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**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.

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 |