How to Go Nonmonotonic

Contents Purpose 1.Introduction 1.1. We are all nonmonotonic 1.2. Recalling some features of classical consequence 1.3. Some misunderstandings and a habit to suspend 1.4. Three ways of getting more conclusions out of your premises 2. Using background assumptions 2.1. From classical consequence to pivotal assumptions 2.2. From pivotal assumptions to default assumptions 2.3. Specializations and generalizations 3. Restricting the set of valuations 3.1. From classical consequence to pivotal valuations 3.2. From pivotal valuations to default valuations 3.3. Specializations and generalizations 4. Using additional rules 4.1. From classical consequence to pivotal rules 4.2. From pivotal rules to default rules 4.3. Generalizations and variants 5. Connections with probabilistic inference 5.1. Basic concepts and axioms for probability 5.2. Probabilistic characterizations of classical consequence 5.3. Supraclassical probabilistic consequence relations 5.4. Bringing probabilistic and qualitative inference closer together Appendix: Proof of Theorem 4.3-1. Acknowledgements References 2 Purpose This overview is directed to those who have heard of nonmonotonic reasoning and would like to get a better idea of what it is all about. What are its driving ideas? In what ways does it differ from classical logic? How does it relate to probability? More generally, how does it behave and how can it be used? We will try to answer these questions as clearly as possible, without undue technicality on the one hand, nor vagueness or hand waving on the other. For the outsider, nonmonotonic logic can be a rather mysterious affair. Even from the inside, it can appear to lack unity, with multiple systems proposed by as many authors going in different directions. The few available textbooks tend to perpetuate this impression. Our main purpose is to take some of the mystery out of the subject and show that it is not as unfamiliar as may at first sight seem. In fact, it is easily accessible to anybody with a minimal background in classical propositional logic and a few basic mathematical tools-provided certain misunderstandings and a tenacious habit, which we will signal in the first sections, are avoided and put aside. As we shall show, there are monotonic logics that act as natural bridges between classical consequence and the principal kinds of nonmonotonic logic to be found in the literature. These logics, which we call paraclassical, are very simple to define and easy to study. They provide three main ways of getting more out of your premises than is allowed by strict deduction, that …