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first figure； and the conclusion that A may possibly belong to some of
the Bs。 For when one of the premisses in the first figure is
problematic； the conclusion also （as we saw） is problematic。 Similarly
if the proposition BC is pure； AC problematic； or if AC is negative；
BC affirmative； no matter which of the two is pure； in both cases
the conclusion will be problematic： for the first figure is obtained
once more； and it has been proved that if one premiss is problematic
in that figure the conclusion also will be problematic。 But if the
minor premiss BC is negative； or if both premisses are negative； no
syllogistic conclusion can be drawn from the premisses as they
stand； but if they are converted a syllogism is obtained as before。
If one of the premisses is universal； the other particular； then
when both are affirmative； or when the universal is negative； the
particular affirmative； we shall have the same sort of syllogisms： for
all are completed by means of the first figure。 So it is clear that we
shall have not a pure but a problematic syllogistic conclusion。 But if
the affirmative premiss is universal； the negative particular； the
proof will proceed by a reductio ad impossibile。 Suppose that B
belongs to all C； and A may possibly not belong to some C： it
follows that may possibly not belong to some B。 For if A necessarily
belongs to all B； and B （as has been assumed） belongs to all C； A will
necessarily belong to all C： for this has been proved before。 But it
was assumed at the outset that A may possibly not belong to some C。
Whenever both premisses are indefinite or particular； no syllogism
will be possible。 The demonstration is the same as was given in the
case of universal premisses； and proceeds by means of the same terms。
22
If one of the premisses is necessary； the other problematic； when
the premisses are affirmative a problematic affirmative conclusion can
always be drawn； when one proposition is affirmative； the other
negative； if the affirmative is necessary a problematic negative can
be inferred； but if the negative proposition is necessary both a
problematic and a pure negative conclusion are possible。 But a
necessary negative conclusion will not be possible； any more than in
the other figures。 Suppose first that the premisses are affirmative；
i。e。 that A necessarily belongs to all C； and B may possibly belong to
all C。 Since then A must belong to all C； and C may belong to some
B； it follows that A may （not does） belong to some B： for so it
resulted in the first figure。 A similar proof may be given if the
proposition BC is necessary； and AC is problematic。 Again suppose
one proposition is affirmative； the other negative； the affirmative
being necessary： i。e。 suppose A may possibly belong to no C； but B
necessarily belongs to all C。 We shall have the first figure once
more： and…since the negative premiss is problematic…it is clear that
the conclusion will be problematic： for when the premisses stand
thus in the first figure； the conclusion （as we found） is problematic。
But if the negative premiss is necessary； the conclusion will be not
only that A may possibly not belong to some B but also that it does
not belong to some B。 For suppose that A necessarily does not belong
to C； but B may belong to all C。 If the affirmative proposition BC
is converted； we shall have the first figure； and the negative premiss
is necessary。 But when the premisses stood thus； it resulted that A
might possibly not belong to some C； and that it did not belong to
some C； consequently here it follows that A does not belong to some B。
But when the minor premiss is negative； if it is problematic we
shall have a syllogism by altering the premiss into its
complementary affirmative； as before； but if it is necessary no
syllogism can be formed。 For A sometimes necessarily belongs to all B；
and sometimes cannot possibly belong to any B。 To illustrate the
former take the terms sleep…sleeping horse…man； to illustrate the
latter take the terms sleep…waking horse…man。
Similar results will obtain if one of the terms is related
universally to the middle； the other in part。 If both premisses are
affirmative； the conclusion will be problematic； not pure； and also
when one premiss is negative； the other affirmative； the latter
being necessary。 But when the negative premiss is necessary； the
conclusion also will be a pure negative proposition； for the same kind
of proof can be given whether the terms are universal or not。 For
the syllogisms must be made perfect by means of the first figure； so
that a result which follows in the first figure follows also in the
third。 But when the minor premiss is negative and universal； if it
is problematic a syllogism can be formed by means of conversion； but
if it is necessary a syllogism is not possible。 The proof will
follow the same course as where the premisses are universal； and the
same terms may be used。
It is clear then in this figure also when and how a syllogism can be
formed； and when the conclusion is problematic； and when it is pure。
It is evident also that all syllogisms in this figure are imperfect；
and that they are made perfect by means of the first figure。
23
It is clear from what has been said that the syllogisms in these
figures are made perfect by means of universal syllogisms in the first
figure and are reduced to them。 That every syllogism without
qualification can be so treated； will be clear presently； when it
has been proved that every syllogism is formed through one or other of
these figures。
It is necessary that every demonstration and every syllogism
should prove either that something belongs or that it does not； and
this either universally or in part； and further either ostensively
or hypothetically。 One sort of hypothetical proof is the reductio ad
impossibile。 Let us speak first of ostensive syllogisms： for after
these have been pointed out the truth of our contention will be
clear with regard to those which are proved per impossibile； and in
general hypothetically。
If then one wants to prove syllogistically A of B； either as an
attribute of it or as not an attribute of it； one must assert
something of something else。 If now A should be asserted of B； the
proposition originally in question will have been assumed。 But if A
should be asserted of C； but C should not be asserted of anything； nor
anything of it； nor anything else of A； no syllogism will be possible。
For nothing necessarily fol