The Complete Aristotle (eng.)

belong to some of the As’; e.g. it is possible
that no man should be white (for it is also possible that every man
should be white), but it is not true to say that it is possible
that no white thing should be a man: for many white things are
necessarily not men, and the necessary (as we saw) other than the
possible.
    Moreover it is not possible to prove the convertibility of these
propositions by a reductio ad absurdum, i.e. by claiming assent to
the following argument: ‘since it is false that B may belong to no
A, it is true that it cannot belong to no A, for the one statement
is the contradictory of the other. But if this is so, it is true
that B necessarily belongs to some of the As: consequently A
necessarily belongs to some of the Bs. But this is impossible.’ The
argument cannot be admitted, for it does not follow that some A is
necessarily B, if it is not possible that no A should be B. For the
latter expression is used in two senses, one if A some is
necessarily B, another if some A is necessarily not B. For it is
not true to say that that which necessarily does not belong to some
of the As may possibly not belong to any A, just as it is not true
to say that what necessarily belongs to some A may possibly belong
to all A. If any one then should claim that because it is not
possible for C to belong to all D, it necessarily does not belong
to some D, he would make a false assumption: for it does belong to
all D, but because in some cases it belongs necessarily, therefore
we say that it is not possible for it to belong to all. Hence both
the propositions ‘A necessarily belongs to some B’ and ‘A
necessarily does not belong to some B’ are opposed to the
proposition ‘A belongs to all B’. Similarly also they are opposed
to the proposition ‘A may belong to no B’. It is clear then that in
relation to what is possible and not possible, in the sense
originally defined, we must assume, not that A necessarily belongs
to some B, but that A necessarily does not belong to some B. But if
this is assumed, no absurdity results: consequently no syllogism.
It is clear from what has been said that the negative proposition
is not convertible.
    This being proved, suppose it possible that A may belong to no B
and to all C. By means of conversion no syllogism will result: for
the major premiss, as has been said, is not convertible. Nor can a
proof be obtained by a reductio ad absurdum: for if it is assumed
that B can belong to all C, no false consequence results: for A may
belong both to all C and to no C. In general, if there is a
syllogism, it is clear that its conclusion will be problematic
because neither of the premisses is assertoric; and this must be
either affirmative or negative. But neither is possible. Suppose
the conclusion is affirmative: it will be proved by an example that
the predicate cannot belong to the subject. Suppose the conclusion
is negative: it will be proved that it is not problematic but
necessary. Let A be white, B man, C horse. It is possible then for
A to belong to all of the one and to none of the other. But it is
not possible for B to belong nor not to belong to C. That it is not
possible for it to belong, is clear. For no horse is a man. Neither
is it possible for it not to belong. For it is necessary that no
horse should be a man, but the necessary we found to be different
from the possible. No syllogism then results. A similar proof can
be given if the major premiss is negative, the minor affirmative,
or if both are affirmative or negative. The demonstration can be
made by means of the same terms. And whenever one premiss is
universal, the other particular, or both are particular or
indefinite, or in whatever other way the premisses can be altered,
the proof will always proceed through the same terms. Clearly then,
if both the premisses are problematic, no syllogism results.
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    But if one premiss is assertoric, the other problematic, if the
affirmative is assertoric and the negative problematic no syllogism
will be possible, whether the premisses are universal or
particular. The proof is the same as above, and by means of the
same terms. But when the affirmative premiss is problematic, and
the negative assertoric, we shall have a syllogism. Suppose A
belongs to no B, but can belong to all C. If the negative
proposition is converted, B will belong to no A. But ex hypothesi
can belong to all C: so a syllogism is made, proving by means of
the first figure that B may belong to no C.