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 universal and necessary, then
the conclusion will be necessary: but whenever the affirmative premiss
is universal, the negative particular, the conclusion will not be
necessary. First then let the negative premiss be both universal and
necessary: let it be possible for no B that A should belong to it, and
let A simply belong to some C. Since the negative statement is
convertible, it will be possible for no A that B should belong to
it: but A belongs to some C; consequently B necessarily does not
belong to some of the Cs. Again let the affirmative premiss be both
universal and necessary, and let the major premiss be affirmative.
If then A necessarily belongs to all B, but does not belong to some C,
it is clear that B will not belong to some C, but not necessarily. For
the same terms can be used to demonstrate the point, which were used
in the universal syllogisms. Nor again, if the negative statement is
necessary but particular, will the conclusion be necessary. The
point can be demonstrated by means of the same terms.

                                11

  In the last figure when the terms are related universally to the
middle, and both premisses are affirmative, if one of the two is
necessary, then the conclusion will be necessary. But if one is
negative, the other affirmative, whenever the negative is necessary
the conclusion also will be necessary, but whenever the affirmative is
necessary the conclusion will not be necessary. First let both the
premisses be affirmative, and let A and B belong to all C, and let
AC be necessary. Since then B belongs to all C, C also will belong
to some B, because the universal is convertible into the particular:
consequently if A belongs necessarily to all C, and C belongs to
some B, it is necessary that A should belong to some B also. For B
is under C. The first figure then is formed. A similar proof will be
given also if BC is necessary. For C is convertible with some A:
consequently if B belongs necessarily to all C, it will belong
necessarily also to some A.

  Again let AC be negative, BC affirmative, and let the negative
premiss be necessary. Since then C is convertible with some B, but A
necessarily belongs to no C, A will necessarily not belong to some B
either: for B is under C. But if the affirmative is necessary, the
conclusion will not be necessary. For suppose BC is affirmative and
necessary, while AC is negative and not necessary. Since then the
affirmative is convertible, C also will belong to some B
necessarily: consequently if A belongs to none of the Cs, while C
belongs to some of the Bs, A will not belong to some of the Bs-but not
of necessity; for it has been proved, in the case of the first figure,
that if the negative premiss is not necessary, neither will the
conclusion be necessary. Further, the point may be made clear by
considering the terms. Let the term A be 'good', let that which B
signifies be 'animal', let the term C be 'horse'. It is possible
then that the term good should belong to no horse, and it is necessary
that the term animal should belong to every horse: but it is not
necessary that some animal should not be good, since it is possible
for every animal to be good. Or if that is not possible, take as the
term 'awake' or 'asleep': for every animal can accept these.