### Abstract

Proof-theory has traditionally been developed based on linguistic (symbolic) representations of logical proofs. Recently, however, logical reasoning based on diagrammatic or graphical representations has been investigated by logicians. Euler diagrams were introduced in the eighteenth century. But it is quite recent (more precisely, in the 1990s) that logicians started to study them from a formal logical viewpoint. We propose a novel approach to the formalization of Euler diagrammatic reasoning, in which diagrams are defined not in terms of regions as in the standard approach, but in terms of topological relations between diagrammatic objects. We formalize the unification rule, which plays a central role in Euler diagrammatic reasoning, in a style of natural deduction. We prove the soundness and completeness theorems with respect to a formal set-theoretical semantics. We also investigate structure of diagrammatic proofs and prove a normal form theorem.

Original language | English |
---|---|

Pages (from-to) | 365-391 |

Number of pages | 27 |

Journal | Journal of Logic, Language and Information |

Volume | 21 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2012 Jul |

### Fingerprint

### Keywords

- Diagrammatic reasoning
- Euler diagram
- Proof-theory

### ASJC Scopus subject areas

- Linguistics and Language
- Philosophy
- Computer Science (miscellaneous)

### Cite this

*Journal of Logic, Language and Information*,

*21*(3), 365-391. https://doi.org/10.1007/s10849-012-9160-6

**A Diagrammatic Inference System with Euler Circles.** / Mineshima, Koji; Okada, Mitsuhiro; Takemura, Ryo.

Research output: Contribution to journal › Article

*Journal of Logic, Language and Information*, vol. 21, no. 3, pp. 365-391. https://doi.org/10.1007/s10849-012-9160-6

}

TY - JOUR

T1 - A Diagrammatic Inference System with Euler Circles

AU - Mineshima, Koji

AU - Okada, Mitsuhiro

AU - Takemura, Ryo

PY - 2012/7

Y1 - 2012/7

N2 - Proof-theory has traditionally been developed based on linguistic (symbolic) representations of logical proofs. Recently, however, logical reasoning based on diagrammatic or graphical representations has been investigated by logicians. Euler diagrams were introduced in the eighteenth century. But it is quite recent (more precisely, in the 1990s) that logicians started to study them from a formal logical viewpoint. We propose a novel approach to the formalization of Euler diagrammatic reasoning, in which diagrams are defined not in terms of regions as in the standard approach, but in terms of topological relations between diagrammatic objects. We formalize the unification rule, which plays a central role in Euler diagrammatic reasoning, in a style of natural deduction. We prove the soundness and completeness theorems with respect to a formal set-theoretical semantics. We also investigate structure of diagrammatic proofs and prove a normal form theorem.

AB - Proof-theory has traditionally been developed based on linguistic (symbolic) representations of logical proofs. Recently, however, logical reasoning based on diagrammatic or graphical representations has been investigated by logicians. Euler diagrams were introduced in the eighteenth century. But it is quite recent (more precisely, in the 1990s) that logicians started to study them from a formal logical viewpoint. We propose a novel approach to the formalization of Euler diagrammatic reasoning, in which diagrams are defined not in terms of regions as in the standard approach, but in terms of topological relations between diagrammatic objects. We formalize the unification rule, which plays a central role in Euler diagrammatic reasoning, in a style of natural deduction. We prove the soundness and completeness theorems with respect to a formal set-theoretical semantics. We also investigate structure of diagrammatic proofs and prove a normal form theorem.

KW - Diagrammatic reasoning

KW - Euler diagram

KW - Proof-theory

UR - http://www.scopus.com/inward/record.url?scp=84862174730&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84862174730&partnerID=8YFLogxK

U2 - 10.1007/s10849-012-9160-6

DO - 10.1007/s10849-012-9160-6

M3 - Article

AN - SCOPUS:84862174730

VL - 21

SP - 365

EP - 391

JO - Journal of Logic, Language and Information

JF - Journal of Logic, Language and Information

SN - 0925-8531

IS - 3

ER -