Burali-Forti as a Purely Logical Paradox

  • Graham Leach-KrouseEmail author


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 \(\mathcal {L} + \mathsf {W}\).


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


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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.


  1. 1.
    Belnapv, N.D. (1962). Tonk, plonk and plink. Analysis, 22(6), 130.CrossRefGoogle Scholar
  2. 2.
    Barwise, J, & Feferman, S. (Eds.). (1985). Model-theoretic logics. Perspectives in mathematical logic. New York: Springer.Google Scholar
  3. 3.
    Cardone, F., & Hindley, J.R. (2009). Lambda-calculus and combinators in the 20th century. In Gabbay, D.M., & Woods, J. (Eds.) Logic from Russell to church, volume 5 of Handbook of the History of Logic. New York: Elsevier.Google Scholar
  4. 4.
    Dummett, M. (1973). Frege: philosophy of language. London: Duckworth. OCLC: 845425340.Google Scholar
  5. 5.
    Dummett, M. (1994). The logical basis of metaphysics. Number 1976 in The William James Lectures. Cambridge: Harvard University Press. Mass, 3. print edition. OCLC: 176860499.Google Scholar
  6. 6.
    Gentzen, G. (1969). The Collected Papers of Gerhard Gentzen. North Holland.Google Scholar
  7. 7.
    Girard, J.-Y. (1971). Une Extension De L’Interpretation De Gödel a L’Analyse, Et Son Application a L’Elimination Des Coupures Dans L’Analyse Et La Theorie Des Types. In Studies in logic and the foundations of mathematics, (Vol. 63 pp. 63–92): Elsevier.Google Scholar
  8. 8.
    Girard, J.-Y. (1990). The system F of variable types, 15 years later. In Huet, G. (Ed.) Logical Foundations of Functional Programming (pp. 87–126). Boston: Addison-Wesley Longman Publishing Co., Inc.Google Scholar
  9. 9.
    Klement, K.C. (2010). Russell, his paradoxes, and Cantor’s theorem: part I philosophy compass.Google Scholar
  10. 10.
    Kotlarski, H. (1978). Some remarks on well-ordered models. Fundamenta Mathematicae, 99, 123–132.CrossRefGoogle Scholar
  11. 11.
    Lawvere, F. (1969). Diagonal arguments and cartesian closed categories. Category theory.Google Scholar
  12. 12.
    Martin, RL. (1977). On a puzzling classical validity. The Philosophical Review.Google Scholar
  13. 13.
    McGee, V. (1996). Logical Operations. Journal of Philosophical Logic, 25(6), 567–580.CrossRefGoogle Scholar
  14. 14.
    Moore, G.H., & Garciadiego, A. (1981). Burali-Forti’s paradox: a reappraisal of its origins. Historia Mathematica, 8(3), 319–350.CrossRefGoogle Scholar
  15. 15.
    Mirimanoff, D. (1917). Les antinomies de Russell et de Burali-Forti et le problème fondamental de la théorie des ensembles. L’Enseignement Mathématique, 19(1), 37–52.Google Scholar
  16. 16.
    Michael, P. (2004). Set theory and its philosophy: a critical introduction. Oxford University Press.Google Scholar
  17. 17.
    Prawitz, D. (1974). On the idea of a general proof theory. Synthese, 27(1/2,).Google Scholar
  18. 18.
    Prawitz, D. (2006). Natural deduction: a proof-theoretical study. Dover books on mathematics. Mineola: Dover Publications. Dover ed edition. OCLC: ocm61296001.Google Scholar
  19. 19.
    Prior, A.N. (1960). The runabout inference-ticket. Analysis, 21(2), 38.CrossRefGoogle Scholar
  20. 20.
    Priest, G.G. (1994). The structure of the paradoxes of self-reference. Mind, 103(409), 25–34.CrossRefGoogle Scholar
  21. 21.
    Ramsey, F.P. (1925). The foundations of mathematics. In The foundations of mathematics and other logical essays (pp. 1–61). Routledge & Kegan Paul LTD.Google Scholar
  22. 22.
    Restall, G. (1994). On logics without contraction. PhD thesis, Department of Philosophy University of Queensland.Google Scholar
  23. 23.
    Rosser, B. (1942). The Burali-Forti paradox. The Journal of Symbolic Logic, 7(01), 1–17.CrossRefGoogle Scholar
  24. 24.
    Russell, B. (1907). On some difficulties in the theory of transfinite numbers and order types. Proceedings of the London Mathematical Society, s2-4(1), 29–53.CrossRefGoogle Scholar
  25. 25.
    Russell, B. (1920). Introduction to mathematical philosophy. London: Allen & Unwin.Google Scholar
  26. 26.
    Steinberger, F. (2009). Harmony and logical inferentialism. PhD thesis, University of Cambridge, Hughes Hall.Google Scholar
  27. 27.
    Steinberger, F. (2011). What harmony could and could not be? Australasian Journal of Philosophy, 89(4), 617–639.CrossRefGoogle Scholar
  28. 28.
    Tait, W.W. (1966). A nonconstructive proof of Gentzen’s Hauptsatz for second order predicate logic. Bulletin of the American Mathematical Society, 72(6), 980–984.CrossRefGoogle Scholar
  29. 29.
    Takeuti, G. (1953). On a generalized logical calculus. Japanese Journal of Mathematics, 23, 39–96.CrossRefGoogle Scholar
  30. 30.
    Tarski, A. (1986). What are logical notions? History and Philosophy of Logic, 7(2), 143–154.CrossRefGoogle Scholar
  31. 31.
    Tennant, N. Inferentialism, logicism, harmony, and a counterpoint. To appear in ed. Alex Miller, essays for Crispin Wright: logic, language and mathematics (in preparation for Oxford University Press: volume 2 of a two-volume Festschrift for Crispin Wright, co-edited with Annalisa Coliva).Google Scholar
  32. 32.
    Tennant, N. (1990). Natural logic. Edinburgh: Edinburgh University Press.Google Scholar
  33. 33.
    Tennant, N. (2012). Cut for core logic. The Review of Symbolic Logic, 5(03), 450–479.CrossRefGoogle Scholar
  34. 34.
    Tennant, N. (2016). Normalizability, cut eliminability and paradox. Synthese.Google Scholar
  35. 35.
    Thompson, J.F. (1962). On some paradoxes. In Butler, R.J. (Ed.), Analytical philosophy (pp. 104–119).Google Scholar
  36. 36.
    Väänänen, J. (2001). Second-order logic and the foundations of mathematics. The Bulletin of Symbolic Logic, 7(4), 504–520.CrossRefGoogle Scholar
  37. 37.
    Yanofsky, N.S. (2003). A universal approach to self-referential paradoxes, incompleteness and fixed points. The Bulletin of Symbolic Logic, 9(3), 362–386.CrossRefGoogle Scholar
  38. 38.
    Zwicker, W.S. (1987). Playing games with games: the hypergame paradox. The American Mathematical Monthly, 94(6), 507.CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of PhilosophyKansas State UniversityManhattanUSA

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