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