Advertisement

Rethinking Revision

  • P. D. Welch
Open Access
Article
  • 39 Downloads

Abstract

We sketch a broadening of the Gupta-Belnap notion of a circular or revision theoretic definition into that of a more generalized form incorporating ideas of Kleene’s generalized or higher type recursion. This thereby connects the philosophically motivated, and derived, notion of a circular definition with an older form of definition by recursion using functionals, that is functions of functions, as oracles. We note that Gupta and Belnap’s notion of ‘categorical in L’ can be formulated in at least one of these schemes.

Keywords

Revision theory Spector class Kleene recursion Theory of definition 

References

  1. 1.
    Belnap, N. (1982). Gupta’s rule of revision theory of truth. Journal of Philosophical Logic, 11, 103–116.CrossRefGoogle Scholar
  2. 2.
    Burgess, J.P. (1986). The truth is never simple. Journal of Symbolic Logic, 51 (3), 663–681.CrossRefGoogle Scholar
  3. 3.
    Field, H. (2003). A revenge-immune solution to the semantic paradoxes. Journal of Philosophical Logic, 32(3), 139–177.CrossRefGoogle Scholar
  4. 4.
    Field, H. (2008) In Beall, J.C. (Ed.), The Revenge of the Liar. Oxford: O.U.P.Google Scholar
  5. 5.
    Gandy, R.O. (1967). Generalized recursive functionals of finite type and hierarchies of functionals. Annales de la Faculté, des Sciences de l’Université de Clermont-Ferrand, 35, 5–24.Google Scholar
  6. 6.
    Gupta, A. (1981). Truth and paradox. Journal of Philosophical Logic, 11, 1–60.CrossRefGoogle Scholar
  7. 7.
    Gupta, A., & Belnap, N. (1993). The revision theory of truth. Cambridge: MIT Press.Google Scholar
  8. 8.
    Hamkins, J.D., & Lewis, A. (2000). Infinite time Turing machines. Journal of Symbolic Logic, 65(2), 567–604.CrossRefGoogle Scholar
  9. 9.
    Herzberger, H.G. (1982). Naive semantics and the Liar paradox. Journal of Philosophy, 79, 479–497.CrossRefGoogle Scholar
  10. 10.
    Herzberger, H.G. (1982). Notes on naive semantics. Journal of Philosophical Logic, 11, 61–102.CrossRefGoogle Scholar
  11. 11.
    Hinman, P. (1978). Recursion-Theoretic Hierarchies. Ω series in mathematical logic. Berlin: Springer.CrossRefGoogle Scholar
  12. 12.
    Kechris, A., & Moschovakis, Y.N. (1977). Recursion in higher types. In Barwise (Ed.) Handbook of Mathematical Logic, Studies in Logic and Foundations of Mathematics, chapter C6, 681–738. North-Holland, Amtserdam.CrossRefGoogle Scholar
  13. 13.
    Kechris, A.S. (1978). On Spector classes. In Kechris, A.S., & Moschovakis, Y.N. (Eds.) Cabal Seminar 76-77, volume 689 of Lecture Notes in Mathematics Series, 245–278. Springer.Google Scholar
  14. 14.
    Kleene, S.C. (1959). Quantification of number-theoretic predicates. Compositio Mathematicae, 14, 23–40.Google Scholar
  15. 15.
    Kleene, S.C. (1963). Recursive quantifiers and functionals of finite type II. Transactions of the American Mathematical Society, 108, 106–142.Google Scholar
  16. 16.
    Kühnberger, K.-U., Löwe, B., Möllerfeld, M., Welch, P.D. (2005). Comparing inductive and circular definitions: parameters, complexity and games. Studia Logica, 81, 79–98.CrossRefGoogle Scholar
  17. 17.
    Löwe, B. (2001). Revision sequences and computers with an infinite amount of time. Journal of Logic and Computation, 11, 25–40.CrossRefGoogle Scholar
  18. 18.
    Löwe, B., & Welch, P.D. (2001). Set-theoretic absoluteness and the revision theory of truth. Studia Logica, 68(1), 21–41.CrossRefGoogle Scholar
  19. 19.
    Moschovakis, Y.N. (1974). Elementary Induction on Abstract structures, volume 77 of Studies in Logic series. North-Holland, Amsterdam.Google Scholar
  20. 20.
    Moschovakis, Y.N. (2009). Descriptive Set Theory. Studies in Logic series. North-Holland, Amsterdam.Google Scholar
  21. 21.
    Soare, R.I. (1987). Recursively enumerable sets and degrees. Perspectives in mathematical logic. Springer.Google Scholar
  22. 22.
    Welch, P.D. (2000). Eventually Infinite Time Turing degrees: infinite time decidable reals. Journal of Symbolic Logic, 65(3), 1193–1203.CrossRefGoogle Scholar
  23. 23.
    Welch, P.D. (2003). On revision operators. Journal of Symbolic Logic, 68(3), 689–711.CrossRefGoogle Scholar
  24. 24.
    Welch, P.D. (2008). Ultimate truth vis à vis stable truth. Review of Symbolic Logic, 1(1), 126–142.CrossRefGoogle Scholar
  25. 25.
    Welch, P.D. (2014). Some observations on truth hierarchies. Review of Symbolic Logic, 7(1), 1–30.CrossRefGoogle Scholar
  26. 26.
    Yaqūb, A. (1993). The liar speaks the truth. a defense of the revision theory of truth. New York: O.U.P.Google Scholar

Copyright information

© The Author(s) 2018

Open AccessThis 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.

Authors and Affiliations

  1. 1.School of MathematicsUniversity of BristolBristolEngland

Personalised recommendations