Skip to main content

Advertisement

Springer Nature Link
Log in

Default consequence relations from topology and measure theory

  • Regular Submission
  • Published:
Annals of Mathematics and Artificial Intelligence Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  1. Abbott, S.: Understanding Analysis, 2nd edn. Springer, Berlin (2015)

    Book  Google Scholar 

  2. Adams, C., Franzosa, R.: Introduction to Topology: Pure and Applied. Pearson, London (2007)

    Google Scholar 

  3. Aiello, M., Pratt-Hartmann, I., van Benthem, J. (eds.): Handbook of Spatial Logics. Springer, Berlin (2007)

  4. Baltag, A., Bezhanishvili, N., Özgün, A., Smets, S.: Justified belief and the topology of evidence. In: Väänȧnen, J.A., Hirvonen, Å., de Queiroz, R.J.G.B. (eds.) Proceedings of WoLLIC 2016 - Logic, Language, Information, and Computation, LNCS, vol. 9803, pp. 83–103. Springer (2016)

  5. Bauldry, W.C.: Introduction to Real Analysis: An Educational Approach. Wiley, Hoboken (2009)

    Book  Google Scholar 

  6. Beierle, C., Kern-Isberner, G.: Semantical investigations into nonmonotonic and probabilistic logics. Ann. Math. Artif. Intell. 65(2-3), 123–158 (2012)

    Article  MathSciNet  Google Scholar 

  7. Bell, J.L., Slomson, A.B.: Models and Ultraproducts. North-Holland, Oxford (1969)

    MATH  Google Scholar 

  8. Bochman, A.: A Logical Theory of Nonmonotonic Inference and Belief Change. Springer, Berlin (2001)

    Book  Google Scholar 

  9. Bressoud, D.M.: Calculus Reordered: a History of the Big Ideas. Princeton University Press, Princeton (2019)

    Book  Google Scholar 

  10. Britz, K., Varzinczak, I.J.: From KLM-style conditionals to defeasible modalities, and back. J. Appl. Non-Class. Log. 28(1), 92–121 (2018)

    Article  MathSciNet  Google Scholar 

  11. Carnielli, W.A., Veloso, P.A.S.: Ultrafilter logic and generic reasoning. In: Gottlob, G., Leitsch, A., Mundici, D. (eds.) Computational Logic and Proof Theory, 5th Kurt Gȯdel Colloquium, KGC’97, Proceedings, LNCS, vol. 1289, pp. 34–53. Springer (1997)

  12. del Cerro, L.F., Herzig, A., Lang, J.: From ordering-based nonmonotonic reasoning to conditional logics. Artif. Intell. 66(2), 375–393 (1994)

    Article  MathSciNet  Google Scholar 

  13. Croom, F.: Principles of Topology. Dover Publications, Mineola (2016)

    Google Scholar 

  14. Cruz-Filipe, L., Rasga, J., Sernadas, A., Sernadas, C.: Complete axiomatization of discrete-measure almost-everywhere quantification. J. Log. Comput. 18 (6), 885–911 (2008)

    Article  MathSciNet  Google Scholar 

  15. Delgrande, J.: What’s in a default?. In: Brewka, G., Marek, V.W., Truszczyński, M. (eds.) NonMonotonic Reasoning, Essays celebrating its 30th anniversary, pp. 89–110. College Publications (2012)

  16. Delgrande, J.P., Renne, B.: On a minimal logic of default conditionals. In: Beierle, C., Brewka, G., Thimm, M. (eds.) Computational Models of Rationality, pp. 73–83. College Publications (2016)

  17. Eiter, T., Lukasiewicz, T.: Default reasoning from conditional knowledge bases: complexity and tractable cases. Artif. Intell. 124(2), 169–241 (2000)

    Article  MathSciNet  Google Scholar 

  18. Engelking, R.: General Topology. Heldermann, Berlin (1989)

    MATH  Google Scholar 

  19. Folland, G.B.: Real Analysis: Modern Techniques and Their Applications, 2nd edn Wiley (1999)

  20. Gabbay, D.: Theoretical foundations for nonmonotonic reasoning in expert systems. In: Apt, K.R. (ed.) Logic and Models of Concurrent Systems, pp. 439–457. Springer (1985)

  21. Gabbay, D., Schlechta, K.: Conditionals and Modularity in General Logics Springer (2011)

  22. Gabbay, D.M., Woods, J. (eds.): Logic and the Modalities in the Twentieth Century Handbook of the History of Logic, vol. 7. Elsevier, Amsterdam (2006)

  23. Goldblatt, R.: Mathematical Modal Logic: a View of its Evolution, pp. 1–98. Vol. 7 of. In: Gabbay, D.M., Woods, J. (eds.) Logic and the Modalities in the Twentieth Century, Handbook of the History of Logic. Elsevier, Amsterdam (2006)

  24. Greever, J.: Theory and Examples of Point-Set Topology Brooks/Cole (1967)

  25. Hawthorne, J.: Nonmonotonic conditionals that behave like conditional probabilities above a threshold. J. Appl. Log. 5(4), 625–637 (2007)

    Article  MathSciNet  Google Scholar 

  26. Hawthorne, J.: A primer on rational consequence relations, Popper functions, and their ranked structures. Stud. Logica. 102(4), 731–749 (2014)

    Article  MathSciNet  Google Scholar 

  27. Hawthorne, J., Makinson, D.: The quantitative/qualitative watershed for rules of uncertain inference. Stud. Logica. 86(2), 247–297 (2007)

    Article  MathSciNet  Google Scholar 

  28. Herzig, A., Besnard, P.: Knowledge Representation: Modalities, Conditionals, and Nonmonotonic Reasoning, pp. 45– 68. Vol. I: Knowledge Representation, Reasoning and Learning of. In: Marquis, P., Papini, O., Prade, H. (eds.) A Guided Tour of Artificial Intelligence Research, vol. I: Knowledge Representation, Reasoning and Learning. Springer (2020) (2020)

  29. Hunter, J.K.: An introduction to Real Analysis. Draft. Dept. of Mathematics. University of California at Davis, Available on the Web (2014)

    Google Scholar 

  30. Jauregui, V.: Modalities, conditionals and nonmonotonic reasoning. Ph.D. thesis, Department of Computer Science and Engineering University of New South Wales (2008)

  31. Just, W., Weese, M.: Discovering Modern Set Theory I: the Basics. American Mathematical Society, Providence (1996)

    MATH  Google Scholar 

  32. Just, W., Weese, M.: Discovering Modern Set Theory II: Set-Theoretic Tools for every Mathematician. American Mathematical Society, Providence (1997)

    MATH  Google Scholar 

  33. Kantor, I., Matoušek, J., Šámal, R.: Mathematics++: Selected Topics Beyond the Basic Courses. American Mathematical Society, Providence (2015)

    Book  Google Scholar 

  34. Koutras, C.D., Liaskos, K., Moyzes, C., Rantsoudis, C.: Default Reasoning via Topology and Mathematical Analysis: a preliminary report. In: Thielscher, M., Toni, F., Wolter, F. (eds.) Principles of Knowledge Representation and Reasoning: Proceedings of KR 2018, pp. 267–276. AAAI Press (2018)

  35. Koutras, C.D., Moyzes, C., Rantsoudis, C.: A Reconstruction of Default Conditionals within Epistemic Logic. Fundamenta Informaticae 166(2), 167–197 (2019)

    Article  MathSciNet  Google Scholar 

  36. Koutras, C.D., Rantsoudis, C.: In all but finitely many possible worlds: Model-theoretic investigations on ‘overwhelming majority’ default conditionals. Journal of Logic Language and Information 26(2), 109–141 (2017)

    Article  MathSciNet  Google Scholar 

  37. Kraus, S., Lehmann, D.J., Magidor, M.: Nonmonotonic Reasoning, Preferential Models and Cumulative Logics. Artif. Intell. 44(1-2), 167–207 (1990)

    Article  MathSciNet  Google Scholar 

  38. Lehmann, D.J., Magidor, M.: What does a conditional knowledge base entail? Artif. Intell. 55(1), 1–60 (1992)

    Article  MathSciNet  Google Scholar 

  39. Makinson, D.: General patterns in nonmonotonic reasoning. In: Gabbay, D.M., Hogger, C.J., Robinson, J.A. (eds.) Handbook of Logic in Artificial Intelligence and Logic Programming, vol. 3, pp. 35–110. Clarendon Press - Oxford (1994)

  40. Makinson, D.: Bridges from Classical to Nonmonotonic Logic College Publications (2005)

  41. Marquis, P., Papini, O., Prade, H. (eds.): A Guided Tour of Artificial Intelligence Research. Knowledge Representation, Reasoning and Learning, vol. I. Springer, Berlin (2020)

  42. Marti, J., Pinosio, R.: Topological Semantics for Conditionals. In: Punčochár, V., Švarný, P. (eds.) The Logica Yearbook 2013. College Publications (2014)

  43. McKinsey, J.C.C., Tarski, A.: The algebra of topology. Annals of Mathematics 45 141–191 (1944)

  44. Moschovakis, Y.N.: Notes on Set Theory, 2nd edn. Undergraduate Texts in Mathematics. Springer New York, New York (2005)

    MATH  Google Scholar 

  45. Parikh, R., Moss, L.S., Steinsvold, C.: Topology and Epistemic Logic, pp. 299–341. In: Aiello, M., Pratt-Hartmann, I., van Benthem, J. (eds.) Handbook of Spatial Logics. Springer (2007)

  46. Paris, J.B., Simmonds, R.: O is not enough. Rev. Symb. Log. 2 (2), 298–309 (2009)

    Article  MathSciNet  Google Scholar 

  47. Pozzato, G.L.: Conditional and Preferential Logics: Proof Methods and Theorem Proving. Frontiers in Artificial Intelligence and Applications. IOS Press, Amsterdam (2010)

    MATH  Google Scholar 

  48. Rasga, J., Lotfallah, W.B., Sernadas, C.: Completeness and interpolation of almost-everywhere quantification over finitely additive measures. Math. Log. Q. 59(4-5), 286–302 (2013)

    Article  MathSciNet  Google Scholar 

  49. Schilling, R.L.: Measures, Integrals and Martingales, 3nd edn. Cambridge University Press, Cambridge (2017)

    MATH  Google Scholar 

  50. Schlechta, K.: Defaults as generalized quantifiers. J. Log. Comput. 5(4), 473–494 (1995)

    Article  MathSciNet  Google Scholar 

  51. Schlechta, K.: Coherent Systems. Elsevier, Amsterdam (2013)

    MATH  Google Scholar 

  52. Shelah, S., Woodin, W.H.: Large cardinals imply that every reasonably definable set of reals is Lebesgue measurable. Israel Journal of Mathematics 70(3), 381–394 (1990)

    Article  MathSciNet  Google Scholar 

  53. Shoham, Y.: A semantical approach to nonmonotonic logics. In: Proceedings of the Symposium on Logic in Computer Science (LICS ’87), pp. 275–279. IEEE Computer Society (1987)

  54. Solovay, R.M.: A model of set-theory in which every set of reals is Lebesgue measurable. Ann. Math. 2(96), 1–56 (1970)

    Article  MathSciNet  Google Scholar 

  55. Steen, L.A., Seebach, J.A.: Counterexamples in Topology. Dover Publications, Mineola (1995)

    MATH  Google Scholar 

  56. Väänänen, J.: Models and Games. Cambridge University Press, Cambridge (2011)

    Book  Google Scholar 

  57. Veloso, P.A.S., Veloso, S.R.M.: On Ultrafilter Logic and Special Functions. Stud. Logica. 78(3), 459–477 (2004)

    Article  MathSciNet  Google Scholar 

  58. Westerståhl, D.: Generalized quantifiers. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy. Stanford University (2016)

Download references

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.

Author information

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

Authors
  1. Costas D. Koutras
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Konstantinos Liaskos
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Christos Moyzes
    View author publications

    You can also search for this author inPubMed Google Scholar

  4. Christos Nomikos
    View author publications

    You can also search for this author inPubMed Google Scholar

  5. Christos Rantsoudis
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Costas D. Koutras.

Ethics declarations

Conflict of Interests

The authors have no relevant financial or non-financial interests to disclose, and no competing interests to declare that are relevant to the content of this article.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10472-021-09779-7

Keywords

Mathematics Subject Classification (2010)