Approximations of System W Between c-Inference, System Z, and Lexicographic Inference

Inductive inference operators have been introduced to formalize the process of completing a conditional belief base to a full inference relation. In this paper, we investigate the approximation of inductive inference operator system W with combinations of system Z (or equivalently rational closure) and c-inference, both of which are known to be extended by system W. 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 which, however, does not satisfy system P. We also construct the least inductive inference operator extending system Z and c-inference that also satisfies system P and show that it is strictly extended by system W. Furthermore, we develop approximations that extend system W and introduce an inductive inference operator that strictly extends system W and that is strictly extended by lexicographic inference. This leads to a map of inference relations between rational closure and c-inference on the one side and lexicographic inference on the other side.