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Synthese

, Volume 196, Issue 12, pp 5205–5229 | Cite as

Paradoxical hypodoxes

  • Alexandre BillonEmail author
Article
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Abstract

Most paradoxes of self-reference have a dual or ‘hypodox’. The Liar paradox (Lr = ‘Lr is false’) has the Truth-Teller (Tt = ‘Tt is true’). Russell’s paradox, which involves the set of sets that are not self-membered, has a dual involving the set of sets which are self-membered, etc. It is widely believed that these duals are not paradoxical or at least not as paradoxical as the paradoxes of which they are duals. In this paper, I argue that some paradox’s duals or hypodoxes are as paradoxical as the paradoxes of which they are duals, and that they raise neglected and interestingly different problems. I first focus on Richard’s paradox (arguably the simplest case of a paradoxical dual), showing both that its dual is as paradoxical as Richard’s paradox itself, and that the classical, Richard-Poincaré solution to the latter does not generalize to the former in any obvious way. I then argue that my argument applies mutatis mutandis to other paradoxes of self-reference as well, the dual of the Liar (the Truth-Teller) proving paradoxical.

Keywords

Paradoxes of self-reference Hypodoxes Richard’s paradox Truth-teller Circularity Definitions 

References

  1. Armour-Garb, B., & Woodbridge, J. A. (2012). Liars, truthtellers and naysayers: A broader view of semantic pathology i. Language and Communication, 32(4), 293–311.Google Scholar
  2. Austin, J. L. (1950). Truth. Aristotelian Society Supplements, 24(1), 111–29.Google Scholar
  3. Beall, J. (2009). Spandrels of truth. Oxford: Oxford University Press.Google Scholar
  4. Belnap, N. (1993). On rigorous definitions. Philosophical Studies, 72(2), 115–146.Google Scholar
  5. Billon, A. (2011). My own truth, relative truth and the semantic pathologies of self-reference. In S. Rahman & G. Primiero (Eds.), Antirealism: The realism/antirealism debate in the age of alternative logics,., Logic, epistemology, and the unity of science Dordrecht: Springer.Google Scholar
  6. Billon, A. (2013). The truth-tellers paradox. Logique Et Analyse, 224, 371–389.Google Scholar
  7. Burge, T. (1979). Semantical paradox. Journal of Philosophy, 76, 169–198.Google Scholar
  8. Cobreros, P., Égré, P., Ripley, D., & Van Rooij, R. (2013). Reaching transparent truth. Mind, 122(488), 841–866.Google Scholar
  9. Eldridge-Smith, P. (2007). Paradoxes and hypodoxes of time travel. In J. L. Jones, P. Campbell, & P. Wylie (Eds.), Art and time (pp. 172–189). Melbourne: Australian Scholarly Publishing.Google Scholar
  10. Eldridge-Smith, P. (2008). The liar paradox and its relatives. Melbourne: The Australian National University.Google Scholar
  11. Eldridge-Smith, P. (2012). A hypodox! a hypodox! a disingeneous hypodox!. The Reasoner, 6(7), 8–10.Google Scholar
  12. Eldridge-Smith, P. (2015). Two paradoxes of satisfaction. Mind, 124(493), 85–119.Google Scholar
  13. Feferman, S. (2005). Predicativity. In S. Shapiro (Ed.), The Oxford handbook of philosophy of mathematics and logic (pp. 590–624).Google Scholar
  14. Gaifman, H. (2004). The easy way to gödel’s proof and related matters. http://www.columbia.edu/~hg17/Diagonal-Cantor-Goedel-05.pdf.
  15. Gaifman, H. (2006). Naming and diagonalization, from cantor to gödel to kleene. Logic Journal of IGPL, 14(5), 709–728.Google Scholar
  16. Goldstein, L. (2000). A unified solution to some paradoxes. In Proceedings of the Aristotelian Society, New Series. The Aristotelian Society, Blackwell Publishing.Google Scholar
  17. Goldstein, L. (2006). Fibonacci, yablo, and the cassationist approach to paradox. Mind, 115(460), 867–890.Google Scholar
  18. Gupta, A. (1993). Minimalism. Philosophical Perspectives, 7, 359–369.Google Scholar
  19. Gupta, A. (1997). Definition and revision: A response to McGee and Martin. Philosophical Issues, 8, 419–443.Google Scholar
  20. Gupta, A. (2015). Definitions. In Zalta, E. N. (Eds.), The stanford encyclopedia of philosophy, Summer 2015 edn.Google Scholar
  21. Gupta, A., & Belnap, N. (1993). The revision theory of truth. Cambridge: MIT Press.Google Scholar
  22. Halbach, V., & Visser, A. (2014). The henkin sentence. In E. Alonso., M. Manzano & I. Sain (Eds.), The life and work of Leon Henkin (pp. 249–263). Springer.Google Scholar
  23. Herzberger, H. G. (1970). Paradoxes of grounding in semantics. The Journal of philosophy, 67(6), 145–167.Google Scholar
  24. Horsten, L. (2011). The Tarskian turn: Deflationism and axiomatic truth. Cambridge: MIT Press.Google Scholar
  25. Kremer, M. (1988). Kripke and the logic of truth. Journal of Philosophical Logic, 17(3), 225–278.Google Scholar
  26. Kremer, P. (2009). The revision theory of truth. In Zalta, E. N. (Eds.), The stanford encyclopedia of philosophy, Spring 2009 edn.Google Scholar
  27. Kripke, S. (1975). Outline of a theory of truth. Journal of Philosophy, 72(19), 690–716.Google Scholar
  28. Lewy, C. (1947). Truth and significance. Analysis, 8(2), 24–27.Google Scholar
  29. Mackie, J. L. (1973). Truth, probability and paradox: Studies in philosophical logic. Oxford: Oxford University Press.Google Scholar
  30. McGee, V. (1991). Truth. Hackett: Vagueness and Paradox.Google Scholar
  31. Poincaré, H. (1906). Les mathématiques et la logique. Revue de métaphysique et de Morale, 14(3), 294–317.Google Scholar
  32. Priest, G. (1994). The structure of the paradoxes of self-reference. Mind, 103(409), 25–34.Google Scholar
  33. Priest, G. (2006a). In contradiction (2nd ed.). New York: Oxford University Press.Google Scholar
  34. Priest, G. (2006b). The paradoxes of denotation. In T. Bolander, V. F. Hendricks, & S. A. Pedersen (Eds.), Self-reference (pp. 137–150). Stanford: CSLI Publications.Google Scholar
  35. Priest, G. (2014). Plurivalent logics. The Australasian Journal of Logic, 11(1). https://ojs.victoria.ac.nz/ajl/article/view/1830.
  36. Priest, G., & Mortensen, C. (1981). The truth-teller paradox. Logique et Analyse, 24, 381–388.Google Scholar
  37. Quine, W. V. O. (1953). Notes on the theory of reference. In From a logical point of view (pp. 130–138). Harper & Row.Google Scholar
  38. Raatikainen, P. (2003). More on Putnam and Tarski. Synthese, 135(1), 37–47.Google Scholar
  39. Read, S. (2008a). Further thoughts on Tarski’s T-scheme and the liar. In S. Rahman, T. Tulenheimo, & E. Genot (Eds.), Unity, truth and the liar (Vol. 8, pp. 205–225)., Logic, epistemology, and the unity of science Dordrecht: Springer.Google Scholar
  40. Read, S. (2008b). The truth schema and the liar. In S. Rahman, T. Tulenheimo, & E. Genot (Eds.), Unity, truth and the liar, logic, epistemology, and the unity of science (Vol. 8, pp. 3–18). Dordrecht: Springer.Google Scholar
  41. Richard, J. (1905). Les principes des mathématiques et le problème des ensembles. Revue générale des Sciences Pures et Appliquées, 16(541), 142–144.Google Scholar
  42. Ripley, D. (2013). Paradoxes and failures of cut. Australasian Journal of Philosophy, 91(1), 139–164.Google Scholar
  43. Routley, R. (1980). Exploring meinong’s jungle and beyond. Camberra: Australian National University.Google Scholar
  44. Russell, B. (1906). Les paradoxes de la logique. Revue de métaphysique et de morale, 14(5), 627–650.Google Scholar
  45. Sainsbury, R. M. (1995). Paradoxes. Cambridge: Cambridge University Press.Google Scholar
  46. Scharp, K. (2013). Replacing truth. Oxford: OUP Oxford.Google Scholar
  47. Simmons, K. (1993). Universality and the liar: an essay on truth and the diagonal argument. Cambridge: Cambridge University Press.Google Scholar
  48. Smith, J. W. (1984). A simple solution to Mortensen and Priest’s truth teller paradox. Logique et Analyse, 27, 217–220.Google Scholar
  49. Soames, S. (1999). Understanding truth. Oxford: Oxford University Press.Google Scholar
  50. Sorensen, R. (2001). Vagueness and contradiction. Oxford: Oxford University Press.Google Scholar
  51. Suppes, P. (1957). Introduction to logic. New York: Dover Publications.Google Scholar
  52. Tarski, A. (1943). The semantic conception of truth: And the foundations of semantics. Philosophy and Phenomenological Research, 4(3), 341–376.Google Scholar
  53. Wagner, P. (2017). Definition. In Kristanek, M. (Eds.), L’encyclopédie Philosophique.Google Scholar
  54. Woodbridge, J. A. (2004). A neglected dimension of semantic pathology. In Logica yearbook (pp. 277–292). Prague. Filosofia: Institute of Philosophy, Academy of Sciences of the Czech Republic.Google Scholar
  55. Woodbridge, J., & Armour-Garb, B. (2005). Semantic pathology and the open pair. Philosophy and Phenomenological Research, 71(3), 695–703.Google Scholar
  56. Yablo, S. (1993). Hop, skip, and jump: The agnostic conception of truth (Corrections in Philosophical Perspectives, 9, 509–506). Philosophical Perspectives, 7, 371–393.Google Scholar
  57. Yaqub, A. M. (1993). The liar speaks the truth: A defense of the revision theory of truth. Oxford: Oxford University Press.Google Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Université Lille Nord de FranceLilleFrance
  2. 2.UdL3, STLVilleneuve d’AscqFrance

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