The Probability of a Possibility: Adding Uncertainty to Default Rules

We present a semantics for adding uncertainty to conditional logics for default reasoning and be­ lief revision. We are able to treat conditional sen­ tences as statements of conditional probability, and express rules for revision such as "If A were believed, then B would be believed to degree p." This method of revision extends conditional­ ization by allowing meaningful revision by sen­ tences whose probability is zero. This is achieved through the use of counterfactual probabilities. Thus, our system accounts for the best properties of qualitative methods of update (in particular, the AGM theory of revision) and probabilistic meth­ ods. We also show how our system can be viewed as a unification of probability theory and possi­ bility theory, highlighting their orthogonality and providing a means for expressing the probability of a possibility. We also demonstrate the connec­ tion to Lewis's method of imaging.