A Qualitative Theory of Dynamic Interactive Belief Revision

  • Alexandru Baltag
  • Sonja Smets
Part of the Springer Graduate Texts in Philosophy book series (SGTP, volume 1)


We present a logical setting that incorporates a belief-revision mechanism within Dynamic-Epistemic logic. As the “static” basis for belief revision, we use epistemic plausibility models, together with a modal language based on two epistemic operators: a “knowledge” modality K (the standard S5, fully introspective, notion), and a “safe belief” modality □ (“weak”, non-negatively-introspective, notion, capturing a version of Lehrer’s “indefeasible knowledge”). To deal with “dynamic” belief revision, we introduce action plausibility models, representing various types of “doxastic events”. Action models “act” on state models via a modified update product operation: the “Action-Priority” Update. This is the natural dynamic generalization of AGM revision, giving priority to the incoming information (i.e., to “actions”) over prior beliefs. We completely axiomatize this logic, and show how our update mechanism can “simulate”, in a uniform manner, many different belief-revision policies.



Sonja Smets’ contribution to this research was made possible by the post-doctoral fellowship awarded to her by the Flemish Fund for Scientific Research. We thank Johan van Benthem for his insights and help, and for the illuminating discussions we had with him on the topic of this paper. His pioneering work on dynamic belief revision acted as the “trigger” for our own. We also thank Larry Moss, Hans van Ditmarsch, Jan van Eijck and Hans Rott for their most valuable feedback. Finally, we thank the editors and the anonymous referees of the LOFT7-proceedings for their useful suggestions and comments.

During the republication of this paper in 2015, the research of Sonja Smets was funded by the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant agreement no.283963.


  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(2), 510–530.CrossRefGoogle Scholar
  2. Aucher, G. (2003). A combined system for update logic and belief revision. Master’s thesis, University of Amsterdam. ILLC Publications MoL-2003-03.Google Scholar
  3. Aumann, R. J. (1999). Interactive epistemology I: Knowledge. International Journal of Game Theory, 28(3), 263–300.CrossRefGoogle Scholar
  4. Baltag, A. (2002). A logic for suspicious players: Epistemic actions and belief updates in games. Bulletin of Economic Research, 54(1), 1–46.CrossRefGoogle Scholar
  5. Baltag, A., & Moss, L. S. (2004). Logics for epistemic programs. Synthese, 139(2), 165–224.CrossRefGoogle Scholar
  6. Baltag, A., Moss, L. S., & Solecki, S. (1998). The logic of public announcements, common knowledge, and private suspicions. In I. Gilboa (Ed.), Proceedings of the 7th Conference on Theoretical Aspects of Rationality and Knowledge (TARK 98), Morgan Kaufmann Publishers Inc. San Francisco, CA, USA, (pp. 43–56).Google Scholar
  7. Baltag, A., & Sadrzadeh, M. (2006). The algebra of multi-agent dynamic belief revision. Electronic Notes in Theoretical Computer Science, 157(4), 37–56.CrossRefGoogle Scholar
  8. Baltag, A., & Smets, S. (2006). Conditional doxastic models: A qualitative approach to dynamic belief revision. Electronic Notes in Theoretical Computer Science, 165, 5–21.CrossRefGoogle Scholar
  9. Baltag, A., & Smets, S. (2006b) Dynamic belief revision over multi-agent plausibility models. In Bonanno et al. (2006) (pp. 11–24).Google Scholar
  10. Baltag, A., & Smets, S. (2006c). The logic of conditional doxastic actions: A theory of dynamic multi-agent belief revision. In S. Artemov, & Parikh, R. (Eds.), Proceedings of ESSLLI Workshop on Rationality and Knowledge, (pp. 13–30). ESSLLI.Google Scholar
  11. Baltag, A., & Smets, S. (2007a). From conditional probability to the logic of doxastic actions. In D. Samet (Ed.), Proceedings of the 11th Conference on Theoretical Aspects of Rationality and Knowledge (TARK), Brussels (pp. 52–61). UCL Presses Universitaires de Louvain.Google Scholar
  12. Baltag, A., & Smets, S. (2007b). Probabilistic dynamic belief revision. In J. F. A. K. van Benthem, S. Ju, & F. Veltman (Eds.), A Meeting of the Minds: Proceedings of the Workshop on Logic, Rationality and Interaction, Beijing, 2007 (Texts in computer science, Vol. 8). London: College Publications.Google Scholar
  13. Battigalli, P., & Siniscalchi, M. (2002). Strong belief and forward induction reasoning. Journal of Economic Theory, 105(2), 356–391.CrossRefGoogle Scholar
  14. Blackburn, P., de Rijke, M., & Venema, Y. (2001). Modal logic (Cambridge tracts in theoretical computer science, Vol. 53). Cambridge: Cambridge University Press.Google Scholar
  15. Board, O. (2002). Dynamic interactive epistemology. Games and Economic Behaviour, 49(1), 49–80.CrossRefGoogle Scholar
  16. Bonanno, G. (2005). A simple modal logic for belief revision. Synthese, 147(2), 193–228.CrossRefGoogle Scholar
  17. Bonanno, G., van der Hoek, W., & Wooldridge, M. (Eds.). (2006). Proceedings of the 7th Conference on Logic and the Foundations of Game and Decision Theory (LOFT7), University of Liverpool UK.Google Scholar
  18. Friedmann, N., & Halpern, J. Y. (1994). Conditional logics of belief revision. In Proceedings of the of 12th National Conference on Artificial Intelligence (AAAI-94), Seattle, 31 July–4 Aug 1994 (pp. 915–921). Menlo Park: AAAI.Google Scholar
  19. Gärdenfors, P. Knowledge in flux: Modelling the dynamics of epistemic states. Gardenfors. 1988, MIT Press, Cambridge/London.Google Scholar
  20. Gerbrandy, J. (1999). Dynamic epistemic logic. In L. S. Moss, J. Ginzburg, & M. de Rijke (Eds.), Logic, language and information (Vol. 2, p. 67–84). Stanford: CSLI Publications/Stanford University.Google Scholar
  21. Gerbrandy, J., & Groeneveld, W. (1997). Reasoning about information change. Journal of Logic, Language and Information, 6(2), 147–169.CrossRefGoogle Scholar
  22. Gerbrandy, J. D. (1999). Bisimulations on planet Kripke. PhD thesis, University of Amsterdam. ILLC Publications, DS-1999-01.Google Scholar
  23. Gettier, E. (1963). Is justified true belief knowledge? Analysis, 23(6), 121–123.CrossRefGoogle Scholar
  24. Gochet, P., & Gribomont, P. (2006). Epistemic logic. In D. M. Gabbay & J. Woods (Eds.), Handbook of the history of logic (Vol. 7, p. 99–195). Oxford: Elsevier.Google Scholar
  25. Grove, A. (1988). Two modellings for theory change. Journal of Philosophical Logic, 17(2), 157–170.CrossRefGoogle Scholar
  26. Hintikka, J. (1962). Knowledge and belief. Ithaca: Cornell University Press.Google Scholar
  27. Katsuno, H., & Mendelzon, A. O. (1992). On the difference between updating a knowledge base and revising it. In P. Gärdenfors (Ed.), Belief revision (Cambridge tracts in theoretical computer science, pp. 183–203). Cambridge/New York: Cambridge University Press.Google Scholar
  28. Klein, P. (1971). A proposed definition of propositional knowledge. Journal of Philosophy, 68(16), 471–482.CrossRefGoogle Scholar
  29. Kooi, B. P. (2003). Probabilistic dynamic epistemic logic. Journal of Logic, Language and Information, 12(4), 381–408.CrossRefGoogle Scholar
  30. Lehrer, K. (1990). Theory of knowledge. London: Routledge.Google Scholar
  31. Lehrer, K., & Paxson, T. Jr. (1969). Knowledge: Undefeated justified true belief. Journal of Philosophy, 66(8), 225–237.CrossRefGoogle Scholar
  32. Meyer, J.-J. Ch. & van der Hoek, W. (1995). Epistemic logic for AI and computer science (Cambridge tracts in theoretical computer science, Vol. 41). Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  33. Pappas, G., & Swain, M. (Eds.). (1978). Essays on knowledge and justification. Ithaca: Cornell University Press.Google Scholar
  34. Plaza, J. A. (1989). Logics of public communications. In M. L. Emrich, M. S. Pfeifer, M. Hadzikadic, & Z. W. Ras (Eds.), Proceedings of the 4th International Symposium on Methodologies for Intelligent Systems Poster Session Program (pp. 201–216). Oak Ridge National Laboratory, ORNL/DSRD-24.Google Scholar
  35. Rott, H. (1989). Conditionals and theory change: Revisions, expansions, and additions. Synthese, 81(1), 91–113.CrossRefGoogle Scholar
  36. Rott, H. (2004). Stability, strength and sensitivity: Converting belief into knowledge. Erkenntnis, 61(2–3), 469–493.CrossRefGoogle Scholar
  37. Ryan, M., & Schobbens, P.-Y. (1997). Counterfactuals and updates as inverse modalities. Journal of Logic, Language and Information, 6(2), 123–146.CrossRefGoogle Scholar
  38. Segerberg, K. (1998). Irrevocable belief revision in dynamic doxastic logic. Notre Dame Journal of Formal Logic, 39(3), 287–306.CrossRefGoogle Scholar
  39. Spohn, W. (1988). Ordinal conditional functions: A dynamic theory of epistemic states. In W. L. Harper & B. Skyrms (Eds.), Causation in decision, belief change, and statistics (Vol. II, pp. 105–134). Dordrecht/Boston: Kluwer AcademicCrossRefGoogle Scholar
  40. Stalnaker, R. (1968). A theory of conditionals. In N. Rescher (Ed.), Studies in logical theory (APQ monograph series, Vol. 2). Oxford: Blackwell.Google Scholar
  41. Stalnaker, R. (1996). Knowledge, belief and counterfactual reasoning in games. Economics and Philosophy, 12, 133–163.CrossRefGoogle Scholar
  42. Stalnaker, R. (2006). On logics of knowledge and belief. Philosophical Studies, 128(1), 169–199.CrossRefGoogle Scholar
  43. van Benthem, J. F. A. K. (2007). Dynamic logic for belief revision. Journal of Applied Non-classical Logics, 17(2), 129–155.CrossRefGoogle Scholar
  44. van Benthem, J. F. A. K., Gerbrandy, J., & Kooi, B. (2006a) Dynamic update with probabilities. In Bonanno et al. (2006) (pp. 237–246).Google Scholar
  45. van Benthem, J. F. A. K., & Liu, F. (2004). Dynamic logic of preference upgrade. Technical report, University of Amsterdam. ILLC Publications, PP-2005-29.Google Scholar
  46. van Benthem, J. F. A. K., van Eijck, J., & Kooi, B. P. (2006b). Logics of communication and change. Information and Computation, 204(11), 1620–1662.CrossRefGoogle Scholar
  47. van der Hoek, W. (1993). Systems for knowledge and beliefs. Journal of Logic and Computation, 3(2), 173–195.CrossRefGoogle Scholar
  48. van Ditmarsch, H. P. (2000). Knowledge games. PhD thesis, University of Groningen. ILLC Pubications, DS-2000-06.Google Scholar
  49. van Ditmarsch, H. P. (2002). Descriptions of game actions. Journal of Logic, Language and Information, 11(3), 349–365.CrossRefGoogle Scholar
  50. van Ditmarsch, H. P. (2005) Prolegomena to dynamic logic for belief revision. Synthese, 147(2), 229–275.CrossRefGoogle Scholar
  51. van Ditmarsch, H. P., & Labuschagne, W. (2007). My beliefs about your beliefs: A case study in theory of mind and epistemic logic. Synthese, 155(2), 191–209.CrossRefGoogle Scholar
  52. van Ditmarsch, H. P., van der Hoek, W., & Kooi, B. P. (2007). Dynamic epistemic logic (Synthese library, Vol. 337). Dordrecht: Springer.CrossRefGoogle Scholar
  53. Voorbraak, F. P. J. M. (1993). As far as I know. PhD thesis, Utrecht University, Utrecht (Quaestiones infinitae, Vol. VII ).Google Scholar
  54. Williamson, T. (2001). Some philosophical aspects of reasoning about knowledge. In J. van Benthem (Ed.), Proceedings of the 8th Conference on Theoretical Aspects of Rationality and Knowledge (TARK’01) (p. 97). San Francisco: Morgan Kaufmann.Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.ILLCUniversity of AmsterdamAmsterdamThe Netherlands

Personalised recommendations