The Complete Aristotle (eng.)

stated, a syllogism
results of necessity; and if there is a syllogism, the terms must
be so related. But it is evident also that all the syllogisms in
this figure are imperfect: for all are made perfect by certain
supplementary statements, which either are contained in the terms
of necessity or are assumed as hypotheses, i.e. when we prove per
impossibile. And it is evident that an affirmative conclusion is
not attained by means of this figure, but all are negative, whether
universal or particular.
6
    But if one term belongs to all, and another to none, of a third,
or if both belong to all, or to none, of it, I call such a figure
the third; by middle term in it I mean that of which both the
predicates are predicated, by extremes I mean the predicates, by
the major extreme that which is further from the middle, by the
minor that which is nearer to it. The middle term stands outside
the extremes, and is last in position. A syllogism cannot be
perfect in this figure either, but it may be valid whether the
terms are related universally or not to the middle term.
    If they are universal, whenever both P and R belong to S, it
follows that P will necessarily belong to some R. For, since the
affirmative statement is convertible, S will belong to some R:
consequently since P belongs to all S, and S to some R, P must
belong to some R: for a syllogism in the first figure is produced.
It is possible to demonstrate this also per impossibile and by
exposition. For if both P and R belong to all S, should one of the
Ss, e.g. N, be taken, both P and R will belong to this, and thus P
will belong to some R.
    If R belongs to all S, and P to no S, there will be a syllogism
to prove that P will necessarily not belong to some R. This may be
demonstrated in the same way as before by converting the premiss
RS. It might be proved also per impossibile, as in the former
cases. But if R belongs to no S, P to all S, there will be no
syllogism. Terms for the positive relation are animal, horse, man:
for the negative relation animal, inanimate, man.
    Nor can there be a syllogism when both terms are asserted of no
S. Terms for the positive relation are animal, horse, inanimate;
for the negative relation man, horse, inanimate-inanimate being the
middle term.
    It is clear then in this figure also when a syllogism will be
possible and when not, if the terms are related universally. For
whenever both the terms are affirmative, there will be a syllogism
to prove that one extreme belongs to some of the other; but when
they are negative, no syllogism will be possible. But when one is
negative, the other affirmative, if the major is negative, the
minor affirmative, there will be a syllogism to prove that the one
extreme does not belong to some of the other: but if the relation
is reversed, no syllogism will be possible. If one term is related
universally to the middle, the other in part only, when both are
affirmative there must be a syllogism, no matter which of the
premisses is universal. For if R belongs to all S, P to some S, P
must belong to some R. For since the affirmative statement is
convertible S will belong to some P: consequently since R belongs
to all S, and S to some P, R must also belong to some P: therefore
P must belong to some R.
    Again if R belongs to some S, and P to all S, P must belong to
some R. This may be demonstrated in the same way as the preceding.
And it is possible to demonstrate it also per impossibile and by
exposition, as in the former cases. But if one term is affirmative,
the other negative, and if the affirmative is universal, a
syllogism will be possible whenever the minor term is affirmative.
For if R belongs to all S, but P does not belong to some S, it is
necessary that P does not belong to some R. For if P belongs to all
R, and R belongs to all S, then P will belong to all S: but we
assumed that it did not. Proof is possible also without reduction
ad impossibile, if one of the Ss be taken to which P does not
belong.
    But whenever the major is affirmative, no syllogism will be
possible, e.g. if P belongs to all S and R does not belong to some
S. Terms for the universal affirmative relation are animate, man,
animal. For the universal negative relation it is not possible to
get terms, if R belongs to some S, and does not belong to some S.
For if P belongs to all S, and R to some S, then P will belong to
some R: but we assumed that it belongs to no R. We must put the
matter as before.’ Since the expression ‘it does