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.
KeywordsSemantic 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).
- 3.Carnap, R. (1937). Logische Syntax der Sprache. London: Routledge.Google Scholar
- 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
- 12.Herzberger, H. (1970). Paradoxes of grounding in semantics. Journal of Philosophical Logic, 67, 145–167.Google Scholar
- 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
- 22.Picollo, L. Reference and truth. Journal of Philosophical Logic (to appear).Google Scholar
- 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.Urbaniak, R. (2009). Leitgeb, “About,” Yablo. Logique et Analyse, 207, 239–254.Google Scholar
- 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.Yablo, S. (1985). Truth and reflexion. Journal of Philosphical Logic, 14, 297–349.Google Scholar
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), 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.