Defining Relative Likelihood in Partially-Ordered Preferential Structures

TL;DR

This paper explores how to extend likelihood orders from individual worlds to sets of worlds within partial orders, analyzing the resulting logic and its implications for default reasoning, building on Lewis's earlier work on total orders.

Contribution

It introduces a natural extension of likelihood orders to sets of worlds in partial orders and examines the associated logical framework and its connection to default reasoning.

Findings

Extended likelihood ordering to sets of worlds in partial orders

Analyzed the logic of relative likelihood in partial orders

Provided insights into default reasoning connections

Abstract

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.