Hostname: page-component-7dd5485656-bvgqh Total loading time: 0 Render date: 2025-10-31T09:15:38.483Z Has data issue: false hasContentIssue false
  • English
  • Français

On revision operators

Published online by Cambridge University Press:  12 March 2014

P. D. Welch
P. D. Welch*
Affiliation:
Department of Mathematics, University of Bristol, Bristol BS8 1TW. England Graduate School of Science & Technology, Kobe University, Rokko-Dai. Nada-Ku, Kobe 657, Japan
*
Current address: Mathematisches Institut der Universität Bonn, Beringstr. 6, D-53115 Bonn, Germany, E-mail: p.welch@bristol.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We look at various notions of a class of definability operations that generalise inductive operations, and are characterised as “revision operations”. More particularly we: (i) characterise the revision theoretically definable subsets of a countable acceptable structure; (ii) show that the categorical truth set of Belnap and Gupta's theory of truth over arithmetic using fully varied revision sequences yields a complete Σ31 set of integers; (iii) the set of stably categorical sentences using their revision operator Ψ is similarly Σ31 and which is complete in GÖdel's universe of constructive sets L; (iv) give an alternative account of a theory of truth—realistic variance that simplifies full variance, whilst at the same time arriving at Kripkean fixed points.

Keywords

revision theorydefinability theoryadmissibility theorydescriptive set theory

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 68 , Issue 2 , June 2003 , pp. 689 - 711
Copyright
Copyright © Association for Symbolic Logic 2003

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

[1]Antonelli, G. A., A revision-theoretic analysis of the arithmetical hierarchy, Notre Dame Journal of Formal Logic, vol. 35 (1994), pp. 204–208.CrossRefGoogle Scholar
[2]Barwise, K. J., Admissible sets and structures, Perspectives in Mathematical Logic, Springer Verlag, Berlin, Heidelberg, 1975.CrossRefGoogle Scholar
[3]Burgess, J. P., The truth is never simple, this Journal, vol. 51 (1986), no. 3, pp. 663–681.Google Scholar
[4]Chapuis, A., Alternate revision theories of truth, Journal of Philosophical Logic, vol. 25 (1996), pp. 399–423.CrossRefGoogle Scholar
[5]Clote, P., A generalization of the limit lemma and clopen games, this Journal, vol. 51 (1986), no. 2, pp. 273–291.Google Scholar
[6]Devlin, K., Constructihility, Perspectives in Mathematical Logic, Springer Verlag, Berlin, Heidelberg, 1984.CrossRefGoogle Scholar
[7]Friedman, H., Minimality in the Δ21 2-degrees, Fundamenta Mathematicae, vol. 81 (1974), pp. 183–192.CrossRefGoogle Scholar
[8]Gandy, R. O., On a proof of Mostowski's conjecture, Bulletin de l'Academie Polonaise des Sciences (sÉrie des sciences mathÉmatique, astronomique etphysique), vol. 8 (1960), pp. 571–575.Google Scholar
[9]Gupta, A. and Belnap, N., The revision theory of truth, MIT Press, Cambridge MA, 1993.CrossRefGoogle Scholar
[10]Hamkins, J. D. and Lewis, A., Infinite time Turing machines, this Journal, vol. 65 (2000), no. 2, pp. 567–604.Google Scholar
[11]Herzberger, H. G., Naive semantics and the Liar paradox, The Journal of Philosophy, vol. 79 (1982), pp. 479–497.CrossRefGoogle Scholar
[12]Herzberger, H. G., Notes on naive semantics, Journal of Philosophical Logic, vol. 11 (1982), pp. 61–102.CrossRefGoogle Scholar
[13]Kremer, P., The Gupta-Belnap systems S# and S* are not axiomatisable, Notre Dame Journal of Formal Logic, vol. 34 (1993), pp. 583–596.CrossRefGoogle Scholar
[14]Kripke, S., Outline of a theory of truth, The Journal of Philosophy, vol. 72 (1975), pp. 690–716.CrossRefGoogle Scholar
[15]Löwe, B., Revision sequences and computers with an infinite amount of time, Proceedings of XVIII Deutscher Kongreβ für Philosophie der AGPD. Okt. 1999, reprinted in Journal of Logic and Computation, vol. 11 (2001), pp. 25–40.Google Scholar
[16]Löwe, B. and Welch, P. D., Set-theoretic absoluteness and the revision theory of truth, Stadia Logica, vol. 68 (2001), no. 1, pp. 25–40.Google Scholar
[17]Moschovakis, Y., Elementary induction on abstract structures, Studies in Logic series, vol. 77, North-Holland, 1974.Google Scholar
[18]Moschovakis, Y., On non-monotone inductive definitions, Fundamenta Mathematicae, vol. 82 (1974), pp. 39–83.CrossRefGoogle Scholar
[19]Rogers, H., Recursive function theory, Series in Higher Mathematics, McGraw-Hill, 1967.Google Scholar
[20]Welch, P. D., On an algorithmic theory of truth, in preparation.Google Scholar
[21]Welch, P. D., On Gupta-Belnap revision theories of truth, Kripkean fixedpoints, and the next stable set, The Bulletin of Symbolic Logic, vol. 7 (2001), no. 3, pp. 345–360.CrossRefGoogle Scholar
[22]Yaqūb, A., The liar speaks the truth. A defense of the revision theory of truth, Oxford University Press, New York, 1993.CrossRefGoogle Scholar