## A logic with conditional probability operators.(English)Zbl 1265.03010

The authors present a sound and strongly complete infinitary axiomatization of reasoning about linear combinations of conditional probabilities. The logical language is rich enough to formalize sentences of the form: The conditional probability of $$\alpha$$ given $$\beta$$ (in the opinion of Agent 1) is at least the sum of the conditional probabilities of $$\alpha$$ given $$\gamma$$ (in the opinion of Agent 2) and two times the conditional probabilities of $$\gamma$$ given $$\delta$$ (in the opinion of Agent 3). Decidability of the logic is proven. The corresponding decision procedure is in PSPACE (in the case of only one agent).

### MSC:

 03B48 Probability and inductive logic 03B25 Decidability of theories and sets of sentences 03B42 Logics of knowledge and belief (including belief change)

### Keywords:

completeness theorem; decidability; PSPACE-complexity
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