From linear possibility distributions to a non-infinitesimal probabilistic semantics of conditional knowledge

The authors have proposed in their previous works to view a set of default information of the form, {open_quotes}generally, from {alpha}{sub i} deduce {beta}{sub i}{close_quotes}, as the family of possibility distributions satisfying constraints expressing that the situations where {alpha}{sub 1} {beta}{sub i} is true are more possible than the situations where {alpha}{sub i} {beta}{sub i} is true. This provides a representation theorem for default reasoning obeying the System P of postulates proposed by Lehmann et al., and for which it also exists a semantics in terms of infinitesimal probabilities. This paper offers a detailed analysis of the structure of this family of possibility distributions by making use of two different orderings between them: the specificity ordering and the refinement ordering. It is shown that from a representation point of view, it is sufficient to consider the subset of linear possibility distributions which corresponds to all the possible completions of the default knowledge in agreement with the constraints. Surprisingly, it is also shown that a standard probabilistic semantics can be equivalently given to System P, without referring to infinitesimals, by using a special family of probability measures, here called acceptance functions, and that has been also recently considered by Snow inmore » that perspective.« less