Journal of Philosophical Logic

, Volume 41, Issue 1, pp 173–200

Bounded Revision: Two-Dimensional Belief Change Between Conservative and Moderate Revision

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Abstract

This paper presents the model of ‘bounded revision’ that is based on two-dimensional revision functions taking as arguments pairs consisting of an input sentence and a reference sentence. The key idea is that the input sentence is accepted as far as (and just a little further than) the reference sentence is ‘cotenable’ with it. Bounded revision satisfies the AGM axioms as well as the Same Beliefs Condition (SBC) saying that the set of beliefs accepted after the revision does not depend on the reference sentence (although the posterior belief state does depend on it). Bounded revision satisfies the Darwiche–Pearl (DP) axioms for iterated belief change. If the reference sentence is fixed to be a tautology or a contradiction, two well-known one-dimensional revision operations result. Bounded revision thus naturally fills the space between conservative revision (also known as natural revision) and moderate revision (also known as lexicographic revision). I compare this approach to the two-dimensional model of ‘revision by comparison’ investigated by Fermé and Rott (Artif Intell 157:5–47, 2004) that satisfies neither the SBC nor the DP axioms. I conclude that two-dimensional revision operations add substantially to the expressive power of qualitative approaches that do not make use of numbers as measures of degrees of belief.

Keywords

Belief revision Two-dimensional belief change Revision by comparison Iterated revision Sphere semantics Epistemic entrenchment 

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References

  1. 1.
    Alchourrón, C. E., Gärdenfors, P., & Makinson, D. (1985). On the logic of theory change: Partial meet contraction and revision functions. Journal of Symbolic Logic, 50, 510–530.CrossRefGoogle Scholar
  2. 2.
    Booth, R., & Meyer, T. (2006). Admissible and restrained revision. Journal of Artificial Intelligence Research, 26, 127–151.Google Scholar
  3. 3.
    Boutilier, C. (1993). Revision sequences and nested conditionals. In R. Bajcsy (Ed.), Proceedings of the thirteenth International Joint Conference on Artificial Intelligence (IJCAI’93) (pp. 519–525).Google Scholar
  4. 4.
    Cantwell, J. (1997). On the logic of small changes in hypertheories. Theoria, 63, 54–89.CrossRefGoogle Scholar
  5. 5.
    Darwiche, A., & Pearl, J. (1997). On the logic of iterated belief revision. Artificial Intelligence, 89, 1–29.CrossRefGoogle Scholar
  6. 6.
    Fermé, E., & Rott, H. (2004). Revision by comparison. Artificial Intelligence, 157, 5–47.CrossRefGoogle Scholar
  7. 7.
    Field, H. (1980). Science without numbers: A defence of nominalism. Princeton/Oxford: Princeton University Press/Blackwell.Google Scholar
  8. 8.
    Gärdenfors, P. (1988). Knowledge in flux: Modeling the dynamics of epistemic states. Cambridge: Bradford Books, MIT Press.Google Scholar
  9. 9.
    Gärdenfors, P., & Makinson, D. (1988). Revisions of knowledge systems using epistemic entrenchment. In M. Vardi (Ed.), TARK’88: Proceedings of the second conference on theoretical aspects of reasoning about knowledge (pp. 83–95). Los Altos: Morgan Kaufmann.Google Scholar
  10. 10.
    Goodman, N. (1955). Fact, fiction, and forecast (4th ed., 1983). Cambridge: Harvard University Press.Google Scholar
  11. 11.
    Gordon, P. (2004). Numerical cognition without words: Evidence from Amazonia. Science, 306(5695), 496–499.CrossRefGoogle Scholar
  12. 12.
    Grove, A. (1988). Two modellings for theory change. Journal of Philosophical Logic, 17, 157–170.CrossRefGoogle Scholar
  13. 13.
    Hellman, G. (1989). Mathematics without numbers: Towards a modal-structural interpretation. Oxford: Clarendon Press.Google Scholar
  14. 14.
    Hild, M., & Spohn, W. (2008). The measurement of ranks and the laws of iterated contraction. Artificial Intelligence, 172, 1195–1218.CrossRefGoogle Scholar
  15. 15.
    Konieczny, S., Medina Grespan, M., & Pino-Pérez, R. (2010). Taxonomy of improvement operators and the problem of minimal change. In F. Lin, U. Sattler, & M. Truszczynski (Eds.), Principles of knowledge representation and reasoning: Proceedings of the twelfth international conference, KR 2010 (pp. 161–170). AAAI Press. Toronto, Canada, 9–13 May 2010.Google Scholar
  16. 16.
    Konieczny, S., & Pino-Pérez, R. (2008). Improvement operators. In G. Brewka, & J. Lang (Eds.), Principles of knowledge representation and reasoning: Proceedings of the eleventh international conference, KR 2008 (pp. 177–187). AAAI Press. Sydney, Australia, 16–19 September 2008.Google Scholar
  17. 17.
    Levi, I. (2003). Contracting from epistemic hell is routine. Synthese, 135, 141–164.CrossRefGoogle Scholar
  18. 18.
    Lewis, D. (1973). Counterfactuals (2nd ed., 1986). Oxford: Blackwell.Google Scholar
  19. 19.
    Lindström, S., & Rabinowicz, W. (1991). Epistemic entrenchment with incomparabilities and relational belief revision. In A. Fuhrmann, & M. Morreau (Eds.), The logic of theory change. LNAI 465 (pp. 93–126). Berlin: Springer.CrossRefGoogle Scholar
  20. 20.
    Nayak, A. (1994). Iterated belief change based on epistemic entrenchment. Erkenntnis, 41, 353–390.CrossRefGoogle Scholar
  21. 21.
    Nayak, A., Pagnucco, M., & Peppas, P. (2003). Dynamic belief revision operators. Artificial Intelligence, 146, 193–228.CrossRefGoogle Scholar
  22. 22.
    Olsson, E. J. (2003). Avoiding epistemic hell: Levi on pragmatism and consistency. Synthese, 135, 119–140.CrossRefGoogle Scholar
  23. 23.
    Pagnucco, M., & Rott, H. (1999). Severe withdrawal—and recovery. Journal of Philosophical Logic, 28, 501–547. (Correct, complete reprint with original pagination in the February 2000 issue; see publisher’s ‘Erratum’ (2000). Journal of Philosophical Logic, 29, 121).Google Scholar
  24. 24.
    Papini, O. (2001). Iterated revision operations stemming from the history of an agent’s observations. In M.-A. Williams, & H. Rott (Eds.), Frontiers in belief revision (pp. 279–301). Dordrecht: Kluwer.Google Scholar
  25. 25.
    Rott, H. (1991). A nonmonotonic conditional logic for belief revision I. In A. Fuhrmann, & M. Morreau (Eds.), The logic of theory change. LNAI (Vol. 465, pp. 135–181). Berlin: Springer-Verlag.CrossRefGoogle Scholar
  26. 26.
    Rott, H. (2001). Change, choice and inference: A study in belief revision and nonmonotonic reasoning. Oxford: Oxford University Press.Google Scholar
  27. 27.
    Rott, H. (2003). Basic entrenchment. Studia Logica, 73, 257–280.CrossRefGoogle Scholar
  28. 28.
    Rott, H. (2003). Coherence and conservatism in the dynamics of belief. Part II: Iterated belief change without dispositional coherence. Journal of Logic and Computation, 13, 111–145.CrossRefGoogle Scholar
  29. 29.
    Rott, H. (2006). Revision by comparison as a unifying framework: Severe withdrawal, irrevocable revision and irrefutable revision. Theoretical Computer Science, 355, 228–242.CrossRefGoogle Scholar
  30. 30.
    Rott, H. (2007). Bounded revision: Two-dimensional belief change between conservatism and moderation. In T. Rønnow-Rasmussen, B. Petersson, J. Josefsson, & D. Egonsson (Eds.), Hommage à Wlodek. philosophical papers dedicated to Wlodek Rabinowicz. Lunds Universitet. Available at http://www.fil.lu.se/hommageawlodek/site/papper/RottHans.pdf. Accessed 5 June 2011
  31. 31.
    Rott, H. (2007). Two-dimensional belief change—an advertisement. In G. Bonanno, J. P. Delgrande, J. Lang, & H. Rott (Eds.), Formal models of belief change in rational agents. IBFI, Schloss Dagstuhl. Available at http://drops.dagstuhl.de/opus/volltexte/2007/1240. Accessed 5 June 2011
  32. 32.
    Rott, H. (2009). Shifting priorities: Simple representations for twenty-seven iterated theory change operators. In D. Makinson, J. Malinowski, & H. Wansing (Eds.), Towards mathematical philosophy. Trends in Logic (pp. 269–296). Berlin: Springer Verlag.Google Scholar
  33. 33.
    Segerberg, K. (1998). Irrevocable belief revision in dynamic doxastic logic. Notre Dame Journal of Formal Logic, 39, 287–306.CrossRefGoogle Scholar
  34. 34.
    Spohn, W. (1988). Ordinal conditional functions: A dynamic theory of epistemic states. In W. Harper, & B. Skyrms (Eds.), Causation in decision, belief change, and statistics (pp. 105–134). Dordrecht: Kluwer.Google Scholar

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© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Department of PhilosophyUniversity of RegensburgRegensburgGermany

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