Burali-Forti as a Purely Logical Paradox

Abstract

Russell’s paradox is purely logical in the following sense: a contradiction can be formally deduced from the proposition that there is a set of all non-self-membered sets, in pure first-order logic—the first-order logical form of this proposition is inconsistent. This explains why Russell’s paradox is portable—why versions of the paradox arise in contexts unrelated to set theory, from propositions with the same logical form as the claim that there is a set of all non-self-membered sets. Burali-Forti’s paradox, like Russell’s paradox, is portable. I offer the following explanation for this fact: Burali-Forti’s paradox, like Russell’s, is purely logical. Concretely, I show that if we enrich the language \(\mathcal {L}\) of first-order logic with a well-foundedness quantifier W and adopt certain minimal inference rules for this quantifier, then a contradiction can be formally deduced from the proposition that there is a greatest ordinal. Moreover, a proposition with the same logical form as the claim that there is a greatest ordinal can be found at the heart of several other paradoxes that resemble Burali-Forti’s. The reductio of Burali-Forti can be repeated verbatim to establish the inconsistency of these other propositions. Hence, the portability of the Burali-Forti’s paradox is explained in the same way as the portability of Russell’s: both paradoxes involve an inconsistent logical form—Russell’s involves an inconsistent form expressible in \(\mathcal {L}\) and Burali-Forti’s involves an inconsistent form expressible in \(\mathcal {L} + \mathsf {W}\).

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Acknowledgments

Thanks to Salvatore Florio, Walter Dean, Richard Kaye, and other members of the Midlands Logic Seminar, as well as Hidenori Kurokawa, for valuable feedback which considerably sharpened this paper. Thanks also to Curtis Franks, Sean Ebbels-Duggan, Landon Elkind, Bernd Buldt, Christopher Pynes, and other attendees of the 2018 ASL North American annual meeting with whom I discussed these ideas. Finally, thanks to two anonymous referees, who read with tremendous care and whose suggestions greatly improved the above.

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Correspondence to Graham Leach-Krouse.

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Leach-Krouse, G. Burali-Forti as a Purely Logical Paradox. J Philos Logic 48, 885–908 (2019). https://doi.org/10.1007/s10992-019-09500-4

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Keywords

  • Burali-Forti
  • Paradoxes
  • Harmony
  • Logical constants
  • Russell
  • Hypergame
  • Mirimanoff
  • Well-foundedness
  • Quantifiers