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some B； it is necessary that A should belong to some B also。 For B
is under C。 The first figure then is formed。 A similar proof will be
given also if BC is necessary。 For C is convertible with some A：
consequently if B belongs necessarily to all C； it will belong
necessarily also to some A。
Again let AC be negative； BC affirmative； and let the negative
premiss be necessary。 Since then C is convertible with some B； but A
necessarily belongs to no C； A will necessarily not belong to some B
either： for B is under C。 But if the affirmative is necessary； the
conclusion will not be necessary。 For suppose BC is affirmative and
necessary； while AC is negative and not necessary。 Since then the
affirmative is convertible； C also will belong to some B
necessarily： consequently if A belongs to none of the Cs； while C
belongs to some of the Bs； A will not belong to some of the Bs…but not
of necessity； for it has been proved； in the case of the first figure；
that if the negative premiss is not necessary； neither will the
conclusion be necessary。 Further； the point may be made clear by
considering the terms。 Let the term A be 'good'； let that which B
signifies be 'animal'； let the term C be 'horse'。 It is possible
then that the term good should belong to no horse； and it is necessary
that the term animal should belong to every horse： but it is not
necessary that some animal should not be good； since it is possible
for every animal to be good。 Or if that is not possible； take as the
term 'awake' or 'asleep'： for every animal can accept these。
If； then； the premisses are universal； we have stated when the
conclusion will be necessary。 But if one premiss is universal； the
other particular； and if both are affirmative； whenever the
universal is necessary the conclusion also must be necessary。 The
demonstration is the same as before； for the particular affirmative
also is convertible。 If then it is necessary that B should belong to
all C； and A falls under C； it is necessary that B should belong to
some A。 But if B must belong to some A； then A must belong to some
B： for conversion is possible。 Similarly also if AC should be
necessary and universal： for B falls under C。 But if the particular
premiss is necessary； the conclusion will not be necessary。 Let the
premiss BC be both particular and necessary； and let A belong to all
C； not however necessarily。 If the proposition BC is converted the
first figure is formed； and the universal premiss is not necessary；
but the particular is necessary。 But when the premisses were thus； the
conclusion （as we proved was not necessary： consequently it is not
here either。 Further； the point is clear if we look at the terms。
Let A be waking； B biped； and C animal。 It is necessary that B
should belong to some C； but it is possible for A to belong to C；
and that A should belong to B is not necessary。 For there is no
necessity that some biped should be asleep or awake。 Similarly and
by means of the same terms proof can be made； should the proposition
AC be both particular and necessary。
But if one premiss is affirmative； the other negative； whenever
the universal is both negative and necessary the conclusion also
will be necessary。 For if it is not possible that A should belong to
any C； but B belongs to some C； it is necessary that A should not
belong to some B。 But whenever the affirmative proposition is
necessary； whether universal or particular； or the negative is
particular； the conclusion will not be necessary。 The proof of this by
reduction will be the same as before； but if terms are wanted； when
the universal affirmative is necessary； take the terms
'waking'…'animal'…'man'； 'man' being middle； and when the
affirmative is particular and necessary； take the terms
'waking'…'animal'…'white'： for it is necessary that animal should
belong to some white thing； but it is possible that waking should
belong to none； and it is not necessary that waking should not
belong to some animal。 But when the negative proposition being
particular is necessary； take the terms 'biped'； 'moving'； 'animal'；
'animal' being middle。
12
It is clear then that a simple conclusion is not reached unless both
premisses are simple assertions； but a necessary conclusion is
possible although one only of the premisses is necessary。 But in
both cases； whether the syllogisms are affirmative or negative； it
is necessary that one premiss should be similar to the conclusion。 I
mean by 'similar'； if the conclusion is a simple assertion； the
premiss must be simple； if the conclusion is necessary； the premiss
must be necessary。 Consequently this also is clear； that the
conclusion will be neither necessary nor simple unless a necessary
or simple premiss is assumed。
13
Perhaps enough has been said about the proof of necessity； how it
comes about and how it differs from the proof of a simple statement。
We proceed to discuss that which is possible； when and how and by what
means it can be proved。 I use the terms 'to be possible' and 'the
possible' of that which is not necessary but； being assumed； results
in nothing impossible。 We say indeed ambiguously of the necessary that
it is possible。 But that my definition of the possible is correct is
clear from the phrases by which we deny or on the contrary affirm
possibility。 For the expressions 'it is not possible to belong'； 'it
is impossible to belong'； and 'it is necessary not to belong' are
either identical or follow from one another； consequently their
opposites also； 'it is possible to belong'； 'it is not impossible to
belong'； and 'it is not necessary not to belong'； will either be
identical or follow from one another。 For of everything the
affirmation or the denial holds good。 That which is possible then will
be not necessary and that which is not necessary will be possible。
It results that all premisses in the mode of possibility are
convertible into one another。 I mean not that the affirmative are
convertible into the negative； but that those which are affirmative in
form admit of conversion by opposition； e。g。 'it is possible to
belong' may be converted into 'it is possible not to belong'； and
'it is possible for A to belong to all B' into 'it is possible for A
to belong to no B' or 'not to all B'； and 'it is possible for A to
belong to some B' into 'it is possible for A not to belong to some B'。
And similarly the other propositions in this mode can be converted。
For since that which is possible is not necessa