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mean not only in being affirmative or negative； but also in being
necessary； pure； problematic。 We must consider also the other forms of
predication。
It is clear also when a syllogism in general can be made and when it
cannot； and when a valid； when a perfect syllogism can be formed；
and that if a syllogism is formed the terms must be arranged in one of
the ways that have been mentioned。
25
It is clear too that every demonstration will proceed through
three terms and no more； unless the same conclusion is established
by different pairs of propositions； e。g。 the conclusion E may be
established through the propositions A and B； and through the
propositions C and D； or through the propositions A and B； or A and C；
or B and C。 For nothing prevents there being several middles for the
same terms。 But in that case there is not one but several
syllogisms。 Or again when each of the propositions A and B is obtained
by syllogistic inference； e。g。 by means of D and E； and again B by
means of F and G。 Or one may be obtained by syllogistic； the other
by inductive inference。 But thus also the syllogisms are many； for the
conclusions are many； e。g。 A and B and C。 But if this can be called
one syllogism； not many； the same conclusion may be reached by more
than three terms in this way； but it cannot be reached as C is
established by means of A and B。 Suppose that the proposition E is
inferred from the premisses A； B； C； and D。 It is necessary then
that of these one should be related to another as whole to part： for
it has already been proved that if a syllogism is formed some of its
terms must be related in this way。 Suppose then that A stands in
this relation to B。 Some conclusion then follows from them。 It must
either be E or one or other of C and D； or something other than these。
（1） If it is E the syllogism will have A and B for its sole
premisses。 But if C and D are so related that one is whole； the
other part； some conclusion will follow from them also； and it must be
either E； or one or other of the propositions A and B； or something
other than these。 And if it is （i） E； or （ii） A or B； either （i） the
syllogisms will be more than one； or （ii） the same thing happens to be
inferred by means of several terms only in the sense which we saw to
be possible。 But if （iii） the conclusion is other than E or A or B；
the syllogisms will be many； and unconnected with one another。 But
if C is not so related to D as to make a syllogism； the propositions
will have been assumed to no purpose； unless for the sake of induction
or of obscuring the argument or something of the sort。
（2） But if from the propositions A and B there follows not E but
some other conclusion； and if from C and D either A or B follows or
something else； then there are several syllogisms； and they do not
establish the conclusion proposed： for we assumed that the syllogism
proved E。 And if no conclusion follows from C and D； it turns out that
these propositions have been assumed to no purpose； and the
syllogism does not prove the original proposition。
So it is clear that every demonstration and every syllogism will
proceed through three terms only。
This being evident； it is clear that a syllogistic conclusion
follows from two premisses and not from more than two。 For the three
terms make two premisses； unless a new premiss is assumed； as was said
at the beginning； to perfect the syllogisms。 It is clear therefore
that in whatever syllogistic argument the premisses through which
the main conclusion follows （for some of the preceding conclusions
must be premisses） are not even in number； this argument either has
not been drawn syllogistically or it has assumed more than was
necessary to establish its thesis。
If then syllogisms are taken with respect to their main premisses；
every syllogism will consist of an even number of premisses and an odd
number of terms （for the terms exceed the premisses by one）； and the
conclusions will be half the number of the premisses。 But whenever a
conclusion is reached by means of prosyllogisms or by means of several
continuous middle terms； e。g。 the proposition AB by means of the
middle terms C and D； the number of the terms will similarly exceed
that of the premisses by one （for the extra term must either be
added outside or inserted： but in either case it follows that the
relations of predication are one fewer than the terms related）； and
the premisses will be equal in number to the relations of predication。
The premisses however will not always be even； the terms odd； but they
will alternate…when the premisses are even； the terms must be odd；
when the terms are even； the premisses must be odd： for along with one
term one premiss is added； if a term is added from any quarter。
Consequently since the premisses were （as we saw） even； and the
terms odd； we must make them alternately even and odd at each
addition。 But the conclusions will not follow the same arrangement
either in respect to the terms or to the premisses。 For if one term is
added； conclusions will be added less by one than the pre…existing
terms： for the conclusion is drawn not in relation to the single
term last added； but in relation to all the rest； e。g。 if to ABC the
term D is added； two conclusions are thereby added； one in relation to
A； the other in relation to B。 Similarly with any further additions。
And similarly too if the term is inserted in the middle： for in
relation to one term only； a syllogism will not be constructed。
Consequently the conclusions will be much more numerous than the terms
or the premisses。
26
Since we understand the subjects with which syllogisms are
concerned； what sort of conclusion is established in each figure；
and in how many moods this is done； it is evident to us both what sort
of problem is difficult and what sort is easy to prove。 For that which
is concluded in many figures and through many moods is easier； that
which is concluded in few figures and through few moods is more
difficult to attempt。 The universal affirmative is proved by means
of the first figure only and by this in only one mood； the universal
negative is proved both through the first figure and through the
second； through the first in one mood； through the second in two。
The particular affirmative is proved through the first and through the
last figure； in one mood through the first； in three moods through the
last。 The particular negative is proved in all the figures； but once
in the first； in two moods in the second； in three moods in the third。
It is clear then that the universal affirmative is most difficult to
establish； most easy to overthrow。 In general； universals are easier
game for the destroyer than particulars： for whether the predicate
belongs to none or not to some； they are destroyed： and the particular
negative is proved in all the figures； the universal negative in
two。 Similarly with universal negatives： the original statement is
destroyed； whether the predicate belongs to all or to some： and this
we found possible in two figures。 But particular statements can be
refuted in one way only…by proving that the predicate belongs either
to all or to none。 But particular statements are easier to
establish： for proof is possible in more figures and through more
moods。 And in general we must not forget that it is possible to refute
statements by means of one another； I mean； universal s