
Prior Analytics  Book II
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
