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they stand； but if the problematic premiss is converted into its
complementary affirmative a syllogism is formed to prove that B may
belong to no C； as before： for we shall again have the first figure。
But if both premisses are affirmative； no syllogism will be
possible。 This arrangement of terms is possible both when the relation
is positive； e。g。 health； animal； man； and when it is negative； e。g。
health； horse； man。
The same will hold good if the syllogisms are particular。 Whenever
the affirmative proposition is assertoric； whether universal or
particular； no syllogism is possible （this is proved similarly and
by the same examples as above）； but when the negative proposition is
assertoric； a conclusion can be drawn by means of conversion； as
before。 Again if both the relations are negative； and the assertoric
proposition is universal； although no conclusion follows from the
actual premisses； a syllogism can be obtained by converting the
problematic premiss into its complementary affirmative as before。
But if the negative proposition is assertoric； but particular； no
syllogism is possible； whether the other premiss is affirmative or
negative。 Nor can a conclusion be drawn when both premisses are
indefinite； whether affirmative or negative； or particular。 The
proof is the same and by the same terms。
19
If one of the premisses is necessary； the other problematic； then if
the negative is necessary a syllogistic conclusion can be drawn； not
merely a negative problematic but also a negative assertoric
conclusion； but if the affirmative premiss is necessary； no conclusion
is possible。 Suppose that A necessarily belongs to no B； but may
belong to all C。 If the negative premiss is converted B will belong to
no A： but A ex hypothesi is capable of belonging to all C： so once
more a conclusion is drawn by the first figure that B may belong to no
C。 But at the same time it is clear that B will not belong to any C。
For assume that it does： then if A cannot belong to any B； and B
belongs to some of the Cs； A cannot belong to some of the Cs： but ex
hypothesi it may belong to all。 A similar proof can be given if the
minor premiss is negative。 Again let the affirmative proposition be
necessary； and the other problematic； i。e。 suppose that A may belong
to no B； but necessarily belongs to all C。 When the terms are arranged
in this way； no syllogism is possible。 For （1） it sometimes turns
out that B necessarily does not belong to C。 Let A be white； B man；
C swan。 White then necessarily belongs to swan； but may belong to no
man； and man necessarily belongs to no swan； Clearly then we cannot
draw a problematic conclusion； for that which is necessary is
admittedly distinct from that which is possible。 （2） Nor again can
we draw a necessary conclusion： for that presupposes that both
premisses are necessary； or at any rate the negative premiss。 （3）
Further it is possible also； when the terms are so arranged； that B
should belong to C： for nothing prevents C falling under B； A being
possible for all B； and necessarily belonging to C； e。g。 if C stands
for 'awake'； B for 'animal'； A for 'motion'。 For motion necessarily
belongs to what is awake； and is possible for every animal： and
everything that is awake is animal。 Clearly then the conclusion cannot
be the negative assertion； if the relation must be positive when the
terms are related as above。 Nor can the opposite affirmations be
established： consequently no syllogism is possible。 A similar proof is
possible if the major premiss is affirmative。
But if the premisses are similar in quality； when they are
negative a syllogism can always be formed by converting the
problematic premiss into its complementary affirmative as before。
Suppose A necessarily does not belong to B； and possibly may not
belong to C： if the premisses are converted B belongs to no A； and A
may possibly belong to all C： thus we have the first figure。 Similarly
if the minor premiss is negative。 But if the premisses are affirmative
there cannot be a syllogism。 Clearly the conclusion cannot be a
negative assertoric or a negative necessary proposition because no
negative premiss has been laid down either in the assertoric or in the
necessary mode。 Nor can the conclusion be a problematic negative
proposition。 For if the terms are so related； there are cases in which
B necessarily will not belong to C； e。g。 suppose that A is white； B
swan； C man。 Nor can the opposite affirmations be established； since
we have shown a case in which B necessarily does not belong to C。 A
syllogism then is not possible at all。
Similar relations will obtain in particular syllogisms。 For whenever
the negative proposition is universal and necessary； a syllogism
will always be possible to prove both a problematic and a negative
assertoric proposition （the proof proceeds by conversion）； but when
the affirmative proposition is universal and necessary； no syllogistic
conclusion can be drawn。 This can be proved in the same way as for
universal propositions； and by the same terms。 Nor is a syllogistic
conclusion possible when both premisses are affirmative： this also may
be proved as above。 But when both premisses are negative； and the
premiss that definitely disconnects two terms is universal and
necessary； though nothing follows necessarily from the premisses as
they are stated； a conclusion can be drawn as above if the problematic
premiss is converted into its complementary affirmative。 But if both
are indefinite or particular； no syllogism can be formed。 The same
proof will serve； and the same terms。
It is clear then from what has been said that if the universal and
negative premiss is necessary； a syllogism is always possible； proving
not merely a negative problematic； but also a negative assertoric
proposition； but if the affirmative premiss is necessary no conclusion
can be drawn。 It is clear too that a syllogism is possible or not
under the same conditions whether the mode of the premisses is
assertoric or necessary。 And it is clear that all the syllogisms are
imperfect； and are completed by means of the figures mentioned。
20
In the last figure a syllogism is possible whether both or only
one of the premisses is problematic。 When the premisses are
problematic the conclusion will be problematic； and also when one
premiss is problematic； the other assertoric。 But when the other
premiss is necessary； if it is affirmative the conclusion will be
neither necessary or assertoric； but if it is negative the syllogism
will result in a negative assertoric proposition； as above。 In these
also we must understand the expression 'possible' in the conclusion in
the same way as before。
First let the premisses be problematic and suppose that both A and B
may possibly belong to every C。 Since then the affirmative proposition
is convertible into a particular； and B may possibly belong to every
C； it follows that C may possibly belong to some B。 So； if A is
possible for every C； and C is possible for some of the Bs； then A
is possible for some of the Bs。 For we have got the first figure。
And A if may possibly belong to no C； but B may possibly belong to all
C； it follows that A may possibly not belong to some B： for we shall
have the first figure again by conversion。 But if both premisses
should be negative no necessary consequence will follow from them as
they are stated； but if the premisses are converted into their
corresponding affirmatives there will be a syllogism as before。 For if
A and B may possibly not belong to C； if 'may possibly belong' is
substituted we shall again have the first figure by means of
conversion。 But if one of the premisses is universal； the other
particular； a syllogism will be possible； or not； under the
arrangement of the terms as in the case of assertoric propositions。
Suppose that A may possibly belong to all C； and B to some C。 We shall
have the first figure again if the particular premiss is converted。
For if A is possible for all C； and C for some of the Bs； then A is
possible for some of the Bs。 Similarly if the proposition BC is
universal。 Likewise also if the proposition AC is negative； and the
proposition BC affirmative： for we shall again have the first figure
by conversion。 But if both premisses should be negative…the one
universal and the other particular…although no