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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
is necessary then that C should not belong to some B。 But originally
it belonged to all B； consequently the hypothesis is false： A then
will belong to no B。
When A does not belong to an B； suppose it does belong to all B； and
to no C。 It is necessary then that C should belong to no B。 But this
is impossible： so that it is true that A does not belong to all B。
It is clear then that all the syllogisms can be formed in the middle
figure。
13
Similarly they can all be formed in the last figure。 Suppose that
A does not belong to some B； but C belongs to all B： then A does not
belong to some C。 If then this is impossible； it is false that A
does not belong to some B； so that it is true that A belongs to all B。
But if it is supposed that A belongs to no B； we shall have a
syllogism and a conclusion which is impossible： but the problem in
hand is not proved： for if the contrary is supposed； we shall have the
same results as before。
But to prove that A belongs to some B； this hypothesis must be made。
If A belongs to no B； and C to some B； A will belong not to all C。
If then this is false； it is true that A belongs to some B。
When A belongs to no B； suppose A belongs to some B； and let it have
been assumed that C belongs to all B。 Then it is necessary that A
should belong to some C。 But ex hypothesi it belongs to no C； so
that it is false that A belongs to some B。 But if it is supposed
that A belongs to all B； the problem is not proved。
But this hypothesis must be made if we are prove that A belongs
not to all B。 For if A belongs to all B and C to some B； then A
belongs to some C。 But this we assumed not to be so； so it is false
that A belongs to all B。 But in that case it is true that A belongs
not to all B。 If however it is assumed that A belongs to some B； we
shall have the same result as before。
It is clear then that in all the syllogisms which proceed per
impossibile the contradictory must be assumed。 And it is plain that in
the middle figure an affirmative conclusion； and in the last figure
a universal conclusion； are proved in a way。
14
Demonstration per impossibile differs from ostensive proof in that
it posits what it wishes to refute by reduction to a statement
admitted to be false； whereas ostensive proof starts from admitted
positions。 Both； indeed； take two premisses that are admitted； but the
latter takes the premisses from which the syllogism starts； the former
takes one of these； along with the contradictory of the original
conclusion。 Also in the ostensive proof it is not necessary that the
conclusion should be known； nor that one should suppose beforehand
that it is true or not： in the other it is necessary to suppose
beforehand that it is not true。 It makes no difference whether the
conclusion is affirmative or negative； the method is the same in
both cases。 Everything which is concluded ostensively can be proved
per impossibile； and that which is proved per impossibile can be
proved ostensively； through the same terms。 Whenever the syllogism
is formed in the first figure； the truth will be found in the middle
or the last figure； if negative in the middle； if affirmative in the
last。 Whenever the syllogism is formed in the middle figure； the truth
will be found in the first； whatever the problem may be。 Whenever
the syllogism is formed in the last figure； the truth will be found in
the first and middle figures； if affirmative in first； if negative
in the middle。 Suppose that A has been proved to belong to no B； or
not to all B； through the first figure。 Then the hypothesis must
have been that A belongs to some B； and the original premisses that
C belongs to all A and to no B。 For thus the syllogism was made and
the impossible conclusion reached。 But this is the middle figure； if C
belongs to all A and to no B。 And it is clear from these premisses
that A belongs to no B。 Similarly if has been proved not to belong
to all B。 For the hypothesis is that A belongs to all B； and the
original premisses are that C belongs to all A but not to all B。
Similarly too； if the premiss CA should be negative： for thus also
we have the middle figure。 Again suppose it has been proved that A
belongs to some B。 The hypothesis here is that is that A belongs to no
B； and the original premisses that B belongs to all C； and A either to
all or to some C： for in this way we shall get what is impossible。 But
if A and B belong to all C； we have the last figure。 And it is clear
from these premisses that A must belong to some B。 Similarly if B or A
should be assumed to belong to some C。
Again suppose it has been proved in the middle figure that A belongs
to all B。 Then the hypothesis must have been that A belongs not to all
B； and the original premisses that A belongs to all C； and C to all B：
for thus we shall get what is impossible。 But if A belongs to all C；
and C to all B； we have the first figure。 Similarly if it has been
proved that A belongs to some B： for the hypothesis then must have
been that A belongs to no B； and the original premisses that A belongs
to all C； and C to some B。 If the syllogism is negative； the
hypothesis must have been that A belongs to some B； and the original
premisses that A belongs to no C； and C to all B； so that the first
figure results。 If the syllogism is not universal； but proof has
been given that A does not belong to some B； we may infer in the
same way。 The hypothesis is that A belongs to all B； the original
premisses that A belongs to no C； and C belongs to some B： for thus we
get the first figure。
Again suppose it has been proved in the third figure that A
belongs to all B。 Then the hypothesis must have been that A belongs
not to all B； and the original premisses that C belongs to all B；
and A belongs to all C； for thus we shall get what is impossible。
And the original premisses form the first figure。 Similarly if the
demonstration establishes a particular proposition： the hypothesis
then must have been that A belongs to no B； and the original premisses
that C belongs to some B； and A to all C。 If the syllogism is
negative； the hypothesis must have been that A belongs to some B；
and the original premisses that C belongs to no A and to all B； and
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