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Whatever problems are proved in more than one figure； if they have
been established in one figure by syllogism； can be reduced to another
figure； e。g。 a negative syllogism in the first figure can be reduced
to the second； and a syllogism in the middle figure to the first；
not all however but some only。 The point will be clear in the
sequel。 If A belongs to no B； and B to all C； then A belongs to no
C。 Thus the first figure； but if the negative statement is
converted； we shall have the middle figure。 For B belongs to no A； and
to all C。 Similarly if the syllogism is not universal but
particular； e。g。 if A belongs to no B； and B to some C。 Convert the
negative statement and you will have the middle figure。
The universal syllogisms in the second figure can be reduced to
the first； but only one of the two particular syllogisms。 Let A belong
to no B and to all C。 Convert the negative statement； and you will
have the first figure。 For B will belong to no A and A to all C。 But
if the affirmative statement concerns B； and the negative C； C must be
made first term。 For C belongs to no A； and A to all B： therefore C
belongs to no B。 B then belongs to no C： for the negative statement is
convertible。
But if the syllogism is particular； whenever the negative
statement concerns the major extreme； reduction to the first figure
will be possible； e。g。 if A belongs to no B and to some C： convert the
negative statement and you will have the first figure。 For B will
belong to no A and A to some C。 But when the affirmative statement
concerns the major extreme； no resolution will be possible； e。g。 if
A belongs to all B； but not to all C： for the statement AB does not
admit of conversion； nor would there be a syllogism if it did。
Again syllogisms in the third figure cannot all be resolved into the
first； though all syllogisms in the first figure can be resolved
into the third。 Let A belong to all B and B to some C。 Since the
particular affirmative is convertible； C will belong to some B： but
A belonged to all B： so that the third figure is formed。 Similarly
if the syllogism is negative： for the particular affirmative is
convertible： therefore A will belong to no B； and to some C。
Of the syllogisms in the last figure one only cannot be resolved
into the first； viz。 when the negative statement is not universal： all
the rest can be resolved。 Let A and B be affirmed of all C： then C can
be converted partially with either A or B： C then belongs to some B。
Consequently we shall get the first figure； if A belongs to all C； and
C to some of the Bs。 If A belongs to all C and B to some C； the
argument is the same： for B is convertible in reference to C。 But if B
belongs to all C and A to some C； the first term must be B： for B
belongs to all C； and C to some A； therefore B belongs to some A。
But since the particular statement is convertible； A will belong to
some B。 If the syllogism is negative； when the terms are universal
we must take them in a similar way。 Let B belong to all C； and A to no
C： then C will belong to some B； and A to no C； and so C will be
middle term。 Similarly if the negative statement is universal； the
affirmative particular： for A will belong to no C； and C to some of
the Bs。 But if the negative statement is particular； no resolution
will be possible； e。g。 if B belongs to all C； and A not belong to some
C： convert the statement BC and both premisses will be particular。
It is clear that in order to resolve the figures into one another
the premiss which concerns the minor extreme must be converted in both
the figures： for when this premiss is altered； the transition to the
other figure is made。
One of the syllogisms in the middle figure can； the other cannot； be
resolved into the third figure。 Whenever the universal statement is
negative； resolution is possible。 For if A belongs to no B and to some
C； both B and C alike are convertible in relation to A； so that B
belongs to no A and C to some A。 A therefore is middle term。 But
when A belongs to all B； and not to some C； resolution will not be
possible： for neither of the premisses is universal after conversion。
Syllogisms in the third figure can be resolved into the middle
figure； whenever the negative statement is universal； e。g。 if A
belongs to no C； and B to some or all C。 For C then will belong to
no A and to some B。 But if the negative statement is particular； no
resolution will be possible： for the particular negative does not
admit of conversion。
It is clear then that the same syllogisms cannot be resolved in
these figures which could not be resolved into the first figure； and
that when syllogisms are reduced to the first figure these alone are
confirmed by reduction to what is impossible。
It is clear from what we have said how we ought to reduce
syllogisms； and that the figures may be resolved into one another。
46
In establishing or refuting； it makes some difference whether we
suppose the expressions 'not to be this' and 'to be not…this' are
identical or different in meaning； e。g。 'not to be white' and 'to be
not…white'。 For they do not mean the same thing； nor is 'to be
not…white' the negation of 'to be white'； but 'not to be white'。 The
reason for this is as follows。 The relation of 'he can walk' to 'he
can not…walk' is similar to the relation of 'it is white' to 'it is
not…white'； so is that of 'he knows what is good' to 'he knows what is
not…good'。 For there is no difference between the expressions 'he
knows what is good' and 'he is knowing what is good'； or 'he can walk'
and 'he is able to walk'： therefore there is no difference between
their contraries 'he cannot walk'…'he is not able to walk'。 If then
'he is not able to walk' means the same as 'he is able not to walk'；
capacity to walk and incapacity to walk will belong at the same time
to the same person （for the same man can both walk and not…walk； and
is possessed of knowledge of what is good and of what is not…good）；
but an affirmation and a denial which are opposed to one another do
not belong at the same time to the same thing。 As then 'not to know
what is good' is not the same as 'to know what is not good'； so 'to be
not…good' is not the same as 'not to be good'。 For when two pairs
correspond； if the one pair are different from one another； the
other pair also must be different。 Nor is 'to be not…equal' the same
as 'not to be equal'： for there is something underlying the one；
viz。 that which is not…equal； and this is the unequal； but there is
nothing underlying the other。 Wherefore not everything is either equal
or unequal； but everything is equal or is not equal。 Further the
expressions 'it is a not…white log' and 'it is not a white log' do not
imply one another's truth。 For if 'it is a not…white log'； it must
be a log： but that which is not a white log need not be a log at
all。 Therefore it is clear that 'it is not…good' is not the denial
of 'it is good'。 If then every single statement may truly be said to
be either an affirmation or a negation； if it is not a negation
clearly it must in a sense be an affirmation。 But every affirmation
has a corresponding negation。 The negation then of 'it is not…good' is
'it is not not…good'。 The relation of these statements to one
another is as follows。 Let A stand for 'to be good'； B for 'not to
be good'； let C stand for 'to be not…good' and be placed under B；
and let D stand for not to be not…good' and be placed under A。 Then