Synthese (2008) 165:179–202 Knowledge, Rationality & Action 143–166 DOI 10.1007/s11229-008-9369-8 Probabilistic dynamic belief revision Alexandru Baltag · Sonja Smets Received: 14 January 2008 / Accepted: 8 July 2008 / Published online: 8 October 2008 © Springer Science+Business Media B.V. 2008 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 plau- sibility models. We also consider a related notion, that of safe belief, which is a weak (non-negatively 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 probabilistic “action models” and proposing a notion of probabilistic update product, that comes together with appropriate reduction laws. Keywords Belief revision · Conditional belief · Dynamic-epistemic logic · Popper functions A. Baltag Oxford University Computing Laboratory, University of Oxford, Parks Road, Oxford OX1 3QD, UK e-mail: alexandru.baltag@comlab.ox.ac.uk A. Baltag GPI, University of Hertfordshire, Hatfield, UK S. Smets (B ) Center for Logic and Philosophy of Science, Vrije Universiteit Brussel, Brussels, Belgium e-mail: sonsmets@vub.ac.be S. Smets IEG, University of Oxford, Oxford, UK 123