I may add that not a few logicians have accepted this view of the universal affirmative proposition. Leibnitz, in his *Difficultates Quædam Logicæ*, adopts it, saying, “Omne A est B; id est æquivalent AB et A, seu A non B est nonens.” Boole employed the logical equation *x* = *xy* concurrently with *x* = *vy*; and Spalding[52] distinctly says that the proposition “all metals are minerals” might be described as an assertion of *partial identity* between the two classes. Hence the name which I have adopted for the proposition. [52] *Encyclopædia Britannica*, Eighth Ed. art. Logic, sect. 37, note. 8vo. reprint, p. 79. *Limited Identities.* An important class of propositions have the form AB = AC, expressing the identity of the class AB with the class AC. In other words, “Within the sphere of the class A, all the B’s are all the C’s;” or again, “The B’s and C’s, which are A’s, are identical.” But it will be observed that nothing is asserted concerning things which are outside of the class A; and thus the identity is of limited extent. It is the proposition B = C limited to the sphere of things called A. Thus we may say, with some approximation to truth, that “Large plants are plants devoid of locomotive power.” A barrister may make numbers of most general statements concerning the relations of persons and things in the course of an argument, but it is of course to be understood that he speaks only of persons and things under the English Law. Even mathematicians make statements which are not true with absolute generality. They say that imaginary roots enter into equations by pairs; but this is only true under the tacit condition that the equations in question shall not have imaginary coefficients.[53] The universe, in short, within which they habitually discourse is that of equations with real coefficients. These implied limitations form part of that great mass of tacit knowledge which accompanies all special arguments. [53] De Morgan, *On the Root of any Function*. Cambridge Philosophical Transactions, 1867, vol. xi. p. 25. To De Morgan is due the remark, that we do usually think and argue in a limited universe or sphere of notions, even when it is not expressly stated.[54] [54] *Syllabus of a proposed System of Logic*, §§ 122, 123. It is worthy of inquiry whether all identities are not really limited to an implied sphere of meaning. When we make such a plain statement as “Gold is malleable” we obviously speak of gold only in its solid state; when we say that “Mercury is a liquid metal” we must be understood to exclude the frozen condition to which it may be reduced in the Arctic regions. Even when we take such a fundamental law of nature as “All substances gravitate,” we must mean by substance, material substance, not including that basis of heat, light, and electrical undulations which occupies space and possesses many wonderful mechanical properties, but not gravity. The proposition then is really of the form