Part III, Chapter 3: General Rules of Simple Incomplex Syllogisms
Rule 5
The conclusion always follows the weaker part, that is to say, if two propositions be negative, it ought to be negative, and if one of them be particular, it ought to be particular.
The proof of this is, that if there be a negative proposition, the middle term is separated from one of the parts of the conclusion, and thus it is incapable of uniting them, which must be done in order to conclude affirmatively.
And if there be one particular proposition, the conclusion cannot be general, for if the conclusion is general affirmative, the subject being universal, it must also be universal in the minor, and consequently its subject,—the attribute being never taken universally in affirmative propositions. Therefore the middle term joined to this subject, will be particular in the minor, and hence it will be general in the major, because otherwise it would be taken twice particularly. It will be, therefore, its subject, and consequently that major term will be also universal, and thus there cannot be a particular proposition in an affirmative argument whose conclusion is general.
This is still more clear in the case of universal negative conclusions, for then it would follow that there ought to be three universal terms in the two premises, according to the first corollary. Now, as there must be an affirmative proposition by the third rule, whose attribute is taken particularly, it follows that the other three terms are taken universally, and, consequently, the two subjects of the two propositions, which makes them universal, Q, E, D.
The particular is inferred from the general. What infers A infers I, what infers E infers O, but what infers the particular does not infer the general. This is a consequent of the preceding rule, and of the first axiom; but it must be remarked that men have thought right to consider the species of syllogism only according to its worthier conclusion, which is the general, so that we do not reckon as a particular species of syllogism that which infers only particularly, when it might have a general conclusion.
Hence there is no syllogism in which the major being A, and the minor E, the conclusion is O, for (by the 5th corollary) the conclusion of a negative universal minor must be always general, so that if we cannot obtain a general conclusion, it will be because we cannot obtain any at all. Thus A, E, O, is never a syllogism separately, but only so far as it may be contained in A, E, E.