denial. As we always speak and think within a general subject or
universe of discourse, it follows that every denial substitutes some
affirmation for the judgment which it denies. The only judgments in
which this is not the case are those called by an unmeaning tradition
Infinite Judgments, _i.e._ judgments in which the negative predicate
{130} includes every determination which has applicability to the
Subject. This is because the attribute denied has no applicability
to the Subject, and therefore all that has applicability is
undiscriminatingly affirmed, in other words, the judgment has no
meaning. “Virtue is not-square.” This suggests no definite positive
quality applicable to virtue, and therefore is idle. You may safely
analyse a significant negative judgment, “A is not B” as = “A is not B
but C,” or as = “A is X, which excludes B.” For X may be undetermined,
“a colour not red.” But then if the meaning is always affirmative or
positive, why do we ever use the negative form?
_Why use Negation?_
3. In the first place, we use it because it indicates exclusion,
and without it we cannot distinguish between mere differents on the
one hand and contraries on the other. If you ask me, “Are you going
to Victoria, London Chatham and Dover station?” and I answer, “I am
going to Victoria, London Brighton and South Coast,” that will not be
satisfactory to you, unless you happen to know beforehand that these
stations are so arranged that if you are at one you are not at the
other. They might be a single station used by different companies, and
called indifferently by the name of either. To make it clear that the
suggestion and the answer are incompatible, I must say, “I am _not_
going to Victoria, London Chatham and Dover,” and I may add or not add,
“I _am_ going to Victoria, London Brighton and South Coast.” That tells
you that the one predicate excludes the other, and that is the first
reason why we use the generalised form of exclusion, _i.e._ negation.
But in the second place, it can give us more, and something absolutely
necessary to our knowledge, and that is not {131} merely exclusion, but
exhaustion. In literal form negation is absolutely exhaustive, that is
to say, contradictory. The Judgment “A is not B” forms an exhaustive
alternative to the Judgment “A is B,” so that no third case beyond
these two is possible, and therefore you can argue from the falsehood
of either to the truth of the other. Now this form is potentially of
immense value for knowledge, and all disjunction consists in applying
it; but as it stands in the abstract it is worthless, because it is
an empty form. “A is red or not-red.” If either of these is false the
other is true. But what do you gain by this? You are not entitled to
put any positive meaning upon not-red; if you do so you slide into mere
contrary negation, and the inference from falsehood becomes a fallacy.
Make an argument, “The soul is red or not-red.” “It is not-red ∴ it is
some other colour than red.” The argument is futile. We have construed
“not-red” as a positive contrary, and that being so, the disjunction is
no longer exhaustive. We had no right to say that the soul is either
red or some other colour; the law of Excluded Middle does not warrant
that.
I pause to say that the proof of the exhaustiveness of negation, _i.e._
that two negatives make an affirmative--that if A is not not-B, it
follows that A is B--is a disputed problem, the problem known as double
negation. How do you know that what is not not-red must be red? I take
the law of Excluded Middle simply as a definition of the bare form
of denial, or the distinction between this and not-this; “not-this”
being the bare abstraction of the other than this. Others say that
every negation presupposes an affirmation; so “A is not-B” presupposes
the affirmation “A is B,” and {132} if you knock down the negative,
the original affirmative is left standing. Sigwart and B. Erdmann say
this. I think it monstrous. I do not believe that you must find an
affirmative standing before you can deny.
_Stage of Significant Negation. Combination of Contrary and
Contradictory_
4. Well, then, the point we have reached is this. What we mean in
denial is always the contrary, something positive. What we say
in denial--in other words, the literal form which we use--always
approaches the contradictory, _i.e._ is pure exclusion. The Contrary
of the diagram denies more than it need, but still its form is that
of exclusion. Now we have seen that in denial, as used in common
speech, we get the benefit of _both affirmation and exclusion_, but in
accurate thought we want to do much more than this; we want to get the
whole benefit of the negative form--that is, to get a positive meaning
together with not only exclusion, but exhaustion.
I will put the three cases in one example, beginning with mere
affirmations of different facts.
_Different Affirmations_
(1) “He goes by this train to-day.” “He goes by that train to-morrow.”
This conjunction, as simply stated, gives no inference from the truth
or falsehood of either statement to the truth or falsehood of the other.
_Contrary Opposition, exclusive_
(2) “He goes by this train,” and “He goes by that train,” with a
meaning equivalent to “No, he goes by that.” If it is true that in
the sense suggested by the context he goes by this train, then it is
not true that he goes by the other, and if it is true, in the sense
explained, that he goes by the other, then he does not go by this. Each
excludes the other, but both may be excluded by a third alternative.
If it is _not_ true that he goes by this {133} train--nothing follows.
There may be any number of trains he might go by, or he might give up
going; _i.e._ your Universe of discourse, your implicit meaning is
not expressly limited. If it is _not_ true to say, “No, he goes by
that”--taking the whole meaning together, and not separating its parts,
for this combination is essential to the “contrary”--nothing follows as
to the truth of the other statement. He may not be going at all, or may
be going by some third train, or by road.
_Combined Contrary and Contradictory Negation_
But if you limit your Universe, or general subject, then you can
combine the value of contrary and contradictory negation. Then you say,
(3) “He goes either by this train or by that.” Then you can infer not
only from “He goes by this train,” that “He does not go by that,” but
from “He does not go by this train” to “He does go by that.”
The alternative between “A is B” and “A is not-B” remains exhaustive,
but not-B has been given a positive value, _because we have limited
the possibilities by definite knowledge_. The processes of accurate
thinking and observation aim almost entirely at giving a positive value
C to not-B, and a positive value B to not-C, under a disjunction,
because it is then that you define exactly where and within what
conditions C which is not B passes into B which is not C. Take the
disjunction, “Sound is either musical or noise.” If the successive
vibrations are of a uniform period it is musical sound; if they are of
irregular periods it is noise. This is a disjunction which assumes the
form,
A is either B or C. That is to say, If it is B it is not C. If it is
not B it is C.
{134} Therefore I think that all “determination is negation”--of
course, however, not bare negation, but significant negation; the
essence of it consists in correcting and confirming our judgment of
the nature of a positive phenomenon by showing that _just when_ its
condition ceases, _just then_ something else begins, and when you have
exhausted the whole operation of the system of conditions in question,
so that from any one phase of their effects you can read off what _it_
is not but the _others_ are, then you have almost all the knowledge
we can get. The “_Just-not_” is the important point, and this is only
given by a positive negation within a definite system. You want to
explain or define the case in which A becomes B. You want observation
of not-B; but almost the whole world is formally or barely not-B, so
that you are lost in chaos. What you must do is to find the point
within A, where A1 which is B passes into A2 which is C, and that will
give you the _just-not-B_ which is the valuable negative instance.
_Negative judgment expressing fact_
5. You will find it said that a Negative Judgment cannot express fact;
_e.g._ that a Judgment of Perception cannot be negative. This is worth
reflecting upon; I hope that what has been said makes clear how far it
is true. The bare form of Negation is not adequate to fact; it contains
mere emptiness or ignorance; we nowhere in our perception come upon a
mere “not-something.” No doubt negation is in this way more subjective
than affirmation. But then as it fills up in meaning, the denial
becomes more and more on a level with the affirmation, till at last
in systematic knowledge both become double-edged--every affirmative
denies, and every negative affirms. When a man who is both a {135}
musician and a physicist says, “this compound tone A is a discord Y,”
he knows exactly how much of a discord, what ratio of vibration makes
it so much of a discord, how much it would have to change to become a
concord (X which is not Y), and what change in the vibration ratio from
a1 to a2 would be needed to make it a concord. To such knowledge as
this, the accurate negation is just as expressive as the affirmation,
and it does not matter whether he says “A is Y,” or “A is by so much
not X.” It becomes, as Venn says, all but impossible to distinguish the
affirmation from the negation. No doubt affirmative terms come in at
this stage, though the meaning is negative. Observe in this connection
how we sometimes use the nearest word we can think of, knowing that the
negative gives the positive indirectly--“He was, I won’t say insolent,”
meaning _just not_ or “_all but_” insolent; or again, “That was not
right,” rather than saying bluntly “wrong.”
_Operation of the denied idea_
6. Every significant negation “A is not B” can be analysed as “A is X
which excludes B.” Of course X may not be a distinct C; _e.g._ we may
be able to see that A is not red, but we may not be able to make out
for certain what colour it is; then the colour X is “an unknown colour
which excludes red.”
How does the rejected idea operate in Judgment? I suppose it operates
by suggesting a Judgment which as you make it destroys some of its
own characteristics. It is really an expression of the confirmatory
negative instance or “just-not.” _Just_ when two parallel straight
lines swing so that they can meet, _just_ then the two interior angles
begin to be less than two right angles, which tells us that the {136}
straight lines are ceasing to be parallel. Just in as much as two
straight lines begin to enclose a space we become aware that one or
other of them is not straight, so that A in turning from Y to X turns
_pari passu_ from A1 to A2, and we are therefore justified in saying
that A, when it is Y, cannot be X.
This lecture may pave the way for Induction, by giving some idea of the
importance of the negative instance which Bacon preached so assiduously.
In a real system of science the conceptions are negative towards each
other merely as defining each other. One of them is not in itself
more negative than another. Such a conception, _e.g._, is that of a
triangle compared with two parallel straight lines which are cut by
a third line. If the parallels are swung so as to meet, they become
a triangle which gains in its third angle what the parallels lose on
the two interior angles, and the total of two right angles remains
the same. Thus in saying that parallels cut by a third straight line
cannot form a triangle, and that the three angles of a triangle are
equal to two right angles, we are expressing the frontier which is at
once the demarcation between two sets of geometrical relations, and the
positive grasp or connection of the one with the other. The negation is
no bar to a positive continuity in the organism of the science, but is
essential to defining its nature and constituent elements. This is the
bearing of significant negation when fully developed.
{137}
LECTURE IX INFERENCE AND THE SYLLOGISTIC FORMS
_Inference in general_ [1]
1. The Problem of Inference is something of a paradox. Inference
consists in asserting as fact or truth, on the ground of certain given
facts or truths, something which is not included in those data. We
have not got inference unless the conclusion, (i.) is necessary from
the premisses, and (ii.) goes beyond the premisses. To put the paradox
quite roughly--we have not got inference unless the conclusion is (i.)
in the premisses, and (ii.) outside the premisses. This is the problem
which exercises Mill so much in the chapter, “Function and Value of the
Syllogism.” We should notice especially his § 7, “the universal type
of the reasoning process.” The point of it is to make the justice of
inference depend upon relations of content, which are judged of by what
he calls induction. That is quite right, but the question still returns
upon us, “What kind of relations of content must we have, in order to
realise the paradox of Inference?” This the “type of inference” rather
shirks. See Mill’s remarks when he is brought face to face with {138}
Induction, Bk. III. ch. f. § 2. An Inference, as he there recognises,
either does not hold at all, or it holds “in all cases of a certain
description,” _i.e._, it depends on universals.
[1] Read for Lectures IX. and X., Mill, Bk. II ch. i., ii.,
iii.; Bk. III. ch. i. and ii. at least; Venn, ch. xiv., xv.;
Jevons, _Lessons_ xv. and xxiv.; De Morgan’s _Budget of
Paradoxes_.
I ought to warn you at once that though we may have novelty in the
conclusion of Inference (as in multiplication of large numbers),
the necessity is more essential than the novelty. In fact, much of
Inference consists in demonstrating the _connection_ of matters that
as _facts_ are pretty familiar. Of course, however, they are always
modified in the process, and in that sense there is always novelty. You
obtain the most vital idea of Inference by starting from the conclusion
as a suggestion, or even as an observation, and asking yourself how it
is proved, or explained, and treating the whole process as a single
mediate judgment, _i.e._ a reasoned affirmation. Take the observation,
“The tide at new and full moon is exceptionally high.” In scientific
inference this is filled out by a middle term. We may profitably think
of the “middle term,” as the copula or grip which holds the conclusion
together, made explicit and definitely stated. Thus the judgment pulls
out like a telescope, exhibiting fresh parts within it, as it passes
into inference. “The tide at new and full moon, _being at these times
the lunar tide plus the solar tide_, is exceptionally high.” This is
the sort of inference which is really commonest in science. Such an
inference would no doubt give us the conclusion if we did not know it
by observation, but it happens in many cases that we do know it by
observation, and what the inference gives us is the connection, which
of course may enable us to correct the observation.
{139} _Conditions of the possibility of Inference_
2. In the strictest formal sense there can be no inference from
particulars to particulars. When there seems to be such inference, it
is merely that the ground of inference is not mentioned, sometimes
because it is obvious, sometimes because it is not clearly specified
in the mind. Suppose we say, “Morley and Harcourt will go for
Disestablishment, and I think, therefore, that Gladstone will.” I do
not _express_ any connecting link, merely because every one sees at
once that I am inferring from the intentions of some Liberal leaders to
those of another. If the terms are really particulars, “X is A, Y is B,
Z is C,” one is helpless; they do not point to anything further at all;
there is no bridge from one to the other.
Inference cannot possibly take place except through the medium of an
identity or universal which acts as a bridge from one case or relation
to another. If each particular was shut up within itself as in the
letters taken as an instance just now, you could never get from one
which is given to another which is not given, or to a connection not
given between two which are given.
Take the simplest conceivable case, which hardly amounts to Inference,
that of producing a given straight line. How is it that this is
possible? Because the direction of the straight line is universal and
self-identical as against possible directions in space, and it acts as
a rule which carries you beyond the given portion of it. This might
fairly be called an “immediate inference.” So I presume that any curve
can be constructed out of a sufficient portion of the curve, although,
except with a circle, this is more than repeating the same line over
again. The content has a nature which {140} is capable of prescribing
its own continuation. A curve is not a direction; a truth which is a
puzzle to the non-mathematician--it is a law of continuous change of
direction.
_System the ultimate condition of Inference_
3. _Ultimately_ the condition of inference is always a system. And
it will help us in getting a vital notion of inference if we think,
to begin with, of the interdependence of relations in space--in
geometrical figures, or, to take a commonplace example, in the
adjustment of a Chinese puzzle or a dissected map. Or any of the
propositions about the properties of triangles are a good example.
How can one property or attribute determine another, so that you can
say, “Given this, there must be that”? This can only be answered by
pointing to the nature of a whole with parts, or a system, which just
means this, a group of relations or properties or things so held
together by a common nature that you can judge from some of them what
the others must be. Not all systems admit of precise calculation and
demonstration, but wherever there is inference at all there is at
least an identity of content which may be more or less developed into
a precise relation between parts. For example, we cannot construct
geometrically the life and character of an individual man; we can argue
from his character to some extent, but the connection of facts in his
personal identity is all that we can infer for certain; and even this
involves a certain context of facts, as in circumstantial evidence.
Yet this simplest linking together of occurrences by personal identity
is enough to give very startling inferences. Thackeray’s story of the
priest is a good instance of inference from mere identity. “An old
abbé, talking among a party of intimate friends, happened {141} to say,
‘A priest has strange experiences; why, ladies, my first penitent was
a murderer.’ Upon this, the principal nobleman of the neighbourhood
enters the room. ‘Ah, Abbé, here you are; do you know, ladies, I was
the Abbé’s first penitent, and I promise you my confession astonished
him!’” Here the inference depends solely on individual identity, which
is, as we saw, a kind of universal.
But in this case was there really an inference? Does not the conclusion
fall inside the premisses? It must in one sense fall inside the
premisses, or it is not true. But it does not fall inside them until we
have brought them into contact by their point of identity and melted
them down into the same judgment. I admit that these inferences from
individual identity, assuming the terms not to be ambiguous, are only
just within the line of rational inference, but, as we see in this
case, they bring together the parts of a very extended universal. What
_is_ the lower limit of inference?
_Immediate Inference_
4. In the doctrine of _immediate Inference_ common Logic treats of
Conversion and the Opposition of judgments.
Is a mere transposition of Subject and Predicate, where the truth
of the new judgment follows from that of the old, an inference? It
is a matter of degree. [1] Does it give anything new? “The Queen
is a woman.” “A woman is the Queen.” If we make a real difference
between the implications of a Subject and a Predicate, we seem to get
something new; but it is a point of little interest. {142} Comparison
or Recognition are more like immediate inferences. Comparison means
that we do not let ourselves perceive freely, but take a particular
content as the means of apperception of another content, _i.e._ as
the medium through which we look at it. I do not merely look at the
second, but I look at it with the first in my mind. And so far I may
be said to infer, without the form of proof, from data of perception
to a relation between them. “You are taller than me,” is a result
obtained by considering your height from the point of view of mine, or
_vice versâ_. Recognition is somewhat similar. It is more than a mere
perception, because it implies reproduction of elements not given,
and an identification with them. I recognise this man _as_ so-and-so,
_i.e._ I see he is identical with the person who did so-and-so. It is
a judgment, but it goes beyond the primary judgment, “He is such and
such,” and is really inferred from it. It is a matter of degree. Almost
every Judgment can be broken up into elements, and recognition fades
gradually into cognition--we “recognise” an example of a law, a right,
a duty, an authority; not that we knew _it_, the special case, before,
but that in analysing it we find a principle which commands our assent,
and with which we identify the particular instance before us.
[1] The collective or general judgment, as commonly explained,
cannot be converted “simply,” because the predicate is “wider”
than the subject, and the same rule is accepted for the relation
of consequent to antecedent. The aim of science, it might almost
be said, is to get beyond the kind of judgment to which this
rule applies.
_Number of Instances_
5. The difference between guess-work and demonstration rests on the
difference between a detached quality or relation striking enough to
suggest something to us, and a system thoroughly known in its parts as
depending on one another. This is so even in recognising an individual
person; it is necessary to know that the quality by which you recognise
him is one that no one else possesses, or else {143} it is guess-work.
Still more is this the case in attempting a scientific connection.
All scientific connection is really by system as between the parts
of the content. A quality is often forced on our attention by being
repeated a great many times in some particular kind of occurrence, but
as long as we do not know its _causal_ connection with the properties
and relations involved in the occurrence it is only guess-work to
treat them as essentially connected. This is a matter very easy to
confuse, and very important. It is easy to confuse, because a number
of instances does help us really in inference, as it always insensibly
gives us an immense command of content; that is to say, without knowing
it we correct and enlarge our idea of the probable connection a little
with every instance. So the connection between the properties that
strike us becomes much larger and also more correct than it is to
people who have only seen a few instances. But this is because the
instances are all a little different, and so correct each other, and
show transitions from more obvious forms to less obvious forms of the
properties in question which lead us up to a true understanding of
them. If the instances were all exactly the same they would not help
us in this way, but our guess would still be a guess, however many
instances might have suggested it.