number is taken as middle, and it is assumed that A belongs to no B,
and B to some C, then A will not belong to some C, which ex
hypothesi is true. And the premiss AB is true, the premiss BC false.
(10) Also if the premiss AB is partially false, and the premiss BC
is false too, the conclusion may be true. For nothing prevents A
belonging to some B and to some C, though B belongs to no C, e.g. if B
is the contrary of C, and both are accidents of the same genus: for
animal belongs to some white things and to some black things, but
white belongs to no black thing. If then it is assumed that A
belongs to all B, and B to some C, the conclusion will be true.
Similarly if the premiss AB is negative: for the same terms arranged
in the same way will serve for the proof.
(11) Also though both premisses are false the conclusion may be
true. For it is possible that A may belong to no B and to some C,
while B belongs to no C, e.g. a genus in relation to the species of
another genus, and to the accident of its own species: for animal
belongs to no number, but to some white things, and number to
nothing white. If then it is assumed that A belongs to all B and B
to some C, the conclusion will be true, though both premisses are
false. Similarly also if the premiss AB is negative. For nothing
prevents A belonging to the whole of B, and not to some C, while B
belongs to no C, e.g. animal belongs to every swan, and not to some
black things, and swan belongs to nothing black. Consequently if it is
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
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 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 possible: this can be
proved, if the same terms as before are transposed. Also the
conclusion may be true if one premiss is negative, the other
affirmative. For since it is possible that B should belong to the
whole of C, and A to some C, and, when they are so, that A should
not belong to all B, therefore it is assumed that B belongs to the
whole of C, and A to no C, the negative premiss is partly false, the
other premiss wholly true, and the conclusion is true. Again since
it has been proved that if A belongs to no C and B to some C, it is
possible that A should not belong to some C, it is clear that if the
premiss AC is wholly true, and the premiss BC partly false, it is
possible that the conclusion should be true. For if it is assumed that
A belongs to no C, and B to all C, the premiss AC is wholly true,
and the premiss BC is partly false.
(5) It is clear also in the case of particular syllogisms that a
true conclusion may come through what is false, in every possible way.
For the same terms must be taken as have been taken when the premisses
are universal, positive terms in positive syllogisms, negative terms
in negative. For it makes no difference to the setting out of the
terms, whether one assumes that what belongs to none belongs to all or
that what belongs to some belongs to all. The same applies to negative
statements.
It is clear then that if the conclusion is false, the premisses of
the argument must be false, either all or some of them; but when the
conclusion is true, it is not necessary that the premisses should be
true, either one or all, yet it is possible, though no part of the
syllogism is true, that the conclusion may none the less be true;
but it is not necessitated. The reason is that when two things are
so related to one another, that if the one is, the other necessarily
is, then if the latter is not, the former will not be either, but if
the latter is, it is not necessary that the former should be. But it
is impossible that the same thing should be necessitated by the
being and by the not-being of the same thing. I mean, for example,
that it is impossible that B should necessarily be great since A is
white and that B should necessarily be great since A is not white. For
whenever since this, A, is white it is necessary that that, B,
should be great, and since B is great that C should not be white, then
it is necessary if is white that C should not be white. And whenever
it is necessary, since one of two things is, that the other should be,
it is necessary, if the latter is not, that the former (viz. A) should
not be. If then B is not great A cannot be white. But if, when A is
not white, it is necessary that B should be great, it necessarily
results that if B is not great, B itself is great. (But this is
impossible.) For if B is not great, A will necessarily not be white.
If then when this is not white B must be great, it results that if B
is not great, it is great, just as if it were proved through three
terms.
5
Circular and reciprocal proof means proof by means of the
conclusion, i.e. by converting one of the premisses simply and
inferring the premiss which was assumed in the original syllogism:
e.g. suppose it has been necessary to prove that A belongs to all C,
and it has been proved through B; suppose that A should now be
proved to belong to B by assuming that A belongs to C, and C to B-so A
belongs to B: but in the first syllogism the converse was assumed,
viz. that B belongs to C. Or suppose it
is necessary to prove that B
belongs to C, and A is assumed to belong to C, which was the
conclusion of the first syllogism, and B to belong to A but the
converse was assumed in the earlier syllogism, viz. that A belongs
to B. In no other way is reciprocal proof possible. If another term is
taken as middle, the proof is not circular: for neither of the
propositions assumed is the same as before: if one of the accepted
terms is taken as middle, only one of the premisses of the first
syllogism can be assumed in the second: for if both of them are
taken the same conclusion as before will result: but it must be
different. If the terms are not convertible, one of the premisses from
which the syllogism results must be undemonstrated: for it is not
possible to demonstrate through these terms that the third belongs
to the middle or the middle to the first. If the terms are
convertible, it is possible to demonstrate everything reciprocally,
e.g. if A and B and C are convertible with one another. Suppose the
proposition AC has been demonstrated through B as middle term, and
again the proposition AB through the conclusion and the premiss BC
converted, and similarly the proposition BC through the conclusion and
the premiss AB converted. But it is necessary to prove both the
premiss CB, and the premiss BA: for we have used these alone without
demonstrating them. If then it is assumed that B belongs to all C, and
C to all A, we shall have a syllogism relating B to A. Again if it
is assumed that C belongs to all A, and A to all B, C must belong to
all B. In both these syllogisms the premiss CA has been assumed
without being demonstrated: the other premisses had ex hypothesi
been proved. Consequently if we succeed in demonstrating this premiss,
all the premisses will have been proved reciprocally. If then it is
assumed that C belongs to all B, and B to all A, both the premisses
assumed have been proved, and C must belong to A. It is clear then
that only if the terms are convertible is circular and reciprocal
demonstration possible (if the terms are not convertible, the matter