belong to all C, it follows that A belongs to some B: for if A
belonged to no B, and B belongs to all C, A would belong to no C:
but (as we stated) it belongs to all C. Similarly also with the rest.

  It is possible also to reduce all syllogisms to the universal
syllogisms in the first figure. Those in the second figure are clearly
made perfect by these, though not all in the same way; the universal
syllogisms are made perfect by converting the negative premiss, each
of the particular syllogisms by reductio ad impossibile. In the
first figure particular syllogisms are indeed made perfect by
themselves, but it is possible also to prove them by means of the
second figure, reducing them ad impossibile, e.g. if A belongs to
all B, and B to some C, it follows that A belongs to some C. For if it
belonged to no C, and belongs to all B, then B will belong to no C:
this we know by means of the second figure. Similarly also
demonstration will be possible in the case of the negative. For if A
belongs to no B, and B belongs to some C, A will not belong to some C:
for if it belonged to all C, and belongs to no B, then B will belong
to no C: and this (as we saw) is the middle figure. Consequently,
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