Default consequence relations from topology and measure theory

• Published in: Annals of mathematics and artificial intelligence Vol. 90; no. 4; pp. 397 - 424
• Main Authors: Koutras, Costas D., Liaskos, Konstantinos, Moyzes, Christos, Nomikos, Christos, Rantsoudis, Christos
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.04.2022; Springer; Springer Nature B.V
• Subjects: Analysis; Artificial Intelligence; Complex Systems; Computer Science; Euclidean space; Mathematical analysis; Mathematics; Numbers; Numerical analysis; Regular Submission; Topology; Default conditionals; 68T27; 03C80; 68T30; Generalized quantifiers; Nonmonotonic Logic
• ISSN: 1012-2443; 1573-7470
• DOI: 10.1007/s10472-021-09779-7

A default consequence relation is a well-behaved collection of conditional assertions ( defaults ). A default conditional α | ∼ β is read as ‘ if α , then normally β ’ and can be given several interpretations, including a ‘size’-oriented one: ‘ in most α - situations , β is also true ’. Typically, this asks for making the set of ( α ∧ β )-worlds a ‘large’ subset of the α -worlds and the set of ( α ∧¬ β )-worlds a ‘small’ subset of the same set. Technically, this is achieved via a ‘ most ’ generalized quantifier (‘ most A s are B s ’) and we proceed to investigate the default consequence relations emerging upon defining such quantifiers with tools from mathematical analysis. Within topology, we identify ‘large’ sets with topologically dense sets : we show that the unrestricted topological interpretation introduces a consequence relation weaker than the K L M preferential relations (system P ) while the restriction to the finite complement topology over infinite sets captures rational consequence (system R ). Measure theory, seemingly the most fitting tool for a ‘size’-oriented treatment of default conditionals, introduces a rather weak consequence relation, in accordance with probabilistic approaches. It turns out however, that our measure-theoretic approach is essentially equivalent to J. Hawthorne’s system O supplemented with negation rationality . Our results in this paper, show that a ‘size’-oriented interpretation of default reasoning is context-sensitive and in ‘most’ cases it departs from the preferential approach.