Human Reasoning with Proportional Quantifiers and Its Support by Diagrams

  • Yuri Sato
  • Koji Mineshima
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9781)


In this paper, we study the cognitive effectiveness of diagrammatic reasoning with proportional quantifiers such as most. We first examine how Euler-style diagrams can represent syllogistic reasoning with proportional quantifiers, building on previous work on diagrams for the so-called plurative syllogism (Rescher and Gallagher, 1965). We then conduct an experiment to compare performances on syllogistic reasoning tasks of two groups: those who use only linguistic material (two sentential premises and one conclusion) and those who are also given Euler diagrams corresponding to the two premises. Our experiment showed that (a) in both groups, the speed and accuracy of syllogistic reasoning tasks with proportional quantifiers like most were worse than those with standard first-order quantifiers such as all and no, and (b) in both standard and non-standard (proportional) syllogisms, speed and accuracy for the group provided with diagrams were significantly better than the group provided only with sentential premises. These results suggest that syllogistic reasoning with proportional quantifiers like most is cognitively complex, yet can be effectively supported by Euler diagrams that represent the proportionality relationships between sets in a suitable way.


Euler diagrams Proportional quantifiers Reasoning Logic and cognition 


  1. 1.
    Adams, E.W.: The Logic of Conditionals: An Application of Probability to Deductive Logic. Springer, Dordrecht (1975)CrossRefzbMATHGoogle Scholar
  2. 2.
    Altham, J.E.J.: The Logic of Plurality. Methuen, London (1971)zbMATHGoogle Scholar
  3. 3.
    Barwise, J., Cooper, R.: Generalized quantifiers and natural language. Linguist. Philos. 4, 159–219 (1981)CrossRefzbMATHGoogle Scholar
  4. 4.
    van Benthem, J.: Essays in Logical Semantics. Reidel, Dordrecht (1986)CrossRefzbMATHGoogle Scholar
  5. 5.
    Chater, N., Oaksford, M.: The probability heuristics model of syllogistic reasoning. Cogn. Psychol. 38, 191–258 (1999)CrossRefGoogle Scholar
  6. 6.
    Chow, S., Ruskey, F.: Drawing area-proportional Venn and Euler diagrams. In: Liotta, G. (ed.) GD 2003. LNCS, vol. 2912, pp. 466–477. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  7. 7.
    Cleveland, W.S., McGill, R.: Graphical perception: theory, experimentation, and application to the development of graphical methods. J. Am. Stat. Assoc. 79, 531–554 (1984)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Endrullis, J., Moss, L.S.: Syllogistic logic with “most”. In: de Paiva, V., de Queiroz, R., Moss, L.S., Leivant, D., de Oliveira, A. (eds.) WoLLIC 2015. LNCS, vol. 9160, pp. 124–139. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  9. 9.
    Geach, P.T.: Reason and Argument. University of California Press, Berkeley (1976)Google Scholar
  10. 10.
    Geurts, B., van Der Slik, F.: Monotonicity and processing load. J. Seman. 22, 97–117 (2005)CrossRefGoogle Scholar
  11. 11.
    MacCartney, B.: Natural Language Inference. Ph.D. thesis, Stanford University (2009)Google Scholar
  12. 12.
    Mineshima, K., Okada, M., Takemura, R.: A diagrammatic reasoning system with Euler circles. J. Logic Lang. Inf. 21, 365–391 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Mineshima, K., Okada, M., Takemura, R.: A generalized syllogistic inference system based on inclusion and exclusion relations. Stud. Logica 100, 753–785 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Mineshima, K., Sato, Y., Takemura, R., Okada, M.: Towards explaining the cognitive efficacy of Euler diagrams in syllogistic reasoning: a relational perspective. J. Vis. Lang. Comput. 25, 156–169 (2014)CrossRefGoogle Scholar
  15. 15.
    Rescher, N., Gallagher, N.A.: Venn diagrams for plurative syllogisms. Philos. Stud. 16, 49–55 (1965)CrossRefGoogle Scholar
  16. 16.
    Sato, Y., Masuda, S., Someya, Y., Tsujii, T., Watanabe, S.: An fMRI analysis of the efficacy of Euler diagrams in logical reasoning. In: VL/HCC 2015, pp. 143–151. IEEE Press (2015)Google Scholar
  17. 17.
    Sato, Y., Mineshima, K.: How diagrams can support syllogistic reasoning: an experimental study. J. Logic Lang. Inf. 24, 409–455 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Sato, Y., Wajima, Y., Ueda, K.: An empirical study of diagrammatic inference process by recording the moving operation of diagrams. In: Dwyer, T., Purchase, H., Delaney, A. (eds.) Diagrams 2014. LNCS, vol. 8578, pp. 190–197. Springer, Heidelberg (2014)Google Scholar
  19. 19.
    Sato, Y., Wajima, Y., Ueda, K.: Visual bias of diagram in logical reasoning. In: CogSci 2014, pp. 2342–2347. Cognitive Science Society, Austin (2014b)Google Scholar
  20. 20.
    Shimojima, A.: On the Efficacy of Representation. Ph.D. thesis, Indiana University (1996)Google Scholar
  21. 21.
    Shimojima, A.: Semantic Properties of Diagrams and Their Cognitive Potentials. CSLI Publications, Stanford (2015)Google Scholar
  22. 22.
    Stapleton, G., Rodgers, P., Howse, J.: A general method for drawing area-proportional Euler diagrams. J. Vis. Lang. Comput. 22, 426–442 (2011)CrossRefGoogle Scholar
  23. 23.
    Szymanik, J., Zajenkowski, M.: Comprehension of simple quantifiers: empirical evaluation of a computational model. Cogn. Sci. 34, 521–532 (2010)CrossRefGoogle Scholar
  24. 24.
    Takemura, R.: Counter-example construction with Euler diagrams. Stud. Logica 103, 669–696 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Thompson, B.: Syllogisms using “few”, “many”, and “most”. Notre Dame J. Form. Logic 23, 75–84 (1982)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Interfaculty Initiative in Information StudiesThe University of TokyoTokyoJapan
  2. 2.Center for Simulation SciencesOchanomizu UniversityTokyoJapan

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