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

  • Koji Mineshima
  • Mitsuhiro Okada
  • Ryo Takemura
Part of the Lecture Notes in Computer Science book series (LNCS, 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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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Koji Mineshima
    • 1
  • Mitsuhiro Okada
    • 1
  • Ryo Takemura
    • 1
  1. 1.Department of PhilosophyKeio UniversityJapan

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