Defining Relative Likelihood in Partially-Ordered Preferential Structures

TL;DR

This paper extends the concept of likelihood ordering from worlds to sets of worlds within partial orders, exploring the logical and axiomatic implications, especially in relation to counterfactuals and default reasoning.

Contribution

It introduces a natural extension of likelihood orders to sets of worlds in partial orders and analyzes the resulting logic and axioms, building on Lewis's earlier total order approach.

Findings

Extended likelihood ordering to sets of worlds in partial orders

Identified subtleties in lifting orders from worlds to sets

Connected likelihood logic with default reasoning

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 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.