  At first sight, it is somewhat perplexing how to formalize this notion of nonmonotonic inference, since it seems to require a circular definition of provability that can’t be replaced with an inductive definition, as in the nonmonotonic case. The difficulty with the early theory of  Sandewall 1972  is that it does not address this difficulty successfully.  McDermott &amp; Doyle 1980  and  Reiter 1980  use fixpoint definitions to solve the problem. In both cases, the logical task is (1) to develop a formalism in which rules like DR can be expressed, and (2) to define the relation between a theory DT (which may incorporate such rules) and the theories E which could count as reasonable consequences of DT. In the terminology that later became standard, we need to define the relation between a theory DT and its extensions.