The Basic Works of Aristotle (Modern Library Classics) (26 page)

17
     (2) In the case of attributes not atomically connected with or disconnected from their subjects, (
a
) (i) as long as the false conclusion is inferred through the ‘appropriate’ middle,
(20)
only the major and not both premisses can be false. By ‘appropriate middle’ I mean the middle term through which the contradictory—i. e. the true—conclusion is inferrible. Thus, let
A
be attributable to
B
through a middle term
C:
then, since to produce a conclusion the premiss
C–B
must be taken affirmatively, it is clear that this premiss must always be true,
(25)
for its quality is not changed. But the major
A–C
is false, for it is by a change in the quality of
A–C
that the conclusion becomes its contradictory—i. e. true. Similarly (ii) if the middle is taken from another series of predication; e. g. suppose
D
to be not only contained within
A
as a part within its whole but also predicable of all
B.
Then the premiss
D–B
must remain unchanged,
(30)
but the quality of
A–D
must be changed; so that
D–B
is always true,
A–D
always false. Such error is practically identical with that which is inferred through the ‘appropriate’ middle. On the other hand, (
b
) if the conclusion is not inferred through the ‘appropriate’ middle—(i) when the middle is subordinate to
A
but is predicable of no
B,
(35)
both premisses must be false, because if there is to be a conclusion both must be posited as asserting the contrary of what is actually the fact, and so posited both become false: e. g. suppose that actually all
D
is
A
but no
B
is
D;
then if these premisses are
changed in quality, a conclusion will follow and both of the new premisses will be false.
(40)
When, however, (ii) the middle
D
is not subordinate to
A, A–D
will be true,
D–B
false—
A–D
true because
A
was not subordinate to
D, D–B
false because if it had been true, the conclusion too would have been true; but it is
ex hypothesi
false.
[81a]

When the erroneous inference is in the second figure,
(5)
both premisses cannot be entirely false; since if
B
is subordinate to
A,
there can be no middle predicable of all of one extreme and of none of the other as was stated before.
¹⁹
One premiss, however, may be false, and it may be either of them. Thus, if
C
is actually an attribute of both
A
and
B,
but is assumed to be an attribute of
A
only and not of
B,
(10)
C–A
will be true,
C–B
false: or again if
C
be assumed to be attributable to
B
but to no
A, C–B
will be true,
C–A
false.

We have stated when and through what kinds of premisses error will result in cases where the erroneous conclusion is negative.
(15)
If the conclusion is affirmative, (
a
) (i) it may be inferred through the ‘appropriate’ middle term. In this case both premisses cannot be false since, as we said before,
²⁰
C–B
must remain unchanged if there is to be a conclusion, and consequently
A–C,
the quality of which is changed, will always be false. This is equally true if (ii) the middle is taken from another series of predication,
(20)
as was stated to be the case also with regard to negative error;
²¹
for
D–B
must remain unchanged, while the quality of
A–D
must be converted, and the type of error is the same as before.

(
b
) The middle may be inappropriate. Then (i) if
D
is subordinate to
A,
(25)
A–D
will be true, but
D–B
false; since
A
may quite well be predicable of several terms no one of which can be subordinated to another. If, however, (ii)
D
is not subordinate to
A,
obviously
A–D,
since it is affirmed, will always be false, while
D–B
may be either true or false; for
A
may very well be an attribute of no
D,
(30)
whereas all
B
is
D,
e. g. no science is animal, all music is science. Equally well
A
may be an attribute of no
D,
and
D
of no
B.
It emerges, then, that if the middle term is not subordinate to the major, not only both premisses but either singly may be false.

Thus we have made it clear how many varieties of erroneous inference are liable to happen and through what kinds of premisses they occur,
(35)
in the case both of immediate and of demonstrable truths.

18
     It is also clear that the loss of any one of the senses entails the loss of a corresponding portion of knowledge, and that, since we
learn either by induction or by demonstration, this knowledge cannot be acquired.
(40)
Thus demonstration develops from universals, induction from particulars; but since it is possible to familiarize the pupil with even the so-called mathematical abstractions only through induction—i. e. only because each subject genus possesses, in virtue of a determinate mathematical character, certain properties which can be treated as separate even though they do not exist in isolation—it is consequently impossible to come to grasp universals except through induction.
[81b]
(5)
But induction is impossible for those who have not sense-perception. For it is sense-perception alone which is adequate for grasping the particulars: they cannot be objects of scientific knowledge, because neither can universals give us knowledge of them without induction, nor can we get it through induction without sense-perception.

19
      Every syllogism is effected by means of three terms.
(10)
One kind of syllogism serves to prove that
A
inheres in
C
by showing that
A
inheres in
B
and
B
in
C;
the other is negative and one of its premisses asserts one term of another, while the other denies one term of another. It is clear, then, that these are the fundamentals and so-called hypotheses of syllogism.
(15)
Assume them as they have been stated, and proof is bound to follow—proof that
A
inheres in
C
through
B,
and again that
A
inheres in
B
through some other middle term, and similarly that
B
inheres in
C.
If our reasoning aims at gaining credence and so is merely dialectical, it is obvious that we have only to see that our inference is based on premisses as credible as possible: so that if a middle term between
A
and
B
is credible though not real,
(20)
one can reason through it and complete a dialectical syllogism. If, however, one is aiming at truth, one must be guided by the real connexions of subjects and attributes. Thus: since there are attributes which are predicated of a subject essentially or naturally and not coincidentally—not,
(25)
that is, in a sense in which we say ‘That white (thing) is a man’, which is not the same mode of predication as when we say ‘The man is white’: the man is white not because he is something else but because he is man, but the white is man because ‘being white’ coincides with ‘humanity’ within one substratum—therefore there are terms such as are naturally subjects of predicates.
(30)
Suppose, then,
C
such a term not itself attributable to anything else as to a subject, but the proximate subject of the attribute
B
—i. e. so that
B–C
is immediate; suppose further
E
related immediately to
F,
and
F
to
B.
The first question is, must this series
terminate, or can it proceed to infinity? The second question is as follows: Suppose nothing is essentially predicated of
A,
but
A
is predicated primarily of
H
and of no intermediate prior term,
(35)
and suppose
H
similarly related to
G
and
G
to
B;
then must this series also terminate, or can it too proceed to infinity? There is this much difference between the questions: the first is, is it possible to start from that which is not itself attributable to anything else but is the subject of attributes, and ascend to infinity? The second is the problem whether one can start from that which is a predicate but not itself a subject of predicates,
(40)
and descend to infinity? A third question is, if the extreme terms are fixed, can there be an infinity of middles? I mean this: suppose for example that
A
inheres in
C
and
B
is intermediate between them, but between
B
and
A
there are other middles,
(5)
and between these again fresh middles; can these proceed to infinity or can they not? This is the equivalent of inquiring, do demonstrations proceed to infinity, i. e. is everything demonstrable?
[82a]
Or do ultimate subject and primary attribute limit one another?

I hold that the same questions arise with regard to negative conclusions and premisses: viz. if
A
is attributable to no
B,
(10)
then either this predication will be primary, or there will be an intermediate term prior to
B
to which
A
is not attributable—
G,
let us say, which is attributable to all
B
—and there may still be another term
H
prior to
G,
which is attributable to all
G.
The same questions arise, I say, because in these cases too either the series of prior terms to which
A
is not attributable is infinite or it terminates.

One cannot ask the same questions in the case of reciprocating terms,
(15)
since when subject and predicate are convertible there is neither primary nor ultimate subject, seeing that all the reciprocals
qua
subjects stand in the same relation to one another, whether we say that the subject has an infinity of attributes or that both subjects and attributes—and we raised the question in both cases—are infinite in number. These questions then cannot be asked—unless, indeed, the terms can reciprocate by two different modes, by accidental predication in one relation and natural predication in the other.
(20)

20
     Now, it is clear that if the predications terminate in both the upward and the downward direction (by ‘upward’ I mean the ascent to the more universal, by ‘downward’ the descent to the more particular), the middle terms cannot be infinite in number. For suppose that
A
is predicated of
F,
and that the intermediates—call them
BB′ B
″ …—are infinite,
(25)
then clearly you might descend from
A
and find one term predicated of another
ad infinitum,
since you have an infinity of terms between you and
F;
and equally, if you ascend from
F,
there are infinite terms between you and
A.
It follows that if these processes are impossible there cannot be an infinity of intermediates between
A
and
F.
(30)
Nor is it of any effect to urge that some terms of the series
AB … F
are contiguous so as to exclude intermediates, while others cannot be taken into the argument at all: whichever terms of the series
B
 … I take, the number of intermediates in the direction either of
A
or of
F
must be finite or infinite: where the infinite series starts, whether from the first term or from a later one,
(35)
is of no moment, for the succeeding terms in any case are infinite in number.

21
     Further, if in affirmative demonstration the series terminates in both directions, clearly it will terminate too in negative demonstration. Let us assume that we cannot proceed to infinity either by ascending from the ultimate term (by ‘ultimate term’ I mean a term such as
F
was, not itself attributable to a subject but itself the subject of attributes), or by descending towards an ultimate from the primary term (by ‘primary term’ I mean a term predicable of a subject but not itself a subject
²²
).
[82b]
If this assumption is justified, the series will also terminate in the case of negation.
(5)
For a negative conclusion can be proved in all three figures. In the first figure it is proved thus: no
B
is
A,
all
C
is
B.
In packing the interval
B–C
we must reach immediate propositions—as is always the case with the minor premiss—since
B–C
is affirmative. As regards the other premiss it is plain that if the major term is denied of a term
D
prior to
B, D
will have to be predicable of all
B,
(10)
and if the major is denied of yet another term prior to
D,
this term must be predicable of all
D.
Consequently, since the ascending series is finite, the descent will also terminate and there will be a subject of which
A
is primarily non-predicable. In the second figure the syllogism is, all
A
is
B,
no
C
is
B,
∴ no
C
is
A.
If proof of this
²³
is required, plainly it may be shown either in the first figure as above,
(15)
in the second as here, or in the third. The first figure has been discussed, and we will proceed to display the second, proof by which will be as follows: all
B
is
D,
no
C
is
D
 …, since it is required that
B
should be a subject of which a predicate is affirmed. Next, since
D
is to be proved not to belong to
C,
then
D
has a further predicate which is denied of
C.
Therefore, since the
succession of predicates affirmed of an ever higher universal terminates,
²⁴
the succession of predicates denied terminates too.
²⁵
(20)

The third figure shows it as follows: all
B
is
A,
some
B
is not
C,
∴ some
A
is not
C.
This premiss, i. e.
C–B,
will be proved either in the same figure or in one of the two figures discussed above.
(25)
In the first and second figures the series terminates. If we use the third figure, we shall take as premisses, all
E
is
B,
some
E
is not
C,
and this premiss again will be proved by a similar prosyllogism. But since it is assumed that the series of descending subjects also terminates, plainly the series of more universal non-predicables will terminate also. Even supposing that the proof is not confined to one method, but employs them all and is now in the first figure, now in the second or third—even so the regress will terminate,
(30)
for the methods are finite in number, and if finite things are combined in a finite number of ways, the result must be finite.

Thus it is plain that the regress of middles terminates in the case of negative demonstration, if it does so also in the case of affirmative demonstration. That in fact the regress terminates in both these cases may be made clear by the following dialectical considerations.
(35)