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It is necessary to recapitulate that quantity (universal and particular) and quality (affirmative and negative) are understood to belong to the terms of propositions which is indeed the correct view.
Thus, in the proposition All Xs are Ys, the subject All Xs is universal-affirmative, the predicate (some) Ys particular-affirmative.
In the proposition, some Xs are Zs, both terms are particular-affirmative.
The proposition No Xs are Zs would in philosophical language be written in the form All Xs are not-Zs. The subject is universal-affirmative, the predicate particular-negative.
In the proposition Some Xs are not-Zs are Ys the subject is universal-negative, the predicate particular-affirmative, and so on.
In a pair of premises there are four terms, viz. two subjects and two predicates; two of these terms, viz. those involving the Y or not-Y may be called the middle terms, the two others the extremes, one of these involving X or not-X, the other Z or not-Z.