Prior Analytics - Book II
be possible. If the contrary is supposed, we shall have a syllogism
and an impossible conclusion, but the problem in hand is not proved.
Suppose that A belongs to all B, and let it have been assumed that C
belongs to all A. It is necessary then that C should belong to all
B. But this is impossible, so that it is false that A belongs to all
B. But we have not yet shown it to be necessary that A belongs to no
B, if it does not belong to all B. Similarly if the other premiss
taken concerns B; we shall have a syllogism and a conclusion which
is impossible, but the hypothesis is not refuted. Therefore it is
the contradictory that we must suppose.
To prove that A does not belong to all B, we must suppose that it
belongs to all B: for if A belongs to all B, and C to all A, then C
belongs to all B; so that if this is impossible, the hypothesis is
false. Similarly if the other premiss assumed concerns B. The same
results if the original proposition CA was negative: for thus also
we get a syllogism. But if the negative proposition concerns B,
nothing is proved. If the hypothesis is that A belongs not to all
but to some B, it is not proved that A belongs not to all B, but
that it belongs to no B. For if A belongs to some B, and C 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