### 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.

Original language | English |
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Title of host publication | Diagrammatic Representation and Inference - 6th International Conference, Diagrams 2010, Proceedings |

Pages | 99-114 |

Number of pages | 16 |

DOIs | |

Publication status | Published - 2010 |

Event | 6th International Conference on the Theory and Application of Diagrams, Diagrams 2010 - Portland, OR, United States Duration: 2010 Aug 9 → 2010 Aug 11 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 6170 LNAI |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 6th International Conference on the Theory and Application of Diagrams, Diagrams 2010 |
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Country | United States |

City | Portland, OR |

Period | 10/8/9 → 10/8/11 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Diagrammatic Representation and Inference - 6th International Conference, Diagrams 2010, Proceedings*(pp. 99-114). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6170 LNAI). https://doi.org/10.1007/978-3-642-14600-8_12