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Conference Papers Year : 2018

Default Reasoning via Topology and Mathematical Analysis: A Preliminary Report


A default consequence relation α|~β (if α, then normally β) can be naturally interpreted via a `most' generalized quantifier: α|~β is valid iff in `most' α-worlds, β is also true. We define various semantic incarnations of this principle which attempt to make the set of (α ∧ β)-worlds `large' and the set of (α ∧ ¬ β)-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.
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hal-03623002 , version 1 (29-03-2022)


  • HAL Id : hal-03623002 , version 1
  • OATAO : 22590


Costas D. Koutras, Konstantinos Liaskos, Christos Moyzes, Christos Rantsoudis. Default Reasoning via Topology and Mathematical Analysis: A Preliminary Report. 16th International Conference on Principles of Knowledge Representation and Reasoning (KR 2018), Oct 2018, Tempe, Arizona, United States. pp.267-276. ⟨hal-03623002⟩
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