Defining Relative Likelihood in Partially-Ordered Preferential
  Structures
  Starting with a likelihood or preference order on worlds, we extend it to a
likelihood ordering on sets of worlds in a natural way, and examine the
resulting logic. Lewis (1973) earlier considered such a notion of relative
likelihood in the context of studying counterfactuals, but he assumed a total
preference order on worlds. Complications arise when examining partial orders
that are not present for total orders. There are subtleties involving the exact
approach to lifting the order on worlds to an order on sets of worlds. In
addition, the axiomatization of the logic of relative likelihood in the case of
partial orders gives insight into the connection between relative likelihood
and default reasoning.
