The Complete Aristotle (eng.)

pure; and also
when one premiss is negative, the other affirmative, the latter
being necessary. But when the negative premiss is necessary, the
conclusion also will be a pure negative proposition; for the same
kind of proof can be given whether the terms are universal or not.
For the syllogisms must be made perfect by means of the first
figure, so that a result which follows in the first figure follows
also in the third. But when the minor premiss is negative and
universal, if it is problematic a syllogism can be formed by means
of conversion; but if it is necessary a syllogism is not possible.
The proof will follow the same course as where the premisses are
universal; and the same terms may be used.
    It is clear then in this figure also when and how a syllogism
can be formed, and when the conclusion is problematic, and when it
is pure. It is evident also that all syllogisms in this figure are
imperfect, and that they are made perfect by means of the first
figure.
23
    It is clear from what has been said that the syllogisms in these
figures are made perfect by means of universal syllogisms in the
first figure and are reduced to them. That every syllogism without
qualification can be so treated, will be clear presently, when it
has been proved that every syllogism is formed through one or other
of these figures.
    It is necessary that every demonstration and every syllogism
should prove either that something belongs or that it does not, and
this either universally or in part, and further either ostensively
or hypothetically. One sort of hypothetical proof is the reductio
ad impossibile. Let us speak first of ostensive syllogisms: for
after these have been pointed out the truth of our contention will
be clear with regard to those which are proved per impossibile, and
in general hypothetically.
    If then one wants to prove syllogistically A of B, either as an
attribute of it or as not an attribute of it, one must assert
something of something else. If now A should be asserted of B, the
proposition originally in question will have been assumed. But if A
should be asserted of C, but C should not be asserted of anything,
nor anything of it, nor anything else of A, no syllogism will be
possible. For nothing necessarily follows from the assertion of
some one thing concerning some other single thing. Thus we must
take another premiss as well. If then A be asserted of something
else, or something else of A, or something different of C, nothing
prevents a syllogism being formed, but it will not be in relation
to B through the premisses taken. Nor when C belongs to something
else, and that to something else and so on, no connexion however
being made with B, will a syllogism be possible concerning A in its
relation to B. For in general we stated that no syllogism can
establish the attribution of one thing to another, unless some
middle term is taken, which is somehow related to each by way of
predication. For the syllogism in general is made out of premisses,
and a syllogism referring to this out of premisses with the same
reference, and a syllogism relating this to that proceeds through
premisses which relate this to that. But it is impossible to take a
premiss in reference to B, if we neither affirm nor deny anything
of it; or again to take a premiss relating A to B, if we take
nothing common, but affirm or deny peculiar attributes of each. So
we must take something midway between the two, which will connect
the predications, if we are to have a syllogism relating this to
that. If then we must take something common in relation to both,
and this is possible in three ways (either by predicating A of C,
and C of B, or C of both, or both of C), and these are the figures
of which we have spoken, it is clear that every syllogism must be
made in one or other of these figures. The argument is the same if
several middle terms should be necessary to establish the relation
to B; for the figure will be the same whether there is one middle
term or many.
    It is clear then that the ostensive syllogisms are effected by
means of the aforesaid figures; these considerations will show that
reductiones ad also are effected in the same way. For all who
effect an argument per impossibile infer syllogistically what is
false, and prove the original conclusion hypothetically when
something impossible results from the assumption of its
contradictory; e.g. that the diagonal of the square is
incommensurate with the side, because odd numbers are equal to
evens