The standard Kolmogorovian notion of probability has been utilized since at least 1950 in order to provide models of conditionals and first order logic (possibly Karl Popper initiated this tradition that has had many contemporary followers - Field, Adams, McGee, Spohn, among others). Nevertheless, recent probabilistic models of non-monotonic notions of inference have appealed either to extremely high probability (Pearl) or to non-standard (infinitesimal) probability (Lehamnn and Magidor). The talk presents first a complete probabilistic characterization of the notion of Rational Consequence R proposed by Lehamnn and Magidor. The model utilizes conditional probability as a primitive notion.. We show that if the underlying language is rich enough (if it contains countably many atoms) the needed notion of conditional probability cannot be countably additive. This and other results indicate that adequate probabilistic models utilize a notion of probability of the type advocated by De Finetti and Savage in decision theory, rather than Kolmogorov's notion. The model is also used in order to build probabilistic models for some recent computational models of abduction. Time permitting we will also consider extensions of probabilistic models capable of dealing with nested conditionals.

Note: The completeness result for R was proved in collaboration with Rohit Parikh.