Conditionals, Infeasible Worlds, and Reasoning with System W

The recently introduced notion of an inductive inference operator captures the process of completing a given conditional belief base to an inference relation. System W is such an inductive inference operator exhibiting some notable properties like extending rational closure and satisfying syntax splitting for inference from conditional belief bases. However, the definition of system W and the shown results regarding its properties only take belief bases into account that satisfy a strong notion of consistency where no worlds may be completely infeasible. In this paper, we lift this limitation and extend the definition of system W to also cover belief bases that force some worlds to be infeasible. We establish the position of the extended system W within a map of other inductive inference operators being able to deal with the presence of infeasible worlds, including system Z and multipreference closure. For placing lexicographic inference in this map, we show that the definition of lexicographic inference must be slightly modified so that it is an inductive inference operator satisfying direct inference even when there are worlds that are infeasible. Furthermore, we show that, like its unextended version, the extended system W enjoys other desirable properties such as still fully complying with syntax splitting.