The Complete Aristotle (eng.)

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.
    If, then, the premisses are universal, we have stated when the
conclusion will be necessary. But if one premiss is universal, the
other particular, and if both are affirmative, whenever the
universal is necessary the conclusion also must be necessary. The
demonstration is the same as before; for the particular affirmative
also is convertible. If then it is necessary that B should belong
to all C, and A falls under C, it is necessary that B should belong
to some A. But if B must belong to some A, then A must belong to
some B: for conversion is possible. Similarly also if AC should be
necessary and universal: for B falls under C. But if the particular
premiss is necessary, the conclusion will not be necessary. Let the
premiss BC be both particular and necessary, and let A belong to
all C, not however necessarily. If the proposition BC is converted
the first figure is formed, and the universal premiss is not
necessary, but the particular is necessary. But when the premisses
were thus, the conclusion (as we proved was not necessary:
consequently it is not here either. Further, the point is clear if
we look at the terms. Let A be waking, B biped, and C animal. It is
necessary that B should belong to some C, but it is possible for A
to belong to C, and that A should belong to B is not necessary. For
there is no necessity that some biped should be asleep or awake.
Similarly and by means of the same terms proof can be made, should
the proposition AC be both particular and necessary.
    But if one premiss is affirmative, the other negative, whenever
the universal is both negative and necessary the conclusion also
will be necessary. For if it is not possible that A should belong
to any C, but B belongs to some C, it is necessary that A should
not belong to some B. But whenever the affirmative proposition is
necessary, whether universal or particular, or the negative is
particular, the conclusion will not be necessary. The proof of this
by reduction will be the same as before; but if terms are wanted,
when the universal affirmative is necessary, take the terms
‘waking’-’animal’-’man’, ‘man’ being middle, and when the
affirmative is particular and necessary, take the terms
‘waking’-’animal’-’white’: for it is necessary that animal should
belong to some white thing, but it is possible that waking should
belong to none, and it is not necessary that waking should not
belong to some animal. But when the negative proposition being
particular is necessary, take the terms ‘biped’, ‘moving’,
‘animal’, ‘animal’ being middle.
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    It is clear then that a simple conclusion is not reached unless
both premisses are simple assertions, but a necessary conclusion is
possible although one only of the premisses is