On natural deduction system for nonmonotonic reasoning

This paper is concerned with the problem of how to construct a natural deduction system for nonmonotonic reasoning. Three possible constructions are investigated, the ones which consists of (1) pure nonmonotonic inference rules; (2) nonmonotonic inference rules plus a nonmonotonic tautology base; or (3) nonmonotonic inference rules plus a nonmonotonic conditional assertion base. Based on a framework of nonmonotonic reasoning, called RN frame, we show that what can be entailed from a given set of premises in the first kind of systems are exactly those can be done in the classical logic. The second sort of systems can entail more only when the premises of an inference relation is consistent with background knowledge, which is generally viewed as a trivial case in nonmonotonic reasoning. A system constructed in the third way could differ from classical logic in non-trivial case, but it is still reducible to a system with the classical derivability and an ordering of knowledge base.