The relationship between KLM and MAK models for nonmonotonic inference operations
The authors discuss the relationship between two variants of preferential models for nonmonotonic inference: the models of Kraus, Lehmann, and Magidor (KLM models) and those of Makinson (MAK models). Models in both variants include a set of states and a binary relation on states. In MAK models a satisfaction relation between states and formulae is used but in KLM models an additional set of worlds and a labelling function associating a non-empty set of worlds with each state is employed. The main idea is to show that it is possible to characterize KLM models in an alternative way by replacing the set of worlds and the labeling function by a background consequence operator and a single satisfaction relation between states and formulae. For that purpose MAK models are amplified by attaching a consequence operator as an additional ingredient. The notion of the core of a KLM model is introduced by substituting a derived satisfaction relation and a consequence operator for the set of worlds and the labeling function. The core fully determines the nonmonotonic inference relation induced by the model. The main result shows that the core of a KLM model can be characterized in two equivalent ways. It is an amplified MAK model whose satisfaction relation may be defined as the intersection of a non-empty family of classically well-behaved satisfaction relations or it is an amplified MAK model whose satisfaction relation fulfills certain syntactic conditions. As a corollary this gives characterizations of classes of KLM models which are called cumulative models and preferential models by Kraus, Lehmann, and Magidor.
scientific article; zbMATH DE number 501110