Before proceeding to applications of circumscription I want to suggest a typology of the uses of nonmonotonic reasoning. Each of the several papers that introduces a mode of nonmonotonic reasoning seems to have a particular application in mind. Perhaps we are looking at different parts of an elephant. The orientation is towards circumscription, but I suppose the considerations apply to other formalisms as well.
Nonmonotonic reasoning has several uses.
1. As a communication convention. Suppose A tells B about a situation involving a bird. If the bird cannot fly, and this is relevant, then A must say so. Whereas if the bird can fly, there is no requirement to mention the fact. For example, if I hire you to build me a bird cage and you don't put a top on it, I can get out of paying for it even if you tell the judge that I never said my bird could fly. However, if I complain that you wasted money by putting a top on a cage I intended for a penguin, the judge will agree with you that if the bird couldn't fly I should have said so.
The proposed Common Business Communication Language (CBCL) (McCarthy 1982) must include nonmonotonic conventions about what may be inferred when a message leaves out such items as the method of delivery.
2. As a database or information storage convention. It may be a convention of a particular database that certain predicates have their minimal extension. This generalizes the closed world assumption. When a database makes the closed world assumption for all predicates it is reasonable to imbed this fact in the programs that use the database. However, when only some predicates are to be minimized, we need to say which ones by appropriate sentences of the database, perhaps as a preamble to the collection of ground sentences that usually constitute the main content.
Neither 1 nor 2 requires that most birds can fly. Should it happen that most birds that are subject to the communication or about which information is requested from the data base cannot fly, the convention may lead to inefficiency but not incorrectness.
3. As a rule of conjecture. This use was emphasized in (McCarthy 1980). The circumscriptions may be regarded as expressions of some probabilistic notions such as ``most birds can fly'' or they may be expressions of standard cases. Thus it is simple to conjecture that there are no relevant present material objects other than those whose presence can be inferred. It is also a simple conjecture that a tool asserted to be present is usable for its normal function. Such conjectures sometimes conflict, but there is nothing wrong with having incompatible conjectures on hand. Besides the possibility of deciding that one is correct and the other wrong, it is possible to use one for generating possible exceptions to the other.
4. As a representation of a policy. The example is Doyle's ``The meeting will be on Wednesday unless another decision is explicitly made''. Again probabilities are not involved.
5. As a very streamlined expression of probabilistic information when numerical probabilities, especially conditional probabilities, are unobtainable. Since circumscription doesn't provide numerical probabilities, its probabilistic interpretation involves probabilities that are either infinitesimal, within an infinitesimal of one, or intermediate -- without any discrimination among the intermediate values. The circumscriptions give conditional probabilities. Thus we may treat the probability that a bird can't fly as an infinitesimal. However, if the rare event occurs that the bird is a penguin, then the conditional probability that it can fly is infinitesimal, but we may hear of some rare condition that would allow it to fly after all.
Why don't we use finite probabilities combined by the usual laws? That would be fine if we had the numbers, but circumscription is usable when we can't get the numbers or find their use inconvenient. Note that the general probability that a bird can fly may be irrelevant, because we are interested in the facts that influence our opinion about whether a particular bird can fly in a particular situation.
Moreover, the use of probabilities is normally considered to require the definition of a sample space, i.e. the space of all possibilities. Circumscription allows one to conjecture that the cases we know about are all that there are. However, when additional cases are found, the axioms don't have to be changed. Thus there is no fixed space of all possibilities.
Notice also that circumscription does not provide for weighing evidence; it is appropriate when the information permits snap decisions. However, many cases nominally treated in terms of weighing information are in fact cases in which the weights are such that circumscription and other defaults work better.
6. Auto-epistemic reasoning. ``If I had an elder brother, I'd know it''. This has been studied by R. Moore. Perhaps it can be handled by circumscription.
7. Both common sense physics and common sense psychology use nonmonotonic rules. An object will continue in a straight line if nothing interferes with it. A person will eat when hungry unless something prevents it. Such rules are open ended about what might prevent the expected behavior, and this is required, because we are always encountering unexpected phenomena that modify the operation of our rules. Science, as distinct from common sense, tries to work with exceptionless rules. However, this means that common sense reasoning has to decide when a scientific model is applicable, i.e. that there are no important phenomena not taken into account by the theories being used and the model of the particular phenomena.
Seven different uses for nonmonotonic reasoning seem too many, so perhaps we can condense later.