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3 Vagueness
Bertrand Russell

Reflection on philosophical problems has convinced me that a much

are
larger number than I used to think , or than is generally thought ,
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. Sincelanguage really
occurs, it obviously has all the properties common to all occurrences,
and to that extent the metaphysic based upon linguistic considerations
are not
may not be erroneous. But language has many properties which
shared by things in general, and when these properties intrude into our
. I do not think that the
metaphysic it becomes altogether misleading
will results in
study of the principles of symbolism yield any positive
a results
metaphysics, but I do think it will yield great many negative
to things .
by enabling us to avoid fallacious inferences from symbols
The influence of symbolism on philosophy is mainly unconscious ; if it
.
were conscious it would do less harm By studying the principles of
influenced
symbolism we can learn not to be unconsciously by
, and in this way can escapea host of erroneous notions .
language
remarks. You
Vagueness, which is my topic tonight } illustrates these " Who
will no doubt think that , in the words of the poet : speaks of
II
vagueness should himself be vague . I propose to prove that all
is but I do not
language is vague and that therefore my language vague,
wish this conclusion to be one that you could derive without the help of
the syllogism . I shall be as little vague as I know how to be if I am to
know that I invented a special
employ the English language. You all
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 .

. 84 - 92 . @ Bertrand
-Journal of- Philosophy and Psychology 1 ( 1923 ) pp
From Australasian
RussellPeaceFoundation. Reprinted by permission.
1. Readbefore the Jowett Society, Oxford (22 November 1922).
62 BertrandRussell
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
becomesless 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 . They 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 also is muddleheaded
. 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. Vaguenessin 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 begin 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 either be bald or not
bald . The law of excluded middle is true when precise symbols are
 Vagueness63

employed , but it is not true when symbols are vague, as, in fact, all
symbols are. All words describing 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 reasonsthe
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 sayan 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 EbenezerWilkes 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 birth the name became attributable .
Death also is 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 ceasesto 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
64 Bertrand Russell

that no word is to be applied in the penumbra , but unfortunately the

penumbra itself is 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 physi -
ological constitution . Stimuli which for various reasonswe believe to be
different produce in us indistinguishable sensations. It is not clear
whether the sensations are really different like their stimuli and only
our power to discriminate between sensations is deficient , or 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 seewith the naked
eye the difference between two glassesof water of which one is wholesome
while the other is full of typhoid bacilli . In this casea microscope
enables us to see the difference , but in the absenceof 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 sensesdo
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 sensesinfects 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
classof 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 , 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 precisemeaning when the symbols
employed - words , perceptions , images, or what not - are themselves
precise. We have seenthat , in practice, this is not the case. It follows that
every proposition that can be framed in practice has a certain degree of
 Vagueness65
vagueness; that is to say, there is not one definite fact necessary and
sufficient for its truth , but a certain region of possible facts, anyone 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 not , 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 imaginea precise meaning for such
words as " or" and " not " . We can, in fact, seeprecisely 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 .
66 BertrandRussell
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" , becauseboth can be represented by the
"
symbol xRy" , by giving suitable meanings to x and Rand y. But this
definition , though easy to understand , does not give the essenceof 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 , becauseit 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
 Vagueness67

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 .2 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
relations , in accordance with the rules of syntax . For
represented " A by " " "
example , precedes B and B precedes A have different meanings,
becausethe 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 casewhich 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 seen 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 vaguenessin 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, becausethere 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 height 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

2. A word is a classof series, and both classesand seriesare logicalfictions. See

Russell1920, ch. 10; Russell1919, ch. 17.
68 Bertrand Russell

precisewhen only one fact would verify it ; it is accuratewhen it is both

precise and true . Precision diminish es the likelihood of truth , but often
increasesthe pragmatic value of a belief if it is true - for example, in the
caseof the water that contained the typhoid bacilli . Scienceis 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 makes scientific truth better
worth having if it can be obtained .
Vaguenessin our knowledge is, I believe, merely a particular caseof
a general law of physics, namely the law that what may be called the
appearancesof 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 distant 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 seemsto me in a position to ignore physics,
and that 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 seeno 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 .
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