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.


Aristotelian logic, or the traditional study of deduction, is based on the following four so-called categorical (or subject-predicate) propositions:

all is (universal affirmative or A proposition), some is (particular affirmative or I proposition), no is (universal negative or E proposition), some is not (particular negative or O proposition).

The categorical propositions A, E, I, and O are designated the mood of a syllogism. is called the subject (or minor) term, and is called the predicate (or major) term of the proposition. A categorical syllogism is a deductive argument about categorical propositions in which a conclusion is inferred from two premises. The term that occurs in both premises is called the middle term. An example of a syllogism is , . There are 256 possible triples of categorical propositions, but only 24 of these are valid syllogisms.

A valid syllogism is found by choosing true premises, which imply true conclusions. To run the Demonstration, click for a new syllogistic form, then move the three circles (using the mouse) into the corresponding configuration. You can vary the size of each circle to enable one to contain another.

To show that the appeared form is not a valid syllogism, move or adopt radii of circles so that the first two propositions (premises) are true and the third (conclusion) is false.


Contributed by: Izidor Hafner (April 2016)
Open content licensed under CC BY-NC-SA



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 Leibniz's use of circles and other diagrammatic methods remained unpublished until 1903 [5, pp. 260–262].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.


[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. Bocheński, A History of Formal Logic, 2nd ed., I. Thomas (trans., ed.), New York: Chelsea Publishing Company, 1970.

[6] Wikipedia, "Euler Diagram." (Mar 30, 2016)

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

[8] Wikipedia, "Categorical Proposition." (Mar 30, 2016)

[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)

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