The first premise states that the intersection of sets S and M is not empty. An X on the border of two compartments means that at least one of the two compartments has members. An X inside a compartment shows that it contains at least one member. Shading a compartment indicates that it has no members. The set of points inside circle S represents all members of class S, and points outside are members of class not- S -and similarly for the other two circles. The syllogism's three terms, S, M, and P, are represented by simple closed curves -most conveniently drawn as circles -that mutually intersect, as in Figure 1. The Venn diagram is best explained by showing how it is used to validate a syllogism. The Euler and Lambert methods, as well as later variants using squares and other types of closed curves, are no longer in use because of the great improvement on their basic conception which was introduced by the English logician John Venn. Organon (1764) used straight lines, in a manner similar to Euler's use of circles, for diagramming syllogisms.

Euler's contemporary Johann Heinrich Lambert, the German mathematician, in his Neues Circles had earlier been employed, by Gottfried Wilhelm Leibniz and others, to diagram syllogisms, but it was Euler who, in 1761, first explained in detail how circles could be manipulated for such purposes.

Leonhard Euler, the Swiss mathematician, was the first to make systematic use of a logic diagram. They are useful teaching devices for strengthening a student's intuitive grasp of logical structure, they can be used for checking results obtained by algebraic methods, and they provide elegant demonstrations of the close relation of logic to topology and set theory. 'Logic diagrams' are geometrical figures that are in some respect isomorphic with the structure of statements in a formal logic and therefore can be manipulated to solve problems in that logic.

Guide to drawing sama logic diagrams