Notre Dame Journal of Formal Logic

A Topological Approach to Yablo's Paradox

Claudio Bernardi


Some years ago, Yablo gave a paradox concerning an infinite sequence of sentences: if each sentence of the sequence is 'every subsequent sentence in the sequence is false', a contradiction easily follows. In this paper we suggest a formalization of Yablo's paradox in algebraic and topological terms. Our main theorem states that, under a suitable condition, any continuous function from 2N to 2N has a fixed point. This can be translated in the original framework as follows. Consider an infinite sequence of sentences, where any sentence refers to the truth values of the subsequent sentences: if the corresponding function is continuous, no paradox arises.

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Notre Dame J. Formal Logic Volume 50, Number 3 (2009), 331-338.

First available in Project Euclid: 10 November 2009

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Primary: 03A05: Philosophical and critical {For philosophy of mathematics, see also 00A30}
Secondary: 03F45: Provability logics and related algebras (e.g., diagonalizable algebras) [See also 03B45, 03G25, 06E25] 54D30: Compactness

fixed point of a continuous function ungrounded sentence


Bernardi, Claudio. A Topological Approach to Yablo's Paradox. Notre Dame J. Formal Logic 50 (2009), no. 3, 331--338. doi:10.1215/00294527-2009-014.

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  • [1] Bernardi, C., "Fixed points and unfounded chains", Annals of Pure and Applied Logic, vol. 109 (2001), pp. 163--78.
  • [2] Bernardi, C., and G. D'Agostino, "Translating the hypergame paradox: Remarks on the set of founded elements of a relation", Journal of Philosophical Logic, vol. 25 (1996), pp. 545--57.
  • [3] Cook, R. T., "Patterns of paradox", The Journal of Symbolic Logic, vol. 69 (2004), pp. 767--74.
  • [4] Goldstein, L., "A Yabloesque paradox in set theory", Analysis, vol. 54 (1994), pp. 223--227.
  • [5] Ketland, J., "Yablo's paradox and $\omega$"-inconsistency, Synthese, vol. 145 (2005), pp. 295--302.
  • [6] Leitgeb, H., "What is a self-referential sentence? Critical remarks on the alleged (non-)circularity" of Yablo's paradox, Logique et Analyse. Nouvelle Série, vol. 45 (2002), pp. 3--14.
  • [7] Leitgeb, H., "What truth depends on", Journal of Philosophical Logic, vol. 34 (2005), pp. 155--92.
  • [8] Priest, G., "Yablo's paradox", Analysis, vol. 57 (1997), pp. 236--42.
  • [9] Schlenker, P., "The elimination of self-reference: Generalized Yablo-series and the theory of truth", Journal of Philosophical Logic, vol. 36 (2007), pp. 251--307.
  • [10] Yablo, S., "Truth and reflection", Journal of Philosophical Logic, vol. 14 (1985), pp. 297--349.
  • [11] Yablo, S., "Paradox without self-reference", Analysis, vol. 53 (1993), pp. 251--52.
  • [12] Yi, B.-U., "Descending chains and the contextualist approach to semantic paradoxes", Notre Dame Journal of Formal Logic, vol. 40 (1999), pp. 554--67.
  • [13] Yuting, S., "Paradox of the class of all grounded classes", The Journal of Symbolic Logic, vol. 18 (1953), p. 114.
  • [14] Zwicker, W. S., "Playing games with games: The hypergame paradox", The American Mathematical Monthly, vol. 94 (1987), pp. 507--14.