A Diagrammatic Inference System with Euler Circles


Authors
Mitsuhiro Okada
Keio University
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.
Keywords Proof-theory  Diagrammatic reasoning  Euler diagram
Categories (categorize this paper)
DOI 10.1007/s10849-012-9160-6
Options
Edit this record
Mark as duplicate
Export citation
Find it on Scholar
Request removal from index
Revision history

Download options

Our Archive


Upload a copy of this paper     Check publisher's policy     Papers currently archived: 44,327
Through your library

References found in this work BETA

Symbolic Logic.John Venn - 1881 - New York: B. Franklin.
Euler’s Visual Logic.Eric Hammer & Sun-Joo Shin - 1998 - History and Philosophy of Logic 19 (1):1-29.
Towards a Model Theory of Diagrams.Hammer Eric & Danner Norman - 1996 - Journal of Philosophical Logic 25 (5):463 - 482.
Symbolic Logic.John Venn - 1881 - Mind 6 (24):574-581.

View all 6 references / Add more references

Citations of this work BETA

How Diagrams Can Support Syllogistic Reasoning: An Experimental Study.Yuri Sato & Koji Mineshima - 2015 - Journal of Logic, Language and Information 24 (4):409-455.
Syllogisms in Rudimentary Linear Logic, Diagrammatically.Ruggero Pagnan - 2013 - Journal of Logic, Language and Information 22 (1):71-113.

Add more citations

Similar books and articles

On Automating Diagrammatic Proofs of Arithmetic Arguments.Mateja Jamnik, Alan Bundy & Ian Green - 1999 - Journal of Logic, Language and Information 8 (3):297-321.
Euler’s Visual Logic.Eric Hammer & Sun-Joo Shin - 1998 - History and Philosophy of Logic 19 (1):1-29.
Diagrams as Sketches.Brice Halimi - 2012 - Synthese 186 (1):387-409.
A Diagrammatic Calculus of Syllogisms.Ruggero Pagnan - 2012 - Journal of Logic, Language and Information 21 (3):347-364.
Syllogisms in Rudimentary Linear Logic, Diagrammatically.Ruggero Pagnan - 2013 - Journal of Logic, Language and Information 22 (1):71-113.
On Frege's Logical Diagrams.Iulian D. Toader - 2004 - In Diagrammatic Representation and Inference. Lecture Notes in Computer Science, vol. 2980,. pp. 22-25.

Analytics

Added to PP index
2012-03-28

Total views
48 ( #175,576 of 2,271,522 )

Recent downloads (6 months)
1 ( #826,846 of 2,271,522 )

How can I increase my downloads?

Downloads

My notes

Sign in to use this feature