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International Conference on Theory and Application of Diagrams

Diagrams 2010: Diagrammatic Representation and Inference pp 99–114Cite as

Two Types of Diagrammatic Inference Systems: Natural Deduction Style and Resolution Style

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Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 6170))

Abstract

Since the 1990s, reasoning with Venn and Euler diagrams has been studied from mathematical and logical viewpoints. The standard approach to a formalization is a “region-based” approach, where a diagram is defined as a set of regions. An alternative is a “relation-based” approach, where a diagram is defined in terms of topological relations (inclusion and exclusion) between circles and points. We compare these two approaches from a proof-theoretical point of view. In general, diagrams correspond to formulas in symbolic logic, and diagram manipulations correspond to applications of inference rules in a certain logical system. From this perspective, we demonstrate the following correspondences. On the one hand, a diagram construed as a set of regions corresponds to a disjunctive normal form formula and the inference system based on such diagrams corresponds to a resolution calculus. On the other hand, a diagram construed as a set of topological relations corresponds to an implicational formula and the inference system based on such diagrams corresponds to a natural deduction system. Based on these correspondences, we discuss advantages and disadvantages of each framework.

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Author information

Authors and Affiliations

  1. Department of Philosophy, Keio University, Japan

    Koji Mineshima, Mitsuhiro Okada & Ryo Takemura

Authors
  1. Koji Mineshima
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  2. Mitsuhiro Okada
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  3. Ryo Takemura
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Editor information

Editors and Affiliations

  1. School of Interactive Computing, Design & Intelligence Laboratory, Georgia Institute of Technology,, 85 Fifth Street, NW 30308, Atlanta, GA, USA

    Ashok K. Goel

  2. Computer Laboratory, University of Cambridge, JJ Thomson Avenue, CB3 0FD, Cambridge, UK

    Mateja Jamnik

  3. Department of Computer Science & Software Engineering, Auburn University, 345 W. Magnolia Street, 36849-5347, Auburn, AL, USA

    N. Hari Narayanan

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Mineshima, K., Okada, M., Takemura, R. (2010). Two Types of Diagrammatic Inference Systems: Natural Deduction Style and Resolution Style. In: Goel, A.K., Jamnik, M., Narayanan, N.H. (eds) Diagrammatic Representation and Inference. Diagrams 2010. Lecture Notes in Computer Science(), vol 6170. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14600-8_12

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  • DOI: https://doi.org/10.1007/978-3-642-14600-8_12

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-14599-5

  • Online ISBN: 978-3-642-14600-8

  • eBook Packages: Computer ScienceComputer Science (R0)

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