The Complete Aristotle (eng.)

in
relation to its species and difference: for animal follows every
man and footed things as a whole, but man does not follow every
footed thing. Consequently if it is assumed that A belongs to the
whole of B, but does not belong to some C, the universal premiss is
true, the particular false, and the conclusion true.
    (6) It is clear too that though both premisses are false they
may yield a true conclusion, since it is possible that A should
belong both to B and to C as wholes, though B does not follow some
C. For if it is assumed that A belongs to no B and to some C, the
premisses are both false, but the conclusion is true. Similarly if
the universal premiss is affirmative and the particular negative.
For it is possible that A should follow no B and all C, though B
does not belong to some C, e.g. animal follows no science but every
man, though science does not follow every man. If then A is assumed
to belong to the whole of B, and not to follow some C, the
premisses are false but the conclusion is true.
4
    In the last figure a true conclusion may come through what is
false, alike when both premisses are wholly false, when each is
partly false, when one premiss is wholly true, the other false,
when one premiss is partly false, the other wholly true, and vice
versa, and in every other way in which it is possible to alter the
premisses. For (1) nothing prevents neither A nor B from belonging
to any C, while A belongs to some B, e.g. neither man nor footed
follows anything lifeless, though man belongs to some footed
things. If then it is assumed that A and B belong to all C, the
premisses will be wholly false, but the conclusion true. Similarly
if one premiss is negative, the other affirmative. For it is
possible that B should belong to no C, but A to all C, and that
should not belong to some B, e.g. black belongs to no swan, animal
to every swan, and animal not to everything black. Consequently if
it is assumed that B belongs to all C, and A to no C, A will not
belong to some B: and the conclusion is true, though the premisses
are false.
    (2) Also if each premiss is partly false, the conclusion may be
true. For nothing prevents both A and B from belonging to some C
while A belongs to some B, e.g. white and beautiful belong to some
animals, and white to some beautiful things. If then it is stated
that A and B belong to all C, the premisses are partially false,
but the conclusion is true. Similarly if the premiss AC is stated
as negative. For nothing prevents A from not belonging, and B from
belonging, to some C, while A does not belong to all B, e.g. white
does not belong to some animals, beautiful belongs to some animals,
and white does not belong to everything beautiful. Consequently if
it is assumed that A belongs to no C, and B to all C, both
premisses are partly false, but the conclusion is true.
    (3) Similarly if one of the premisses assumed is wholly false,
the other wholly true. For it is possible that both A and B should
follow all C, though A does not belong to some B, e.g. animal and
white follow every swan, though animal does not belong to
everything white. Taking these then as terms, if one assumes that B
belongs to the whole of C, but A does not belong to C at all, the
premiss BC will be wholly true, the premiss AC wholly false, and
the conclusion true. Similarly if the statement BC is false, the
statement AC true, the conclusion may be true. The same terms will
serve for the proof. Also if both the premisses assumed are
affirmative, the conclusion may be true. For nothing prevents B
from following all C, and A from not belonging to C at all, though
A belongs to some B, e.g. animal belongs to every swan, black to no
swan, and black to some animals. Consequently if it is assumed that
A and B belong to every C, the premiss BC is wholly true, the
premiss AC is wholly false, and the conclusion is true. Similarly
if the premiss AC which is assumed is true: the proof can be made
through the same terms.
    (4) Again if one premiss is wholly true, the other partly false,
the conclusion may be true. For it is possible that B should belong
to all C, and A to some C, while A belongs to some B, e.g. biped
belongs to every man, beautiful not to every man, and beautiful to
some bipeds. If then it is assumed that both A and B belong to the
whole of C, the premiss BC is wholly true, the premiss AC partly
false, the conclusion true. Similarly if of the premisses assumed
AC is true and BC partly false, a true conclusion is