### Aristotle - The Organon ANALYTICA PRIORIA Book 2 Part 13

## ... in the third figure

1.
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.

2.
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.

3.
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.

4.
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 all B. If however it is assumed that A belongs to some B, we
shall have the same result as before.

5.
It is clear then that in all the syllogisms which proceed per
impossibile the contradictory must be assumed.
And it is
plain that in
the middle figure an affirmative conclusion, and in the last figure
a universal conclusion, are proved in a way.

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