Conditioning on Disjunctive Knowledge: Defaults and Probabilities

Many writers have observed that default logics appear to contain the "lottery paradox" of probability theory. This arises when a default "proof by contradiction" lets us conclude that a typical X is not a Y where Y is an unusual subclass of X. We show that there is a similar problem with default "proof by cases" and construct a setting where we might draw a different conclusion knowing a disjunction than we would knowing any particular disjunct. Though Reiter's original formalism is capable of representing this distinction, other approaches are not. To represent and reason about this case, default logicians must specify how a "typical" individual is selected. The problem is closely related to Simpson's paradox of probability theory. If we accept a simple probabilistic account of defaults based on the notion that one proposition may favour or increase belief in another, the "multiple extension problem" for both conjunctive and disjunctive knowledge vanishes.