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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 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
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。
12
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 mus