each figure when the conclusion is converted; when a result contrary
to the premiss, and when a result contradictory to the premiss, is
obtained. It is clear that in the first figure the syllogisms are
formed through the middle and the last figures, and the premiss
which concerns the minor extreme is alway refuted through the middle
figure, the premiss which concerns the major through the last
figure. In the second figure syllogisms proceed through the first
and the last figures, and the premiss which concerns the minor extreme
is always refuted through the first figure, the premiss which concerns
the major extreme through the last. In the third figure the refutation
proceeds through the first and the middle figures; the premiss which
concerns the major is always refuted through the first figure, the
premiss which concerns the minor through the middle figure.
11
It is clear then what conversion is, how it is effected in each
figure, and what syllogism results. The syllogism per impossibile is
proved when the contradictory of the conclusion stated and another
premiss is assumed; it can be made in all the figures. For it
resembles conversion, differing only in this: conversion takes place
after a syllogism has been formed and both the premisses have been
taken, but a reduction to the impossible takes place not because the
contradictory has been agreed to already, but because it is clear that
it is true. The terms are alike in both, and the premisses of both are
taken in the same way. For example if A belongs to all B, C being
middle, then if it is supposed that A does not belong to all B or
belongs to no B, but to all C (which was admitted to be true), it
follows that C belongs to no B or not to all B. But this is
impossible: consequently the supposition is false: its contradictory
then is true. Similarly in the other figures: for whatever moods admit
of conversion admit also of the reduction per impossibile.
All the problems can be proved per impossibile in all the figures,
excepting the universal affirmative, which is proved in the middle and
third figures, but not in the first. Suppose that A belongs not to all
B, or to no B, and take besides another premiss concerning either of
the terms, viz. that C belongs to all A, or that B belongs to all D;
thus we get the first figure. If then it is supposed that A does not
belong to all B, no syllogism results whichever term the assumed
premiss concerns; but if it is supposed that A belongs to no B, when
the premiss BD is assumed as well we shall prove syllogistically
what is false, but not the problem proposed. For if A belongs to no B,
and B belongs to all D, A belongs to no D. Let this be impossible:
it is false then A belongs to no B. But the universal affirmative is
not necessarily true if the universal negative is false. But if the
premiss CA is assumed as well, no syllogism results, nor does it do so
when it is supposed that A does not belong to all B. Consequently it
is clear that the universal affirmative cannot be proved in the
first figure per impossibile.
But the particular affirmative and the universal and particular
negatives can all be proved. Suppose that A belongs to no B, and let
it have been assumed that B belongs to all or to some C. Then it is
necessary that A should belong to no C or not to all C. But this is
impossible (for let it be true and clear that A belongs to all C):
consequently if this is false, it is necessary that A should belong to
some B. But if the other premiss assumed relates to A, no syllogism
will be possible. Nor can a conclusion be drawn when the contrary of
the conclusion is supposed, e.g. that A does not belong to some B.
Clearly then we must suppose the contradictory.
Again suppose that A belongs to some B, and let it have been assumed
that C belongs to all A. It is necessary then that C should belong
to some B. But let this be impossible, so that the supposition is
false: in that case it is true that A belongs to no B. We may
proceed in the same way if the proposition CA has been taken as
negative. But if the premiss assumed concerns B, no syllogism will
and an impossible conclusion, but the problem in hand is not proved.
Suppose that A belongs to all B, and let it have been assumed that C
belongs to all A. It is necessary then that C should belong to all
B. But this is impossible, so that it is false that A belongs to all
B. But we have not yet shown it to be necessary that A belongs to no
B, if it does not belong to all B. Similarly if the other premiss
taken concerns B; we shall have a syllogism and a conclusion which
is impossible, but the hypothesis is not refuted. Therefore it is
the contradictory that we must suppose.
To prove that A does not belong to all B, we must suppose that it
belongs to all B: for if A belongs to all B, and C to all A, then C
belongs to all B; so that if this is impossible, the hypothesis is
false. Similarly if the other premiss assumed concerns B. The same
results if the original proposition CA was negative: for thus also
we get a syllogism. But if the negative proposition concerns B,
nothing is proved. If the hypothesis is that A belongs not to all
but to some B, it is not proved that A belongs not to all B, but
that it belongs to no B. For if A belongs to some B, and C to all A,
then C will belong to some B. If then this is impossible, it is
false that A belongs to some B; consequently it is true that A belongs
to no B. But if this is proved, the truth is refuted as well; for
the original conclusion was that A belongs to some B, and does not
belong to some B. Further the impossible does not result from the
hypothesis: for then the hypothesis would be false, since it is
impossible to draw a false conclusion from true premisses: but in fact
it is true: for A belongs to some B. Consequently we must not
suppose that A belongs to some B, but that it belongs to all B.
Similarly if we should be proving that A does not belong to some B:
for if 'not to belong to some' and 'to belong not to all' have the
same meaning, the demonstration of both will be identical.
It is clear then that not the contrary but the contradictory ought
to be supposed in all the syllogisms. For thus we shall have necessity
of inference, and the claim we make is one that will be generally
accepted. For if of everything one or other of two contradictory
statements holds good, then if it is proved that the negation does not
hold, the affirmation must be true. Again if it is not admitted that
the affirmation is true, the claim that the negation is true will be
generally accepted. But in neither way does it suit to maintain the
contrary: for it is not necessary that if the universal negative is
false, the universal affirmative should be true, nor is it generally
accepted that if the one is false the other is true.
12
It is clear then that in the first figure all problems except the
universal affirmative are proved per impossibile. But in the middle
and the last figures this also is proved. Suppose that A does not
belong to all B, and let it have been assumed that A belongs to all C.
If then A belongs not to all B, but to all C, C will not belong to all
B. But this is impossible (for suppose it to be clear that C belongs
to all B): consequently the hypothesis is false. It is true then
that A belongs to all B. But if the contrary is supposed, we shall
have a syllogism and a result which is impossible: but the problem
in hand is not proved. For if A belongs to no B, and to all C, C
will belong to no B. This is impossible; so that it is false that A
belongs to no B. But though this is false, it does not follow that
it is true that A belongs to all B.
When A belongs to some B, suppose that A belongs to no B, and let
A belong to all C. It is necessary then that C should belong to no
B. Consequently, if this is impossible, A must belong to some B. But
if it is supposed that A does not belong to some B, we shall have
the same results as in the first figure.
Again suppose that A belongs to some B, and let A belong to no C. It
is necessary then that C should not belong to some B. But originally
it belonged to all B, consequently the hypothesis is false: A then
will belong to no B.
When A does not belong to an B, suppose it does belong to all B, and
to no C. It is necessary then that C should belong to no B. But this
is impossible: so that it is true that A does not belong to all B.
It is clear then that all the syllogisms can be formed in the middle
figure.
13
Similarly they can all be formed in the last figure. Suppose that
A does not belong to some B, but C belongs to all B: then A does not
belong to some C. If then this is impossible, it is false that A
does not belong to some B; so that it is true that A belongs to all B.
But if it is supposed that A belongs to no B, we shall have a
syllogism and a conclusion which is impossible: but the problem in
hand is not proved: for if the contrary is supposed, we shall have the
same results as before.
But to prove that A belongs to some B, this hypothesis must be made.
If A belongs to no B, and C to some B, A will belong not to all C.
If then this is false, it is true that A belongs to some B.
When A belongs to no B, suppose A belongs to some B, and let it have
been assumed that C belongs to all B. Then it is necessary that A
should belong to some C. But ex hypothesi it belongs to no C, so
that it is false that A belongs to some B. But if it is supposed
that A belongs to all B, the problem is not proved.
But this hypothesis must be made if we are prove that A belongs
not to all B. For if A belongs to all B and C to some B, then A
belongs to some C. But this we assumed not to be so, so it is false
that A belongs to all B. But in that case it is true that A belongs
not to all B. If however it is assumed that A belongs to some B, we
shall have the same result as before.
It is clear then that in all the syllogisms which proceed per
impossibile the contradictory must be assumed. And it is plain that in
the middle figure an affirmative conclusion, and in the last figure
a universal conclusion, are proved in a way.
14
Demonstration per impossibile differs from ostensive proof in that
it posits what it wishes to refute by reduction to a statement
admitted to be false; whereas ostensive proof starts from admitted
positions. Both, indeed, take two premisses that are admitted, but the
latter takes the premisses from which the syllogism starts, the former
takes one of these, along with the contradictory of the original
conclusion. Also in the ostensive proof it is not necessary that the
conclusion should be known, nor that one should suppose beforehand
that it is true or not: in the other it is necessary to suppose
beforehand that it is not true. It makes no difference whether the
conclusion is affirmative or negative; the method is the same in
both cases. Everything which is concluded ostensively can be proved
per impossibile, and that which is proved per impossibile can be
proved ostensively, through the same terms. Whenever the syllogism
is formed in the first figure, the truth will be found in the middle
or the last figure, if negative in the middle, if affirmative in the
last. Whenever the syllogism is formed in the middle figure, the truth
will be found in the first, whatever the problem may be. Whenever
the syllogism is formed in the last figure, the truth will be found in
the first and middle figures, if affirmative in first, if negative
in the middle. Suppose that A has been proved to belong to no B, or
not to all B, through the first figure. Then the hypothesis must
have been that A belongs to some B, and the original premisses that
C belongs to all A and to no B. For thus the syllogism was made and
the impossible conclusion reached. But this is the middle figure, if C
belongs to all A and to no B. And it is clear from these premisses
that A belongs to no B. Similarly if has been proved not to belong
to all B. For the hypothesis is that A belongs to all B; and the
original premisses are that C belongs to all A but not to all B.
Similarly too, if the premiss CA should be negative: for thus also
we have the middle figure. Again suppose it has been proved that A
belongs to some B. The hypothesis here is that is that A belongs to no
B; and the original premisses that B belongs to all C, and A either to
all or to some C: for in this way we shall get what is impossible. But
if A and B belong to all C, we have the last figure. And it is clear
from these premisses that A must belong to some B. Similarly if B or A
should be assumed to belong to some C.
Again suppose it has been proved in the middle figure that A belongs
to all B. Then the hypothesis must have been that A belongs not to all
B, and the original premisses that A belongs to all C, and C to all B:
for thus we shall get what is impossible. But if A belongs to all C,
and C to all B, we have the first figure. Similarly if it has been
proved that A belongs to some B: for the hypothesis then must have
been that A belongs to no B, and the original premisses that A belongs
to all C, and C to some B. If the syllogism is negative, the
hypothesis must have been that A belongs to some B, and the original
premisses that A belongs to no C, and C to all B, so that the first
figure results. If the syllogism is not universal, but proof has
been given that A does not belong to some B, we may infer in the
same way. The hypothesis is that A belongs to all B, the original
premisses that A belongs to no C, and C belongs to some B: for thus we
get the first figure.
Again suppose it has been proved in the third figure that A
belongs to all B. Then the hypothesis must have been that A belongs
not to all B, and the original premisses that C belongs to all B,
and A belongs to all C; for thus we shall
get what is impossible.
And the original premisses form the first figure. Similarly if the
demonstration establishes a particular proposition: the hypothesis
then must have been that A belongs to no B, and the original premisses
that C belongs to some B, and A to all C. If the syllogism is
negative, the hypothesis must have been that A belongs to some B,
and the original premisses that C belongs to no A and to all B, and
this is the middle figure. Similarly if the demonstration is not
universal. The hypothesis will then be that A belongs to all B, the
premisses that C belongs to no A and to some B: and this is the middle
figure.
It is clear then that it is possible through the same terms to prove
each of the problems ostensively as well. Similarly it will be
possible if the syllogisms are ostensive to reduce them ad impossibile
in the terms which have been taken, whenever the contradictory of
the conclusion of the ostensive syllogism is taken as a premiss. For
the syllogisms become identical with those which are obtained by means
of conversion, so that we obtain immediately the figures through which
each problem will be solved. It is clear then that every thesis can be
proved in both ways, i.e. per impossibile and ostensively, and it is
not possible to separate one method from the other.
15
In what figure it is possible to draw a conclusion from premisses
which are opposed, and in what figure this is not possible, will be
made clear in this way. Verbally four kinds of opposition are
possible, viz. universal affirmative to universal negative,
universal affirmative to particular negative, particular affirmative
to universal negative, and particular affirmative to particular
negative: but really there are only three: for the particular
affirmative is only verbally opposed to the particular negative. Of
the genuine opposites I call those which are universal contraries, the
universal affirmative and the universal negative, e.g. 'every
science is good', 'no science is good'; the others I call