A Diagrammatic Inference System with Euler Circles

Journal of Logic, Language and Information 21 (3):365-391 (2012)
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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.

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Citations of this work

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.
A Bitstring Semantics for Calculus CL.Fabien Schang & Jens Lemanski - 2022 - In Ioannis M. Vandoulakis & Jean-Yves Beziau (eds.), The Exoteric Square of Opposition. Cham: Springer. pp. 171–193.

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References found in this work

Symbolic Logic.John Venn - 1881 - New York: B. Franklin.
Symbolic Logic.John Venn - 1881 - Mind 6 (24):574-581.
Logic and Visual Information.Eric Hammer - 1995 - CSLI Publications.
[Introduction].O. H. Mitchell & J. Venn - 1884 - Mind 9 (34):321-322.
Euler’s Visual Logic.Eric Hammer & Sun-Joo Shin - 1998 - History and Philosophy of Logic 19 (1):1-29.

View all 6 references / Add more references