The Complete Aristotle (eng.)

figure.
11
    It is clear then what conversion is, how it is effected in each
figure, and what syllogism results. The syllogism per impossibile
is proved when the contradictory of the conclusion stated and
another premiss is assumed; it can be made in all the figures. For
it resembles conversion, differing only in this: conversion takes
place after a syllogism has been formed and both the premisses have
been taken, but a reduction to the impossible takes place not
because the contradictory has been agreed to already, but because
it is clear that it is true. The terms are alike in both, and the
premisses of both are taken in the same way. For example if A
belongs to all B, C being middle, then if it is supposed that A
does not belong to all B or belongs to no B, but to all C (which
was admitted to be true), it follows that C belongs to no B or not
to all B. But this is impossible: consequently the supposition is
false: its contradictory then is true. Similarly in the other
figures: for whatever moods admit of conversion admit also of the
reduction per impossibile.
    All the problems can be proved per impossibile in all the
figures, excepting the universal affirmative, which is proved in
the middle and third figures, but not in the first. Suppose that A
belongs not to all B, or to no B, and take besides another premiss
concerning either of the terms, viz. that C belongs to all A, or
that B belongs to all D; thus we get the first figure. If then it
is supposed that A does not belong to all B, no syllogism results
whichever term the assumed premiss concerns; but if it is supposed
that A belongs to no B, when the premiss BD is assumed as well we
shall prove syllogistically what is false, but not the problem
proposed. For if A belongs to no B, and B belongs to all D, A
belongs to no D. Let this be impossible: it is false then A belongs
to no B. But the universal affirmative is not necessarily true if
the universal negative is false. But if the premiss CA is assumed
as well, no syllogism results, nor does it do so when it is
supposed that A does not belong to all B. Consequently it is clear
that the universal affirmative cannot be proved in the first figure
per impossibile.
    But the particular affirmative and the universal and particular
negatives can all be proved. Suppose that A belongs to no B, and
let it have been assumed that B belongs to all or to some C. Then
it is necessary that A should belong to no C or not to all C. But
this is impossible (for let it be true and clear that A belongs to
all C): consequently if this is false, it is necessary that A
should belong to some B. But if the other premiss assumed relates
to A, no syllogism will be possible. Nor can a conclusion be drawn
when the contrary of the conclusion is supposed, e.g. that A does
not belong to some B. Clearly then we must suppose the
contradictory.
    Again suppose that A belongs to some B, and let it have been
assumed that C belongs to all A. It is necessary then that C should
belong to some B. But let this be impossible, so that the
supposition is false: in that case it is true that A belongs to no
B. We may proceed in the same way if the proposition CA has been
taken as negative. But if the premiss assumed concerns B, no
syllogism will 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