Part II, Chapter 19: Of the Nature of Negative Propositions
Axiom 6
The attribute of a negative proposition is always taken generally; which may also be expressed more distinctly thus: All the subjects of the one idea, which is denied of the other, are also denied of that other idea; that is to say, that an idea is always denied according to its whole extension. If triangle is denied of square, all that is contained in triangle will be denied of square. This rule is commonly expressed in the schools in these terms, which mean the same thing: If the genus is denied, the species also is denied; for the species is subject to the genus. Man is a subject of animal, because he is contained in its extension.
Not only do negative propositions separate the attribute from the subject, according to the whole extension of the attribute, but they separate also this attribute from the subject according to the whole extension which the subject has in the proposition; that is to say, they separate it universally, if the subject is universal,—and particularly, if the subject is particular. If I say that no vicious man is happy, I separate all the happy persons from all the vicious persons; and if I say that some doctor is not learned, I separate learned from some doctor. And hence we may obtain this axiom: