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.



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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