The Complete Aristotle (eng.)

B
belongs to all that to which C belongs, it is necessary that A
should belong to all that to which C belongs, and this cannot be
false: for then the same thing will belong and not belong at the
same time. So A is posited as one thing, being two premisses taken
together. The same holds good of negative syllogisms: it is not
possible to prove a false conclusion from true premisses.
    But from what is false a true conclusion may be drawn, whether
both the premisses are false or only one, provided that this is not
either of the premisses indifferently, if it is taken as wholly
false: but if the premiss is not taken as wholly false, it does not
matter which of the two is false. (1) Let A belong to the whole of
C, but to none of the Bs, neither let B belong to C. This is
possible, e.g. animal belongs to no stone, nor stone to any man. If
then A is taken to belong to all B and B to all C, A will belong to
all C; consequently though both the premisses are false the
conclusion is true: for every man is an animal. Similarly with the
negative. For it is possible that neither A nor B should belong to
any C, although A belongs to all B, e.g. if the same terms are
taken and man is put as middle: for neither animal nor man belongs
to any stone, but animal belongs to every man. Consequently if one
term is taken to belong to none of that to which it does belong,
and the other term is taken to 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 premiss BC is wholly false, a true syllogism will be possible:
for nothing prevents A belonging to all B and to all C, though B
belongs to no C, e.g. these being species of the same genus which
are not subordinate one to the other: for animal belongs both to
horse and to man, but horse to no man. If then it is assumed that A
belongs to all B and B to all C, the conclusion will be true,
although the premiss BC is wholly false. Similarly if the premiss
AB is negative. For it is possible that A should belong neither to
any B nor to any C, and that B should not belong to any C, e.g. a
genus to species of another genus: for animal belongs neither to
music nor to the art of healing, nor does music belong to the art
of healing. If then it is assumed that A belongs to no B, and B to
all C, the conclusion will be true.
    (6) And if the premiss BC is not wholly false but in part only,
even so the conclusion may be true. For nothing prevents A
belonging to the whole of B and of C, while B belongs