## Reasoning about knowledge and probability: Preliminary report.(English)Zbl 0699.03010

Theoretical aspects of reasoning about knowledge, Proc. 2nd Conf., Pacific Grove/CA (USA) 1988, 277-293 (1988).
Summary: [For the entire collection see Zbl 0699.00012.]
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 $$\phi$$ holds with probability at least $$\alpha$$”. 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.

### MSC:

 03B60 Other nonclassical logic 03B48 Probability and inductive logic 68T99 Artificial intelligence

Zbl 0699.00012