Approximations of system W for inference from strongly and weakly consistent belief bases

In this article, we investigate approximations of the inductive inference operator system W that has been shown to exhibit desirable inference properties and to extend both system Z, and thus rational closure, and c-inference. For versions of these inference operators that are extended to also cover inference from belief bases that are only weakly consistent, we first show that extended system Z and extended c-inference are captured by extended system W. Then we introduce general functions for generating inductive inference operators: the combination of two inductive inference operators by union, and the completion of an inductive inference operator by an arbitrary set of axioms. We construct the least inductive inference operator extending system Z and c-inference that is closed under system P and show that it is still strictly extended by extended system W. Furthermore, we introduce an inductive inference operator that strictly extends extended system W and that is strictly extended by lexicographic inference. This leads to a comprehensive map of inference relations between rational closure and extended c-inference on the one side and lexicographic inference on the other side with extended system W and its approximations at its centre, where all relationships also hold for the unextended versions. • Introducing the union of inductive inference operators and the closure of inductive inference under a set of postulates. • Showing that system W, extended to cover weakly consistent belief bases, captures extended system Z and extended c-Inference. • Showing that extended system W is captured by adapted lexicographic inference. • Introducing the minimal inductive inference operator extending c-inference and system Z as approximation of system W. • Development of a comprehensive map of inductive inference operators that approximate (extended) system W.