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assumed that A belongs to no B； and B to some C； then A does not
belong to some C。 The conclusion then is true； but the premisses arc
false。
3
In the middle figure it is possible in every way to reach a true
conclusion through false premisses； whether the syllogisms are
universal or particular； viz。 when both premisses are wholly false；
when each is partially false； when one is true； the other wholly false
（it does not matter which of the two premisses is false）； if both
premisses are partially false； if one is quite true； the other
partially false； if one is wholly false； the other partially true。 For
（1） if A belongs to no B and to all C； e。g。 animal to no stone and
to every horse； then if the premisses are stated contrariwise and it
is assumed that A belongs to all B and to no C； though the premisses
are wholly false they will yield a true conclusion。 Similarly if A
belongs to all B and to no C： for we shall have the same syllogism。
（2） Again if one premiss is wholly false； the other wholly true： for
nothing prevents A belonging to all B and to all C； though B belongs
to no C； e。g。 a genus to its co…ordinate species。 For animal belongs
to every horse and man； and no man is a horse。 If then it is assumed
that animal belongs to all of the one； and none of the other； the
one premiss will be wholly false； the other wholly true； and the
conclusion will be true whichever term the negative statement
concerns。
（3） Also if one premiss is partially false； the other wholly true。
For it is possible that A should belong to some B and to all C； though
B belongs to no C； e。g。 animal to some white things and to every
raven； though white belongs to no raven。 If then it is assumed that
A belongs to no B； but to the whole of C； the premiss AB is
partially false； the premiss AC wholly true； and the conclusion
true。 Similarly if the negative statement is transposed： the proof can
be made by means of the same terms。 Also if the affirmative premiss is
partially false； the negative wholly true； a true conclusion is
possible。 For nothing prevents A belonging to some B； but not to C
as a whole； while B belongs to no C； e。g。 animal belongs to some white
things； but to no pitch； and white belongs to no pitch。 Consequently
if it is assumed that A belongs to the whole of B； but to no C； the
premiss AB is partially false； the premiss AC is wholly true； and
the conclusion is true。
（4） And if both the premisses are partially false； the conclusion
may be true。 For it is possible that A should belong to some B and
to some C； and B to no C； e。g。 animal to some white things and to some
black things； though white belongs to nothing black。 If then it is
assumed that A belongs to all B and to no C； both premisses are
partially false； but the conclusion is true。 Similarly； if the
negative premiss is transposed； the proof can be made by means of
the same terms。
It is clear also that our thesis holds in particular syllogisms。 For
（5） nothing prevents A belonging to all B and to some C； though B does
not belong to some C； e。g。 animal to every man and to some white
things； though man will not belong to some white things。 If then it is
stated that A belongs to no B and to some C； the universal premiss
is wholly false； the particular premiss is true； and the conclusion is
true。 Similarly if the premiss AB is affirmative： for it is possible
that A should belong to no B； and not to some C； though B does not
belong to some C； e。g。 animal belongs to nothing lifeless； and does
not belong to some white things； and lifeless will not belong to
some white things。 If then it is stated that A belongs to all B and
not to some C； the premiss AB which is universal is wholly false；
the premiss AC is true； and the conclusion is true。 Also a true
conclusion is possible when the universal premiss is true； and the
particular is false。 For nothing prevents A following neither B nor
C at all； while B does not belong to some C； e。g。 animal belongs to no
number nor to anything lifeless； and number does not follow some
lifeless things。 If then it is stated that A belongs to no B and to
some C； the conclusion will be true； and the universal premiss true；
but the particular false。 Similarly if the premiss which is stated
universally is affirmative。 For it is possible that should A belong
both to B and to C as wholes； though B does not follow some C； e。g。
a genus 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 whic