The Complete Aristotle (eng.)

the conclusion
will be problematic, not pure; and a syllogism will be possible
under the same arrangement of the terms as before. First let the
premisses be affirmative: suppose that A belongs to all C, and B
may possibly belong to all C. If the proposition BC is converted,
we shall have the first figure, and the conclusion that A may
possibly belong to some of the Bs. For when one of the premisses in
the first figure is problematic, the conclusion also (as we saw) is
problematic. Similarly if the proposition BC is pure, AC
problematic; or if AC is negative, BC affirmative, no matter which
of the two is pure; in both cases the conclusion will be
problematic: for the first figure is obtained once more, and it has
been proved that if one premiss is problematic in that figure the
conclusion also will be problematic. But if the minor premiss BC is
negative, or if both premisses are negative, no syllogistic
conclusion can be drawn from the premisses as they stand, but if
they are converted a syllogism is obtained as before.
    If one of the premisses is universal, the other particular, then
when both are affirmative, or when the universal is negative, the
particular affirmative, we shall have the same sort of syllogisms:
for all are completed by means of the first figure. So it is clear
that we shall have not a pure but a problematic syllogistic
conclusion. But if the affirmative premiss is universal, the
negative particular, the proof will proceed by a reductio ad
impossibile. Suppose that B belongs to all C, and A may possibly
not belong to some C: it follows that may possibly not belong to
some B. For if A necessarily belongs to all B, and B (as has been
assumed) belongs to all C, A will necessarily belong to all C: for
this has been proved before. But it was assumed at the outset that
A may possibly not belong to some C.
    Whenever both premisses are indefinite or particular, no
syllogism will be possible. The demonstration is the same as was
given in the case of universal premisses, and proceeds by means of
the same terms.
22
    If one of the premisses is necessary, the other problematic,
when the premisses are affirmative a problematic affirmative
conclusion can always be drawn; when one proposition is
affirmative, the other negative, if the affirmative is necessary a
problematic negative can be inferred; but if the negative
proposition is necessary both a problematic and a pure negative
conclusion are possible. But a necessary negative conclusion will
not be possible, any more than in the other figures. Suppose first
that the premisses are affirmative, i.e. that A necessarily belongs
to all C, and B may possibly belong to all C. Since then A must
belong to all C, and C may belong to some B, it follows that A may
(not does) belong to some B: for so it resulted in the first
figure. A similar proof may be given if the proposition BC is
necessary, and AC is problematic. Again suppose one proposition is
affirmative, the other negative, the affirmative being necessary:
i.e. suppose A may possibly belong to no C, but B necessarily
belongs to all C. We shall have the first figure once more:
and-since the negative premiss is problematic-it is clear that the
conclusion will be problematic: for when the premisses stand thus
in the first figure, the conclusion (as we found) is problematic.
But if the negative premiss is necessary, the conclusion will be
not only that A may possibly not belong to some B but also that it
does not belong to some B. For suppose that A necessarily does not
belong to C, but B may belong to all C. If the affirmative
proposition BC is converted, we shall have the first figure, and
the negative premiss is necessary. But when the premisses stood
thus, it resulted that A might possibly not belong to some C, and
that it did not belong to some C; consequently here it follows that
A does not belong to some B. But when the minor premiss is
negative, if it is problematic we shall have a syllogism by
altering the premiss into its complementary affirmative, as before;
but if it is necessary no syllogism can be formed. For A sometimes
necessarily belongs to all B, and sometimes cannot possibly belong
to any B. To illustrate the former take the terms sleep-sleeping
horse-man; to illustrate the latter take the terms sleep-waking
horse-man.
    Similar results will obtain if one of the terms is related
universally to the middle, the other in part. If both premisses are
affirmative, the conclusion will be problematic, not