Probabilistic Modeling of Default Reasoning

Abstract

This chapter addresses the main approaches to formalization of default reasoning in first-order framework. The first part of the chapter overviews the historical development of nonmonotonic reasoning systems. After discussing three wellknown early formalisms (Default Logic, Nonmonotonic modal logic and Circumscription), we focus on nonmonotonic consequence relations and their close connections with conditional logics and probabilistic reasoning systems. The logic LP,Iωω ww occupies the central position in this chapter. This logic extends classical first-order logic with a list of Keisler-style conditional probability quantifiers with the intention to express statistical knowledge as well as approximate probabilities. We provide a sound and complete deductive system with respect to semantics based on first-order structures endowed with suitable probability spaces. In addition, we describe two decidable fragments of LP,Iωω ww that seem to be rich enough for many practical purposes. Finally, we discuss how the logic LP,Iωω ww and its suitable extensions can be used to model default reasoning and analyze some properties of nonmonotonic consequence relations in the light of probabilistic inference.