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made。 That whatever is a man is not musical is proved destructively in
the three ways mentioned。
In general whenever A and B are such that they cannot belong at
the same time to the same thing； and one of the two necessarily
belongs to everything； and again C and D are related in the same
way； and A follows C but the relation cannot be reversed； then D
must follow B and the relation cannot be reversed。 And A and D may
belong to the same thing； but B and C cannot。 First it is clear from
the following consideration that D follows B。 For since either C or
D necessarily belongs to everything； and since C cannot belong to that
to which B belongs； because it carries A along with it and A and B
cannot belong to the same thing； it is clear that D must follow B。
Again since C does not reciprocate with but A； but C or D belongs to
everything； it is possible that A and D should belong to the same
thing。 But B and C cannot belong to the same thing； because A
follows C； and so something impossible results。 It is clear then
that B does not reciprocate with D either； since it is possible that D
and A should belong at the same time to the same thing。
It results sometimes even in such an arrangement of terms that one
is deceived through not apprehending the opposites rightly； one of
which must belong to everything； e。g。 we may reason that 'if A and B
cannot belong at the same time to the same thing； but it is
necessary that one of them should belong to whatever the other does
not belong to： and again C and D are related in the same way； and
follows everything which C follows： it will result that B belongs
necessarily to everything to which D belongs'： but this is false。
'Assume that F stands for the negation of A and B； and again that H
stands for the negation of C and D。 It is necessary then that either A
or F should belong to everything： for either the affirmation or the
denial must belong。 And again either C or H must belong to everything：
for they are related as affirmation and denial。 And ex hypothesi A
belongs to everything ever thing to which C belongs。 Therefore H
belongs to everything to which F belongs。 Again since either F or B
belongs to everything； and similarly either H or D； and since H
follows F； B must follow D： for we know this。 If then A follows C； B
must follow D'。 But this is false： for as we proved the sequence is
reversed in terms so constituted。 The fallacy arises because perhaps
it is not necessary that A or F should belong to everything； or that F
or B should belong to everything： for F is not the denial of A。 For
not good is the negation of good： and not…good is not identical with
'neither good nor not…good'。 Similarly also with C and D。 For two
negations have been assumed in respect to one term。
Book II
1
WE have already explained the number of the figures； the character
and number of the premisses； when and how a syllogism is formed；
further what we must look for when a refuting and establishing
propositions； and how we should investigate a given problem in any
branch of inquiry； also by what means we shall obtain principles
appropriate to each subject。 Since some syllogisms are universal；
others particular； all the universal syllogisms give more than one
result； and of particular syllogisms the affirmative yield more than
one； the negative yield only the stated conclusion。 For all
propositions are convertible save only the particular negative： and
the conclusion states one definite thing about another definite thing。
Consequently all syllogisms save the particular negative yield more
than one conclusion； e。g。 if A has been proved to to all or to some B；
then B must belong to some A： and if A has been proved to belong to no
B； then B belongs to no A。 This is a different conclusion from the
former。 But if A does not belong to some B； it is not necessary that B
should not belong to some A： for it may possibly belong to all A。
This then is the reason common to all syllogisms whether universal
or particular。 But it is possible to give another reason concerning
those which are universal。 For all the things that are subordinate
to the middle term or to the conclusion may be proved by the same
syllogism； if the former are placed in the middle； the latter in the
conclusion； e。g。 if the conclusion AB is proved through C； whatever is
subordinate to B or C must accept the predicate A： for if D is
included in B as in a whole； and B is included in A； then D will be
included in A。 Again if E is included in C as in a whole； and C is
included in A； then E will be included in A。 Similarly if the
syllogism is negative。 In the second figure it will be possible to
infer only that which is subordinate to the conclusion； e。g。 if A
belongs to no B and to all C； we conclude that B belongs to no C。 If
then D is subordinate to C； clearly B does not belong to it。 But
that B does not belong to what is subordinate to A is not clear by
means of the syllogism。 And yet B does not belong to E； if E is
subordinate to A。 But while it has been proved through the syllogism
that B belongs to no C； it has been assumed without proof that B
does not belong to A； consequently it does not result through the
syllogism that B does not belong to E。
But in particular syllogisms there will be no necessity of inferring
what is subordinate to the conclusion （for a syllogism does not result
when this premiss is particular）； but whatever is subordinate to the
middle term may be inferred； not however through the syllogism； e。g。
if A belongs to all B and B to some C。 Nothing can be inferred about
that which is subordinate to C； something can be inferred about that
which is subordinate to B； but not through the preceding syllogism。
Similarly in the other figures。 That which is subordinate to the
conclusion cannot be proved； the other subordinate can be proved； only
not through the syllogism； just as in the universal syllogisms what is
subordinate to the middle term is proved （as we saw） from a premiss
which is not demonstrated： consequently either a conclusion is not
possible in the case of universal syllogisms or else it is possible
also in the case of particular syllogisms。
2
It is possible for the premisses of the syllogism to be true； or
to be false； or to be the one true； the other false。 The conclusion is
either true or false necessarily。 From true premisses it is not
possible to draw a false conclusion； but a true conclusion may be
drawn from false premisses； true however only in respect to the
fact； not to the reason。 The reason cannot be established from false
premisses： why this is so will be explained in the sequel。
First then that it is not possible to draw a false conclusion from
true premisses； is made clear by this consideration。 If it is
necessary that B should be when A is； it is necessary that A should
not be when B is not。 If then A is true； B must be true： otherwise
it will turn out that the same thing both is and is not at the same
time。 But this is impossible。 Let it not； because A is laid down as
a single term； be supposed that it is possible； when a single fact
is given； that something should necessarily result。 For that is not
possible。 For what results necessarily is the conclusion； and the
means by which this comes about are at the least three terms； and
two relations of subject and predicate or premisses。 If then it is
true that A belongs to all that to which B belongs； and that B belongs
to all that to which C belongs； it is necessary that A should belong
to all that to which C belongs； and this cannot be false： for then the
same thing will belong and not belong at the same time。 So A is
posited as one thing； being two premisses taken together。 The same
holds good of negative syllogisms： it is not possible to prove a false
conclusion from true premisses。
But from what is false a true conclusion may be drawn； whether
both the premisses are false or only one； provided that this is not
either of the premisses indifferently； if it is taken as wholly false：
but if the premiss is not taken as wholly false； it does not matter
which of the two is false。 （1） Let A belong to the whole of C； but
to none of the Bs； neither let B belong to C。 This is possible； e。g。
animal belongs to no stone； nor stone to any man。 If then A is taken