Prior Analytics - Book II
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.
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