Alethic Reference

  • Lavinia PicolloEmail author
Open Access


I put forward precise and appealing notions of reference, self-reference, and well-foundedness for sentences of the language of first-order Peano arithmetic extended with a truth predicate. These notions are intended to play a central role in the study of the reference patterns that underlie expressions leading to semantic paradox and, thus, in the construction of philosophically well-motivated semantic theories of truth.


Semantic paradoxes Reference Self-reference Well-foundedness 



I am deeply indebted to Volker Halbach, with whom I had countless fruitful discussions on reference and self-reference over the last seven years. I would also like to particularly thank Dan Waxman for extremely helpful comments on the final drafts, Thomas Schindler, for great suggestions and encouragement, and two anonymous referees for serious improvements in clarity and exposition. I should mention as well Eduardo Barrio, Catrin Campbell-Moore, Luca Castaldo, Roy T. Cook, Benedict Eastaugh, Martin Fischer, Hannes Leitgeb, Øystein Linnebo, Carlo Nicolai, Graham Priest, Johannes Stern, Albert Visser, the Buenos Aires Logic Group, the MCMP logic community, and the Oxford logic group. Finally, I would like to thank the Alexander von Humboldt Foundation and, especially, the Deutsche Forschungsgemeinschaft (DFG) for generously funding the research projects “Reference patterns of paradox” (PI 1294/1-1) and “The Logics of Truth: Operational and Substructural Approaches” (GZ HJ 5/1-1, AOBJ 617612).


  1. 1.
    Beall, J.C. (2001). Is Y ablo’s P aradox non-circular? Analysis, 61, 176–187.CrossRefGoogle Scholar
  2. 2.
    Boolos, G., Burgess, J., Jeffrey, R. (2007). Computability and logic, 1edn. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  3. 3.
    Carnap, R. (1937). Logische Syntax der Sprache. London: Routledge.Google Scholar
  4. 4.
    Cook, R.T. (2006). There are Non-circular Paradoxes (but Yablo’s Isn’t One of Them!). The Monist, 89(1), 118–149.CrossRefGoogle Scholar
  5. 5.
    Gödel, K. (1931). Über formal unentscheidebarre Sätze der Principia Mathematica und verwandler S ystem I. Monathshefte für Mathematik und Physik, 38, 173–198.CrossRefGoogle Scholar
  6. 6.
    Goodman, N. (1961). About. Mind, 70, 1–24.CrossRefGoogle Scholar
  7. 7.
    Halbach, V., & Visser, A. (2014). Self-reference in Arithmetic I. Review of Symbolic Logic, 7, 671–691.CrossRefGoogle Scholar
  8. 8.
    Halbach, V., & Visser, A. (2014). Self-reference in Arithmetic II. Review of Symbolic Logic, 7, 692–712.CrossRefGoogle Scholar
  9. 9.
    Hardy, J. (1995). Is Y ablo’s P aradox liar-like? Analysis, 55(3), 197–198.CrossRefGoogle Scholar
  10. 10.
    Heck, R.K. (2007). Self-reference and the languages of Arithmetic. Philosophia Mathematica III (pp. 1–29). Originally published under the name “Richard G. Heck Jr”.Google Scholar
  11. 11.
    Henkin, L. (1952). A problem concerning provability. Journal of Symbolic Logic, 17, 160.CrossRefGoogle Scholar
  12. 12.
    Herzberger, H. (1970). Paradoxes of grounding in semantics. Journal of Philosophical Logic, 67, 145–167.Google Scholar
  13. 13.
    Jeroslow, R.G. (1973). Redundancies in the Hilbert-Bernays derivability conditions for Gödel’s second incompleteness theorem. Journal of Symbolic Logic, 38, 359–367.CrossRefGoogle Scholar
  14. 14.
    Ketland, J. (2004). Bueno and C olyvan on Y ablo’s P aradox. Analysis, 64, 165–172.CrossRefGoogle Scholar
  15. 15.
    Ketland, J. (2005). Yablo’s P aradox and ω-inconsistency. Synthese, 145, 295–307.CrossRefGoogle Scholar
  16. 16.
    Kreisel, G. (1953). On a problem of H enkin’s. Indagationes Mathematicae, 15, 405–406.CrossRefGoogle Scholar
  17. 17.
    Kripke, S. (1975). Outline of a theory of truth. Journal of Philosphy, 72, 690–716.CrossRefGoogle Scholar
  18. 18.
    Leitgeb, H. (2002). What is a self-referential sentence? Critical remarks on the alleged (non)-circularity of Y ablo’s P aradox. Logique et Analyse, 177–178, 3–14.Google Scholar
  19. 19.
    Milne, P. (2007). On Godel̈ sentences and what they say. Philosophia Mathematica, III(15), 193–226.CrossRefGoogle Scholar
  20. 20.
    Montague, R. (1962). Theories incomparable with respect to relative interpretability. Journal of Symbolic Logic, 27, 195–211.CrossRefGoogle Scholar
  21. 21.
    Picollo, L. (2018). Reference in Arithmetic. Review of Symbolic Logic, 11, 573–603.CrossRefGoogle Scholar
  22. 22.
    Picollo, L. Reference and truth. Journal of Philosophical Logic (to appear).Google Scholar
  23. 23.
    Priest, G. (1997). Yablo’s Paradox. Analysis, 57, 236–242.CrossRefGoogle Scholar
  24. 24.
    Putnam, H. (1958). Formalization of the C oncept of “A bout”. Philosophy of Science, 25, 125–130.CrossRefGoogle Scholar
  25. 25.
    Putnam, H. (1980). Models and Reality. Journal of Symbolic Logic, 45, 464–482.CrossRefGoogle Scholar
  26. 26.
    Smoryński, C. (1991). The development of self-reference: Lob’s̈ theorem. In Drucker, T. (Ed.) Perspectives on the history of mathematical logic (pp. 110–133). Boston: Birkhäuser.Google Scholar
  27. 27.
    Urbaniak, R. (2009). Leitgeb, “About,” Yablo. Logique et Analyse, 207, 239–254.Google Scholar
  28. 28.
    Visser, A. (1989). Semantics and the liar paradox. In Gabbay, D. M., & Günthner, F. (Eds.) Handbook of philosophical logic, (Vol. 4 pp. 617–706). Dordrecht: Reidel.Google Scholar
  29. 29.
    Yablo, S. (1985). Truth and reflexion. Journal of Philosphical Logic, 14, 297–349.Google Scholar
  30. 30.
    Yablo, S. (1993). Paradox without self-reference. Analysis, 53, 251–252.CrossRefGoogle Scholar
  31. 31.
    Yablo, S. (2014). Aboutness. Princeton: Princeton University Press.CrossRefGoogle Scholar

Copyright information

© The Author(s) 2019

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of PhilosophyUniversity College LondonLondonUK

Personalised recommendations