the conclusion is retained； B will belong to some C： for we obtain the
first figure and A is middle。 But if the syllogism is negative； it
is not possible to prove the universal premiss； for the reason given
above。 But it is possible to prove the particular premiss； if the
proposition AB is converted as in the universal syllogism； i。e 'B
belongs to some of that to some of which A does not belong'： otherwise
no syllogism results because the particular premiss is negative。
6
In the second figure it is not possible to prove an affirmative
proposition in this way； but a negative proposition may be proved。
An affirmative proposition is not proved because both premisses of the
new syllogism are not affirmative （for the conclusion is negative） but
an affirmative proposition is （as we saw） proved from premisses
which are both affirmative。 The negative is proved as follows。 Let A
belong to all B； and to no C： we conclude that B belongs to no C。 If
then it is assumed that B belongs to all A； it is necessary that A
should belong to no C： for we get the second figure； with B as middle。
But if the premiss AB was negative； and the other affirmative； we
shall have the first figure。 For C belongs to all A and B to no C；
consequently B belongs to no A： neither then does A belong to B。
Through the conclusion； therefore； and one premiss； we get no
syllogism； but if another premiss is assumed in addition； a
syllogism will be possible。 But if the syllogism not universal； the
universal premiss cannot be proved； for the same reason as we gave
above； but the particular premiss can be proved whenever the universal
statement is affirmative。 Let A belong to all B； and not to all C： the
conclusion is BC。 If then it is assumed that B belongs to all A； but
not to all C； A will not belong to some C； B being middle。 But if
the universal premiss is negative； the premiss AC will not be
demonstrated by the conversion of AB： for it turns out that either
both or one of the premisses is negative； consequently a syllogism
will not be possible。 But the proof will proceed as in the universal
syllogisms； if it is assumed that A belongs to some of that to some of
which B does not belong。
7
In the third figure； when both premisses are taken universally； it
is not possible to prove them reciprocally： for that which is
universal is proved through statements which are universal； but the
conclusion in this figure is always particular； so that it is clear
that it is not possible at all to prove through this figure the
universal premiss。 But if one premiss is universal； the other
particular； proof of the latter will sometimes be possible；
sometimes not。 When both the premisses assumed are affirmative； and
the universal concerns the minor extreme； proof will be possible；
but when it concerns the other extreme； impossible。 Let A belong to
all C and B to some C： the conclusion is the statement AB。 If then
it is assumed that C belongs to all A； it has been proved that C
belongs to some B； but that B belongs to some C has not been proved。
And yet it is necessary； if C belongs to some B； that B should
belong to some C。 But it is not the same that this should belong to
that； and that to this： but we must assume besides that if this
belongs to some of that； that belongs to some of this。 But if this
is assumed the syllogism no longer results from the conclusion and the
other premiss。 But if B belongs to all C； and A to some C； it will
be possible to prove the proposition AC； when it is assumed that C
belongs to all B； and A to some B。 For if C belongs to all B and A
to some B； it is necessary that A should belong to some C； B being
middle。 And whenever one premiss is affirmative the other negative；
and the affirmative is universal； the other premiss can be proved。 Let
B belong to all C； and A not to some C： the conclusion is that A
does not belong to some B。 If then it is assumed further that C
belongs to all B； it is necessary that A should not belong to some
C； B being middle。 But when the negative premiss is universal； the
other premiss is not except as before； viz。 if it is assumed that that
belongs to some of that； to some of which this does not belong； e。g。
if A belongs to no C； and B to some C： the conclusion is that A does
not belong to some B。 If then it is assumed that C belongs to some
of that to some of which does not belong； it is necessary that C
should belong to some of the Bs。 In no other way is it possible by
converting the universal premiss to prove the other： for in no other
way can a syllogism be formed。
It is clear then that in the first figure reciprocal proof is made
both through the third and through the first figure…if the
conclusion is affirmative through the first； if the conclusion is
negative through the last。 For it is assumed that that belongs to
all of that to none of which this belongs。 In the middle figure；
when the syllogism is universal； proof is possible through the
second figure and through the first； but when particular through the
second and the last。 In the third figure all proofs are made through
itself。 It is clear also that in the third figure and in the middle
figure those syllogisms which are not made through those figures
themselves either are not of the nature of circular proof or are
imperfect。
8
To convert a syllogism means to alter the conclusion and make
another syllogism to prove that either the extreme cannot belong to
the middle or the middle to the last term。 For it is necessary； if the
conclusion has been changed into its opposite and one of the premisses
stands； that the other premiss should be destroyed。 For if it should
stand； the conclusion also must stand。 It makes a difference whether
the conclusion is converted into its contradictory or into its
contrary。 For the same syllogism does not result whichever form the
conversion takes。 This will be made clear by the sequel。 By
contradictory opposition I mean the opposition of 'to all' to 'not
to all'； and of 'to some' to 'to none'； by contrary opposition I
mean the opposition of 'to all' to 'to none'； and of 'to some' to 'not
to some'。 Suppose that A been proved of C； through B as middle term。
If then it should be assumed that A belongs to no C； but to all B； B
will belong to no C。 And if A belongs to no C； and B to all C； A
will belong； not to no B at all； but not to all B。 For （as we saw） the
universal is not proved through the last figure。 In a word it is not
possible to refute universally by conversion the premiss which
concerns the major extreme： for the refutation always proceeds through
the third since it is necessary to take both premisses in reference to
the minor extreme。 Similarly if the syllogism is negative。 Suppose
it has been proved that A belongs to no C through B。 Then if it is
assumed that A belongs to all C； and to no B； B will belong to none of
the Cs。 And if A and B belong to all C； A will belong to some B： but
in the original premiss it belonged to no B。
If the conclusion is converted into its contradictory； the
syllogisms will be contradictory and not universal。 For one premiss is
particular； so that the conclusion also will be particular。 Let the
syllogism be affirmative； and let it be converted as stated。 Then if A
belongs not to all C； but to all B； B will belong not to all C。 And if
A belongs not to all C； but B belongs to all C； A will belong not to
all B。 Similarly if the syllogism is negative。 For if A belongs to
some C； and to no B； B will belong； not to no C at all； but…not to
some C。 And if A belongs to some C； and B to all C； as was
originally assumed； A will belong to some B。
In particular syllogisms when the conclusion is converted into its
contradictory； both premisses may be refuted； but when it is converted
into its contrary； neither。 For the result is no longer； as in the
universal syllogisms； refutation in which the conclusion reached by O；
conversion lacks universality； but no refutation at all。 Suppose
that A has been proved of some C。 If then it is assumed that A belongs
to no C； and B to some C； A will not belong to some B： and if A
belongs to no C； but to all B； B will belong to no C。 Thus both
premisses are refuted。 But neither can be refuted if the conclusion is
converted into its contrary。 For if A does not