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antecedents of E by G； and attributes which cannot belong to E by H。
If then one of the Cs should be identical with one of the Fs； A must
belong to all E： for F belongs to all E； and A to all C；
consequently A belongs to all E。 If C and G are identical； A must
belong to some of the Es： for A follows C； and E follows all G。 If F
and D are identical； A will belong to none of the Es by a
prosyllogism： for since the negative proposition is convertible； and F
is identical with D； A will belong to none of the Fs； but F belongs to
all E。 Again； if B and H are identical； A will belong to none of the
Es： for B will belong to all A； but to no E： for it was assumed to
be identical with H； and H belonged to none of the Es。 If D and G
are identical； A will not belong to some of the Es： for it will not
belong to G； because it does not belong to D： but G falls under E：
consequently A will not belong to some of the Es。 If B is identical
with G； there will be a converted syllogism： for E will belong to
all A since B belongs to A and E to B （for B was found to be identical
with G）： but that A should belong to all E is not necessary； but it
must belong to some E because it is possible to convert the
universal statement into a particular。
It is clear then that in every proposition which requires proof we
must look to the aforesaid relations of the subject and predicate in
question： for all syllogisms proceed through these。 But if we are
seeking consequents and antecedents we must look for those which are
primary and most universal； e。g。 in reference to E we must look to
KF rather than to F alone； and in reference to A we must look to KC
rather than to C alone。 For if A belongs to KF； it belongs both to F
and to E： but if it does not follow KF； it may yet follow F。 Similarly
we must consider the antecedents of A itself： for if a term follows
the primary antecedents； it will follow those also which are
subordinate； but if it does not follow the former； it may yet follow
the latter。
It is clear too that the inquiry proceeds through the three terms
and the two premisses； and that all the syllogisms proceed through the
aforesaid figures。 For it is proved that A belongs to all E；
whenever an identical term is found among the Cs and Fs。 This will
be the middle term； A and E will be the extremes。 So the first
figure is formed。 And A will belong to some E； whenever C and G are
apprehended to be the same。 This is the last figure： for G becomes the
middle term。 And A will belong to no E； when D and F are identical。
Thus we have both the first figure and the middle figure； the first；
because A belongs to no F； since the negative statement is
convertible； and F belongs to all E： the middle figure because D
belongs to no A； and to all E。 And A will not belong to some E；
whenever D and G are identical。 This is the last figure： for A will
belong to no G； and E will belong to all G。 Clearly then all
syllogisms proceed through the aforesaid figures； and we must not
select consequents of all the terms； because no syllogism is
produced from them。 For （as we saw） it is not possible at all to
establish a proposition from consequents； and it is not possible to
refute by means of a consequent of both the terms in question： for the
middle term must belong to the one； and not belong to the other。
It is clear too that other methods of inquiry by selection of middle
terms are useless to produce a syllogism； e。g。 if the consequents of
the terms in question are identical； or if the antecedents of A are
identical with those attributes which cannot possibly belong to E；
or if those attributes are identical which cannot belong to either
term： for no syllogism is produced by means of these。 For if the
consequents are identical； e。g。 B and F； we have the middle figure
with both premisses affirmative： if the antecedents of A are identical
with attributes which cannot belong to E； e。g。 C with H； we have the
first figure with its minor premiss negative。 If attributes which
cannot belong to either term are identical； e。g。 C and H； both
premisses are negative； either in the first or in the middle figure。
But no syllogism is possible in this way。
It is evident too that we must find out which terms in this
inquiry are identical； not which are different or contrary； first
because the object of our investigation is the middle term； and the
middle term must be not diverse but identical。 Secondly； wherever it
happens that a syllogism results from taking contraries or terms which
cannot belong to the same thing； all arguments can be reduced to the
aforesaid moods； e。g。 if B and F are contraries or cannot belong to
the same thing。 For if 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。
29
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 establ