Following is an excerpt from

*MATHEMATICS: The Science of Patterns*

by Keith Devlin

The nineteenth-century English mathematician John Venn
invented a simple, geometric method for checking the
validity of syllogisms, known nowadays as the method of
Venn diagrams. In Venn's method, the syllogism is
represented by means of three overlapping circles, as shown.
The idea is that the region inside the circle marked *S*
represents all objects of type *S*, and analogously for circles
marked *P* and *M*. The procedure used to verify the syllogism
is to see what the two premises say about various regions in
the diagram, numbered 1 to 7.

To illustrate the method, consider the simple example given in the text, namely, the syllogism

MaPSaM---SaP

The major premise, *All M are P*, says that regions 3
and 5 are empty. (All the objects in *M* are in *P*, that is to
say, are in the regions marked 2 and 4.) The minor premise,
*All S are M*, says that regions 1 and 7 are empty. Thus, the
combined effect of the two premises is to say that regions 1,
3, 5, and 7 are empty.

The goal now is to construct a proposition involving *S*
and *P* that is consistent with this information about the
various regions. Regions 1, 3, and 7 are empty, so anything
in *S* Must be in region 4, and is therefore in *P*. In other
words, *all S are P*. And that verifies this particular syllogism.

Though not all syllogisms are as easy to analyze as this example, all the other valid syllogisms can be verified in the same way.

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