Part II, Chapter 19: Of the Nature of Negative Propositions
THE nature of negative propositions cannot be expressed more clearly than by saying, that it is the conceiving that one thing is not another; but, in order that one thing be not another, it is not necessary that it should have nothing in common with it; it is enough that it has not all which the other has, as it is enough, in order that a beast be not a man, that it should not have all that a man has, and it is not necessary that it should have nothing of what is in man. Whence we may obtain this axiom: —
Axiom 5
The negative proposition does not separate from the subject all the parts contained in the comprehension of the attribute, but it separates only the total and complete idea composed of all these attributes united.
If I say that matter is not a substance that thinks, I should not, therefore, say that it is not a substance, but I say that it is not a thinking substance, which is the total and complete idea that I deny of matter.
It is quite the reverse with the extension of idea, for the negative proposition separates from the subject the idea of the attribute, according to the whole of its extension; and the reason of this is clear, for, to be the subject of an idea, and to be contained in its extension, is nothing else but to include that idea; and, consequently, when we say that one idea does not include another, which is termed denying, we say that it is not one of the subjects of that idea.
Thus, if I say that man is not an insensible being, I mean to say that he is not among the number of the insensible beings; and I, therefore, separate them all from him. Whence we may obtain this other axiom.