One form of argument is called a syllogistic argument. It is also known as a syllogism. The validity of a syllogistic argument is determined by using Euler (pronounced “oiler”) circles.

Syllogistic arguments make use of the quantifiers

All . . . are . . . Some . . . are . . .

None . . . are . . . Some . . .are not. . .

An example of a syllogistic argument is

All German shepherds are dogs.

All dogs bark.

\All German shepherds bark.

Euler diagrams are used to determine whether an argument is valid or invalid. To diagram the statement

All German shepherds are dogs,

draw two circles, one representing dogs and the other representing German shepherds. Since All German shepherds are dogs, the circle representing German shepherds must lie completely inside the circle representing dogs.

The premise All dogs bark requires that a circle representing all animals that bark contains the circle representing dogs.

Putting together the two diagrams results in this diagram.

As a result, it is clear that the conclusion All German shepherds bark, is valid because the circle representing all German shepherds lies entirely inside the circle representing all animals that bark.