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.
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References
Barwise, J., Seligman, J.: Information Flow: The Logic of Distributed Systems. Cambridge University Press (1997)
Buss, S.R.: An Introduction to Proof Theory. In: Buss, S.R. (ed.) Handbook Proof Theory. Elsevier, Amsterdam (1998)
Gurr, C.A., Lee, J., Stenning, K.: Theories of diagrammatic reasoning: Distinguishing component problems. Minds and Machines 8(4), 533–557 (1998)
Howse, J., Stapleton, G., Taylor, J.: Spider Diagrams. LMS Journal of Computation and Mathematics 8, 145–194 (2005)
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
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)
Robinson, J.A.: A Machine-Oriented Logic Based on the Resolution Principle. Journal of the ACM 12(1), 23–41 (1965)
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)
Shimojima, A.: On the Efficacy of Representation, Ph.D. thesis, Indiana University (1996)
Shin, S.-J.: The Logical Status of Diagrams. Cambridge University Press (1994)
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)
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
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Takemura, R. (2012). Proof-Theoretical Investigation of Venn Diagrams: A Logic Translation and Free Rides. In: Cox, P., Plimmer, B., Rodgers, P. (eds) Diagrammatic Representation and Inference. Diagrams 2012. Lecture Notes in Computer Science(), vol 7352. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31223-6_17
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DOI: https://doi.org/10.1007/978-3-642-31223-6_17
Publisher Name: Springer, Berlin, Heidelberg
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