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

Paradoxical hypodoxes

  • Alexandre BillonEmail author


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.


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


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© 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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