The Complete Aristotle (eng.)

related,
there are cases in which B necessarily will not belong to C; e.g.
suppose that A is white, B swan, C man. Nor can the opposite
affirmations be established, since we have shown a case in which B
necessarily does not belong to C. A syllogism then is not possible
at all.
    Similar relations will obtain in particular syllogisms. For
whenever the negative proposition is universal and necessary, a
syllogism will always be possible to prove both a problematic and a
negative assertoric proposition (the proof proceeds by conversion);
but when the affirmative proposition is universal and necessary, no
syllogistic conclusion can be drawn. This can be proved in the same
way as for universal propositions, and by the same terms. Nor is a
syllogistic conclusion possible when both premisses are
affirmative: this also may be proved as above. But when both
premisses are negative, and the premiss that definitely disconnects
two terms is universal and necessary, though nothing follows
necessarily from the premisses as they are stated, a conclusion can
be drawn as above if the problematic premiss is converted into its
complementary affirmative. But if both are indefinite or
particular, no syllogism can be formed. The same proof will serve,
and the same terms.
    It is clear then from what has been said that if the universal
and negative premiss is necessary, a syllogism is always possible,
proving not merely a negative problematic, but also a negative
assertoric proposition; but if the affirmative premiss is necessary
no conclusion can be drawn. It is clear too that a syllogism is
possible or not under the same conditions whether the mode of the
premisses is assertoric or necessary. And it is clear that all the
syllogisms are imperfect, and are completed by means of the figures
mentioned.
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    In the last figure a syllogism is possible whether both or only
one of the premisses is problematic. When the premisses are
problematic the conclusion will be problematic; and also when one
premiss is problematic, the other assertoric. But when the other
premiss is necessary, if it is affirmative the conclusion will be
neither necessary or assertoric; but if it is negative the
syllogism will result in a negative assertoric proposition, as
above. In these also we must understand the expression ‘possible’
in the conclusion in the same way as before.
    First let the premisses be problematic and suppose that both A
and B may possibly belong to every C. Since then the affirmative
proposition is convertible into a particular, and B may possibly
belong to every C, it follows that C may possibly belong to some B.
So, if A is possible for every C, and C is possible for some of the
Bs, then A is possible for some of the Bs. For we have got the
first figure. And A if may possibly belong to no C, but B may
possibly belong to all C, it follows that A may possibly not belong
to some B: for we shall have the first figure again by conversion.
But if both premisses should be negative no necessary consequence
will follow from them as they are stated, but if the premisses are
converted into their corresponding affirmatives there will be a
syllogism as before. For if A and B may possibly not belong to C,
if ‘may possibly belong’ is substituted we shall again have the
first figure by means of conversion. But if one of the premisses is
universal, the other particular, a syllogism will be possible, or
not, under the arrangement of the terms as in the case of
assertoric propositions. Suppose that A may possibly belong to all
C, and B to some C. We shall have the first figure again if the
particular premiss is converted. For if A is possible for all C,
and C for some of the Bs, then A is possible for some of the Bs.
Similarly if the proposition BC is universal. Likewise also if the
proposition AC is negative, and the proposition BC affirmative: for
we shall again have the first figure by conversion. But if both
premisses should be negative-the one universal and the other
particular-although no syllogistic conclusion will follow from the
premisses as they are put, it will follow if they are converted, as
above. But when both premisses are indefinite or particular, no
syllogism can be formed: for A must belong sometimes to all B and
sometimes to no B. To illustrate the affirmative relation take the
terms animal-man-white; to illustrate the negative, take the terms
horse-man-white—white being the middle term.
21
    If one premiss is pure, the other problematic,