Defining Relative Likelihood in Partially-Ordered Preferential Structures
This work addresses foundational issues in formal logic and AI for researchers in non-monotonic reasoning, but it is incremental as it builds on Lewis's prior work on total orders.
The paper tackles the problem of extending a likelihood or preference order from individual worlds to sets of worlds in partially-ordered structures, revealing complications not present in total orders, and it axiomatizes the resulting logic to connect relative likelihood with default reasoning.
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.