A number of penetrating papers in Brachman et al. fall under our broader construal of metatheory:
Andrew Baker's ``Non-monotonic Reasoning in the Framework of Situation Calculus'' presents a modification of standard circumscription allowing this technique to solve the famous ``Yale Shooting Problem''.In a nutshell, the problem is that knowing that shooting a loaded gun will result in Fred's death, along with knowledge that the gun is loaded and that Fred is alive in the initial situation, doesn't allow, even when conjoined with axioms urged by minimization techniques, for the system to conclude that firing the gun sees to Fred's demise - because it's possible that the gun mysteriously becomes unloaded while waiting, with Fred surviving.
Doyle and Michael Wellman, in their fascinating ``Impediments to Universal Preference-Based Default Theories'', investigate the prospect of a formal theory of rational inference and preference that could unify the various families of approaches to revisable reasoning - and find that such unification is probably not possible. (This paper will be of special interest to philosophers: It ranges over topics as diverse as Pascal's wager and ``mental societies'', composite agents composed of individual selves.)
No discussion of the LAI approach would be complete without noting that this approach is in large part driven by certain gem-like problems - problems that, upon analysis, explode with inspiring and guiding challenges. A perfect example of such a problem is the Lottery Paradox, discussed by David Poole in his ``The Effect of Knowledge on Belief: Conditioning, Specificity and the Lottery Paradox in Default Reasoning''. The paradox runs as follows:We express the paradox in a way that parallel's Pollock's (1995) succint formulation. For a more difficult ``gem'', and a discussion of how it threatens to derail the LAI approach, see Pollock's (1995) discussion of the Paradox of the Preface.
Suppose you hold one ticket ( , for some ) in a fair lottery consisting of 1 million tickets, and suppose it is known that one and only one ticket will win. Since the probability is only .000001 of 's being drawn, it seems reasonable to believe that will not win. (After all, isn't it true that you believe that your head won't be mashed by a meteorite when you walk outside today? And don't you so believe because the probability that your head will be mashed is very low?) By the same reasoning, it seems reasonable to believe that will not win, that will not win, , and that will not win. Therefore, it is reasonable to believe
But we know that
So we have an outright contradiction.
Poole does an excellent job of systematically laying out the various maneuvers taken by LAIniks in an attempt to resolve the Lottery Paradox, and the costs associated with these moves. (How would you design an artificial reasoner capable of escaping the paradox? Grappling with focussed metatheoretical questions such as this one are really, at bottom, the best way to truly understand the LAI approach.)For what it's worth, we happen to like Pollock's (1995) proposed solution to the Lottery Paradox. The core of this solution is the phenomenon of collective defeat, the schema for which is as follows: Suppose that you are warranted in believing r and that you have equally good prima facie reasons for
where is inconsistent but no proper subset of is inconsistent with r. Then, for every :
In this case, we have equally strong support for each and each , so they collectively defeat one another, and so we are not entitled to believe ``either way''. In order to get the proposed solution to the Lottery Paradox, instantiate r to the proposition that the lottery is fair and one ticket will win, and instantiate to the proposition that ticket will not win.