablo, S. (2006). Circularity and paradox. In Bolander, T., Hendricks, V. F., & Pedersen, S. A., editors. Self-Reference, Stanford: CSLI Publications, pp. 139–157.Google Scholar