按键盘上方向键 ← 或 → 可快速上下翻页,按键盘上的 Enter 键可回到本书目录页,按键盘上方向键 ↑ 可回到本页顶部!
————未阅读完?加入书签已便下次继续阅读!
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
contradictories。
In the first figure no syllogism whether affirmative or negative can
be made out of opposed premisses: no affirmative syllogism is possible
because both premisses must be affirmative; but opposites are; the one
affirmative; the other negative: no negative syllogism is possible
because opposites affirm and deny the same predicate of the same
subject; and the middle term in the first figure is not predicated
of both extremes; but one thing is denied of it; and it is affirmed of
something else: but such premisses are not opposed。
In the middle figure a syllogism can be made both
oLcontradictories and of contraries。 Let A stand for good; let B and C
stand for science。 If then one assumes that every science is good; and
no science is good; A belongs to all B and to no C; so that B
belongs to no C: no science then is a science。 Similarly if after
taking 'every science is good' one took 'the science of medicine is
not good'; for A belongs to all B but to no C; so that a particular
science will not be a science。 Again; a particular science will not be
a science if A belongs to all C but to no B; and B is science; C
medicine; and A supposition: for after taking 'no science is
supposition'; one has assumed that a particular science is
supposition。 This syllogism differs from the preceding because the
relations between the terms are reversed: before; the affirmative
statement concerned B; now it concerns C。 Similarly if one premiss
is not universal: for the middle term is always that which is stated
negatively of one extreme; and affirmatively of the other。
Consequently it is possible that contradictories may lead to a
conclusion; though not always or in every mood; but only if the
terms subordinate to the middle are such that they are either
identical or related as whole to part。 Otherwise it is impossible: for
the premisses cannot anyhow be either contraries or contradictories。
In the third figure an affirmative syllogism can never be made out
of opposite premisses; for the reason given in reference to the
first figure; but a negative syllogism is possible whether the terms
are universal or not。 Let B and C stand for science; A for medicine。
If then one should assume that all medicine is science and that no
medicine is science; he has assumed that B belongs to all A and C to
no A; so that a particular science will not be a science。 Similarly if
the premiss BA is not assumed universally。 For if some medicine is
science and again no medicine is science; it results that some science
is not science; The premisses are contrary if the terms are taken
universally; if one is particular; they are contradictory。
We must recognize that it is possible to take opposites in the way
we said; viz。 'all science is good' and 'no science is good' or
'some science is not good'。 This does not usually escape notice。 But
it is possible to establish one part of a contradiction through
other premisses; or to assume it in the way suggested in t