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belong to all C； not that A does belong to all C： and it is perfect；
not imperfect： for it is completed directly through the original
premisses。
But if the premisses are not similar in quality； suppose first
that the negative premiss is necessary； and let necessarily A not be
possible for any B； but let B be possible for all C。 It is necessary
then that A belongs to no C。 For suppose A to belong to all C or to
some C。 Now we assumed that A is not possible for any B。 Since then
the negative proposition is convertible； B is not possible for any
A。 But A is supposed to belong to all C or to some C。 Consequently B
will not be possible for any C or for all C。 But it was originally
laid down that B is possible for all C。 And it is clear that the
possibility of belonging can be inferred； since the fact of not
belonging is inferred。 Again； let the affirmative premiss be
necessary； and let A possibly not belong to any B； and let B
necessarily belong to all C。 The syllogism will be perfect； but it
will establish a problematic negative； not an assertoric negative。 For
the major premiss was problematic； and further it is not possible to
prove the assertoric conclusion per impossibile。 For if it were
supposed that A belongs to some C； and it is laid down that A possibly
does not belong to any B； no impossible relation between B and C
follows from these premisses。 But if the minor premiss is negative；
when it is problematic a syllogism is possible by conversion； as
above； but when it is necessary no syllogism can be formed。 Nor
again when both premisses are negative； and the minor is necessary。
The same terms as before serve both for the positive
relation…white…animal…snow； and for the negative
relation…white…animal…pitch。
The same relation will obtain in particular syllogisms。 Whenever the
negative proposition is necessary； the conclusion will be negative
assertoric： e。g。 if it is not possible that A should belong to any
B； but B may belong to some of the Cs； it is necessary that A should
not belong to some of the Cs。 For if A belongs to all C； but cannot
belong to any B； neither can B belong to any A。 So if A belongs to all
C； to none of the Cs can B belong。 But it was laid down that B may
belong to some C。 But when the particular affirmative in the
negative syllogism； e。g。 BC the minor premiss； or the universal
proposition in the affirmative syllogism； e。g。 AB the major premiss；
is necessary； there will not be an assertoric conclusion。 The
demonstration is the same as before。 But if the minor premiss is
universal； and problematic； whether affirmative or negative； and the
major premiss is particular and necessary； there cannot be a
syllogism。 Premisses of this kind are possible both where the relation
is positive and necessary； e。g。 animal…white…man； and where it is
necessary and negative； e。g。 animal…white…garment。 But when the
universal is necessary； the particular problematic； if the universal
is negative we may take the terms animal…white…raven to illustrate the
positive relation； or animal…white…pitch to illustrate the negative；
and if the universal is affirmative we may take the terms
animal…white…swan to illustrate the positive relation； and
animal…white…snow to illustrate the negative and necessary relation。
Nor again is a syllogism possible when the premisses are indefinite；
or both particular。 Terms applicable in either case to illustrate
the positive relation are animal…white…man： to illustrate the
negative； animal…white…inanimate。 For the relation of animal to some
white； and of white to some inanimate； is both necessary and
positive and necessary and negative。 Similarly if the relation is
problematic： so the terms may be used for all cases。
Clearly then from what has been said a syllogism results or not from
similar relations of the terms whether we are dealing with simple
existence or necessity； with this exception； that if the negative
premiss is assertoric the conclusion is problematic； but if the
negative premiss is necessary the conclusion is both problematic and
negative assertoric。 ＇It is clear also that all the syllogisms are
imperfect and are perfected by means of the figures above mentioned。＇
17
In the second figure whenever both premisses are problematic； no
syllogism is possible； whether the premisses are affirmative or
negative； universal or particular。 But when one premiss is assertoric；
the other problematic； if the affirmative is assertoric no syllogism
is possible； but if the universal negative is assertoric a
conclusion can always be drawn。 Similarly when one premiss is
necessary； the other problematic。 Here also we must understand the
term 'possible' in the conclusion； in the same sense as before。
First we must point out that the negative problematic proposition is
not convertible； e。g。 if A may belong to no B； it does not follow that
B may belong to no A。 For suppose it to follow and assume that B may
belong to no A。 Since then problematic affirmations are convertible
with negations； whether they are contraries or contradictories； and
since B may belong to no A； it is clear that B may belong to all A。
But this is false： for if all this can be that； it does not follow
that all that can be this： consequently the negative proposition is
not convertible。 Further； these propositions are not incompatible；
'A may belong to no B'； 'B necessarily does not belong to some of
the As'； e。g。 it is possible that no man should be white （for it is
also possible that every man should be white）； but it is not true to
say that it is possible that no white thing should be a man： for
many white things are necessarily not men； and the necessary （as we
saw） other than the possible。
Moreover it is not possible to prove the convertibility of these
propositions by a reductio ad absurdum； i。e。 by claiming assent to the
following argument： 'since it is false that B may belong to no A； it
is true that it cannot belong to no A； for the one statement is the
contradictory of the other。 But if this is so； it is true that B
necessarily belongs to some of the As： consequently A necessarily
belongs to some of the Bs。 But this is impossible。' The argument
cannot be admitted； for it does not follow that some A is
necessarily B； if it is not possible that no A should be B。 For the
latter expression is used in two senses； one if A some is
necessarily B； another if some A is necessarily not B。 For it is not
true to say that that which necessarily does not belong to some of the
As may possibly