Default Reasoning via Topology and Mathematical Analysis: A Preliminary Report.

A default consequence relation alpha vertical bar similar to beta (if alpha, then normally beta) can be naturally interpreted via a 'most' generalized quantifier: alpha vertical bar similar to beta is valid iff in 'most' ffworlds, beta is also true. We define various semantic incarnations of this principle which attempt to make the set of (alpha boolean AND beta)-worlds 'large' and the set of (alpha boolean AND (sic)beta)-worlds 'small'. The straightforward implementation of this idea on finite sets is via 'clear majority'. We proceed to examine different 'majority' interpretations of normality which are defined upon notions of classical mathematics which formalize aspects of 'size'. We define default consequence using the notion of asymptotic density from analytic number theory. Asymptotic density provides a way to measure the size of integer sequences in a way much more fine-grained and accurate than set cardinality. Further on, in a topological setting, we identify 'large' sets with dense sets and 'negligibly small' sets with nowhere dense sets. Finally, we define default consequence via the concept of measure, classically developed in mathematical analysis for capturing 'size' through a generalization of the notions of length, area and volume. The logics defined via asymptotic density and measure are weaker than the KLM system P, the so-called 'conservative core' of nonmonotonic reasoning, and they resemble to probabilistic consequence. Topology goes a longer way towards system P but it misses Cautious Monotony (CM) and AND. Our results show that a 'size'-oriented interpretation of default reasoning is context-sensitive and in 'most' cases it departs from the preferential approach.