The Basic Works of Aristotle (Modern Library Classics) (16 page)

23
     
[40b]
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.
(20)
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,
(25)
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,
(30)
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.
(35)
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,
(40)
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.
[41a]
For in general we stated
⁵²
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,
(5)
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,
(10)
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
),
(15)
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.
(20)

It is clear then that the ostensive syllogisms are effected by means of the aforesaid figures; these considerations will show that
reductiones ad impossibile
also are effected in the same way. For all who
effect an argument
per impossibile
infer syllogistically what is false,
(25)
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,
(30)
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,
(35)
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.
⁵³
[41b]
(40)
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.
(5)

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.
(10)
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.
(15)
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,
(20)
unless he also states that when equals are taken from equals the remainders are equal.
⁵⁴

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,
(25)
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, pure, or problematic. We must consider also the other forms of predication.
(30)

It is clear also when a syllogism in general can be made and when it cannot; and when a valid,
⁵⁵
when a perfect syllogism can be formed; and that if a syllogism is formed the terms must be arranged in one of the ways that have been mentioned.
(35)

25
     It is clear too that every demonstration will proceed through three terms and no more, unless the same conclusion is established by different pairs of propositions; e. g. the conclusion
E
may be established through the propositions
A
and
B,
and through the propositions
C
and
D,
or through the propositions
A
and
B,
or
A
and
C,
(40)
or
B
and
C.
For nothing prevents there being several middles for the same terms. But in that case there is not one but several syllogisms.
[42a]
Or again when each of the propositions
A
and
B
is obtained by syllogistic inference, e. g.
A
by means of
D
and
E,
and again
B
by means of
F
and
G.
Or one may be obtained by syllogistic, the other by inductive inference. But thus also the syllogisms are many; for the conclusions are many,
(5)
e. g.
A
and
B
and
C.
But if this can be called one syllogism, not many, the same conclusion may be reached by more than three terms in this way, but it cannot be reached as
C
is established by means of
A
and
B.
Suppose that the proposition
E
is inferred from the premisses
A, B, C,
and
D.
It is necessary then that of these one should be related to another as whole to part: for it has already been proved that if a syllogism is formed some of its terms must be related in this way.
⁵⁶
(10)
Suppose then that
A
stands in this relation to
B.
Some conclusion then follows from them. It must either be
E
or one or other of
C
and
D,
or something other than these.
(15)

(1) If it is
E
the syllogism will have
A
and
B
for its sole premisses. But if
C
and
D
are so related that one is whole, the other part, some conclusion will follow from them also; and it must be either
E,
or one or other of the propositions
A
and
B,
or something other than these. And if it is (i)
E,
or (ii)
A
or
B,
either (i) the syllogisms will be more than one, or (ii) the same thing happens to be inferred by means of several terms only in the sense which we saw to be possible.
⁵⁷
But if (iii) the conclusion is other than
E
or
A
or
B,
(20)
the syllogisms will be many, and unconnected with one another. But if
C
is not so related to
D
as to make a syllogism, the propositions will have been assumed to no purpose, unless for the sake of induction or of obscuring the argument or something of the sort.

(2) But if from the propositions
A
and
B
there follows not
E
but some other conclusion,
(25)
and if from
C
and
D
either
A
or
B
follows or something else, then there are several syllogisms, and
they do not establish the conclusion proposed: for we assumed that the syllogism proved
E.
And if no conclusion follows from
C
and
D,
it turns out that these propositions have been assumed to no purpose, and the syllogism does not prove the original proposition.
(30)

So it is clear that every demonstration and every syllogism will proceed through three terms only.

This being evident, it is clear that a syllogistic conclusion follows from two premisses and not from more than two. For the three terms make two premisses, unless a new premiss is assumed, as was said at the beginning,
⁵⁸
to perfect the syllogisms. It is clear therefore that in whatever syllogistic argument the premisses through which the main conclusion follows (for some of the preceding conclusions must be premisses) are not even in number,
(35)
this argument either has not been drawn syllogistically or it has assumed more than was necessary to establish its thesis.
(40)

If then syllogisms are taken with respect to their main premisses, every syllogism will consist of an even number of premisses and an odd number of terms (for the terms exceed the premisses by one), and the conclusions will be half the number of the premisses.
[42b]
(5)
But whenever a conclusion is reached by means of prosyllogisms or by means of several continuous middle terms, e. g. the proposition
AB
by means of the middle terms
C
and
D,
the number of the terms will similarly exceed that of the premisses by one (for the extra term must either be added outside or inserted: but in either case it follows that the relations of predication are one fewer than the terms related), and the premisses will be equal in number to the relations of predication.
(10)
The premisses however will not always be even, the terms odd; but they will alternate—when the premisses are even, the terms must be odd; when the terms are even, the premisses must be odd: for along with one term one premiss is added, if a term is added from any quarter. Consequently since the premisses were (as we saw) even, and the terms odd,
(15)
we must make them alternately even and odd at each addition. But the conclusions will not follow the same arrangement either in respect to the terms or to the premisses. For if one term is added, conclusions will be added less by one than the pre-existing terms: for the conclusion is drawn not in relation to the single term last added, but in relation to all the rest,
(20)
e. g. if to
ABC
the term
D
is added, two conclusions are thereby added, one in relation to
A,
the other in relation to
B.
Similarly
with any further additions. And similarly too if the term is inserted in the middle: for in relation to one term only, a syllogism will not be constructed.
(25)
Consequently the conclusions will be much more numerous than the terms or the premisses.