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Default consequence relations from topology and measure theory

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Abstract

A default consequence relation is a well-behaved collection of conditional assertions (defaults). A default conditional \(\alpha \mathrel |\joinrel \sim \beta \) is read as ‘if α, then normally β’ and can be given several interpretations, including a ‘size’-oriented one: ‘in mostα-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 KLM 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.

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Acknowledgements

We wish to thank the anonymous referees of the journal for the careful reading and the valuable suggestions that greatly improved the paper. In particular, we are grateful to the referee who suggested that we compare our approach with J. Hawthorne’s system O [25, 27, 46]. Our research has been initially triggered by an insightful comment of Panos Rondogiannis, back to the Fall of 2015; we wish to thank him for being a constant source of productive suggestions and sharp questions. We wish also to thank the referees of KR 2018 who reviewed [34] and gave us a wealth of comments, suggestions and ideas. A preliminary version of (some of) the results of this paper appeared in the KR 2018 paper [34], which includes also other approaches that were singled out for a separate treatment (i.e., the use of asymptotic density). The results of Section ?? are new and were obtained when the fourth author (C.N.) joined our effort with new ideas and renewed effort(s) to pin down the ‘correct’ topology and/or characterize the logic of the standard topology of ℝ.

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Authors and Affiliations

  1. College of Engineering and Technology, American University of the Middle East, Egaila, Kuwait

    Costas D. Koutras

  2. Department of Mathematics, National and Kapodistrian University of Athens, Athens, Greece

    Konstantinos Liaskos

  3. Department of Computer Science, University of Liverpool, Liverpool, UK

    Christos Moyzes

  4. Department of Computer Science & Engineering, University of Ioannina, Ioannina, Greece

    Christos Nomikos

  5. School of Computing Science, Simon Fraser University, Vancouver, Canada

    Christos Rantsoudis

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  1. Costas D. Koutras
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Correspondence to Costas D. Koutras.

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Koutras, C.D., Liaskos, K., Moyzes, C. et al. Default consequence relations from topology and measure theory. Ann Math Artif Intell 90, 397–424 (2022). https://doi.org/10.1007/s10472-021-09779-7

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