(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).

Two classes of objects may differ in qualities, and yet they may agree in number. This point strongly confirms me in the opinion which I have already expressed, that all inference really depends upon equations, not differences.

The Logical Alphabet thus enables us to make a complete analysis of any numerical problem, and though the symbolical statement may sometimes seem prolix, I conceive that it really represents the course which the mind must follow in solving the question. Although thought may outstrip the rapidity with which the symbols can be written down, yet the mind does not really follow a different course from that indicated by the symbols. For a fuller explanation of this natural system of Numerically Definite Reasoning, with more abundant illustrations and an analysis of De Morgan’s Numerically Definite Syllogism, I must refer the reader to the paper‍93 in the Memoirs of the Manchester Literary and Philosophical Society, already mentioned, portions of which, however, have been embodied in the present section.

The reader may be referred, also, to Boole’s writings upon the subject in the Laws of Thought, chap. xix. p. 295, and in a paper on “Propositions Numerically Definite,” communicated by De Morgan, in 1868, to the Cambridge Philosophical Society, and printed in their Transactions, vol. xi. part ii.