The following problem illustrates the expression for the common part of any three classes:—The number of paupers who are blind males, is equal to the excess, if any, of the sum of the whole number of blind persons, added to the whole number of male persons, added to the number of those who being paupers are neither blind nor males, above the sum of the whole number of paupers added to the number of those who, not being paupers, are blind, and to the number of those who, not being paupers, are male.
The reader is requested to prove the truth of the above statement, (1) by his own unaided common sense; (2) by the Aristotelian Logic; (3) by the method of numerical logic just expounded; and then to decide which method is most satisfactory.
Numerical meaning of Logical Conditions.
In many cases classes of objects may exist under special logical conditions, and we must consider how these conditions can be interpreted numerically. Every logical proposition gives rise to a corresponding numerical equation. Sameness of qualities occasions sameness of numbers. Hence if
A = B
denotes the identity of the qualities of A and B, we may conclude that
(A) = (B).
It is evident that exactly those objects, and those objects only, which are comprehended under A must be comprehended under B. It follows that wherever we can draw an equation of qualities, we can draw a similar equation of numbers. Thus, from
A = B = C
we infer