Prior Analytics - Book II
belong to all of that to which it does not belong, though both the
premisses are false the conclusion will be true. (2) A similar proof
may be given if each premiss is partially false.
(3) But if one only of the premisses is false, when the first
premiss is wholly false, e.g. AB, the conclusion will not be true, but
if the premiss BC is wholly false, a true conclusion will be possible.
I mean by 'wholly false' the contrary of the truth, e.g. if what
belongs to none is assumed to belong to all, or if what belongs to all
is assumed to belong to none. Let A belong to no B, and B to all C. If
then the premiss BC which I take is true, and the premiss AB is wholly
false, viz. that A belongs to all B, it is impossible that the
conclusion should be true: for A belonged to none of the Cs, since A
belonged to nothing to which B belonged, and B belonged to all C.
Similarly there cannot be a true conclusion if A belongs to all B, and
B to all C, but while the true premiss BC is assumed, the wholly false
premiss AB is also assumed, viz. that A belongs to nothing to which
B belongs: here the conclusion must be false. For A will belong to all
C, since A belongs to everything to which B belongs, and B to all C.
It is clear then that when the first premiss is wholly false,
whether affirmative or negative, and the other premiss is true, the
conclusion cannot be true.
(4) But if the premiss is not wholly false, a true conclusion is
possible. For if A belongs to all C and to some B, and if B belongs to
all C, e.g. animal to every swan and to some white thing, and white to
every swan, then if we take as premisses that A belongs to all B,
and B to all C, A will belong to all C truly: for every swan is an
animal. Similarly if the statement AB is negative. For it is
possible that A should belong to some B and to no C, and that B should
belong to all C, e.g. animal to some white thing, but to no snow,
and white to all snow. If then one should assume that A belongs to
no B, and B to all C, then will belong to no C.
(5) But if the premiss AB, which is assumed, is wholly true, and the