The Complete Aristotle (eng.)

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.
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    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