Possibilistic and standard probabilistic semantics of conditional knowledge bases

Default pieces of information of the form, 'generally, if alpha then beta', can be modelled by constraints expressing that, when alpha is true, beta is more plausible than its negation. In previous works, the authors have cast this view in the framework of comparative possibility theory, showing that a set of default rules is equivalent to a set of comparative possibility distributions, each encoding an epistemic state. A representation theorem in terms of this semantics, for default reasoning obeying the System P of postulates proposed by Kraus, Lehmann and Magidor, has been obtained. This paper offers a detailed analysis of the structure of comparative possibility distributions representing default knowledge, by laying bare two different relations between epistemic states: the specificity ordering and the informativeness ordering. It is shown that the representation theorem still holds when restricting to linear comparative possibility distributions. They correspond to all the possible completions of the default knowledge by means of a so-called completion rule of inference. As a consequence of this result we provide a standard probabilistic semantics to System P, without referring to infinitesimals (used in Adams' semantics, revisited by Pearl). It relies on a special family of probability measures, that we call big-stepped probabilities, recently considered by Snow.