1. Circumscription is not a ``nonmonotonic logic''. It is a form of nonmonotonic reasoning augmenting ordinary first order logic. Of course, sentence schemata are not properly handled by most present general purpose resolution theorem provers. Even fixed schemata of mathematical induction when used for proving programs correct usually require human intervention or special heuristics, while here the program would have to use new schemata produced by circumscription. In (McCarthy 1979b) we treat some modalities in first order logic instead of in modal logic. In our opinion, it is better to avoid modifying the logic if at all possible, because there are many temptations to modify the logic, and it would be very difficult to keep them compatible.
2. The default case reasoning provided in many systems is less general than circumscription. Suppose, for example, that a block x is considered to be on a block y only if this is explicitly stated, i.e. the default is that x is not on y. Then for each individual block x, we may be able to conclude that it isn't on block A, but we will not be able to conclude, as circumscription would allow, that there are no blocks on A. That would require a separate default statement that a block is clear unless something is stated to be on it.
3. The conjunct in the premiss of (1) is the result of suggestions by Ashok Chandra (1979) and Patrick Hayes (1979) whom I thank for their help. Without it, circumscribing a disjunction, as in the second example in Section 4, would lead to a contradiction.
4. The most direct way of using circumscription in AI is in a heuristic reasoning program that represents much of what it believes by sentences of logic. The program would sometimes apply circumscription to certain predicates in sentences. In particular, when it wants to perform an action that might be prevented by something, it circumscribes the prevention predicate in a sentence A representing the information being taken into account.
Clearly the program will have to include domain dependent heuristics for deciding what circumscriptions to make and when to take them back.
5. In circumscription it does no harm to take irrelevant facts into account. If these facts do not contain the predicate symbol being circumscribed, they will appear as conjuncts on the left side of the implication unchanged. Therefore, the original versions of these facts can be used in proving the left side.
6. Circumscription can be used in other formalisms than first order logic. Suppose for example that a set a satisfies a formula A(a) of set theory. The circumscription of this formula can be taken to be
If a occurs in A(a) only in expressions of the form , then its mathematical properties should be analogous to those of predicate circumscription. We have not explored what happens if formulas like occur.
7. The results of circumscription depend on the set of predicates used to express the facts. For example, the same facts about the blocks world can be axiomatized using the relation on or the relation above considered in section 4 or also in terms of the heights and horizontal positions of the blocks. Since the results of circumscription will differ according to which representation is chosen, we see that the choice of representation has epistemological consequences if circumscription is admitted as a rule of conjecture. Choosing the set of predicates in terms of which to axiomatize a set of facts, such as those about blocks, is like choosing a co-ordinate system in physics or geography. As discussed in (McCarthy 1979a), certain concepts are definable only relative to a theory. What theory admits the most useful kinds of circumscription may be an important criterion in the choice of predicates. It may also be possible to make some statements about a domain like the blocks world in a form that does not depend on the language used.
8. This investigation was supported in part by ARPA Contract MDA-903-76-C-0206, ARPA Order No. 2494, in part by NSF Grant MCS 78-00524, in part by the IBM 1979 Distinguished Faculty Program at the T. J. Watson Research Center, and in part by the Center for Advanced Study in the Behavioral Sciences.