The Complete Aristotle (eng.)

to all A, then C will belong to some B. If then
this is impossible, it is false that A belongs to some B;
consequently it is true that A belongs to no B. But if this is
proved, the truth is refuted as well; for the original conclusion
was that A belongs to some B, and does not belong to some B.
Further the impossible does not result from the hypothesis: for
then the hypothesis would be false, since it is impossible to draw
a false conclusion from true premisses: but in fact it is true: for
A belongs to some B. Consequently we must not suppose that A
belongs to some B, but that it belongs to all B. Similarly if we
should be proving that A does not belong to some B: for if ‘not to
belong to some’ and ‘to belong not to all’ have the same meaning,
the demonstration of both will be identical.
    It is clear then that not the contrary but the contradictory
ought to be supposed in all the syllogisms. For thus we shall have
necessity of inference, and the claim we make is one that will be
generally accepted. For if of everything one or other of two
contradictory statements holds good, then if it is proved that the
negation does not hold, the affirmation must be true. Again if it
is not admitted that the affirmation is true, the claim that the
negation is true will be generally accepted. But in neither way
does it suit to maintain the contrary: for it is not necessary that
if the universal negative is false, the universal affirmative
should be true, nor is it generally accepted that if the one is
false the other is true.
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    It is clear then that in the first figure all problems except
the universal affirmative are proved per impossibile. But in the
middle and the last figures this also is proved. Suppose that A
does not belong to all B, and let it have been assumed that A
belongs to all C. If then A belongs not to all B, but to all C, C
will not belong to all B. But this is impossible (for suppose it to
be clear that C belongs to all B): consequently the hypothesis is
false. It is true then that A belongs to all B. But if the contrary
is supposed, we shall have a syllogism and a result which is
impossible: but the problem in hand is not proved. For if A belongs
to no B, and to all C, C will belong to no B. This is impossible;
so that it is false that A belongs to no B. But though this is
false, it does not follow that it is true that A belongs to all
B.
    When A belongs to some B, suppose that A belongs to no B, and
let A belong to all C. It is necessary then that C should belong to
no B. Consequently, if this is impossible, A must belong to some B.
But if it is supposed that A does not belong to some B, we shall
have the same results as in the first figure.
    Again suppose that A belongs to some B, and let A belong to no
C. It is necessary then that C should not belong to some B. But
originally it belonged to all B, consequently the hypothesis is
false: A then will belong to no B.
    When A does not belong to an B, suppose it does belong to all B,
and to no C. It is necessary then that C should belong to no B. But
this is impossible: so that it is true that A does not belong to
all B. It is clear then that all the syllogisms can be formed in
the middle figure.
13
    Similarly they can all be formed in the last figure. Suppose
that A does not belong to some B, but C belongs to all B: then A
does not belong to some C. If then this is impossible, it is false
that A does not belong to some B; so that it is true that A belongs
to all B. But if it is supposed that A belongs to no B, we shall
have a syllogism and a conclusion which is impossible: but the
problem in hand is not proved: for if the contrary is supposed, we
shall have the same results as before.
    But to prove that A belongs to some B, this hypothesis must be
made. If A belongs to no B, and C to some B, A will belong not to
all C. If then this is false, it is true that A belongs to some
B.
    When A belongs to no B, suppose A belongs to some B, and let it
have been assumed that C belongs to all B. Then it is necessary
that A should belong to some C. But ex hypothesi it belongs to no
C, so that it is false that A belongs to some B. But if it is
supposed that A belongs to all B, the problem is not proved.
    But this hypothesis must be made if we are prove that A belongs
not to all B. For if A belongs to all B and C to some B, then A
belongs to some C. But this we assumed not to be so, so it is false
that A belongs to all B. But in that case it is true that A belongs
not to