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 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.
