Proof-Theoretical Investigation of Venn Diagrams: A Logic Translation and Free Rides

  • Ryo Takemura
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7352)

Abstract

In the literature on diagrammatic reasoning, Venn diagrams are abstractly formalized in terms of minimal regions. In view of the cognitive process to recognize Venn diagrams, we modify slightly the formalization by distinguishing conjunctive, negative, and disjunctive regions among possible regions in Venn diagrams. Then we study a logic translation of the Venn diagrammatic system with the aim of investigating how our inference rules are rendered to resolution calculus. We further investigate the free ride property of the Venn diagrammatic system. Free ride is one of the most basic properties of diagrammatic systems and it is mainly discussed in cognitive science literature as an account of the inferential efficacy of diagrams. The soundness of our translation shows that a free ride occurs between the Venn diagrammatic system and resolution calculus. Furthermore, our translation provides a more in-depth analysis of the free ride. In particular, we calculate how many pieces of information are obtained in the manipulation of Venn diagrams.

Keywords

Inference Rule Venn Diagram Free Ride Natural Deduction Proof Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Barwise, J., Seligman, J.: Information Flow: The Logic of Distributed Systems. Cambridge University Press (1997)Google Scholar
  2. 2.
    Buss, S.R.: An Introduction to Proof Theory. In: Buss, S.R. (ed.) Handbook Proof Theory. Elsevier, Amsterdam (1998)Google Scholar
  3. 3.
    Gurr, C.A., Lee, J., Stenning, K.: Theories of diagrammatic reasoning: Distinguishing component problems. Minds and Machines 8(4), 533–557 (1998)CrossRefGoogle Scholar
  4. 4.
    Howse, J., Stapleton, G., Taylor, J.: Spider Diagrams. LMS Journal of Computation and Mathematics 8, 145–194 (2005)MathSciNetMATHGoogle Scholar
  5. 5.
    Mineshima, K., Okada, M., Takemura, R.: A Diagrammatic Inference System with Euler Circles. Journal of Logic, Language and Information (to appear), A preliminary version is available at: http://abelard.flet.keio.ac.jp/person/takemura/index.html
  6. 6.
    Mineshima, K., Okada, M., Takemura, R.: Two Types of Diagrammatic Inference Systems: Natural Deduction Style and Resolution Style. In: Goel, A.K., Jamnik, M., Narayanan, N.H. (eds.) Diagrams 2010. LNCS (LNAI), vol. 6170, pp. 99–114. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  7. 7.
    Robinson, J.A.: A Machine-Oriented Logic Based on the Resolution Principle. Journal of the ACM 12(1), 23–41 (1965)MATHCrossRefGoogle Scholar
  8. 8.
    Sato, Y., Mineshima, K., Takemura, R.: The Efficacy of Euler and Venn Diagrams in Deductive Reasoning: Empirical Findings. In: Goel, A.K., Jamnik, M., Narayanan, N.H. (eds.) Diagrams 2010. LNCS (LNAI), vol. 6170, pp. 6–22. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  9. 9.
    Shimojima, A.: On the Efficacy of Representation, Ph.D. thesis, Indiana University (1996)Google Scholar
  10. 10.
    Shin, S.-J.: The Logical Status of Diagrams. Cambridge University Press (1994)Google Scholar
  11. 11.
    Stapleton, G.: A survey of reasoning systems based on Euler diagrams. In: Proc. of Euler 2004. Electronic Notes in Theoretical Computer Science, vol. 134(1), pp. 127–151 (2005)Google Scholar
  12. 12.
    Takemura, R.: Proof theory for reasoning with Euler diagrams: a logic translation and normalization. Studia Logica (to appear), A preliminary version is available at: http://abelard.flet.keio.ac.jp/person/takemura/index.html

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Ryo Takemura
    • 1
  1. 1.Nihon UniversityJapan

Personalised recommendations