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 8Th Street Latina or other of two contradictory
statements holds good, then if it is 8Th Street Latina proved that the negation does not
hold, the affirmation must 8Th Street Latina be true. Again if 8Th Street Latina it 8Th Street Latina is not
admitted that
8Th Street Latina the affirmation is true, the claim that the negation is true will be
8Th Street Latina 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.
12
It is clear then that in the first figure all problems except 8Th Street Latina the
universal affirmative are proved per 8Th Street Latina impossibile. But in the middle
and the last figures this also is proved. Suppose that 8Th Street Latina A does not
belong to all B, and 8Th Street Latina let it have been assumed that A belongs to all 8Th Street Latina C.
If then A belongs not to all B, but to all C, C will not belong to all
B. But this 8Th Street Latina 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. 8Th Street Latina 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 8Th Street Latina A belongs
8Th Street Latina 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 8Th Street Latina 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 8Th Street Latina A belong to no C. It
is necessary then that C should not belong to some B. But originally
it belonged 8Th Street Latina to all B, consequently the hypothesis is false: A then
will belong to 8Th Street Latina 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. 8Th Street Latina but this
is impossible: so that it is true that A does 8Th Street Latina not belong to all B.