Skip to main content
SpringerLink
Log in

Revision Without Revision Sequences: Circular Definitions

  • Published:
Journal of Philosophical Logic Aims and scope Submit manuscript

Abstract

The classical theory of definitions bans so-called circular definitions, namely, definitions of (say) a unary predicate P, based on stipulations of the form

$Px =_{\mathsf {Df}} \phi (P, x),$

where ϕ is a formula of a fixed first-order language and the definiendumP occurs into the definiensϕ. In their seminal book The Revision Theory of Truth (Gupta and Belnap 1993), Gupta and Belnap claim that “General theories of definitions are possible within which circular definitions [...] make logical and semantic sense” [p. IX]. In order to sustain their claim, they develop in this book one general theory of definitions (in some variants) based on revision sequences, namely, ordinal-length iterations of the operator which is induced by the (possibly circular) definition of the predicate. Gupta-Belnap’s approach to circular definitions has been criticised, among others, by D. Martin (Philosophical Issues, 8, 407–418, 1997) and V. McGee (Philosophical Issues, 8, 387–406, 1997). Their criticisms point to the logical complexity of revision sequences, to their relations with ordinary mathematical practice, and to their merits relative to alternative approaches. I will present an alternative general theory of definitions, based on a combination of supervaluation and ω-length revision, which aims to address some criticisms raised against revision sequences, while preserving the philosophical and mathematical core of revision.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aczel, P. (1977). An introduction to inductive definitions. In Barwise J. (Ed.) Handbook of mathematical logic (pp. 739–782). Elsevier.

  2. Antonelli, G. A. (2000). Virtuous circles. From fixed points to revision rules. In Chapuis, A., & Gupta A. (Eds.) Circularity, definition and truth Indian Council of Philosophical Research. New Delhi.

  3. Asmus, C.M. (2013). Vagueness and revision sequences. Synthese, 190, 953–974.

    Article  Google Scholar 

  4. Belnap, N. (1993). On rigorous definitions. Philosophical Studies, 72, 115–146.

    Article  Google Scholar 

  5. Chang, C. C., & Keisler, H.J. (1990). Model theory. In Studies in logic and the foundations of mathematics 73, 3rd edn. North-Holland.

  6. Chapuis, A. (1996). Alternative revision theories of truth. Journal of Philosophical Logic, 25, 399–423.

    Article  Google Scholar 

  7. Fitting, M. (1986). Notes on the mathematical aspects of Kripke’s theory of truth. Notre Dame Journal of Formal Logic, 27, 75–88.

    Article  Google Scholar 

  8. Gupta, A. (1989). Remarks on definitions and the concept of truth. Proceedings of the Aristotelian Society, 89, 227–246.

    Article  Google Scholar 

  9. Gupta, A. (1997). Definition and revision: a response to McGee and Martin. Philosophical Issues, 8, 419–443.

    Article  Google Scholar 

  10. Gupta, A., & Belnap, N. (1993). The revision theory of truth. A Bradford Book. Cambridge: MIT Press.

    Google Scholar 

  11. Hansen, C.S. (2015). Supervaluation on trees for Kripke’s theory of truth. The Review of Symbolic Logic, 8, 46–74.

    Article  Google Scholar 

  12. Herzberger, H.G. (1982). Notes on Naive Semantics. Journal of Philosophical Logic, 11(1), 61–102.

    Article  Google Scholar 

  13. Horsten, L., & Halbach, V. (2014). Truth and paradox. In Horsten, L., & Pettigrew R. (Eds.) The Bloomsbury companion to philosophical logic. Bloomsbury Publishing.

  14. Kremer, P. (2009). Comparing fixed-point and revision theories of truth. Journal of Philosophical Logic, 38, 363–403.

    Article  Google Scholar 

  15. Kripke, S. (1975). Outline of a theory of truth. Journal of Philosophy, 72, 690–716.

    Article  Google Scholar 

  16. Kühnberger, K-U., & et al. (2005). Comparing inductive and circular definitions: parameters, complexity and games. Studia Logica, 81, 79–98.

    Article  Google Scholar 

  17. Kuratowski, K., & Mostowski, A. (1976). Set theory. With an introduction to descriptive set theory. Amsterdam: North-Holland.

    Google Scholar 

  18. Martin, D. (1997). Revision and its rivals. Philosophical Issues, 8, 407–418.

    Article  Google Scholar 

  19. McGee, V. (1997). Revision. Philosophical Issues, 8, 387–406.

    Article  Google Scholar 

  20. Meadows, T. (2013). Truth, dependence and supervaluation: living with the ghost. Journal of Philosophical Logic, 42, 221–240.

    Article  Google Scholar 

  21. Moschovakis, Y.N. (1974). Elementary induction on abstract structures. Amsterdam: North-Holland.

    Google Scholar 

  22. Orilia, F., & Varzi, A. (1996). Truth and circular definitions. Mind and Machines, 6, 124–129.

    Google Scholar 

  23. Rivello, E. (2018). Revision without revision sequences: Self-referential truth. Journal of Philosophical Logic, in print.

  24. Sheard, M. (1994). A guide to truth predicates in the modern era. The Journal of Symbolic Logic, 59, 1032–1054.

    Article  Google Scholar 

  25. Tarski, A. (1955). A lattice-theoretical fixpoint theorem and its applications. Pacific Journal of Mathematics, 5, 285–309.

    Article  Google Scholar 

  26. Visser, A. (1989). Semantics and the Liar Paradox. In Gabbay D. M., & Günthner F. (Eds.) Handbook of Philosophical Logic, (Vol. IV pp. 617–706). Dordrecht: Reidel.

  27. Welch, P. (2003). On revision operators. The Journal of Symbolic Logic, 68, 689–711.

    Article  Google Scholar 

  28. Yablo, S. (1993). Definitions—consistent and inconsistent. Philosophical Studies, 72, 147–175.

    Article  Google Scholar 

  29. Yaqub, A.M. (1993). The liar speaks the truth. A defense of the revision theory of truth. Oxford: Oxford University Press.

    Google Scholar 

Download references

Acknowledgements

First, let me thank the editors Riccardo and Shawn for inviting me to contribute to this special issue of the Journal of Philosophical Logic. The idea of revision-theoretic supervaluation was originally presented at the Panhellenic Logic Symposium held in Samos in June 2015, and its application to the theory of definitions was later proposed at the Logic Colloquium held in Leeds in August 2016: I wish to thank both audiences of these two events. I also want to mention the efforts of an anonymous referee helping me to improve the quality of my paper. Finally, let me dedicate this work to the memory of Aldo Antonelli (1962 - 2015).

Author information

Authors and Affiliations

  1. Department of Philosophy and Educational Sciences, University of Torino, Turin, Italy

    Edoardo Rivello

Authors
  1. Edoardo Rivello
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Edoardo Rivello.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rivello, E. Revision Without Revision Sequences: Circular Definitions. J Philos Logic 48, 57–85 (2019). https://doi.org/10.1007/s10992-018-9481-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10992-018-9481-2

Keywords

Search

Navigation