In the original syllogism (α) let X (universal) and Y (particular) prove Z (particular), the minor, middle, and major terms being S M, and P, respectively. Then we are to have another syllogism (β) in which X and Z1 (the sub-contrary of Z) prove Y1 (the sub-contrary of Y). In β, S or P will be the middle term.
It is clear that only one term can be distributed in α if the conclusion is affirmative, and only two if the conclusion is negative. Hence S cannot be distributed in α, and it follows that it cannot be distributed in the premisses of β. The middle term of β must therefore be P, and as X must consequently contain P it must be the major premiss of α and Y the minor premiss.
Z must be either SiP or SoP. First, let Z = SiP. Then it is clear that X = MaP, Z1 = SoP, Y1 = SoM, Y = SiM. Secondly, let Z = SoP. Then Z1 = SiP, X = PaM or MeP or PeM (since it must distribute P), Y1 = SiM (if X is affirmative) or SoM (if X is negative), Y = SoM or SiM accordingly.
Hence we have four syllogisms satisfying the required conditions as follows:—

MaPMePPeMPaM
SiMSiMSiMSoM
⎯⎯ ⎯⎯⎯⎯⎯⎯
SiPSoPSoPSoP

It will be observed that these are all the moods of the first and second figures, in which one premiss is particular.

354. Is it possible that there should be a valid syllogism such that, each of the premisses being converted, a new syllogism is obtainable giving a conclusion in which the old major and minor terms have changed places? Prove the correctness of your answer by general reasoning, and if it is in the 408 affirmative, determine the syllogism or syllogisms fulfilling the given conditions. [K.]

If such a syllogism be possible, it cannot have two affirmative premisses, or (since A can only be converted per accidens) we should have two particular premisses in the new syllogism.
Therefore, the original syllogism must have one negative premiss. This cannot be O, since O is inconvertible.
Therefore, one premiss of the original syllogism must be E.
First, let this be the major premiss. Then the minor premiss must be affirmative, and its converse (being a particular affirmative), will not distribute either of its terms. But this converse will be the major premiss of the new syllogism, which also must have a negative conclusion. We should then have illicit major in the new syllogism; and hence the above supposition will not give us the desired result.
Secondly, let the minor premiss of the original syllogism be E. The major premiss in order to distribute the old major term must be A, with the major term as subject. We get then the following, satisfying the given conditions:—

All P is M,
No M is S, or No S is M,
therefore, No S is P, or Some S is not P ;

that is, we really have four syllogisms, such that both premisses being converted, thus,

No S is M, or No M is S,
Some M is P,

we have a new syllogism yielding a conclusion in which the old major and minor terms have changed places, namely,

Some P is not S.