A logic for reasoning about probabilities.

*(English)*Zbl 0811.03014Summary: We consider a language for reasoning about probability which allows us to make statements such as “the probability of \(E_ 1\) is less than \({1\over 3}\)” and “the probability of \(E_ 1\) is at least twice the probability of \(E_ 2\)”, where \(E_ 1\) and \(E_ 2\) are arbitrary events. We consider the case where all events are measurable (i.e., represent measurable sets) and the more general case, which is also of interest in practice, where they may not be measurable. The measurable case is essentially a formalization of (the propositional fragment of) Nilsson’s probabilistic logic. As we show elsewhere, the general (nonmeasurable) case corresponds precisely to replacing probability measures by Dempster-Shafer belief functions. In both cases, we provide a complete axiomatization and show that the problem of deciding satisfiability is NP-complete, no worse than that of propositional logic. As a tool for proving our complete axiomatizations, we give a complete axiomatization for reasoning about Boolean combinations of linear inequalities, which is of independent interest. This proof and others make crucial use of results from the theory of linear programming. We then extend the language to allow reasoning about conditional probability and show that the resulting logic is decidable and completely axiomatizable, by making use of the theory of real closed fields.

##### MSC:

03B48 | Probability and inductive logic |

68T27 | Logic in artificial intelligence |

68Q25 | Analysis of algorithms and problem complexity |

03B25 | Decidability of theories and sets of sentences |

##### Keywords:

language for reasoning about probability; measurable sets; probabilistic logic; complete axiomatization; satisfiability; reasoning about Boolean combinations of linear inequalities; linear programming; reasoning about conditional probability; real closed fields
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\textit{R. Fagin} et al., Inf. Comput. 87, No. 1--2, 78--128 (1990; Zbl 0811.03014)

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