From conditional probability to the logic of doxastic actions

ABSTRACT

We investigate the discrete (finite) case of the Popper-Renyi theory of conditional probability, introducing discrete conditional probabilistic models for knowledge and conditional belief, and comparing them with the more standard plausibility models. We also consider a related notion, that of safe belief, which is a weak (nonnegatively introspective) type of "knowledge". We develop a probabilistic version of this concept ("degree of safety") and we analyze its role in games. We completely axiomatize the logic of conditional belief, knowledge and safe belief over conditional probabilistic models. We develop a theory of probabilistic dynamic belief revision, introducing "action models" and a notion of probabilistic update product, that comes together with appropriate reduction laws.

References

1. C. E. Alchourrón, P. Gärdenfors, and D. Makinson. On the logic of theory change: partial meet contraction and revision functions. Journal of Symbolic Logic, 50:510--530, 1985.
2. H. Arlo-Costa and R. Parikh. Conditional probability and defeasible inference. Journal of Philosophical Logic, 34:97--119, 2005.
3. G. Aucher. A combined system for update logic and belief revision. Master's thesis, ILLC, University of Amsterdam, Amsterdam, the Netherlands, 2003.
4. R. J. Aumann. Backwards induction and common knowledge of rationality. Games and Economic Behavior, 8:6--19, 1995.
5. A. Baltag. A logic for suspicious players: epistemic actions and belief updates in games. Bulletin Of Economic Research, 54(1):1--46, 2002.
6. A. Baltag and L. S. Moss. Logics for epistemic programs. Synthese, 139:165--224, 2004. Knowledge, Rationality & Action 1--60.
7. A. Baltag, L. S. Moss, and S. Solecki. The logic of common knowledge, public announcements, and private suspicions. In I. Gilboa, editor, Proceedings of the 7th Conference on Theoretical Aspects of Rationality and Knowledge (TARK 98), pages 43--56, 1998.
8. A. Baltag and S. Smets. Conditional doxastic models: a qualitative approach to dynamic belief revision. Electronic Notes in Theoretical Computer Science, 165:5--21, 2006.
9. A. Baltag and S. Smets. Dynamic belief revision over multi-agent plausibility models. In W. van der Hoek and M. Wooldridge, editors, Proceedings of LOFT'06, pages 11--24. Liverpool, 2006.
10. A. Baltag and S. Smets. The logic of conditional doxastic actions: a theory of dynamic multi-agent belief revision. In Proceedings of ESSLLI Workshop on Rationality and Knowledge. 2006.
11. A. Baltag and S. Smets. A qualitative theory of dynamic interactive belief revision. In W. van der Hoek G. Bonanno and M. Wooldridge, editors, Texts in Logic and Games. Amsterdam University Press, 2007. To appear.
12. O. Board. Dynamic interactive epistemology. Games and Economic Behaviour, 49:49--80, 2004.
13. C. Boutilier. On the revision of probabilistic belief states. Notre Dame Journal of Formal Logic, 36(1):158--183, 1995.
14. B. C. Van Fraassen. Representational of conditional probabilities. Journal of Philosophical Logic, 5:417--430, 1976.
15. B. C. Van Fraassen. Fine-grained opinion, probability, and the logic of full belief. Journal of Philosophical Logic, 24:349--377, 1995.
16. P. Gärdenfors. Knowledge in Flux: Modeling the Dynamics of Epistemic States. Bradford Books, MIT Press, Cambridge, MA, 1988.
17. J. Gerbrandy. Dynamic epistemic logic. Logic, Language and Information, 2, 1999. CSLI Publications, Stanford University.
18. J. Y. Halpern. Lexicographic probability, conditional probability, and nonstandard probability. In Proceedings of the Eighth Conference on Theoretical Aspects of Rationality and Knowledge (TARK 8), pages 17--30, 2001.
19. J. Y. Halpern. Reasoning about Uncertainty. MIT Press, Cambridge MA, 2003.
20. H. Katsuno and A. Mendelzon. On the difference between updating a knowledge base and revising it. In Cambridge Tracts in Theoretical Computer Science, pages 183--203, 1992.
21. B. P. Kooi. Probabilistic dynamic epistemic logic. Journal of Logic, Language and Information, 12:381--408, 2003.
22. G. Pappas and M. Swain, editors. Essays on Knowledge and Justification. Cornell Univ. Press, Ithaca, NY, 1978.
23. K. R. Popper. The Logic of Scientific Discovery (revised Edition). London Hutchison, 1968. First Ed. 1934.
24. A. Renyi. On a new axiomatic theory of probability. Acta Mathematica Academiae Scientiarum Hungaricae, 6:285--335, 1955.
25. A. Renyi. Sur les espaces simples des probabilites conditionnelles. Annales de L'institut Henri Poincare: section B: calcul des probabilities et statistique, B1:3--21, 1964.
26. R. Stalnaker. Knowledge, belief and counterfactual reasoning in games. Economics and Philosophy, 12:133--163, 1996.
27. J. van Benthem. Conditional probability meets update logic. Journal of Philosophical Logic, 12(4):409--421, 2003.
28. J. van Benthem. Dynamic logic of belief revision. Technical Report, ILLC, DARE electronic archive, University of Amsterdam, 2006.
29. J. F. A. K. van Benthem, J. Gerbrandy, and B. P. Kooi. Dynamic update with probabilities. In W. van der Hoek and M. Wooldridge, editors, Proceedings of LOFT'06. Liverpool, 2006.
30. J. F. A. K. van Benthem and F. Liu. Dynamic logic of preference upgrade. Technical report, 2004. ILLC Research Report PP-2005-29.
31. J. F. A. K. van Benthem, J. van Eijck, and B. P. Kooi. Logics of communication and change. Information and Computation, 204(11):1620--1662, 2006.
32. H. P. van Ditmarsch. Prolegomena to dynamic logic for belief revision. Synthese (Knowledge, Rationality & Action), 147:229--275, 2005.

• Theory of computation