Prior Analytics - Book II   

impossibile, and when the original hypothesis is so related to the

impossible conclusion, that the conclusion results indifferently

whether the hypothesis is made or not. The most obvious case of the

irrelevance of an assumption to a conclusion which is false is when

a syllogism drawn from middle terms to an impossible conclusion is

independent of the hypothesis, as we have explained in the Topics. For

to put that which is not the cause as the cause, is just this: e.g. if

a man, wishing to prove that the diagonal of the square is

incommensurate with the side, should try to prove Zeno's theorem

that motion is impossible, and so establish a reductio ad impossibile:

for Zeno's false theorem has no connexion at all with the original

assumption. Another case is where the impossible conclusion is

connected with the hypothesis, but does not result from it. This may

happen whether one traces the connexion upwards or downwards, e.g.

if it is laid down that A belongs to B, B to C, and C to D, and it

should be false that B belongs to D: for if we eliminated A and

assumed all the same that B belongs to C and C to D, the false

conclusion would not depend on the original hypothesis. Or again trace

the connexion upwards; e.g. suppose that A belongs to B, E to A and

F to E, it being false that F belongs to A. In this way too the

impossible conclusion would result, though the original hypothesis

were eliminated. But the impossible conclusion ought to be connected

with the original terms: in this way it will depend on the hypothesis,

e.g. when one traces the connexion downwards, the impossible

conclusion must be connected with that term which is predicate in

the hypothesis: for if it is impossible that A should belong to D, the

false conclusion will no longer result after A has been eliminated. If

one traces the connexion upwards, the impossible conclusion must be

connected with that term which is subject in the hypothesis: for if it

is impossible that F should belong to B, the impossible conclusion

will disappear if B is eliminated. Similarly when the syllogisms are

negative.