Diagrammatic Reasoning System with Euler Circles: Theory and Experiment Design

  • Koji Mineshima
  • Mitsuhiro Okada
  • Yuri Sato
  • Ryo Takemura
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5223)

Abstract

In this paper we are concerned with logical and cognitive aspects of reasoning with Euler circles. We give a proof-theoretical analysis of diagrammatic reasoning with Euler circles involving unification and deletion rules. Diagrammatic syllogistic reasoning is characterized as a particular class of the general diagrammatic proofs. Given this proof-theoretical analysis, we present some conjectures on cognitive aspects of reasoning with Euler diagrams. Then we propose a design of experiment for a cognitive psychological study.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ando, J., Shikishima, C., Hiraishi, K., Sugimoto, Y., Takemura, R., Okada, M.: At the crossroads of logic, psychology, and behavioral genetics. In: Okada, M., et al. (eds.) Reasoning and Cognition, pp. 19–36. Keio University Press (2006)Google Scholar
  2. 2.
    Braine, M., O’Brien, D.: Mental logic. Lawrence Erlbaum Associates, Mahwah (1998)Google Scholar
  3. 3.
    Blackett, D.: Elementary Topology. Academic Press, London (1983)Google Scholar
  4. 4.
    Calvillo, P.D., DeLeeuw, K., Revlin, R.: Deduction with Euler Circles. In: Baker-Plummer, D., et al. (eds.) Diagrams 2006. LNCS (LNAI), vol. 4045, pp. 199–203. Springer, Heidelberg (2006)Google Scholar
  5. 5.
    Chater, N., Oakford, M.: The probability heuristics model of syllogistic reasoning. Cognitive Psychology 38, 191–258 (1999)CrossRefGoogle Scholar
  6. 6.
    Dickstein, L.S.: The effect of figure on syllogistic reasoning. Memory and Cognition 6, 76–83 (1978)Google Scholar
  7. 7.
    Euler, L.: Lettres à une Princesse d’Allemagne sur Divers Sujets de Physique et de Philosophie. De l’Académie des Sciences, Saint-Pétersbourg (1768)Google Scholar
  8. 8.
    Evans, J., Newstead, S.E., Byrne, R.: Human Reasoning. Lawrence Erlbaum Associates, Mahwah (1993)Google Scholar
  9. 9.
    Hammer, E.: Logic and Visual Information. CSLI Publications (1995)Google Scholar
  10. 10.
    Hammer, E., Shin, S.: Euler’s visual logic. History and Philosophy of Logic 19, 1–29 (1998)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Meyer, B.: Diagrammatic evaluation of visual mathematical notations. In: Anderson, M., Meyer, B., Olivier, P. (eds.) Diagrammatic Representation and Reasoning, pp. 261–277. Springer, Heidelberg (2001)Google Scholar
  12. 12.
    Peirce, C.S.: Collected Papers IV. Harvard University Press (1897/1933)Google Scholar
  13. 13.
    Rips, L.J.: The Psychology of Proof. MIT Press, Cambridge (1994)MATHGoogle Scholar
  14. 14.
    Rizzo, A., Palmonari, M.: The mediating role of artifacts in deductive reasoning. In: Poster in the 27th Annual Conference of the Cognitive Science Society (2005)Google Scholar
  15. 15.
    Sato, Y., Takemura, R., Mineshima, K., Shikishima, C., Sugimoto, Y., Ando, J., Okada, M.: Some remarks on deductive syllogistic reasoning studies. In: Okada, M., Takemura, R., Ando, J. (eds.) Reports on Interdisciplinary Logic Inference Studies, pp. 3–32. Keio University Press (2008)Google Scholar
  16. 16.
    Shin, S.-J.: The Logical Status of Diagrams. Cambridge University Press, Cambridge (1994)MATHGoogle Scholar
  17. 17.
    Stapleton, G.: A survey of reasoning systems based on Euler diagrams. In: Euler Diagrams 2004. ENTCS, vol. 134, pp. 127–151. Elsevier, Amsterdam (2005)Google Scholar
  18. 18.
    Stapleton, G., Rodgers, P., Howse, J., Taylor, J.: Properties of Euler diagrams. In: Stapleton, G., Rodgers, P., Howse, J. (eds.) Proc. Layout of Software Engineering Diagrams. ECEASST, vol. 7, pp. 2–16 (2007)Google Scholar
  19. 19.
    Stenning, K.: Seeing Reason. Oxford University Press, Oxford (2002)Google Scholar
  20. 20.
    Stenning, K., Oberlander, J.: A cognitive theory of graphical and linguistic reasoning. Cognitive Science 19, 97–140 (1995)CrossRefGoogle Scholar
  21. 21.
    Stenning, K., van Lambalgen, M.: Human Reasoning and Cognitive Science. MIT Press, Cambridge (2007)Google Scholar
  22. 22.
    Swoboda, N., Allwein, G.: Using DAG transformations to verify Euler/Venn homogeneous and Euler/Venn FOL heterogeneous rules of inference. Journal on Software and System Modeling 3(2), 136–149 (2004)CrossRefGoogle Scholar
  23. 23.
    Swoboda, N., Allwein, G.: Heterogeneous reasoning with Euler/Venn diagrams containing named constants and FOL. ENTCS, vol. 134, pp. 153–187. Elsevier, Amsterdam (2005)Google Scholar
  24. 24.
    Venn, J.: Symbolic Logic. Macmillan, Basingstoke (1881)Google Scholar
  25. 25.
    Zhang, J., Norman, D.A.: Representations in distributed cognitive tasks. Cognitive Science 18(1), 87–122 (1994)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Koji Mineshima
    • 1
  • Mitsuhiro Okada
    • 1
  • Yuri Sato
    • 1
  • Ryo Takemura
    • 1
  1. 1.Department of PhilosophyKeio UniversityTokyoJapan

Personalised recommendations