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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 if the syllogism is negative； it
is not possible to prove the universal premiss； for the reason given
above。 But it is possible to prove the particular premiss； if the
proposition AB is converted as in the universal syllogism； i。e 'B
belongs to some of that to some of which A does not belong'： otherwise
no syllogism results because the particular premiss is negative。
6
In the second figure it is not possible to prove an affirmative
proposition in this way； but a negative proposition may be proved。
An affirmative proposition is not proved because both premisses of the
new syllogism are not affirmative （for the conclusion is negative） but
an affirmative proposition is （as we saw） proved from premisses
which are both affirmative。 The negative is proved as follows。 Let A
belong to all B； and to no C： we conclude that B belongs to no C。 If
then it is assumed that B belongs to all A； it is necessary that A
should belong to no C： for we get the second figure； with B as middle。
But if the premiss AB was negative； and the other affirmative； we
shall have the first figure。 For C belongs to all A and B to no C；
consequently B belongs to no A： neither then does A belong to B。
Through the conclusion； therefore； and one premiss； we get no
syllogism； but if another premiss is assumed in addition； a
syllogism will be possible。 But if the syllogism not universal； the
universal premiss cannot be proved； for the same reason as we gave
above； but the particular premiss can be proved whenever the universal
statement is affirmative。 Let A belong to all B； and not to all C： the
conclusion is BC。 If then it is assumed that B belongs to all A； but
not to all C； A will not belong to some C； B being middle。 But if
the universal premiss is negative； the premiss AC will not be
demonstrated by the conversion of AB： for it turns out that either
both or one of the premisses is negative； consequently a syllogism
will not be possible。 But the proof will proceed as in the universal
syllogisms； if it is assumed that A belongs to some of that to some of
which B does not belong。
7
In the third figure； when both premisses are taken universally； it
is not possible to prove them reciprocally： for that which is
universal is proved through statements which are universal； but the
conclusion in this figure is always particular； so that it is clear
that it is not possible at all to prove through this figure the
universal premiss。 But if one premiss is universal； the other
particular； proof of the latter will sometimes be possible；
sometimes not。 When both the premisses assumed are affirmative； and
the universal concerns the minor extreme； proof will be possible；
but when it concerns the other extreme； impossible。 Let A b