(McCarthy 1986) discusses several formalizations, proposing those based on nonmonotonic reasoning as improvements of earlier ones. Here are some.
1. Inheritance with exceptions. Birds normally fly, but there are exceptions, e.g. ostriches and birds whose feet are encased in concrete. The first exception might be listed in advance, but the second has to be derived or verified when mentioned on the basis of information about the mechanism of flying and the properties of concrete.
There are many ways of nonmonotonically axiomatizing the facts about which birds can fly. The following axioms using a predicate ab standing for ``abnormal'' seem to me quite straightforward.
Unless an object is abnormal in aspect1, it can't fly.
It wouldn't work to write ab(x) instead of ab(aspect1(x)), because we don't want a bird that is abnormal with respect to its ability to fly to be automatically abnormal in other respects. Using aspects limits the effects of proofs of abnormality.
Unless a bird is abnormal in aspect2, it can fly.
When these axioms are combined with other facts about the problem, the predicate ab is then to be circumscribed, i.e. given its minimal extent compatible with the facts being taken into account. This has the effect that a bird will be considered to fly unless other axioms imply that it is abnormal in aspect2. (2) is called a cancellation of inheritance axiom, because it explicitly cancels the general presumption that objects don't fly. This approach works fine when the inheritance hierarchy is given explicitly. More elaborate approaches, some of which are introduced in (McCarthy 1986) and improved in (Haugh 1988), are required when hierarchies with indefinite numbers of sorts are considered.
2. (McCarthy 1986) contains a similar treatment of the effects of actions like moving and painting blocks using the situation calculus. Moving and painting are axiomatized entirely separately, and there are no axioms saying that moving a block doesn't affect the positions of other blocks or the colors of blocks. A general ``common-sense law of inertia''
asserts that a fact p that holds in a situation s is presumed to hold in the situation result(e,s) that results from an event e unless there is evidence to the contrary. Unfortunately, Lifschitz (1985 personal communication) and Hanks and McDermott (1986) showed that simple treatments of the common-sense law of inertia admit unintended models. Several authors have given more elaborate treatments, but in my opinion, the results are not yet entirely satisfactory. The best treatment so far seems to be that of (Lifschitz 1987).
