Reasoning about knowledge and probability

We provide a model for reasoning about knowledge and probability together. We allow explicit mention of probabilities in formulas, so that our language has formulas that essentially say "according to agent i, formula φ holds with probability at least α." The language is powerful enough to allow reasoning about 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, and consider various properties that might hold of the interrelationship between agents' subjective probability spaces at different states. We provide a complete axiomatization for reasoning about knowledge and probability, prove a small model property, and obtain decision procedures. We then consider the effects of adding common knowledge and a probabilistic variant of common knowledge to the language.

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