I've already said something about circumscription conditions here. We develop circumscription conditions because there are too many things to think about; so we simply don't consider some possibilities because they are a waste of our time. To do this without being completely arbitrary, we formulate or presuppose a circumscription condition. A case of a circumscription condition would be something like this: the objects that can be shown to have a certain property P (e.g., flight) by reasoning from certain facts A (e.g., the possession of wings) are all the objects that satisfy P (in other words, to find flying things, we ignore anything that does not have wings).
A default assumption is an assumption about what is typically true. For instance, we take it to be typically true that birds can fly. Therefore our default assumption is that when Tweetie is a bird, and we know of nothing that makes it unable to fly, Tweetie can fly. If some new fact comes along suggesting that Tweetie is unable to fly (Tweetie's wings are clipped, or too short, or whatever else) this makes us revise our inference, but our default remains the same. Likewise, we take it to be typically true that animals that fly that are not insects are birds. There are other things that fly that are not insects (bats, for instance), but if something is an animal the flies, and we have reason to think it is not an insect, it is a reasonable (albeit defeasible) conclusion that it is a bird. An example of a type of default assumption very important for rational searches is a closed world assumption. Closed world assumptions basically say that what is not recognizably true (in some way) is false. A closed world assumption is essential for one of the most important types of rational searches, the negation-as-failure search. In a negation-as-failure search, failure to find something is taken as proof that it does not exist. Is there a cat on the desk in front of me? I haven't run every possible test to determine whether there is, but I don't see one. I have a default assumption that if something is a cat, it is visible; given this, I form a plausible circumscription condition that the only cats that can be said to exist are those that can be seen. I perform a search by viewing the desk. I can't see a cat. Therefore there is no cat on the desk in front of me. It's not impossible that there's an invisible cat (an exception to the typical case) on the desk in front of me; it's not impossible that the reason I can't see a cat on the desk is not that there isn't one, and not that it's invisible, but that there's something wrong with my vision. But it's entirely reasonable to conclude that there's no cat on the desk if I can't see one, because the default assumption and the circumscription condition are both good ones. So this is a good, reasonable negation-as-failure search.
Philosophy is full of negation-as-failure searches, put in one form or another; a good example is to be had in Berkeley (Principles 10):
But I desire any one to reflect and try whether he can, by any abstraction of thought, conceive the extension and motion of a body without all other sensible qualities. For my own part, I see evidently that it is not in my power to frame an idea of a body extended and moving, but I must withal give it some colour or other sensible quality which is acknowledged to exist only in the mind. In short, extension, figure, and motion, abstracted from all other qualities, are inconceivable.
This is a challenge: perform a rational search (by reflecting) for the ability to conceive extension without other sensible qualities among all abilities to frame ideas; if your search fails, the reasonable conclusion is that you have no such ability. Indeed, on the basis of his own search, Berkeley says that he sees evidently, i.e., clearly, that he has no such ability. Of course, this reasoning makes perfect sense in a framework based on a closed world assumption (namely, that if you aren't able to recognize your possession of an ability simply by reflecting, you don't have that ability).
Another philosophical example of a negation-as-failure search is found in the problem of evil. The problem of evil, as usually presented, is effectively based on a challenge like Berkeley's, but with different parameters. The challenge is this: Perform a rational search among possible reasons for allowing the existence of such-and-such evils. If you can't find one, we can presume that none exists (negation as failure). Then, typically, one goes on to use this result (no reason exists) to challenge the existence of God (on the assumption that there would have to be a reason, given that God is good and wise).
Several classical responses to the problem of evil focus on the challenge. An example is the omniscience objection. (The omniscience objection is based on the point that the challenger has to show not merely that we wouldn't have a reason, but that an omniscience, omnipotent, omnibenevolent being wouldn't have a reason. In other words, we have to search all possible reasons that are available to an omniscient, omnipotent being.) The search among all possible reasons is an immense one. When we are dealing with finite creatures like human beings, we can circumscribe this search using assumptions about human beings. When we are talking about an omniscient, omnipotent, omnibenevolent being, however, we can't use those circumscription conditions. Someone who puts forward the omniscience objection in a sense regards the challenge stage of the problem of evil as a bluff: the challenger says, "If we run a rational search, we find no reason"; the omniscience objector says, "You have not run the relevant rational search, because you are assuming circumscription conditions that are not relevant to this problem, and the circumscription conditions that are relevant are not sufficient to make the search manageable." Another (related, but different) sort of response is the ignorance response, which attacks the closed world assumption itself. One form of the ignorance response, the skeptical form, simply denies that we have any reason to think that a negation-as-failure would yield a correct answer in this sort of case. Another form, that associated with Bishop Butler, takes a different route. Butler argues not from skepticism about our ability to know but from our knowledge about what we don't know: the claim is that we know that we don't know a lot of things that are relevant to the success of the search. Of course, not every formulation of the problem of evil that implicitly makes use of a rational search uses the same assumptions and conditions; so it does not follow from a given response's being successful against one formulation that it will be successful against every such formulation. And we need not assume that every formulation of the problem of evil makes use of a rational search (although all the most common ones seem to do so). But a review of formulations of the problem of evil will easily turn up formulations that fall to objections like these, because the search to which they appeal is not of the right sort, or is improperly formulated, or whatever else.
There are lots of other cases where these points are relevant. (Of course, there has been some fascinating work recently on circumscription and default logics, particularly with regard to artificial intelligence; obviously the above is related to these, but it should be kept in mind that the focus of such research is formal systems, which is a much narrower focus than anything I am discussing above.)