# Euler Circles for Categorical Syllogisms

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This Demonstration shows representations of categorical syllogisms [10] by Euler circles (or Euler diagrams). These are equivalent to Venn diagrams, except that Euler circles for disjoint sets do not touch. The Demonstration produces a random syllogistic form, and the user must show whether the form is a valid syllogism or not by moving circles and adapting their radii. The Demonstration simultaneously calculates values of propositions.

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Contributed by: Izidor Hafner (April 2016)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

The so-called figure of a categorical syllogism is determined by the possible position of a middle term. There are four figures:

where is , , , or .

Representing syllogistic moods by geometric figures was familiar to the ancient commentators. The use of circles is usually ascribed to Euler [9]. Leibniz's use of circles and other diagrammatic methods remained unpublished until 1903 [5, pp. 260–262].

In [7, pp. 203], it is asserted that this technique is less sophisticated than Venn diagrams.

References

[1] R. Audi, ed., *The Cambridge Dictionary of Philosophy*, Cambridge: Cambridge University Press, 1995 pp. 780–782.

[2] L. Borkowski, *Elementy Logiki Formalnej* (in Polish), Warsaw: Polish Scientific Publishers, 1976.

[3] L. Carroll, *Symbolic Logic and the Game of Logic*, New York: Dover, 1958.

[4] I. M. Copi and C. Cohen, *Introduction to Logic*, 9th ed., New York: Macmillan, 1994 pp. 214–218.

[5] I. M. BocheÅ„ski, *A History of Formal Logic*, 2nd ed., I. Thomas (trans., ed.), New York: Chelsea Publishing Company, 1970.

[6] Wikipedia, "Euler Diagram." (Mar 30, 2016)en.wikipedia.org/wiki/Euler_diagram.

[7] E. J. Borowski and J. M. Borwein, *Collins Dictionary of Mathematics*, New York: HarperCollins, 1991.

[8] Wikipedia, "Categorical Proposition." (Mar 30, 2016) en.wikipedia.org/wiki/Categorical_proposition.

[9] L. Euler, *Lettres à une princesse d'Allemagne*, Saint Petersburg: De l'Imprimerie de l'Academie impériale des sciences, 1768.

[10] G. Kemerling, "Categorical Syllogisms." (Mar 30, 2016) www.philosophypages.com/lg/e08a.htm.

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