The Complete Aristotle (eng.)

these are taken, a syllogism will
be formed to prove that A belongs to none of the Es, not however
from the premisses taken but in the aforesaid mood. For B will
belong to all A and to no E. Consequently B must be identical with
one of the Hs. Again, if B and G cannot belong to the same thing,
it follows that A will not belong to some of the Es: for then too
we shall have the middle figure: for B will belong to all A and to
no G. Consequently B must be identical with some of the Hs. For the
fact that B and G cannot belong to the same thing differs in no way
from the fact that B is identical with some of the Hs: for that
includes everything which cannot belong to E.
    It is clear then that from the inquiries taken by themselves no
syllogism results; but if B and F are contraries B must be
identical with one of the Hs, and the syllogism results through
these terms. It turns out then that those who inquire in this
manner are looking gratuitously for some other way than the
necessary way because they have failed to observe the identity of
the Bs with the Hs.
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    Syllogisms which lead to impossible conclusions are similar to
ostensive syllogisms; they also are formed by means of the
consequents and antecedents of the terms in question. In both cases
the same inquiry is involved. For what is proved ostensively may
also be concluded syllogistically per impossibile by means of the
same terms; and what is proved per impossibile may also be proved
ostensively, e.g. that A belongs to none of the Es. For suppose A
to belong to some E: then since B belongs to all A and A to some of
the Es, B will belong to some of the Es: but it was assumed that it
belongs to none. Again we may prove that A belongs to some E: for
if A belonged to none of the Es, and E belongs to all G, A will
belong to none of the Gs: but it was assumed to belong to all.
Similarly with the other propositions requiring proof. The proof
per impossibile will always and in all cases be from the
consequents and antecedents of the terms in question. Whatever the
problem the same inquiry is necessary whether one wishes to use an
ostensive syllogism or a reduction to impossibility. For both the
demonstrations start from the same terms, e.g. suppose it has been
proved that A belongs to no E, because it turns out that otherwise
B belongs to some of the Es and this is impossible-if now it is
assumed that B belongs to no E and to all A, it is clear that A
will belong to no E. Again if it has been proved by an ostensive
syllogism that A belongs to no E, assume that A belongs to some E
and it will be proved per impossibile to belong to no E. Similarly
with the rest. In all cases it is necessary to find some common
term other than the subjects of inquiry, to which the syllogism
establishing the false conclusion may relate, so that if this
premiss is converted, and the other remains as it is, the syllogism
will be ostensive by means of the same terms. For the ostensive
syllogism differs from the reductio ad impossibile in this: in the
ostensive syllogism both remisses are laid down in accordance with
the truth, in the reductio ad impossibile one of the premisses is
assumed falsely.
    These points will be made clearer by the sequel, when we discuss
the reduction to impossibility: at present this much must be clear,
that we must look to terms of the kinds mentioned whether we wish
to use an ostensive syllogism or a reduction to impossibility. In
the other hypothetical syllogisms, I mean those which proceed by
substitution, or by positing a certain quality, the inquiry will be
directed to the terms of the problem to be proved-not the terms of
the original problem, but the new terms introduced; and the method
of the inquiry will be the same as before. But we must consider and
determine in how many ways hypothetical syllogisms are
possible.
    Each of the problems then can be proved in the manner described;
but it is possible to establish some of them syllogistically in
another way, e.g. universal problems by the inquiry which leads up
to a particular conclusion, with the addition of an hypothesis. For
if the Cs and the Gs should be identical, but E should be assumed
to belong to the Gs only, then A would belong to every E: and again
if the Ds and the Gs should be identical, but E should be
predicated of the Gs only, it follows that A will belong to none of
the Es. Clearly then we must consider the matter in this way also.
The method is the same whether the relation is necessary