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always necessary。 Let A be 'moving'； B 'science'； C 'man'。 A then will
belong to no B； but B is possible for all C。 And the conclusion will
not be necessary。 For it is not necessary that no man should move；
rather it is not necessary that any man should move。 Clearly then
the conclusion establishes that one term does not necessarily belong
to any instance of another term。 But we must take our terms better。
If the minor premiss is negative and indicates possibility； from the
actual premisses taken there can be no syllogism； but if the
problematic premiss is converted； a syllogism will be possible； as
before。 Let A belong to all B； and let B possibly belong to no C。 If
the terms are arranged thus； nothing necessarily follows： but if the
proposition BC is converted and it is assumed that B is possible for
all C； a syllogism results as before： for the terms are in the same
relative positions。 Likewise if both the relations are negative； if
the major premiss states that A does not belong to B； and the minor
premiss indicates that B may possibly belong to no C。 Through the
premisses actually taken nothing necessary results in any way； but
if the problematic premiss is converted； we shall have a syllogism。
Suppose that A belongs to no B； and B may possibly belong to no C。
Through these comes nothing necessary。 But if B is assumed to be
possible for all C （and this is true） and if the premiss AB remains as
before； we shall again have the same syllogism。 But if it be assumed
that B does not belong to any C； instead of possibly not belonging；
there cannot be a syllogism anyhow； whether the premiss AB is negative
or affirmative。 As common instances of a necessary and positive
relation we may take the terms white…animal…snow： of a necessary and
negative relation； white…animal…pitch。 Clearly then if the terms are
universal； and one of the premisses is assertoric； the other
problematic； whenever the minor premiss is problematic a syllogism
always results； only sometimes it results from the premisses that
are taken； sometimes it requires the conversion of one premiss。 We
have stated when each of these happens and the reason why。 But if
one of the relations is universal； the other particular； then whenever
the major premiss is universal and problematic； whether affirmative or
negative； and the particular is affirmative and assertoric； there will
be a perfect syllogism； just as when the terms are universal。 The
demonstration is the same as before。 But whenever the major premiss is
universal； but assertoric； not problematic； and the minor is
particular and problematic； whether both premisses are negative or
affirmative； or one is negative； the other affirmative； in all cases
there will be an imperfect syllogism。 Only some of them will be proved
per impossibile； others by the conversion of the problematic
premiss； as has been shown above。 And a syllogism will be possible
by means of conversion when the major premiss is universal and
assertoric； whether positive or negative； and the minor particular；
negative； and problematic； e。g。 if A belongs to all B or to no B；
and B may possibly not belong to some C。 For if the premiss BC is
converted in respect of possibility； a syllogism results。 But whenever
the particular premiss is assertoric and negative； there cannot be a
syllogism。 As instances of the positive relation we may take the terms
white…animal…snow； of the negative； white…animal…pitch。 For the
demonstration must be made through the indefinite nature of the
particular premiss。 But if the minor premiss is universal； and the
major particular； whether either premiss is negative or affirmative；
problematic or assertoric； nohow is a syllogism possible。 Nor is a
syllogism possible when the premisses are particular or indefinite；
whether problematic or assertoric； or the one problematic； the other
assertoric。 The demonstration is the same as above。 As instances of
the necessary and positive relation we may take the terms
animal…white…man； of the necessary and negative relation；
animal…white…garment。 It is evident then that if the major premiss
is universal； a syllogism always results； but if the minor is
universal nothing at all can ever be proved。
16
Whenever one premiss is necessary； the other problematic； there will
be a syllogism when the terms are related as before； and a perfect
syllogism when the minor premiss is necessary。 If the premisses are
affirmative the conclusion will be problematic； not assertoric；
whether the premisses are universal or not： but if one is affirmative；
the other negative； when the affirmative is necessary the conclusion
will be problematic； not negative assertoric； but when the negative is
necessary the conclusion will be problematic negative； and
assertoric negative； whether the premisses are universal or not。
Possibility in the conclusion must be understood in the same manner as
before。 There cannot be an inference to the necessary negative
proposition： for 'not necessarily to belong' is different from
'necessarily not to belong'。
If the premisses are affirmative； clearly the conclusion which
follows is not necessary。 Suppose A necessarily belongs to all B；
and let B be possible for all C。 We shall have an imperfect
syllogism to prove that A may belong to all C。 That it is imperfect is
clear from the proof： for it will be proved in the same manner as
above。 Again； let A be possible for all B； and let B necessarily
belong to all C。 We shall then have a syllogism to prove that A may
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