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