Vagueness
Read before the Jowett Society, Oxford,
25 November 1922.
First published in The Australasian Journal of Psychology and Philosophy,
1 (June 1923): 84--92.
This text taken from Collected Papers, vol. 9, pp. 147--154.
Reflection on philosophical problems has
convinced me that a much larger number than I used to think, or than is generally
thought, are connected with the principles of symbolism, that is to say, with
the relation between what means and what is meant. In dealing with highly abstract
matters it is much easier to grasp the symbols (usually words) than it is to
grasp what they stand for. The result of this is that almost all thinking that
purports to be philosophical or logical consists in attributing to the world
the properties of language. Since language really occurs, it obviously has all
the properties common to all occurrences, and to that extent the metaphysic
based upon linguistic considerations may not be erroneous. But language has
many properties which are not shared by things in general, and when these properties
intrude into our metaphysic it becomes altogether misleading. I do not think
that the study of the principles of symbolism will yield any positive
results in metaphysics, but I do think it will yield a great many negative results
by enabling us to avoid fallacious inferences from symbols to things. The influence
of symbolism on philosophy is mainly unconscious; if it were conscious it would
do less harm. By studying the principles of symbolism we can learn not to be
unconsciously influenced by language, and in this way can escape a host of erroneous
notions.
Vagueness, which is my topic tonight, illustrates
these remarks. You will no doubt think that, in the words of the poet: "Who
speaks of vagueness should himself be vague." I propose to prove that all language
is vague, and that therefore my language is vague, but I do not wish this conclusion
to be one that you could derive without the help of the symbolism. I shall be
as little vague as I know how to be if I am to employ the English language.
You all know that I invented a special language with a view to avoiding vagueness,
but unfortunately it is unsuited for public occasions. I shall, therefore, though
regretfully, address you in English, and whatever vagueness is to be found in
my words must be attributed to our ancestors for not having been predominantly
interested in logic.
There is a certain tendency in those who
have realized that words are vague to infer that things also are vague. We hear
a great deal about the flux and the continuum and the unanalysability of the
Universe, and it is often suggested that as our language becomes more precise,
it becomes less adapted to represent the primitive chaos out of which man is
supposed to have evolved the cosmos. This seems to me precisely a case of the
fallacy of verbalism --- the fallacy that consists in mistaking the properties
of words for the properties of things. Vagueness and precision alike are characteristics
which can only belong to a representation, of which language is an example.
The have to do with the relation between a representation and that which it
represents. Apart from representation, whether cognitive or mechanical, there
can be no such thing as vagueness or precision; things are what they are, and
there is an end of it. Nothing is more or less what it is, or to a certain extent
possessed of the properties which it possesses. Idealism has produced habits
of confusion even in the minds of those who think that they have rejected it.
Ever since Kant there has been a tendency in philosophy to confuse knowledge
with what is known. It is thought that there must be some kind of identity between
the knower and the known, and hence the knower infers that the known is also
muddle-headed. All this identity of knower and known, and all this supposed
intimacy of the relation of knowing, seems to me a delusion. Knowing is an occurrence
having a certain relation to some other occurrence, or groups of occurrences,
or characteristic of a group of occurrences, which constitutes what is said
to be known. When knowledge is vague, this does not apply to the knowing as
an occurrence; as an occurrence it is incapable of being either vague or precise,
just as all other occurrences are. Vagueness in a cognitive occurrence is a
characteristic of its relation to that which is known, not a characteristic
of the occurrence in itself.
Let us consider the various ways in which
common words are vague, and let us being with such a word as "red". It is perfectly
obvious, since colours form a continuum, that there are shades of colour concerning
which we shall be in doubt whether to call them red or not, not because we are
ignorant of the meaning of the word "red", but because it is a word the extent
of whose application is essentially doubtful. This, of course, is the answer
to the old puzzle about the man who went bald. It is supposed that at first
he was not bald, that he lost his hairs one by one, and that in the end he was
bald; therefore, it is argued, there must have been one hair the loss of which
converted him into a bald man. This, of course, is absurd. Baldness is a vague
conception; some men are certainly bald, some are certainly not bald, while
between them there are men of whom it is not true to say they must be either
be bald or not bald. The law of excluded middle is true when precise symbols
are employed, but it is not true when symbols are vague, as, in fact, all symbols
are. All words denoting sensible qualities have the same kind of vagueness which
belongs to the word "red". This vagueness exists also, though in a lesser degree,
in the quantitative words which science has tried hardest to make precise, such
as a metre or a second. I am not going to invoke Einstein for the purpose of
making these words vague. The metre, for example, is defined
as the distance between two marks on a certain rod in Paris, when that rod is
at a certain temperature. Now the marks are not points, but patches of a
finite size, so that the distance between them is not a precise conception.
Moreover, temperature cannot be measured with more than a certain degree of
accuracy, and the temperature of a rod is never quite uniform. For all these
reasons the conception of a metre is lacking in precision. The same applies
to a second. The second is defined by relation to the rotation
of the earth, but the earth is not a rigid body, and two parts of the earth's
surface do not take exactly the same time to rotate; moreover all observations
have a margin of error. There are some occurrences of which we can say that
they take less than a second to happen, and others of which we can say that
they take more, but between the two there will be a number of occurrences of
which we believe that they do not all last equally long, but of none of which
we can say whether they last more or less than a second. Therefore, when we
say an occurrence lasts a second, all that it is worth while to mean is that
no possible accuracy of observation will show whether it lasts more or less
than a second.
Now let us take proper names. I pass by
the irrelevant fact that the same proper name often belongs to many people.
I once knew a man called Ebenezer Wilkes Smith, and I decline to believe that
anybody else ever had this name. You might say, therefore, that here at last
we have discovered an unambiguous symbol. This, however, would be a mistake.
Mr. Ebenezer Wilkes Smith was born, and being born is a gradual process. It
would seem natural to suppose that the name was not attributable before birth;
if so, there was doubt, while birth was taking place, whether the name was attributable
or not. If it be said that the name was attributable before birth, the ambiguity
is even more obvious, since no one can decide how long before before the name
became attributable. Death is also a process: even when it is what is called
instantaneous, death must occupy a finite time. If you continue to apply the
name to the corpse, there must gradually come a stage in decomposition when
the name ceases to be attributable, but no one can say precisely when this stage
has been reached. The fact is that all words are attributable without doubt
over a certain area, but become questionable within a penumbra, outside which
they are again certainly not attributable. Someone might seek to obtain precision
in the use of words by saying that no word is to be applied in the penumbra,
but unfortunately the penumbra is itself not accurately definable, and all the
vaguenesses which apply to the primary use of words apply also when we try to
fix a limit to their indubitable applicability. This has a reason in our physiological
constitution. Stimuli which for various reasons we believe to be different produce
in us indistinguishable sensations. It is not clear whether the sensations themselves
are sometimes identical in relevant respects even when the stimuli differ in
relevant respects. This is a kind of question which the theory of quanta at
some much later stage in its development may be able to answer, but for the
present it may be left in doubt. For our purpose it is not the vital question.
What is clear is that the knowledge that we can obtain through our sensations
is not as fine-grained as the stimuli to those sensations. We cannot see with
the naked eye the difference between two glasses of water of which one is wholesome
while the other is full of typhoid bacilli. In this case a microscope enables
us to see the difference, but in the absence of a microscope the difference
is only inferred from the differing effects of things which are sensibly indistinguishable.
It is this fact that things which our senses do not distinguish produce different
effects --- as, for example, one glass of water gives you typhoid while the
other does not --- that has led us to regard the knowledge derived from the
senses as vague. And the vagueness of the knowledge derived from the senses
infects all words in the definition of which there is a sensible element. This
includes all words which contain geographical or chronological constituents,
such as "Julius Caesar", "the twentieth century", or "the solar system".
There remains a more abstract class of
words: first, words which apply to all parts of time and space, such as "matter"
or "causality"; secondly, the words of pure logic. I shall leave out of discussion
the first class of words, since all of them raise great difficulties, and I
can scarcely imagine a human being who would deny that they are all more or
less vague. I come therefore to the words of pure logic, words such as "or"
and "not". Are these words also vague or have they a precise meaning?
Words such as "or" and "not" might seem,
at first sight, to have a perfectly precise meaning: "p or q"
is true when p is true, when q is true, and false when both
are false. But the trouble is that this involves the notions of "true" and "false";
and it will be found, I think, that all the concepts of logic involve these
notions, directly or indirectly. Now "true" and "false" can only have a precise
meaning when the symbols employed - words, perceptions, images, or what not
- are themselves precise. We have seen that, in practice, this is not the
case. It follows that every proposition that can be framed in practice has a
certain degree of vagueness; that is to say, there is not one definite fact
necessary and sufficient for its truth, but a certain region of possible facts,
any one of which would make it true. And this region is itself ill-defined:
we cannot assign to it a definite boundary. This is the difference between vagueness
and generality. A proposition involving a general concept - e.g. "This is
a man" - will be verified by a number of facts, such as "This" being Brown
or Jones or Robinson. But if "man" were a precise idea, the set of possible
facts that would verify "this is a man" would be quite definite. Since, however,
the conception "man" is more or less vague, it is possible to discover prehistoric
specimens concerning which there is no, even in theory, a definite answer to
the question "Is this a man?" As applied to such specimens, the proposition
"this is a man" is neither definitely true nor definitely false. Since all non-logical
words have this kind of vagueness, it follows that the conceptions of truth
and falsehood, as applied to propositions composed of or containing non-logical
words, are themselves more or less vague. Since propositions containing non-logical
words are the substructure on which logical propositions are built, it follows
that logical propositions also, so far as we can know them, become vague through
the vagueness of "truth" and "falsehood". We can see an ideal of precision,
to which we can approximate indefinitely; but we cannot attain this ideal. Logical
words, like the rest, when used by human beings, share the vagueness of all
other words. There is, however, less vagueness about logical words than about
the words of daily life, because logical words apply essentially to symbols,
and may be conceived as applying rather to possible than to actual symbols.
We are capable of imagining what a precise symbolism would be, though we cannot
actually construct such a symbolism. Hence we are able to imagine a
precise meaning for such words as "or" and "not". We can, in fact, see precisely
what they would mean if our symbolism were precise. All traditional logic habitually
assumes that precise symbols are being employed. It is therefore not applicable
to this terrestrial life, but only to an imagined celestial existence. Where,
however, this celestial existence would differ from ours, so far as logic is
concerned, would be not in the nature of what is known, but only in the accuracy
of our knowledge. Therefore, if the hypothesis of a precise symbolism enables
us to draw any inferences as to what is symbolized, there is no reason to distrust
such inferences merely on the ground that our actual symbolism is not precise.
We are able to conceive precision; indeed, if we could not do so, we could not
conceive vagueness, which is merely the contrary of precision. This is one reason
why logic takes us nearer to heaven than most other studies. On this point I
agree with Plato. But those who dislike logic will, I fear, find my heaven disappointing.
It is now time to tackle the definition
of vagueness. Vagueness, though it applies primarily to what is cognitive, is
a conception applicable to every kind of representation - for example, a photograph,
or a barograph. But before defining vagueness it is necessary
to define accuracy. One of the most easily intelligible definitions of accuracy
is as follows: One structure is an accurate representation of another when the
words describing the one will also describe the other by being given new meanings.
For example, "Brutus killed Caesar" has the same structure as "Plato loved Socrates",
because both can be represented by the symbol "xRy", by giving suitable
meanings to x and R and y. But this definition, though
easy to understand, does not give the essence of the matter, since the introduction
of words describing the two systems is irrelevant. The exact definition is as
follows: One system of terms related in various ways is an accurate representation
of another system of terms related in various other ways if there is a one-one
relation of the terms of the one to the terms of the other, and likewise a one-one
relation of the relations of the one to the relations of the other, such that,
when two or more terms in the one system have a relation belonging to that system,
the corresponding terms of the other system have the corresponding relation
belonging to the other system. Maps, charts, photographs, catalogues, etc. all
come within this definition in so far as they are accurate.
Per contra, a representation is
vague when the relation of the representing system to the represented
system is not one-one, but one-many. For example, a photograph which is so smudged
that it might equally represent Brown or Jones or Robinson is vague. A small-scale
map is usually vaguer than a large-scale map, because it does not show all the
turns and twists of the roads, rivers, etc. so that various slightly different
courses are compatible with the representation that it gives. Vagueness, clearly,
is a matter of degree, depending upon the extent of the possible differences
between different systems represented by the same representation. Accuracy,
on the contrary, is an ideal limit.
Passing from representation in general
to the kinds of representation that are specially interesting to the logician,
the representing system will consist of words, perceptions, thoughts, or something
of the kind, and the would-be one-one relation between the representing system
and the represented system will be meaning. In an accurate language,
meaning would be a one-one relation; no word would have two meanings, and no
two words would have the same meaning. In actual languages, as we have seen,
meaning is one-many. (It happens often that two words have the same meaning,
but this is easily avoided, and can be assumed not to happen without injuring
the argument.) That is to say, there is not only one object that a word means,
and not only one possible fact that will verify a proposition. The fact that
meaning is a one-many relation is the precise statement of the fact that all
language is more or less vague. There is, however, a complication
about language as a method of representing a system, namely that words which
mean relations are not themselves relations, but just as substantial or unsubstantial
as other words. In this respect a map, for instance, is superior to language,
since the fact that one place is to the west of another is represented by the
fact that the corresponding place on the map is to the left of the other; that
is to say, a relation is represented by a relation. But in language this is
not the case. Certain relations of higher order are represented by relations,
in accordance with the rules of syntax. For example, "A precedes B"
and "B precedes A" have different meanings, because the order
of the words is an essential part of the meaning of the sentence. But this does
not hold of elementary relations; the word "precedes", though it means a relation,
is not a relation. I believe that this simple fact is at the bottom of the hopeless
muddle which has prevailed in all schools of philosophy as to the nature
of relations. It would, however, take me too far from my present theme to pursue
this line of thought.
It may be said: How do you know that all
knowledge is vague, and what does it matter if it is? The case which I took
before, of two glasses of water, one of which is wholesome while the other gives
you typhoid, will illustrate both points. Without calling in the microscope
, (it is obvious that you cannot distinguish the wholesome
glass of water from the one that will give you typhoid, just as, without calling
in the telescope,) it is obvious that what you see of a man who is
200 yards away is vague compared to what you see of a man who is 2 feet away;
that is to say, many men who look quite different when see close at hand look
indistinguishable at a distance, while men who look different at a distance
never look indistinguishable when seen close at hand. Therefore, according to
the definition, there is less vagueness in the near appearance than in the distant
one. There is still less vagueness about the appearance under the microscope.
It is perfectly ordinary facts of this kind that prove the vagueness of most
of our knowledge, and lead us to infer the vagueness of all of it.
It would be a great mistake to suppose
that vague knowledge must be false. On the contrary, a vague belief has a much
better chance of being true than a precise one, because there are more possible
facts that would verify it. If I believe that so-and-so is tall, I am more likely
to be right than if I believe that his heigh is between 6 ft. 2 in. and 6 ft.
3 in. In regard to beliefs and propositions, though not in regard to single
words, we can distinguish between accuracy and precision. A belief is precise
when only one fact would verify it; it is accurate when it is both
precise and true. Precision diminishes the likelihood of truth, but often increases
the pragmatic value of a belief if it is true --- for example, in the case of
the water that contained the typhoid bacilli. Science is perpetually trying
to substitute more precise beliefs for vague ones; this makes it harder for
a scientific proposition to be true than for the vague beliefs of uneducated
persons to be true, but it makes scientific truth better worth having if it
can be obtained.
Vagueness in our knowledge is, I believe,
merely a particular case of a general law of physics, namely that law that what
may be called the appearances of a thing at different places are less and less
differentiated as we get further away from the thing. When I speak of "appearances"
I am speaking of something purely physical --- the sort of thing, in fact, that,
if it is visual, can be photographed. From a close-up photograph it is possible
to infer a photograph of the same object at a distance, while the contrary inference
is much more precarious. That is to say, there is a one-many relation between
distant and close-up appearances. Therefore the distance appearance, regarded
as a representation of the close-up appearance, is vague according to our definition.
I think all vagueness in language and thought is essentially analogous to this
vagueness which may exist in a photograph. My own belief is that most of the
problems of epistemology, in so far as they are genuine, are really problems
of physics and physiology; moreover, I believe that physiology is only a complicated
branch of physics. The habit of treating knowledge as something mysterious and
wonderful seems to me unfortunate. People do not say that a
barometer "knows" when it is going to rain; but I doubt if there is any essential
difference in this respect between the barometer and the meteorologist who observes
it. There is only one philosophical theory which seems to me in a position
to ignore physics, and this is solipsism. If you are willing to believe that
nothing exists except what you directly experience, no other person can prove
that you are wrong, and probably no valid arguments exist against your view.
But if you are going to allow any inferences from what you directly experience
to other entities, then physics supplies the safest form of such inferences.
And I believe that (apart from illegitimate problems derived from misunderstood
symbolism) physics, in its modern forms, supplies materials for answers to all
philosophical problems that are capable of being answered, except the one problem
raised by solipsism, namely: Is there any valid inference ever from an entity
experienced to one inferred? On this problem, I see no refutation of the sceptical
position. But the sceptical philosophy is so short as to be uninteresting; therefore
it is natural for a person who has learnt to philosophize to work out other
alternatives, even if there is no very good ground for regarding them as preferable.
Notes:
(1) The metre is now
(1995) defined optically, as so many wavelengths of a certain frequency of light,
and not by the standard metre stick; this hardly effects the Russell's point,
however. [CRS]
(2) Again, the second
is now defined in terms of the vibrations of the cesium atom; but, as with the
re-defined metre, these changes don't signify for the purpose at hand. [CRS]
(3) According to the
on-line Webster's, "a recording barometer." [CRS]
(4) A word is a class
of series, and both classes and series are logical fictions. See Analysis of
Mind, Chap. x; Introduction to Mathematical Philosophy, Ch. xvii. [Russell's
note.]
(5) There is an obvious
gap in the text here, both in the published version and the unpublished revised
typescript. To quote the editors of the Collected Papers, ``since Russell's
meaning is obvious, the missing lines have been reconstructed here.'' Following
their lead, I have enclosed the inserted words in angle-brackets. [CRS]
(6) Actually, laboratory
scientists say things like this all the time; and this way of speaking has become
much more common with the spread of computers. Whether it is anything more than
metaphorical is, however, a vexed question. Surely one difference between the
barometer and the meteorologist is that the meteorologist has a representation
of rain at a future time, which the barometer (so far as I can see) does not?
[CRS]
On
Denoting (1905)
Vagueness (1923)
Icarus
or The Future of Science (1924)
What is an Agnostic
Knowledge and
Wisdom
Why I am
not a Christian (06.03.1927)
In
Praise of Idleness (1932)
Of Co-Operation
(18.05.1932)
On Sales Resistance
(22.06.1932)
On Modern
Uncertainty (20.07.1932)
What is the Soul?
(28.09.1932)
On youthful
Cynism (1930)
Philosophical
Consequences of Relativity (1626)
On Astrologers
How
to become a Man of Genius
Education
and Disciple
What
Desires are politically important? (Speech at the Nobel Award, 11.12.1950)
Prolog to his Autobiography: What
I have lived for