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converted into its contrary。 For if A does not belong to some C； but
to all B； then B will not belong to some C。 But the original premiss
is not yet refuted： for it is possible that B should belong to some C；
and should not belong to some C。 The universal premiss AB cannot be
affected by a syllogism at all： for if A does not belong to some of
the Cs； but B belongs to some of the Cs； neither of the premisses is
universal。 Similarly if the syllogism is negative： for if it should be
assumed that A belongs to all C； both premisses are refuted： but if
the assumption is that A belongs to some C； neither premiss is
refuted。 The proof is the same as before。
9
In the second figure it is not possible to refute the premiss
which concerns the major extreme by establishing something contrary to
it； whichever form the conversion of the conclusion may take。 For
the conclusion of the refutation will always be in the third figure；
and in this figure （as we saw） there is no universal syllogism。 The
other premiss can be refuted in a manner similar to the conversion：
I mean； if the conclusion of the first syllogism is converted into its
contrary； the conclusion of the refutation will be the contrary of the
minor premiss of the first； if into its contradictory； the
contradictory。 Let A belong to all B and to no C： conclusion BC。 If
then it is assumed that B belongs to all C； and the proposition AB
stands； A will belong to all C； since the first figure is produced。 If
B belongs to all C； and A to no C； then A belongs not to all B： the
figure is the last。 But if the conclusion BC is converted into its
contradictory； the premiss AB will be refuted as before； the
premiss； AC by its contradictory。 For if B belongs to some C； and A to
no C； then A will not belong to some B。 Again if B belongs to some
C； and A to all B； A will belong to some C； so that the syllogism
results in the contradictory of the minor premiss。 A similar proof can
be given if the premisses are transposed in respect of their quality。
If the syllogism is particular； when the conclusion is converted
into its contrary neither premiss can be refuted； as also happened
in the first figure；' if the conclusion is converted into its
contradictory； both premisses can be refuted。 Suppose that A belongs
to no B； and to some C： the conclusion is BC。 If then it is assumed
that B belongs to some C； and the statement AB stands； the
conclusion will be that A does not belong to some C。 But the
original statement has not been refuted： for it is possible that A
should belong to some C and also not to some C。 Again if B belongs
to some C and A to some C； no syllogism will be possible： for
neither of the premisses taken is universal。 Consequently the
proposition AB is not refuted。 But if the conclusion is converted into
its contradictory； both premisses can be refuted。 For if B belongs
to all C； and A to no B； A will belong to no C： but it was assumed
to belong to some C。 Again if B belongs to all C and A to some C； A
will belong to some B。 The same proof can be given if the universal
statement is affirmative。
10
In the third figure when the conclusion is converted into its
contrary； neither of the premisses can be refuted in any of the
syllogisms； but when the conclusion is converted into its
contradictory； both premisses may be refuted and in all the moods。
Suppose it has been proved that A belongs to some B； C being taken
as middle； and the premisses being universal。 If then it is assumed
that A does not belong to some B； but B belongs to all C； no syllogism
is formed about A and C。 Nor if A does not belong to some B； but
belongs to all C； will a syllogism be possible about B and C。 A
similar proof can be given if the premisses are not universal。 For
either both premisses arrived at by the conversion must be particular；
or the universal premiss must refer to the minor extreme。 But we found
that no syllogism is possible thus either in the first or in the
middle figure。 But if the conclusion is converted into its
contradictory； both the premisses can be refuted。 For if A belongs
to no B； and B to all C； then A belongs to no C： again if A belongs to
no B； and to all C； B belongs to no C。 And similarly if one of the
premisses is not universal。 For if A belongs to no B； and B to some C；
A will not belong to some C： if A belongs to no B； and to C； B will
belong to no C。
Similarly if the original syllogism is negative。 Suppose it has been
proved that A does not belong to some B； BC being affirmative； AC
being negative： for it was thus that； as we saw； a syllogism could
be made。 Whenever then the contrary of the conclusion is assumed a
syllogism will not be possible。 For if A belongs to some B； and B to
all C； no syllogism is possible （as we saw） about A and C。 Nor； if A
belongs to some B； and to no C； was a syllogism possible concerning
B and C。 Therefore the premisses are not refuted。 But when the
contradictory of the conclusion is assumed； they are refuted。 For if A
belongs to all B； and B to C； A belongs to all C： but A was supposed
originally to belong to no C。 Again if A belongs to all B； and to no
C； then B belongs to no C： but it was supposed to belong to all C。 A
similar proof is possible if the premisses are not universal。 For AC
becomes universal and negative； the other premiss particular and
affirmative。 If then A belongs to all B； and B to some C； it results
that A belongs to some C： but it was supposed to belong to no C。 Again
if A belongs to all B； and to no C； then B belongs to no C： but it was
assumed to belong to some C。 If A belongs to some B and B to some C；
no syllogism results： nor yet if A belongs to some B； and to no C。
Thus in one way the premisses are refuted； in the other way they are
not。
From what has been said it is clear how a syllogism results in
each figure when the conclusion is converted； when a result contrary
to the premiss； and when a result contradictory to the premiss； is
obtained。 It is clear that in the first figure the syllogisms are
formed through the middle and the last figures； and the premiss
which concerns the minor extreme is alway refuted through the middle
figure； the premiss which concerns the major through the last
figure。 In the second figure syllogisms proceed through the first
and the last figures； and the premiss which concerns the minor extreme
is always refuted through the first figure； the premiss which concerns
the major extreme through the last。 In the third figure the refutation
proceeds through the first and the middle figures； the premiss which
concerns the major is always refuted through the first figure； the
premiss which concerns the minor through the middle 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