Infinitesimal Theories of Uncertainty for Plausible Reasoning

An important feature of human reasoning is its ability to draw conclusions from available information (which is often incomplete). The conclusions drawn this way are just plausible and can be revised in the light of new information. This kind of reasoning is called plausible or default reasoning (e.g. [Reiter, 1980]). In this chapter, we are interested in how uncertainty models such as probability theory, possibility theory [Zadeh, 1978; Dubois and Prade, 1988] or evidence theory [Shafer, 1976; Smets, 1988] can be used to deal with default information. Default information considered here are rules of the form ‘generally, if α then β’, where α and β are propositional formulas; these rules are then subject to exceptions. A typical example of a default information is ‘generally, birds fly’. Of course, a default reasoning system should be nonmonotonic (the addition of formulas to the knowledge base can make the set of plausible conclusions decrease). For instance, given the default rule ‘generally, birds fly’, and knowing that Tweety is a bird, then we intend to conclude that it flies. If we later learn that it is a penguin, however, we should withdraw this conclusion. Among the various approaches used to tackle this problem, we focus on the use of infinitesimal uncertainty values (where the uncertainty model is either probability or evidence theory), where values committed to each proposition of the language are either close to 1 or close 0, to do plausible reasoning. This chapter is neither intended to be an overview of non-standard analysis (See [Robinson, 1966; Weydert, 1995] and [Lehmann and Magidor, 1992, Appendix B] for an exposition on non-standard analysis) nor an overview of default reasoning systems. For an overview of works on default reasoning systems see [Lea Sombe, 1988; Brewka et al., 1991]. However the approaches presented in this chapter do not appear in these two overviews.