The conversion of the second proposition into the third is usually made
by an indirect demonstration, in the following manner. If possible, let
there be one B which is not A, (2) being true. Then there is one thing
which is not A and is B; but every thing not A is not B; therefore there
is one thing which is B and is not B: which is absurd. It is then absurd
that there should be one single B which is not A; or, Every B is A.
The following proposition contains a method which is of frequent use.
HYPOTHESIS.—Let there be any number of propositions or assertions,—three
for instance, A, B, and C,—of which it is the property that one or the
other must be true, _and one only_. Let there be three other
propositions, P, Q, and R, of which it is also the property that one,
and one only, must be true. Let it also be a connexion of those
assertions, that
When A is true, P is true
When B is true, Q is true
When C is true, R is true