Part II, Chapter 17: Of the Conversion of Propositions, and first, of the Nature of Affirmation
[ The following Chapters are somewhat difficult to comprehend, and are not necessary in practice. Hence those who do not wish to tire the mind with things of little practical use, may pass them over. ]
WE have refrained till now from speaking of the conversion of propositions, because the foundation of all argumentation, of which we are to speak in the following part, depends on it; and thus it is better that this matter should not be far removed from what we have to say of reasoning, although, to treat well of it, we must reproduce some part of what we have already said of affirmation and negation, and explain thoroughly the nature of both.
It is certain that we cannot express a proposition to others, except by employing two ideas, one for the subject, the other for the attribute, and another word which denotes the union which our mind conceives between them. That union cannot be better expressed than by the words themselves which we employ for affirming, when we say that one thing is another thing.
Hence it is clear that the nature of affirmation is to unite and identify, if we may so speak, the subject with the attribute, and this is what is signified by the word is.
And it follows, also, that it is the nature of the affirmation to place the attribute in all that is expressed in the subject, according to the extension which it has in the proposition: as, when I say, all man is an animal—I mean to say, and I express, that everything that is man is also animal; but if I say, simply, some man is just, I do not place just on all men, but only on some men.
But we must also, in like manner, remember here what we have already said, that in ideas it is necessary to distinguish the COMPREHENSION from the EXTENSION, and that the comprehension denotes the attributes contained IN an idea and the extension the subjects (or classes) which contain that idea. Hence it follows that an idea is always affirmed according to its comprehension, because, in taking away any one of its essential attributes, we utterly destroy and annihilate it, so that it is no longer the same idea; and consequently, when it is affirmed, it is always affirmed in relation to everything which it comprehends within itself. Thus when I say that a rectangle is a parallelogram, I affirm of rectangle everything that is comprised in the idea of parallelogram. For if there were any part of this idea that did not belong to a rectangle, it would follow that the whole idea did not belong to it, but only a part of that idea; and thus the word parallelogram, which signifies the whole idea, ought to be denied and not affirmed of the rectangle. We shall see that this is the principle of all affirmative arguments.
And it follows, on the contrary, that the idea of the attribute is not taken according to the whole extension, at least when its extension is not greater than that of the subject, for if I say that all dissolute men will be damned, I do not say that they alone will be damned, but that they will be among the number of the accursed.
Thus the affirmation, placing the idea of the attribute in the subject, is properly that which determines the extension of the attribute in the affirmative proposition, and the identity which it denotes, considers the attribute as restricted to an extension equal to that of the subject, and does not take in all its generality, if that be greater than the subject, for it is true that all lions are animals, that is to say, that every lion contains the idea of animal, but it is not true that they alone are animals.
I said that the attribute is not taken in all its generality, if it is greater than the subject, for being restrained only by the subject, if the subject is as general as the attribute, it is clear that the attribute remains in all its generality, since it will have as much as the subject, and we suppose that by its nature it can have no more.
Whence we may collect these four indubitable axioms:—AXIOM 1.
The attribute is placed in the subject by the affirmative proposition, according to the whole extension which the subject has in the proposition; that is to say, if the subject is universal, the attribute is conceived in the whole extension of the subject, and if the subject is particular, the attribute is conceived only in a part of the extension of the subject. There are examples of this above.
Axiom 2
The attribute of an affirmative proposition is affirmed according to the whole proposition; that is to say, according to all its attributes. The proof of this is above.