Studia Logica

, Volume 91, Issue 2, pp 239–271 | Cite as

Jump Liars and Jourdain’s Card via the Relativized T-scheme

  • Ming HsiungEmail author


A relativized version of Tarski’s T-scheme is introduced as a new principle of the truth predicate. Under the relativized T-scheme, the paradoxical objects, such as the Liar sentence and Jourdain’s card sequence, are found to have certain relative contradictoriness. That is, they are contradictory only in some frames in the sense that any valuation admissible for them in these frames will lead to a contradiction. It is proved that for any positive integer n, the n-jump liar sentence is contradictory in and only in those frames containing at least an n-jump odd cycle. In particular, the Liar sentence is contradictory in and only in those frames containing at least an odd cycle. The Liar sentence is also proved to be less contradictory than Jourdain’s card sequence: the latter must be contradictory in those frames where the former is so, but not vice versa. Generally, the relative contradictoriness is the common characteristic of the paradoxical objects, but different paradoxical objects may have different relative contradictoriness.


graph theory Jourdain’s card paradox Liar paradox relativized T-scheme revision theory of truth 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Boolos G.: Logic of Provability. Cambridge University Press, Cambridge (1993)Google Scholar
  2. 2.
    Chihara C.: ‘The semantic paradoxes: A diagnostic investigation’. The Philosophical Review 88(4), 590–618 (1979)CrossRefGoogle Scholar
  3. 3.
    Diestel R.: Graph Theory, 2nd. edn., Graduate Texts in Mathematics. Springer-Verlag, New York (2000)Google Scholar
  4. 4.
    Gupta, A., ‘Truth and paradox’, Journal of Philosophical Logic, 11 (1982), 1–60. Reprinted in [11], pp. 175-235.Google Scholar
  5. 5.
    Gupta A., Belnap N.: The Revision Theory of Truth. MIT Press, Cambridge (1993)Google Scholar
  6. 6.
    Halbach V., Leitgeb H., Welch P.: ‘Possible-worlds semantics for modal notions conceived as predicates’. Journal of Philosophical Logic 32(2), 179–223 (2003)CrossRefGoogle Scholar
  7. 7.
    Herzberger H.G.: ‘Naive semantics and the Liar paradox’. Journal of Philosophy 79, 479–497 (1982)CrossRefGoogle Scholar
  8. 8.
    Herzberger, H. G., ‘Notes on naive semantics’, Journal of Philosophical Logic, 11 (1982), 61–102. Reprinted in [11], pp. 133-174.Google Scholar
  9. 9.
    Kripke, S. A., ‘Outline of a theory of truth’, Journal of Philosophy, 72 (1975), 19, 690–712. Reprinted in [11], pp. 53-82.Google Scholar
  10. 10.
    Leitgeb H.: ‘Truth as translation – part A, part B’. Journal of Philosophical Logic 30(281–307), 309–328 (2001)CrossRefGoogle Scholar
  11. 11.
    Martin R.L.: Recent Essays on Truth and the Liar Paradox. Oxford University Press, Oxford (1984)Google Scholar
  12. 12.
    Priest G.: ‘Yablo’s paradox’. Analysis 57(4), 236–242 (1997)CrossRefGoogle Scholar
  13. 13.
    William N., William N.: ‘Necessity predicates and operators’. Journal of Philosophical Logic 9(4), 437–450 (1980)CrossRefGoogle Scholar
  14. 14.
    Tarski, A., ‘The concept of truth in formalized languages’, Studia Philosophica, 1 (1936), 261–405. Also in [15], pp. 152-278.Google Scholar
  15. 15.
    Tarski A.: Logic, semantics, metamathematics: papers from 1923 to 1938. Clarendon Press, Oxford (1956)Google Scholar
  16. 16.
    Tarski, A., ‘The semantic conception of truth and the foundations of semantics’, in S. Blackburn, and K. Simmons, (eds.), Truth, Oxford Readings in Philosophy, Oxford University Press, New York, 1999, pp. 115–143.Google Scholar
  17. 17.
    Yablo S.: ‘Paradox without self-reference’. Analysis 53, 251–252 (1993)CrossRefGoogle Scholar
  18. 18.
    Yablo, S., ‘Circularity and paradox’, in V. F. Hendricks T. Bolander, and S. A. Pedersen, (eds.), Self-Reference, CSLI Publications, 2004, pp. 139–157.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.School of Mathematics and Computational ScienceSun Yat-Sen UniversityGuangzhouP. R. China
  2. 2.School of Politics and AdministrationSouth China Normal UniversityGuangzhouP. R. China

Personalised recommendations