This paper explores nonmonotonic reasoning, focusing on the properties that define it positively rather than negatively. The authors study various families of nonmonotonic consequence relations, both proof-theoretic and semantic, aiming to provide a general framework for understanding and comparing different systems. They introduce five families of nonmonotonic consequence relations, characterized by representation theorems that relate the proof-theoretic and semantic perspectives. One of these families, preferential relations, is studied in detail, with the proposed preferential models being more powerful than previous probabilistic semantics. The paper uses propositional logic as the basic language and discusses the expressive power of different systems, comparing them to conditional logic. It also introduces a language for expressing conditional assertions and compares its expressive power to that of other approaches. The paper concludes with a discussion of cumulative reasoning, defining cumulative models and showing how they define cumulative consequence relations. The authors provide soundness and completeness results, demonstrating that every cumulative consequence relation can be defined by a suitable model.This paper explores nonmonotonic reasoning, focusing on the properties that define it positively rather than negatively. The authors study various families of nonmonotonic consequence relations, both proof-theoretic and semantic, aiming to provide a general framework for understanding and comparing different systems. They introduce five families of nonmonotonic consequence relations, characterized by representation theorems that relate the proof-theoretic and semantic perspectives. One of these families, preferential relations, is studied in detail, with the proposed preferential models being more powerful than previous probabilistic semantics. The paper uses propositional logic as the basic language and discusses the expressive power of different systems, comparing them to conditional logic. It also introduces a language for expressing conditional assertions and compares its expressive power to that of other approaches. The paper concludes with a discussion of cumulative reasoning, defining cumulative models and showing how they define cumulative consequence relations. The authors provide soundness and completeness results, demonstrating that every cumulative consequence relation can be defined by a suitable model.