Reiter asks about representing, ``Quakers are normally pacifists and Republicans are normally non-pacifists. How about Nixon, who is both a Quaker and a Republican?'' Systems of nonmonotonic reasoning that use non-provability as a basis for inferring negation will infer that Nixon is neither a pacifist nor a non-pacifist. Combining these conclusions with the original premiss leads to a contradiction. We use
When we circumscribe using these three sentences as A(ab,pacifist), we will only be able to conclude that Nixon is either abnormal in aspect1 or in aspect2, and we will not be able to say whether he is a pacifist. Of course, this is the same conclusion as would be reached without circumscription. The point is merely that we avoid contradiction.
Reiter's second example is that a person normally lives in the same city as his wife and in the same city as his employer. But A's wife lives in Vancouver and A's employer is in Toronto. We write
If we have
we will again only be able to conclude that A lives either in Toronto or Vancouver. In this circumscription, the function city must be taken as variable. This might be considered not entirely satisfactory. If one knows that a person either doesn't live in the same city as his wife or doesn't live in the same city as his employer, then there is an increased probability that he doesn't live in the same city as either. A system that did reasoning of this kind would seem to require a larger body of facts and perhaps more explicitly metamathematical reasoning. Not knowing how to do that, we might want to use in both (26) and (27). Then we would conclude nothing about his city once we knew that he wasn't in the same city as both.