put P = “Some,” meaning by the indefinite adjective “Some,” one or more of the undefined properties of triangles with three equal angles, and take
then the knowledge implied is expressed in the two propositions
These may also be thrown into the form of one proposition, namely,
but these alternatives cannot be reduced, and the proposition is quite different from
Illustrations of the Indirect Method.
A great variety of arguments and logical problems might be introduced here to show the comprehensive character and powers of the Indirect Method. We can treat either a single premise or a series of premises.
Take in the first place a simple definition, such as “a triangle is a three-sided rectilinear figure.” Let
then the definition is of the form
If we take the series of eight combinations of three letters in the Logical Alphabet (p. 94) and strike out those which are inconsistent with the definition, we have the following result:—
For the description of the class C we have
that is, “a rectilinear figure is either a triangle and three-sided, or not a triangle and not three-sided.”
For the class b we have
To the second side of this we may apply the process of simplification by abstraction described in the last section; for by the Law of Duality
and as