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！
third。 But when the minor premiss is negative and universal； if it
is problematic a syllogism can be formed by means of conversion； but
if it is necessary a syllogism is not possible。 The proof will
follow the same course as where the premisses are universal； and the
same terms may be used。
It is clear then in this figure also when and how a syllogism can be
formed； and when the conclusion is problematic； and when it is pure。
It is evident also that all syllogisms in this figure are imperfect；
and that they are made perfect by means of the first figure。
23
It is clear from what has been said that the syllogisms in these
figures are made perfect by means of universal syllogisms in the first
figure and are reduced to them。 That every syllogism without
qualification can be so treated； will be clear presently； when it
has been proved that every syllogism is formed through one or other of
these figures。
It is necessary that every demonstration and every syllogism
should prove either that something belongs or that it does not； and
this either universally or in part； and further either ostensively
or hypothetically。 One sort of hypothetical proof is the reductio ad
impossibile。 Let us speak first of ostensive syllogisms： for after
these have been pointed out the truth of our contention will be
clear with regard to those which are proved per impossibile； and in
general hypothetically。
If then one wants to prove syllogistically A of B； either as an
attribute of it or as not an attribute of it； one must assert
something of something else。 If now A should be asserted of B； the
proposition originally in question will have been assumed。 But if A
should be asserted of C； but C should not be asserted of anything； nor
anything of it； nor anything else of A； no syllogism will be possible。
For nothing necessarily follows from the assertion of some one thing
concerning some other single thing。 Thus we must take another
premiss as well。 If then A be asserted of something else； or something
else of A； or something different of C； nothing prevents a syllogism
being formed； but it will not be in relation to B through the
premisses taken。 Nor when C belongs to something else； and that to
something else and so on； no connexion however being made with B； will
a syllogism be possible concerning A in its relation to B。 For in
general we stated that no syllogism can establish the attribution of
one thing to another； unless some middle term is taken； which is
somehow related to each by way of predication。 For the syllogism in
general is made out of premisses； and a syllogism referring to this
out of premisses with the same reference； and a syllogism relating
this to that proceeds through premisses which relate this to that。 But
it is impossible to take a premiss in reference to B； if we neither
affirm nor deny anything of it； or again to take a premiss relating
A to B； if we take nothing common； but affirm or deny peculiar
attributes of each。 So we must take something midway between the
two； which will connect the predications； if we are to have a
syllogism relating this to that。 If then we must take something common
in relation to both； and this is possible in three ways （either by
predicating A of C； and C of B； or C of both； or both of C）； and these
are the figures of which we have spoken； it is clear that every
syllogism must be made in one or other of these figures。 The
argument is the same if several middle terms should be necessary to
establish the relation to B； for the figure will be the same whether
there is one middle term or many。
It is clear then that the ostensive syllogisms are effected by means
of the aforesaid figures； these considerations will show that
reductiones ad also are effected in the same way。 For all who effect
an argument per impossibile infer syllogistically what is false； and
prove the original conclusion hypothetically when something impossible
results from the assumption of its contradictory； e。g。 that the
diagonal of the square is incommensurate with the side； because odd
numbers are equal to evens if it is supposed to be commensurate。 One
infers syllogistically that odd numbers come out equal to evens； and
one proves hypothetically the incommensurability of the diagonal；
since a falsehood results through contradicting this。 For this we
found to be reasoning per impossibile； viz。 proving something
impossible by means of an hypothesis conceded at the beginning。
Consequently； since the falsehood is established in reductions ad
impossibile by an ostensive syllogism； and the original conclusion
is proved hypothetically； and we have already stated that ostensive
syllogisms are effected by means of these figures； it is evident
that syllogisms per impossibile also will be made through these
figures。 Likewise all the other hypothetical syllogisms： for in
every case the syllogism leads up to the proposition that is
substituted for the original thesis； but the original thesis is
reached by means of a concession or some other hypothesis。 But if this
is true； every demonstration and every syllogism must be formed by
means of the three figures mentioned above。 But when this has been
shown it is clear that every syllogism is perfected by means of the
first figure and is reducible to the universal syllogisms in this
figure。
24
Further in every syllogism one of the premisses must be affirmative；
and universality must be present： unless one of the premisses is
universal either a syllogism will not be possible； or it will not
refer to the subject proposed； or the original position will be
begged。 Suppose we have to prove that pleasure in music is good。 If
one should claim as a premiss that pleasure is good without adding
'all'； no syllogism will be possible； if one should claim that some
pleasure is good； then if it is different from pleasure in music； it
is not relevant to the subject proposed； if it is this very
pleasure； one is assuming that which was proposed at the outset to
be proved。 This is more obvious in geometrical proofs； e。g。 that the
angles at the base of an isosceles triangle are equal。 Suppose the
lines A and B have been drawn to the centre。 If then one should assume
that the angle AC is equal to the angle BD； without claiming generally
that angles of semicircles are equal； and again if one should assume
that the angle C is equal to the angle D； without the additional
assumption that every angle of a segment is equal to every other angle
of the same segment； and further if one should assume that when
equal angles are taken from the whole angles； which are themselves
equal； the remainders E and F are equal； he will beg the thing to be
proved； unless he also states that when equals are taken from equals
the remainders are equal。
It is clear then that in every syllogism there must be a universal
premiss； and that a universal statement is proved only when all the
premisses are universal； while a particular statement is proved both
from two universal premisses and from one only： consequently if the
conclusion is universal； the premisses also must be universal； but
if the premisses are universal it is possible that the conclusion
may not be universal。 And it is clear also that in every syllogism
either both or one of the premisses must be like the conclusion。 I
mean not only in being affirmative or negative； but also in being
necessary；