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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 the first
syllogism can be assumed in the second： for if both of them are
taken the same conclusion as before will result： but it must be
different。 If the terms are not convertible； one of the premisses from
which the syllogism results must be undemonstrated： for it is not
possible to demonstrate through these terms that the third belongs
to the middle or the middle to the first。 If the terms are
convertible； it is possible to demonstrate everything reciprocally；
e。g。 if A and B and C are convertible with one another。 Suppose the
proposition AC has been demonstrated through B as middle term； and
again the proposition AB through the conclusion and the premiss BC
converted； and similarly the proposition BC through the conclusion and
the premiss AB converted。 But it is necessary to prove both the
premiss CB； and the premiss BA： for we have used these alone without
demonstrating them。 If then it is assumed that B belongs to all C； and
C to all A； we shall have a syllogism relating B to A。 Again if it
is assumed that C belongs to all A； and A to all B； C must belong to
all B。 In both these syllogisms the premiss CA has been assumed
without being demonstrated： the other premisses had ex hypothesi
been proved。 Consequently if we succeed in demonstrating this premiss；
all the premisses will have been proved reciprocally。 If then it is
assumed that C belongs to all B； and B to all A； both the premisses
assumed have been proved； and C must belong to A。 It is clear then
that only if the terms are convertible is circular and reciprocal
demonstration possible （if the terms are not convertible； the matter
stands as we said above）。 But it turns out in these also that we use
for the demonstration the very thing that is being proved： for C is
proved of B； and B of by assuming that C is said of and C is proved of
A through these premisses； so that we use the conclusion for the
demonstration。
In negative syllogisms reciprocal proof is as follows。 Let B
belong to all C； and A to none of the Bs： we conclude that A belongs
to none of the Cs。 If again it is necessary to prove that A belongs to
none of the Bs （which was previously assumed） A must belong to no C；
and C to all B： thus the previous premiss is reversed。 If it is
necessary to prove that B belongs to C； the proposition AB must no
longer be converted as before： for the premiss 'B belongs to no A'
is identical with the premiss 'A belongs to no B'。 But we must
assume that B belongs to all of that to none of which longs。 Let A
belong to none of the Cs （which was the previous conclusion） and
assume that B belongs to all of that to none of which A belongs。 It is
necessary then that B should belong to all C。 Consequently each of the
three propositions has been made a conclusion； and this is circular
demonstration； to assume the conclusion and the converse of one of the
premisses； and deduce the remaining premiss。
In particular syllogisms it is not possible to demonstrate the
universal premiss through the other propositions； but the particular
premiss can be demonstrated。 Clearly it is impossible to demonstrate
the universal premiss： for what is universal is proved through
propositions which are universal； but the conclusion is not universal；
and the proof must start from the conclusion and the other premiss。
Further a syllogism cannot be made at all if the other premiss is
converted： for the result is that both premisses are particular。 But
the particular premiss may be proved。 Suppose that A has been proved
of some C through B。 If then it is assumed that B belongs to all A and
the conclusion is retained； B will belong to some C： for we obtain the
first figure and A is middle。 But i