
Prior Analytics  Book II
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 belong to
all C and B to some C: the conclusion is the statement AB. If then
it is assumed that C belongs to all A, it has been proved that C
belongs to some B, but that B belongs to some C has not been proved.
And yet it is necessary, if C belongs to some B, that B should
belong to some C. But it is not the same that this should belong to
that, and that to this: but we must assume besides that if this
belongs to some of that, that belongs to some of this. But if this
is assumed the syllogism no longer results from the conclusion and the
other premiss. But if B belongs to all C, and A to some C, it will
be possible to prove the proposition AC, when it is assumed that C
belongs to all B, and A to some B. For if C belongs to all B and A
to some B, it is necessary that A should belong to some C, B being
middle. And whenever one premiss is affirmative the other negative,
and the affirmative is universal, the other premiss can be proved. Let
B belong to all C, and A not to some C: the conclusion is that A
does not belong to some B. If then it is assumed further that C
belongs to all B, it is necessary that A should not belong to some
C, B being middle. But when the negative premiss is universal, the
other premiss is not except as before, viz. if it is assumed that that
belongs to some of that, to some of which this does not belong, e.g.
