
Prior Analytics  Book II
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
