Reasoning with system W and infeasible worlds

Abstract System W is an inference method for conditional belief bases with some notable properties like capturing system Z while, in contrast to system Z, avoiding the drowning problem . This paper further investigates the properties of system W. We show how system W behaves with respect to postulates put forward for inference relations. We develop postulates ensuring compliance with syntax splitting for inference operators based on a full strict partial order on worlds. By observing that system W satisfies these axioms, it is proven that system W satisfies syntax splitting. We also explore how syntax splitting affects the strict partial order underlying system W and exploit this for answering certain types of queries without having to determine this strict partial order completely. However, the original definition of system W and the results above only consider inference from belief bases satisfying a strong notion of consistency. In the second part of this paper, we lift this limitation and extend system W to also cover inference from belief bases that only satisfy a weaker notion of consistency. We investigate the properties of the such extended system W. Especially, it is shown that extended system W complies with syntax splitting and retains the desireable properties of system W. Furthermore, we give an overview of the relations of extended system W to other inductive inference operators.