REASONING ABOUT KNOWLEDGE AND PROBABILITY: Preliminary Report

A b s t r a c t : We provide a model for reasoning about knowledge anti probability together. We a.llow explicit mention of probabilities in formulas, so that our language has formulas tha.t essentia.lly say "a.ccording to agent i, formula. (p holds with probability a.t least o~." The language is powerfid enough to allow reasoning a~bout higher-order probabilities, as well as allowing explicit comparisons of the probabilities an agent places on distinct events. We present a general framework for interpreting such formulas, a.nd consider various properties that might hold of the interrelationship between agents' subjective probability spaces at different states. We provide a. complete a.xiomatiza.tion for rea.soning about knowledge a.nd probability, prove a. small model property, and obtain decision procedures. We then consider the effects of adding common knowledge and a. probabilistic va.ria.nt of common knowledge to the language.