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A logic with conditional probabilities. (English) Zbl 1111.68688
Alferes, José Júlio (ed.) et al., Logics in artificial intelligence. 9th European conference, JELIA 2004, Lisbon, Portugal, September 27–30, 2004. Proceedings. Berlin: Springer (ISBN 3-540-23242-7/pbk). Lecture Notes in Computer Science 3229. Lecture Notes in Artificial Intelligence, 226-238 (2004).
Summary: The paper presents a logic which enriches propositional calculus with three classes of probabilistic operators which are applied to propositional formulas: $$P_{\geq s}(\alpha)$$, $$CP_{=s}(\alpha,\beta)$$ and $$CP_{\geq s}(\alpha,\beta)$$, with the intended meaning “the probability of $$\alpha$$ is at least $$s$$”, “the conditional probability of $$\alpha$$ given $$\beta$$ is $$s$$”, and “the conditional probability of $$\alpha$$ given $$\beta$$ is at least $$s$$”, respectively. Possible-worlds semantics with a probability measure on sets of worlds is defined and the corresponding strong completeness theorem is proved for a rather simple set of axioms. This is achieved at the price of allowing infinitary rules of inference. One of these rules enables us to syntactically define the range of the probability function. This range is chosen to be the unit interval of a recursive nonarchimedean field, making it possible to define another probabilistic operator $$CP_{\approx 1}(\alpha,\beta)$$ with the intended meaning “probabilities of $$\alpha\wedge\beta$$ and $$\beta$$ are almost the same”. This last operator may be used to model default reasoning.
For the entire collection see [Zbl 1056.68002].

##### MSC:
 68T27 Logic in artificial intelligence 68T37 Reasoning under uncertainty in the context of artificial intelligence 03B48 Probability and inductive logic
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