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admit of proof。
In general then we have explained fairly well how we must select
premisses： we have discussed the matter accurately in the treatise
concerning dialectic。
31
It is easy to see that division into classes is a small part of
the method we have described： for division is； so to speak； a weak
syllogism； for what it ought to prove； it begs； and it always
establishes something more general than the attribute in question。
First； this very point had escaped all those who used the method of
division； and they attempted to persuade men that it was possible to
make a demonstration of substance and essence。 Consequently they did
not understand what it is possible to prove syllogistically by
division； nor did they understand that it was possible to prove
syllogistically in the manner we have described。 In demonstrations；
when there is a need to prove a positive statement； the middle term
through which the syllogism is formed must always be inferior to and
not comprehend the first of the extremes。 But division has a
contrary intention： for it takes the universal as middle。 Let animal
be the term signified by A； mortal by B； and immortal by C； and let
man； whose definition is to be got； be signified by D。 The man who
divides assumes that every animal is either mortal or immortal： i。e。
whatever is A is all either B or C。 Again； always dividing； he lays it
down that man is an animal； so he assumes A of D as belonging to it。
Now the true conclusion is that every D is either B or C； consequently
man must be either mortal or immortal； but it is not necessary that
man should be a mortal animal…this is begged： and this is what ought
to have been proved syllogistically。 And again； taking A as mortal
animal； B as footed； C as footless； and D as man； he assumes in the
same way that A inheres either in B or in C （for every mortal animal
is either footed or footless）； and he assumes A of D （for he assumed
man； as we saw； to be a mortal animal）； consequently it is necessary
that man should be either a footed or a footless animal； but it is not
necessary that man should be footed： this he assumes： and it is just
this again which he ought to have demonstrated。 Always dividing then
in this way it turns out that these logicians assume as middle the
universal term； and as extremes that which ought to have been the
subject of demonstration and the differentiae。 In conclusion； they
do not make it clear； and show it to be necessary； that this is man or
whatever the subject of inquiry may be： for they pursue the other
method altogether； never even suspecting the presence of the rich
supply of evidence which might be used。 It is clear that it is neither
possible to refute a statement by this method of division； nor to draw
a conclusion about an accident or property of a thing； nor about its
genus； nor in cases in which it is unknown whether it is thus or thus；
e。g。 whether the diagonal is incommensurate。 For if he assumes that
every length is either commensurate or incommensurate； and the
diagonal is a length； he has proved that the diagonal is either
incommensurate or commensurate。 But if he should assume that it is
incommensurate； he will have assumed what he ought to have proved。
He cannot then prove it： for this is his method； but proof is not
possible by this method。 Let A stand for 'incommensurate or
commensurate'； B for 'length'； C for 'diagonal'。 It is clear then that
this method of investigation is not suitable for every inquiry； nor is
it useful in those cases in which it is thought to be most suitable。
From what has been said it is clear from what elements
demonstrations are formed and in what manner； and to what points we
must look in each problem。
32
Our next business is to state how we can reduce syllogisms to the
aforementioned figures： for this part of the inquiry still remains。 If
we should investigate the production of the syllogisms and had the
power of discovering them； and further if we could resolve the
syllogisms produced into the aforementioned figures； our original
problem would be brought to a conclusion。 It will happen at the same
time that what has been already said will be confirmed and its truth
made clearer by what we are about to say。 For everything that is
true must in every respect agree with itself First then we must
attempt to select the two premisses of the syllogism （for it is easier
to divide into large parts than into small； and the composite parts
are larger than the elements out of which they are made）； next we must
inquire which are universal and which particular； and if both
premisses have not been stated； we must ourselves assume the one which
is missing。 For sometimes men put forward the universal premiss； but
do not posit the premiss which is contained in it； either in writing
or in discussion： or men put forward the premisses of the principal
syllogism； but omit those through which they are inferred； and
invite the concession of others to no purpose。 We must inquire then
whether anything unnecessary has been assumed； or anything necessary
has been omitted； and we must posit the one and take away the other；
until we have reached the two premisses： for unless we have these；
we cannot reduce arguments put forward in the way described。 In some
arguments it is easy to see what is wanting； but some escape us； and
appear to be syllogisms； because something necessary results from what
has been laid down； e。g。 if the assumptions were made that substance
is not annihilated by the annihilation of what is not substance； and
that if the elements out of which a thing is made are annihilated；
then that which is made out of them is destroyed： these propositions
being laid down； it is necessary that any part of substance is
substance； this has not however been drawn by syllogism from the
propositions assumed； but premisses are wanting。 Again if it is
necessary that animal should exist； if man does； and that substance
should exist； if animal does； it is necessary that substance should
exist if man does： but as yet the conclusion has not been drawn
syllogistically： for the premisses are not in the shape we required。
We are deceived in such cases because something necessary results from
what is assumed； since the syllogism also is necessary。 But that which
is necessary is wider than the syllogism： for every syllogism is
necessary； but not everything which is necessary is a syllogism。
Consequently； though something results when certain prop