
Prior Analytics  Book II
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
