This is evident upon a comparison of terms in all possible positions, but it can be more easily demonstrated with the help of the preceding canons. The premisses cannot both be particular and yield a conclusion without breaking one or other of those canons.
Suppose both are affirmative, II, the Middle is not distributed in either premiss.
Suppose one affirmative and the other negative, IO, or OI. Then, whatever the Figure may be, that is, whatever the order of the terms, only one term can be distributed, namely, the predicate of O. This (Canon II.) must be the Middle. But in that case there must be Illicit Process of the Major (Canon III.), for one of the premisses being negative, the conclusion is negative (Canon V.), and P its predicate is distributed. Briefly, in a negative mood, both Major and Middle must be distributed, and if both premisses are particular this cannot be.
Canon VII. If one Premiss is particular the conclusion is particular.
This canon is sometimes combined with what we have given as Canon V., in a single rule: "The conclusion follows the weaker premiss".
It can most compendiously be demonstrated with the help of the preceding canons.
Suppose both premisses affirmative, then, if one is particular, only one term can be distributed in the premisses, namely, the subject of the Universal affirmative premiss. By Canon II., this must be the Middle, and the Minor, being undistributed in the Premisses, cannot be distributed in the conclusion. That is, the conclusion cannot be Universal—must be particular.
Suppose one Premiss negative, the other affirmative. One premiss being negative, the conclusion must be negative, and P must be distributed in the conclusion. Before, then, the conclusion can be universal, all three terms, S, M, and P, must, by Canons II. and III., be distributed in the premisses. But whatever the Figure of the premisses, only two terms can be distributed. For if one of the Premisses be O, the other must be A, and if one of them is E, the other must be I. Hence the conclusion must be particular, otherwise there will be illicit process of the Minor, or of the Major, or of the Middle.
The argument may be more briefly put as follows: In an affirmative mood, with one premiss particular, only one term can be distributed in the premisses, and this cannot be the Minor without leaving the Middle undistributed. In a negative mood, with one premiss particular, only two terms can be distributed, and the Minor cannot be one of them without leaving either the Middle or the Major undistributed.
Armed with these canons, we can quickly determine, given any combination of three propositions in one of the Figures, whether it is or is not a valid Syllogism.