Part III, Chapter 3: General Rules of Simple Incomplex Syllogisms
Rule 2
The terms of the conclusion cannot be taken more universally in the conclusion than they are in the premises.
Hence, when either term is taken universally in the conclusion, the reasoning will be false if it is taken particularly in the two first propositions.
The reason is, that we cannot conclude anything from the particular to the general (according to the first axiom), for, from the fact that some man is black, we cannot say that all men are black.
First COROLLARY.
There must always be in the premises one universal term more than in the conclusion, for every term which is general in the conclusion ought to be so also in the premises, and besides, the middle term must be taken at least once generally.
2d COROLLARY.
When the conclusion is negative, the greater term must necessarily be taken generally in the major, for it is taken generally in the negative conclusion (by the fourth axiom), and, consequently, it must be taken generally in the major (by the second rule).
3d COROLLARY.
The major (proposition) of an argument whose conclusion is negative, can never be a particular affirmative, for the subject and attribute of an affirmative proposition are both taken particularly (by the second and third axioms), and thus the greater term would be taken only particularly, contrary to the second corollary.
Fourth COROLLARY.
The lesser term is always in the conclusion as in the premises, that is to say, that as it can be only particular in the conclusion, as it is particular in the premises, it may, on the contrary, be always general in the conclusion when it is so in the premises; for the lesser term could not be general in the minor when it is the subject of it, unless it be generally united to the middle; and it cannot be the attribute, and be taken generally in it, unless the proposition be negative, because the attribute of an affirmative proposition is always taken particularly. Now, negative propositions denote that the attribute, taken in its full extension, is separated from the subject.
And, consequently, a proposition in which the lesser term in general denotes either a union of the middle term with the whole of the lesser term, or a separation of the middle from the whole lesser term.
Now if, through this union of the middle with the lesser term, we conclude that another idea is joined to this lesser term, we ought to conclude that it is joined to all the lesser term, and not to a part alone, for, the middle being joined to all the lesser term, nothing can be proved by this union of one part, which cannot also be proved of the others, since it is joined to them all.
In the same way, if the separation of the middle term from the lesser term, prove anything of any part of that lesser term, it proves the same of all the parts, since it is equally separated from them all.
5th COROLLARY. When the minor is a universal negative, if we wish to obtain a legitimate conclusion, it must always be general. This is a consequent of the preceding corollary, for the smaller term must be taken generally in the minor, when it is a universal negative, whether it be its subject (by the second axiom), or whether it be the attribute of it (by the fourth axiom).