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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 if it is supposed to be commensurate。 One
infers syllogistically that odd numbers come out equal to evens； and
one proves hypothetically the incommensurability of the diagonal；
since a falsehood results through contradicting this。 For this we
found to be reasoning per impossibile； viz。 proving something
impossible by means of an hypothesis conceded at the beginning。
Consequently； since the falsehood is established in reductions ad
impossibile by an ostensive syllogism； and the original conclusion
is proved hypothetically； and we have already stated that ostensive
syllogisms are effected by means of these figures； it is evident
that syllogisms per impossibile also will be made through these
figures。 Likewise all the other hypothetical syllogisms： for in
every case the syllogism leads up to the proposition that is
substituted for the original thesis； but the original thesis is
reached by means of a concession or some other hypothesis。 But if this
is true； every demonstration and every syllogism must be formed by
means of the three figures mentioned above。 But when this has been
shown it is clear that every syllogism is perfected by means of the
first figure and is reducible to the universal syllogisms in this
figure。
24
Further in every syllogism one of the premisses must be affirmative；
and universality must be present： unless one of the premisses is
universal either a syllogism will not be possible； or it will not
refer to the subject proposed； or the original position will be
begged。 Suppose we have to prove that pleasure in music is good。 If
one should claim as a premiss that pleasure is good without adding
'all'； no syllogism will be possible； if one should claim that some
pleasure is good； then if it is different from pleasure in music； it
is not relevant to the subject proposed； if it is this very
pleasure； one is assuming that which was proposed at the outset to
be proved。 This is more obvious in geometrical proofs； e。g。 that the
angles at the base of an isosceles triangle are equal。 Suppose the
lines A and B have been drawn to the centre。 If then one should assume
that the angle AC is equal to the angle BD； without claiming generally
that angles of semicircles are equal； and again if one should assume
that the angle C is equal to the angle D； without the additional
assumption that every angle of a segment is equal to every other angle
of the same segment； and further if one should assume that when
equal angles are taken from the whole angles； which are themselves
equal； the remainders E and F are equal； he will beg the thing to be
proved； unless he also states that when equals are taken from equals
the remainders are equal。
It is clear then that in every syllogism there must be a universal
premiss； and that a universal statement is proved only when all the
premisses are universal； while a particular statement is proved both
from two universal premisses and from one only： consequently if the
conclusion is universal； the premisses also must be universal； but
if the premisses are universal it is possible that the conclusion
may not be universal。 And it is clear also that in every syllogism
either both or one of the premisses must be like the conclusion。 I
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