Higher-Order Syllogistics

  • Thomas F. IcardIII
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8612)

Abstract

We propose a distinction between bottom-up and top-down systems of natural logic, with the classical syllogism epitomizing the first and the Monotonicity Calculus the second. We furthermore suggest it useful to view top-down systems as higher-order generalizations of broadly syllogistic systems. We illustrate this view by proving a result of independent interest: we axiomatize the first-order/single-type fragment of a higher-order calculus for reasoning about inclusion and exclusion (MacCartney and Manning, 2009; Icard, 2012). We show this logic is equivalent to a syllogistic logic with All and nominal complementation, in fact a fragment of a system recently studied (Moss, 2010b).

Keywords

syllogistics natural logic exclusion surface reasoning 

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References

  1. van Benthem, J.: Questions about quantifiers. Journal of Symbolic Logic 49(2), 443–466 (1984)CrossRefMATHMathSciNetGoogle Scholar
  2. van Benthem, J.: Language in Action: Categories, Lambdas, and Dynamic Logic. Studies in Logic, vol. 130. Elsevier, Amsterdam (1991)MATHGoogle Scholar
  3. van Benthem, J.: A brief history of natural logic. In: Chakraborty, M., Löwe, B., Mitra, M.N., Sarukkai, S. (eds.) Logic, Navya-Nyaya and Applications, Homage to Bimal Krishna Matilal, College Publications, London (2008)Google Scholar
  4. Bernardi, R.: Reasoning with Polarity in Categorial Type Logic. PhD thesis, University of Utrecht (2002)Google Scholar
  5. Dowty, D.: The role of negative polarity and concord marking in natural language reasoning. In: Proceedings of Semantics and Linguistic Theory (SALT) IV (1994)Google Scholar
  6. van Eijck, J.: Generalized quantifiers and traditional logic. In: van Benthem, J., ter Meulen, A. (eds.) Generalized Quantifiers, Theory, and Applications. Foris, Dordrecht (1985)Google Scholar
  7. van Eijck, J.: Syllogistics = monotonicity + symmetry + existential import. Technical Report SEN-R0512, CWI, Amsterdam (2005)Google Scholar
  8. Endrullis, J., Moss, L.S.: Syllogistic logic with “Most”. Unpublished ms (2014)Google Scholar
  9. Geurts, B., van der Slik, F.: Monotonicity and processing load. Journal of Semantics 22 (2005)Google Scholar
  10. Icard, T.F.: Inclusion and exclusion in natural language. Studia Logica 100(4), 705–725 (2012)CrossRefMATHMathSciNetGoogle Scholar
  11. Icard, T.F., Moss, L.S.: A complete calculus of monotone and antitone higher-order functions. Unpublished ms (2013)Google Scholar
  12. Icard, T.F., Moss, L.S.: Recent progress on monotonicity. Linguistic Issues in Language Technology 9 (2014)Google Scholar
  13. Kant, I.: Critique of Pure Reason. Cambridge University Press (1997)Google Scholar
  14. Keenan, E.L., Faltz, L.M.: Boolean Semantics for Natural Language. Springer (1984)Google Scholar
  15. MacCartney, B., Manning, C.D.: Natural logic for textual inference. In: Proceedings of the ACL Workshop on Textual Entailment and Paraphrasing (2007)Google Scholar
  16. MacCartney, B., Manning, C.D.: An extended model of natural logic. In: Proceedings of the Eighth International Conference on Computational Semantics, IWCS-8 (2009)Google Scholar
  17. McAllester, D.A., Givan, R.: Natural language syntax and first-order inference. Artificial Intelligence 56, 1–20 (1992)CrossRefMATHMathSciNetGoogle Scholar
  18. Moss, L.S.: Logics for natural language inference. ESSLLI 2010 Course Notes (2010a)Google Scholar
  19. Moss, L.S.: Syllogistic logic with complements. In: van Benthem, J., Gupta, A., Pacuit, E. (eds.) Games, Norms, and Reasons: Logic at the Crossroads, pp. 185–203. Springer (2010b)Google Scholar
  20. Pratt-Hartmann, I.: Fragments of language. Journal of Logic, Language, and Information 13, 207–223 (2004)CrossRefMATHMathSciNetGoogle Scholar
  21. Pratt-Hartmann, I., Moss, L.S.: Logics for the relational syllogistic. The Review of Symbolic Logic 2(4), 647–683 (2009)CrossRefMATHMathSciNetGoogle Scholar
  22. Sánchez-Valencia, V.: Studies on Natural Logic and Categorial Grammar. PhD thesis, Universiteit van Amsterdam (1991)Google Scholar
  23. Westerståhl, D.: Some results on quantifiers. Notre Dame Journal of Formal Logic 25(2), 152–170 (1984)CrossRefMATHMathSciNetGoogle Scholar
  24. Zwarts, F.: Negatief polaire uitdrukkingen I. GLOT 4, 35–132 (1981)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Thomas F. IcardIII
    • 1
  1. 1.Department of PhilosophyStanford UniversityUSA

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