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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
either A or B will belong to everything； but they will never belong to
the same thing； and either C or D will belong to everything； but
they will never belong to the same thing。 And B must belong to
everything to which C belongs。 For if it is true to say 'it is a
not…white'； it is true also to say 'it is not white'： for it is
impossible that a thing should simultaneously be white and be
not…white； or be a not…white log and be a white log； consequently if
the affirmation does not belong； the denial must belong。 But C does
not always belong to B： for what is not a log at all； cannot be a
not…white log either。 On the other hand D belongs to everything to
which A belongs。 For either C or D belongs to everything to which A
belongs。 But since a thing cannot be simultaneously not…white and
white； D must belong to everything to which A belongs。 For of that
which is white it is true to say that it is not not…white。 But A is
not true of all D。 For of that which is not a log at all it is not
true to say A； viz。 that it is a white log。 Consequently D is true；
but A is not true； i。e。 that it is a white log。 It is clear also
that A and C cannot together belong to the same thing； and that B
and D may possibly belong to the same thing。
Privative terms are similarly related positive ter terms respect
of this arrangement。 Let A stand for 'equal'； B for 'not equal'； C for
'unequal'； D for 'not unequal'。
In many things also； to some of which something belongs which does
not belong to others； the negation may be true in a similar way；
viz。 that all are not white or that each is not white； while that each
is not…white or all are not…white is false。 Similarly also 'every
animal is not…white' is not the negation of 'every animal is white'
（for both are false）： the proper negation is 'every animal is not
white'。 Since it is clear that 'it is not…white' and 'it is not white'
mean different things； and one is an affirmation； the other a
denial； it is evident that the method of proving each cannot be the
same； e。g。 that whatever is an animal is not white or may not be
white； and that it is true to call it not…white； for this means that
it is not…white。 But we may prove that it is true to call it white
or not…white in the same way for both are proved constructively by
means of the first figure。 For the expression 'it is true' stands on a
similar footing to 'it is'。 For the negation of 'it is true to call it
white' is not 'it is true to call it not…white' but 'it is not true to
call it white'。 If then it is to be true to say that whatever is a man
is musical or is not…musical； we must assume that whatever is an
animal either is musical or is not…musical； and the proof has been
made。 That whatever is a man is not musical is prov