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since all syllogisms in the middle figure can be reduced to
universal syllogisms in the first figure； and since particular
syllogisms in the first figure can be reduced to syllogisms in the
middle figure； it is clear that particular syllogisms can be reduced
to universal syllogisms in the first figure。 Syllogisms in the third
figure； if the terms are universal； are directly made perfect by means
of those syllogisms； but； when one of the premisses is particular；
by means of the particular syllogisms in the first figure： and these
（we have seen） may be reduced to the universal syllogisms in the first
figure： consequently also the particular syllogisms in the third
figure may be so reduced。 It is clear then that all syllogisms may
be reduced to the universal syllogisms in the first figure。
We have stated then how syllogisms which prove that something
belongs or does not belong to something else are constituted； both how
syllogisms of the same figure are constituted in themselves； and how
syllogisms of different figures are related to one another。
8
Since there is a difference according as something belongs；
necessarily belongs； or may belong to something else （for many
things belong indeed； but not necessarily； others neither
necessarily nor indeed at all； but it is possible for them to belong）；
it is clear that there will be different syllogisms to prove each of
these relations； and syllogisms with differently related terms； one
syllogism concluding from what is necessary； another from what is； a
third from what is possible。
There is hardly any difference between syllogisms from necessary
premisses and syllogisms from premisses which merely assert。 When
the terms are put in the same way； then； whether something belongs
or necessarily belongs （or does not belong） to something else； a
syllogism will or will not result alike in both cases； the only
difference being the addition of the expression 'necessarily' to the
terms。 For the negative statement is convertible alike in both
cases； and we should give the same account of the expressions 'to be
contained in something as in a whole' and 'to be predicated of all
of something'。 With the exceptions to be made below； the conclusion
will be proved to be necessary by means of conversion； in the same
manner as in the case of simple predication。 But in the middle
figure when the universal statement is affirmative； and the particular
negative； and again in the third figure when the universal is
affirmative and the particular negative； the demonstration will not
take the same form； but it is necessary by the 'exposition' of a
part of the subject of the particular negative proposition； to which
the predicate does not belong； to make the syllogism in reference to
this： with terms so chosen the conclusion will necessarily follow。 But
if the relation is necessary in respect of the part taken； it must
hold of some of that term in which this part is included： for the part
taken is just some of that。 And each of the resulting syllogisms is in
the appropriate figure。
9
It happens sometimes also that when one premiss is necessary the
conclusion is necessary； not however when either premiss is necessary；
but only when the major is； e。g。 if A is taken as necessarily
belonging or not belonging to B； but B is taken as simply belonging to
C： for if the premisses are taken in this way； A will necessarily
belong or not belong to C。 For since necessarily belongs； or does
not belong； to every B； and since C is one of the Bs； it is clear that
for C also the positive or the negative relation to A will hold
necessarily。 But if the major premiss is not necessary； but the
minor is necessary； the conclusion will not be necessary。 For if it
were； it would result both through the first figure and through the
third that A belongs necessarily to some B。 But this is false； for B
may be such that it is possible that A should belong to none of it。
Further； an example also makes it clear that the conclusion not be
necessary； e。g。 if A were movement； B animal； C man： man is an
animal necessarily； but an animal does not move necessarily； nor
does man。 Similarly also if the major premiss is negative； for the
proof is the same。
In particular syllogisms； if the universal premiss is necessary；
then the conclusion will be necessary； but if the particular； the
conclusion will not be necessary； whether the universal premiss is
negative or affirmative。 First let the universal be necessary； and let
A belong to all B necessarily； but let B simply belong to some C： it
is necessary then that A belongs to some C necessarily： for C falls
under B； and A was assumed to belong necessarily to all B。 Similarly
also if the syllogism should be negative： for the proof will be the
same。 But if the particular premiss is necessary； the conclusion
will not be necessary： for from the denial of such a conclusion
nothing impossible results； just as it does not in the universal
syllogisms。 The same is true of negative syllogisms。 Try the terms
movement； animal； white。
10
In the second figure； if the negative premiss is necessary； then the
conclusion will be necessary； but if the affirmative； not necessary。
First let the negative be necessary； let A be possible of no B； and
simply belong to C。 Since then the negative statement is
convertible； B is possible of no A。 But A belongs to all C；
consequently B is possible of no C。 For C falls under A。 The same
result would be obtained if the minor premiss were negative： for if
A is possible be of no C； C is possible of no A： but A belongs to
all B； consequently C is possible of none of the Bs： for again we have
obtained the first figure。 Neither then is B possible of C： for
conversion is possible without modifying the relation。
But if the affirmative premiss is necessary； the conclusion will not
be necessary。 Let A belong to all B necessarily； but to no C simply。
If then the negative premiss is converted； the first figure results。
But it has been proved in the case of the first figure that if the
negative major premiss is not necessary the conclusion will not be
necessary either。 Therefore the same result will obtain here。 Further；
if the conclusion is necessary； it follows that C necessarily does not
belong to some A。 For if B necessarily belongs to no C； C will
necessarily belong to no B。 But B at any rate must belong to some A；
if it is true （as was assumed） that A necessarily belongs to all B。
Consequently it is necessary that C does not belong to some A。 But
nothing prevents such an A being taken that it is possible for C to
belong to all of it。 Further one might show by an exposition of
terms that the conclusion is not necessary without qualification；
though it is a necessary conclusion from the premisses。 For example
let A be animal； B man； C white； and let the premisses be assumed to
correspond to what we had before： it is possible that animal should
belong to nothing white。 Man then will not belong to anything white；
but not necessarily： for it is possible for man to be born white；
not however so long as animal belongs to nothing white。 Consequently
under these conditions the conclusion will be necessary； but it is not
necessary without qualification。
Similar results will obtain also in particular syllogisms。 For
whenever the negative premiss is both un