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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 not belong to any A； just as it is not true to say
that what necessarily belongs to some A may possibly belong to all
A。 If any one then should claim that because it is not possible for
C to belong to all D； it necessarily does not belong to some D； he
would make a false assumption： for it does belong to all D； but
because in some cases it belongs necessarily； therefore we say that it
is not possible for it to belong to all。 Hence both the propositions
'A necessarily belongs to some B' and 'A necessarily does not belong
to some B' are opposed to the proposition 'A belongs to all B'。
Similarly also they are opposed to the proposition 'A may belong to no
B'。 It is clear then that in relation to what is possible and not
possible； in the sense originally defined； we must assume； not that
A necessarily belongs to some B； but that A necessarily does not
belong to some B。 But if this is assumed； no absurdity results：
consequently no syllogism。 It is clear from what has been said that
the negative proposition is not convertible。
This being proved； suppose it possible that A may belong to no B and
to all C。 By means of conversion no syllogism will result： for the
major premiss； as has been said； is not convertible。 Nor can a proof
be obtained by a reductio ad absurdum： for if it is assumed that B can
belong to all C； no false consequence results： for A may belong both
to all C and to no C。 In general； if there is a syllogism； it is clear
that its conclusion will be problematic because neither of the
premisses is assertoric； and this must be either affirmative or
negative。 But neither is possible。 Suppose the conclusion is
affirmative： it will be proved by an example that the predicate cannot
belong to the subject。 Suppose the conclusion is negative： it will
be proved that it is not problematic but necessary。 Let A be white；
B man； C horse。 It is possible then for A to belong to all of the
one and to none of the other。 But it is not possible for B to belong
nor not to belong to C。 That it is not possible for it to belong； is
clear。 For no horse is a man。 Neither is it possible for it not to
belong。 For it is necessary that no horse should be a man； but the
necessary we found to be different from the possible。 No syllogism
then results。 A similar proof can be given if the major premiss is
negative； the minor affirmative； or if both are affirmative or
negative。 The demonstration can be made by means of the same terms。
And whenever one premiss is universal； the other particular； or both
are particular or indefinite； or in whatever other way the premisses
can be altered； the proof will always proceed through the same
terms。 Clearly then； if both the premisses are problematic； no
syllogism results。