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all C； and A to no C； A will not belong to some B： and the
conclusion is true； though the premisses are false。
（2） Also if each premiss is partly false； the conclusion may be
true。 For nothing prevents both A and B from belonging to some C while
A belongs to some B； e。g。 white and beautiful belong to some
animals； and white to some beautiful things。 If then it is stated that
A and B belong to all C； the premisses are partially false； but the
conclusion is true。 Similarly if the premiss AC is stated as negative。
For nothing prevents A from not belonging； and B from belonging； to
some C； while A does not belong to all B； e。g。 white does not belong
to some animals； beautiful belongs to some animals； and white does not
belong to everything beautiful。 Consequently if it is assumed that A
belongs to no C； and B to all C； both premisses are partly false；
but the conclusion is true。
（3） Similarly if one of the premisses assumed is wholly false； the
other wholly true。 For it is possible that both A and B should
follow all C； though A does not belong to some B； e。g。 animal and
white follow every swan； though animal does not belong to everything
white。 Taking these then as terms； if one assumes that B belongs to
the whole of C； but A does not belong to C at all； the premiss BC will
be wholly true； the premiss AC wholly false； and the conclusion
true。 Similarly if the statement BC is false； the statement AC true；
the conclusion may be true。 The same terms will serve for the proof。
Also if both the premisses assumed are affirmative； the conclusion may
be true。 For nothing prevents B from following all C； and A from not
belonging to C at all； though A belongs to some B； e。g。 animal belongs
to every swan； black to no swan； and black to some animals。
Consequently if it is assumed that A and B belong to every C； the
premiss BC is wholly true； the premiss AC is wholly false； and the
conclusion is true。 Similarly if the premiss AC which is assumed is
true： the proof can be made through the same terms。
（4） Again if one premiss is wholly true； the other partly false； the
conclusion may be true。 For it is possible that B should belong to all
C； and A to some C； while A belongs to some B； e。g。 biped belongs to
every man； beautiful not to every man； and beautiful to some bipeds。
If then it is assumed that both A and B belong to the whole of C；
the premiss BC is wholly true； the premiss AC partly false； the
conclusion true。 Similarly if of the premisses assumed AC is true
and BC partly false； a true conclusion is possible： this can be
proved； if the same terms as before are transposed。 Also the
conclusion may be true if one premiss is negative； the other
affirmative。 For since it is possible that B should belong to the
whole of C； and A to some C； and； when they are so； that A should
not belong to all B； therefore it is assumed that B belongs to the
whole of C； and A to no C； the negative premiss is partly false； the
other premiss wholly true； and the conclusion is true。 Again since
it has been proved that if A belongs to no C and B to some C； it is
possible that A should not belong to some C； it is clear that if the
premiss AC is wholly true； and the premiss BC partly false； it is
possible that the conclusion should be true。 For if it is assumed that
A belongs to no C； and B to all C； the premiss AC is wholly true；
and the premiss BC is partly false。
（5） It is clear also in the case of particular syllogisms that a
true conclusion may come through what is false； in every possible way。
For the same terms must be taken as have been taken when the premisses
are universal； positive terms in positive syllogisms； negative terms
in negative。 For it makes no difference to the setting out of the
terms； whether one assumes that what belongs to none belongs to all or
that what belongs to some belongs to all。 The same applies to negative
statements。
It is clear then that if the conclusion is false； the premisses of
the argument must be false； either all or some of them； but when the
conclusion is true； it is not necessary that the premisses should be
true； either one or all； yet it is possible； though no part of the
syllogism is true； that the conclusion may none the less be true；
but it is not necessitated。 The reason is that when two things are
so related to one another； that if the one is； the other necessarily
is； then if the latter is not； the former will not be either； but if
the latter is； it is not necessary that the former should be。 But it
is impossible that the same thing should be necessitated by the
being and by the not…being of the same thing。 I mean； for example；
that it is impossible that B should necessarily be great since A is
white and that B should necessarily be great since A is not white。 For
whenever since this； A； is white it is necessary that that； B；
should be great； and since B is great that C should not be white； then
it is necessary if is white that C should not be white。 And whenever
it is necessary； since one of two things is； that the other should be；
it is necessary； if the latter is not； that the former （viz。 A） should
not be。 If then B is not great A cannot be white。 But if； when A is
not white； it is necessary that B should be great； it necessarily
results that if B is not great； B itself is great。 （But this is
impossible。） For if B is not great； A will necessarily not be white。
If then when this is not white B must be great； it results that if B
is not great； it is great； just as if it were proved through three
terms。
5
Circular and reciprocal proof means proof by means of the
conclusion； i。e。 by converting one of the premisses simply and
inferring the premiss which was assumed in the original syllogism：
e。g。 suppose it has been necessary to prove that A belongs to all C；
and it has been proved through B； suppose that A should now be
proved to belong to B by assuming that A belongs to C； and C to B…so A
belongs to B： but in the first syllogism the converse was assumed；
viz。 that B belongs to C。 Or suppose it is necessary to prove that B
belongs to C； and A is assumed to belong to C； which was the
conclusion of the first syllogism； and B to belong to A but the
converse was assumed in the earlier syllogism； viz。 that A belongs
to B。 In no other way is reciprocal proof possible。 If another term is
taken as middle； the proof is not circular： for neither of the
propositions assumed is the same as before： if one of the accepted
terms is taken as middle； only one of the premisses of th