As a second instance, let us take the following question:--The whole number of voters in a borough is *a*; the number against whom objections have been lodged by liberals is *b*; and the number against whom objections have been lodged by conservatives is *c*; required the number, if any, who have been objected to on both sides. Taking A = voter, B = objected to by liberals, C = objected to by conservatives, then we require the value of (ABC). Now the following equation is identically true-- (ABC) = (AB) + (AC) + (A*bc*) - (A). (1) For if we develop all the terms on the second side we obtain (ABC) = (ABC) + (AB*c*) + (ABC) + (A*b*C) + (A*bc*) - (ABC) - (AB*c*) - (A*b*C) - (A*bc*); and striking out the corresponding positive and negative terms, we have left only (ABC) = (ABC). Since then (1) is necessarily true, we have only to insert the known values, and we have (ABC) = *b* + *c* - *a* + (A*bc*). Hence the number who have received objections from both sides is equal to the excess, if any, of the whole number of objections over the number of voters together with the number of voters who have received no objection (A*bc*). 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 A = C; and similarly from (A) = (B) = (C), meaning that the numbers of A’s and C’s are equal to the number of B’s, we can infer (A) = (C). But, curiously enough, this does not apply to negative propositions and inequalities. For if A = B ~ D means that A is identical with B, which differs from D, it does not follow that (A) = (B) ~ (D).