Analytic_OnDenoting_1905.txt

Mind Association

On Denoting
Author(s): Bertrand Russell
Source: Mind, New Series, Vol. 14, No. 56 (Oct., 1905), pp. 479-493
Published by: Oxford University Press on behalf of the Mind Association
Stable URL: https://www.jstor.org/stable/2248381
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II.-ON DENOTING.
By BERTRAND RUSSELL.
By a " denioting phrase " I mean a phrase such as any one

of the following: a man, some man, any man, every man,

all men, the present King of England, the present King of

France, the centre of mass of the Solar System at the first
instant of the twentieth century, the revolution of the earth
round the sun, the revolution of the sun round the earth.
Thus a phrase is denoting solely in virtue of its form. We
may distinguish three cases: (1) A phrase may be denoting,
and yet not denote anything; e.g., "{the present King of
France (2) A phrase may denote one definite object; e.g.,
"the present King of England " denotes a certain man. (3)
A phrase may denote ambiguously; e.g., " a man " denotes
not many men, but an ambiguous man. The interpretation
of such phrases is a matter of considerable difficultv; indeed,
it is very hard to frame any theory not susceptible of formal
refutation. All the difficulties with which I am acquainted
are met, so far as I can discover, by the theory which I am
about to explain.
The subject of denoting is of verv great importance, not
only in logic and matheinatics, but also in theory of know-

ledge. For example, we know that the centre of m-ass of the
Solar System at a definite instant is some definite point, and
we can affirm a number of propositions abouit it; but we
have no immediate acquaintance with this point, which is
only known to us by description. The distinction between
acquaintance and knowledge about is the distinction between
the things we have presentations of, and the things we only
reach by means of denoting phrases. It often happens that

we know that a certain phrase denotes unambiguously, although we have no acquaintance with what it denotes; this
occurs in the above case of the centre of mass. In perception we have acquaintance with the objects of perception,
and in thought we have acquaintance with objects of a more
abstract logical character; but we do not necessarily have
acquaintance with the objects denoted by phrases composed

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480 BERTRAND RUSSELL:

of words with whose meanings we are acquainted. To take
a very important instance: There seems no reason to believe
that we are ever acquainted with other people's minds, seeing
that these are not directly perceived; hence what we know
about them is obtained through denoting. All thinking has
to start from acquaintance; but it succeeds in thinking about
many things with which we have no acquaintance.

The course of my argument will be as follows. I shall
begin by stating the theory I intend to advocate; I I shall
then discuss the theories of Frege and Meinong, showing
why neither of them satisfies me; then I shall give the
grounds in favour of my theory; and finally I shall briefly
indicate the philosophical consequences of my theory.
My theory, briefly, is as follows. I take the notion of the
variable as fundamental; I use " C (x) " to mean a proposition2 in which x is a constituent, where x, the variable, is
essentially and wholly undetermined. Then we can consider
the two notions "C (x) is always true" and " C (x) is sometimes true ".3 Then everything and nothing and something
(which are the most primitive of denoting phrases) are to
be interpreted as follows:
C (everything) means " C (x) is always true";

C (nothing) means "'C (x) is false' is always true";
C (something) means "It is false that ' C (x) is false' is
always true ".

Here the notion " C (x) is always true" is taken as ultimate
and indefinable, and the others are defined by means of it.
Everything, nothing, and something, are not assumed to have any

meaning in isolation, but a meaning is assigned to every proposition in which they occur. This is the principle of the
theory of denoting I wish to advocate: that denoting phrases
never have any meaning in themselves, but that every proppsition in whose verbal expression they occur has a mean-

ing. The difficulties concerning denoting are, I believe, atl
the result o*f a wrong analysis of propositions whose verbal
expressions contain denoting phrases. The proper analysis,

if I am not mistaken, may be further set forth as follows.

I I have discussed this subject in Principles of Mathematics, chapter
v., and ? 476. The theory there advocated is very nearly the same as
Frege's, and is quite different from the theory to be advocated in what

follows.

2 More exactly, a propositional function.
3 The second of these can be defined by mieans of the first, if we take

it to mean, "It is not true that 'C (x) is false' is always true ".
4I shall sometimes use, instead of this complicated phrase, the phrase
"C (x) is not always false," or " C (x) is sometimes true," supposed defined
to mean the same as the complicated phrase.

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ON

DENOTING.

481

Suppose now we wish to interpret the proposition, " I met
a man ". If this is true,. I met some definite man; but that
is not what I affirm. What I affirm is, according to the theory
I advocate:
" ' I met x, and x is human' is not always false ".
Generally, defining the class of men as the class of objects
having the predicate human, we say that:
4'C (a man)" means "' C (x) and x is human' is not always
false ".
This leaves "a man," by itself, wholly destitute of meaning,
but gives a meaning to every proposition in whose verbal
expression "' a man " occurs.
Consider next the proposition " all men are mortal".
This proposition 1 is really hypothetical and states that if anything is a man, it is mortal. That is, it states that if x is
a man, x is mortal, whatever x may be. Hence, substituting
'x is human' for' x is a man,' we find:"All men are mortal " means "' If x is human, x is mortal'
is always true ".

This is what is expressed in symbolic logic by saying that
" all men are mortal" means "'x is human' implies 'x is
mortal' for all values of x". More generally, we say:
4C C(all men) ' means "' If x is human, then C (x) is true' is
always true
Similarly
"C (no men)" means "' If x is human, then C (x) is false'
is always true".

"C (some men) " will mean the same as " C (a man)," 2 and
''C (a man) " means " It is false that ' C (x) and x is human'
is always false ".
"C (every man)" will mean the same as " C (all men)
It remains to interpret phrases containing the. These are
by far the most interesting and difficult of denoting phrases.
;Take as an instance " the father of Charles II. was executed ".
This asserts that there was an x who was the father of
Charles II. and was executed. Now the, when it is strictly
used, involves uniqueness; we do, it is true, speak of " the son
,of So-and-so " even when So-and-so has several sons, but it
would be more correct to say " a son of So-and-so ". Thus
for our purposes we take the as involving uniqueness. Thus
when we say " x was the father of Charles II." we not only
.assert that x had a certain relation to Charles II., but also
I As has been ably argued in Mr. Bradley's Logic, book i., chap. ii.

2Psychologically " C (a man) " has a suggestion of only onie, and " C

(some men) " has a suggestion of mnore than one; but we may neglect
*these suggestions in a preliminary sketch.

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482 BERTRAND RUSSELL:

that nothing else had this relation. The relation in question, without the assumption of uniqueness, and without any
denoting phrases, is expressed by " x begat Charles I.". To

get an equivalent of " x was the father of Charles II.," we
mNust add, " If y is other than x, y did niot beget Charles II)"

or, what is equivalent, " If y begat Charles II., y is identical
with x ". Hence " x is the father of Charles II." becomnes
" begat Charles IL; and ' it y begat Charles IL, y is identical

with x' is always true of y"

Thus " the father of Charles IM was executed " becomues
It is not always false of x that x begat Charles II. and that

x was executed and that 'if y begat Charles IL, y is

identical with x' is always true of y".
This mnay seem a somewhat incredible interpretation; but

I am not at present giving reasons, I am merely stating the

theory.
To interpret " C (the father of Charles II.)," where C
stands for any statement about him, we have only to substitute C (x) for " x was executed" in the above. Observe
that, accordina to the above interpretation, whatever state-

ment C may be, " C (the father of Charles II.)" implies:
"It is not always false of x that ' if y begat Charles II., y is
identical with x' is always true of y,"

which is what is expressed in common language by " Charles
II. had one father and no more". Consequently if this con(ition fails, every proposition of the formn " C (the father of
Charles II.)" is false. Thus e.g. every proposition of the
formn " C (the present Kinig of France) " is false. This is a
great advantage in the present theory. I shall show later

that it is not contrary to the law of contradiction, as light

be at first supposed.
The above gives a reduction of all propositions in which
lenoting phrases occur to forms in which no such phrases

occur. Why it is imperative to effect such a reduction, the

subsequent discussion will endeavour to show.
The evidence for the above theory is derived from the
difficulties which seem unavoidable if we regard denoting
phrases as standing for genuine constituents of the propositions in whose verbal expressions they occur. Of the pos-

sible theories which admit such constituents the simplest is
that of Meinong.1 This theory regards any grammatically
correct denoting phrase as standing for an object. Thus

" the present King of France," " the round square," etc., are

I See L'ntersw.Atuinqe.n xur (egenstandatheorie utnd Psychologie, Leip
zig, 1904, the first three articles (by Ml eiuiong, Amaeseder and Mally r
spectively).

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ON DENOTING. 483

supposed to be genuine objects. It is admitted that such
objects do not su6bsist, but nevertheless they are supposed to

be objects. This is in itself a difficult view; but the chief

objection is that such objects, admittedly, are apt to infrioge
the law of contradiction. It is contended, for example, that
the existent present King of France exists, and also does not
exist; that the round square is round, and also not round;
etc. But this is intolerable; and if any theory can be found
to avoid this result, it is surely to be preferred.

The above breach of the law of contradiction is avoided by

Frege's theory. He distinguishes, in a denoting phrase, two

elements, which we may call the meaninzg and the denotation.'

Thus " the centre of mass of the Solar System at the begin-

ning of the twentieth century " is highly complex in meaning,
but its denotation is a certain point, which is simple. The
Solar System, the twentieth century, etc., are constituents of
the meaning; but the denotation has no constituents at all.2
One advantage of this distinction is that it shows why it is
often worth while to assert identity. If we say " Scott is
the author of Waverley," we assert an identity of denotation
with a difference of meaning. I shall, however, not repeat
the grounds in favour of this theory, as I have urged its
claims elsewhere (loc. cit.), and am now concerned to dispute
those claims.

One of the first difficulties that confront us, when we adopt
the view that denoting phrases express a meaning and denote
a denotation,3 concerns the cases in which the denotation

appears to be absent. If we say " the King of England is
bald," that is, it would seem, not a statement about the
complex meaning "the King of England," but .about the
actual man denoted by the meaning. But now consider
" the King of France is bald ". By parity of form, this also
ouight to be about the denotation of the phrase " the King of
France ". But this phrase, though it has a neaning provided
' See his " Ueber Sinn und Bedeutung," Zeitschrift filr Phil. untd Phil.

Kritik, vol. 100.

2Frege distinguishes the two elements of meaning and denotation
everywhere, and not only in complex denoting phrases. Thus it is the
meanings of the constituents of a denoting complex that enter into its
meaning, not their denotation. In the proposition " Mont Blanc is over

1,000 metres high," it is, according to him, the meaning of "c Mont Blanc,"
not the actual mountain, that is a constituent of the nmeaning of the proposition.

3In this theory, we shall say that the denoting phrase expresses a,
meaning; and we shall say both of the phrase and of the meaning that
they denote a denotation. In the other theory, which I advocate, there

is no meaning, and only sometimes a denotation.

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484

BERTRAND

RUSSELL:

" the King of England " has a meaning, has certainly no denotation, at least in any obvious sense. Hence one would
suppose that "the King of France is bald" ought to be
nonsense; but it is not nonsense, since it is plainly false.
Or again consider such a proposition as the following: " If u
is a class which has only one member, then that one member
is a member of u," or, as we may state it, " If u is a unit
class, the u is a u ". This proposition ought to be always
true, since the conclusion is true whenever the hypothesis is
true. But " the u " is a denoting phrase, and it is the denotation, not the meaning, that is said to be a u. Now if u
is not a unit class, "the u " seems to denote nothing; hence
our proposition would seem to become nonsense as soo-n as

mu is not a unit class.
Now it is plain that such propositions do not become
nonsense merely because their hypotheses are false. The

King in " The Tempest " might say, " If Ferdinand is not
drowned, Ferdinand is my only son ". Now " my only son "
is a denoting phrase, which, on the face of it, has a denotation when, and only when, I have exactly one son. But the
above statement would nevertheless have remained true if
Ferdinand had been in fact drowned. Thus we must either
provide a denotation in cases in which it is at first sight
absent, or we must abandon the view that the denotation is
what is concerned in propositions which contain denoting
phrases. The latter is the course that I advocate. The
former course may be taken, as by Meinong, by admitting
objects which do not subsist, and denying that they obey
the law of contradiction; this, however, is to be avoided if
possible. Another way of taking the same course (so far as
our present alternative is concerned) is adopted by Frege,
who provides by definition some purely conventional denotation for the cases in which otherwise there would be none.
Thus "the King of France," is to denote the null-class;

" the only son of Mr. So-and-so " (who has a fine family of
ten), is to.denote the class of all his sons; and so on. But
this procedure, though it may not lead to actual logical error,
is plainly artificial, and does not give an exact analysis of
the matter. Thus if we allow that denoting phrases, in
general, have the two sides of meaning and denotation, the
cases where there seems to be no denotation cause difficulties
both on the assumption that there really is a denotation and
on the assumption that there really is none.
A logical theory may be tested by its capacity for dealing
with puzzles, and it is a wholesome plan, in thinking about
logic, to stock the mind with as many puzzles as possible,

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ON

DENOTING.

485

since these serve much the same purpose as is served by
experiments in physical science. I shall therefore state three
puzzles which a theory as to denoting ought to be able to
solve; and I shall show later that my theory solves them.
(1) If a is identical with b, whatever is true of the one is
true of the other, and either may be substituted for the other
in any proposition without altering the truth or falsehood of
that proposition. Now George IV. wished to know whether
Scott was the author of Wcaverley; and in fact Scott was
the author of Waverley. Hence we may substitute Scott for
the author of " TWaverley," and thereby prove that George IV..
wished to know whether Scott was Scott. Yet an interest
in the law of identity can hardly be attributed to the first.
gentleman of Europe.
(2) By the law of excluded middle, either "A is B " or
"A is not B" must be true. Hence either "the present.
King of France is bald " or " the present King of France is
not bald" must be true. Yet if we enumerated the things
that are bald, and then the things that are not bald, we
should not find the present King of France in either list.
Hegelians, who love a synthesis, will probably conclude that
he wears a wig.
(3) Consider the proposition " A differs from B ". If this
is true, there is a difference between A and B, which fact
may be expressed in the form " the difference between A and
B subsists ". But if it is false that A differs from B, then
there is no difference between A and B, which fact may be
expressed in the form " the difference between A and B does.
not subsist ". But how can a non-entity be the subject of
a proposition ? "I think, therefore I am " is no more evident
than "I am the subject of a proposition, therefore I am,"
provided "I am" is taken to assert subsistence or being,1not existence. Hence, it would appear, it must always be
self-contradictory to deny the being of anything; but we
have seen, in connexion with Meinong, that to admit being
also sometimes leads to contradictions. Thus if A and B
do not differ, to suppose either that there is, or that there is.
not, such an object as "the difference between A and B"
seems equally impossible.
The relation of the meaning to the denotation involves,
certain rather curious difficulties, which seem in themselves
sufficient to prove that the theory which leads to such diffi-culties must be wrong.

When we wish to speak about the meaning of a denoting
I I use these as synonyms.

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486

BERTRAND

RUSSELL:

phrase, as opposed to its denotation, the natural mode of
doing so is by inverted commas. Thus we say:
The centre of mass of the Solar System is a point, not a
denoting complex;
"The centre of mass of the Solar System" is a denoting
complex, not a point.
Or again,
The first line of Gray's Elegy states a proposition.
" The first line of Gray's Elegy " does not state a proposition. Thus taking any denoting phrase, say C, we wish to
consider the relation between C and " C," where the difference of the two is of the kind exemplified in the above two
instances.
We say, to begin with, that when C occurs it is the
denotation that we are speaking about; but when " C " occurs,

it is the meaning. Now the relation of meaning and denota-

tion is not merely linguistic through the phrase: there must
be a logical relation involved, which we express by saving
that the meaning denotes the denotation. But the difficulty
which confronts us is that we cannot succeed in both preserving the connexion of meaning and denotation and preventing
them from being one and the same; also that the meaning
cannot be got at except by means of denoting phrases. This
happens as follows.
The one phrase C was to have both meaning and denotation. But if we speak of " the meaning of C," that gives us
the meaning (if any) of the denotation. "The meaning of
the first line of Gray's Elegy " is the same as "The meaning
of ' The curfew tolls the knell of parting day,' and is not the
same as " The meaning of 'the first line of Gray's Elegy "'.
Thus in order to get the meaning we want, we must speak
not of "the meaning of C," but of "the meaning of ' C,' "
which is the same as " C " by itself. Similarly " the denotation of C " does not mean the denotation we want, but means
something which, if it denotes at all, denotes what is denoted

by the denotation we want. For example, let " C " be " the

denoting complex occurring in the second of the above instances". Then
C =" the first line of Gray's Elegy," and
the denotation of C = The curfew tolls the knell of parting day.
But what we meant ta have as the denotation was " the first
line of Gray's Elegy". Thus we have failed to get what
we wanted.

The difficulty in speaking of the meaning of a denoting,
complex may be stated thus; The moment we put ihe coinplex in a proposition, the proposition is about the denotation;

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ON DENOTING. 487

and if we make a proposition in which the subject is " the
meaning of C," then the subject is the meaning (if any) of
the denotation, which was not intended. This leads us to
say that, when we distinguish meaning and denotation, we
must be dealing with the meaning: the meaning has denotation and is a complex, and there is not something other than
the meaning, which can be called the complex, and be said
to have both meaning and denotation. The right phrase,
on the view in question, is that some meanings have denotations.
But this only makes our difficulty in speaking of meanings
more evident. For suppose C is our complex; then we are
to say that C is the meaning of the complex. Nevertheless,
whenever C occurs without inverted commas, what is said
is not true of the meaning, but only of the denotation, as
when we say: The centre of mass of the Solar System is a
point. Thus to speak of C itself, i.e., to make a proposition
about the meaning, our subject must not be C, but something

which denotes C. Thus " C," which is what we use when
we want to speak of the meaning, must be not the meaning,
but something which denotes the meaning. And C must not
be a constituent of this complex (as it is of " the meaning of
C "); for if C occurs in the complex, it will be its denotation,
not its meaning, that will occur, and there is no backward
road from denotations to meanings, because every object can
be denoted by an infinite number of different denoting phrases.
Thus it would seem that " C " and C are different entities,
such that " C " denotes C; but this cannot be an explanation,
because the relation of " C " to C remains wholly mysterious;
and where are we to find the denoting complex " C " which
is to denote C ? Moreover, when C occurs in a proposition,
it is not only the denotation that occurs (as we shall see in
the next paragraph); yet, on the view in question, C is only
the denotation, the meaning being wholly relegated to " C '.
This is an inextricable tangle, and seems to prove that the
whole distinction of meaning and denotation has been wrongly
conceived.
That the meaning is relevant when a denoting phrase
occurs in a proposition is formally proved by the puzzle
about the author of Waverley. The proposition " Scott was
the author of Wtverley" has a property not possessed by
" Scott was Scott," namely the property that George IV.
wished to know whether it was true. Thus the two are not
identical propositions; hence the meaning of " the author of

Wcaverley" must be relevant as well as the denotation, if we

adhere to the point of view to which this distinction belongs.

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488 BERTRAND RUSSELL:

Yet, as we have just seen, so long as we adhere to this point,
of view, we are compelled to hold that only the denotation
can be relevant. Thus the point of view in question must
be abandoned.
It remains to show how all the puzzles we have been considering are solved by the theory explained at the beginning
of this article.
According to the view which I advocate, a denoting phrase

is essentially part of a sentence, and does not, like most

single words, have any significance on its own account. If

I say " Scott was a man," that is a statement of the formn
" x was a man," and it has " Scott " for its subject. But
if I say " the author of Wcaverley was a man," that is not a,
statement of the form "x was a man," and does not have " the
author of Waverley " for its subject. Abbreviating the statement made at the beginning of this article, we may put, in
place of "the author of Waverley was a man," the following: " One and only one entity wrote Waverley, and that
one was a man". (This is not so strictly what is meant as
what was said earlier; but it is easier to follow.) And speaking generally, suppose we wish to say that the author of

Waverley had the property 0, what we wish to say is equi

lent to " One and only one entity wrote Waverley, and that

one had the property 0 ".
The explanation of denotation is now, as follows. Every
proposition in which " the author of Waverley " occurs
being explained as above, the proposition "Scott was the
author of Waverley" (i.e. " Scott was identical with the
author of Waverley") becomes " One and only one entity
wrote Waverley, and Scott was identical with that one"; or,
reverting to the wholly explicit form: "It is not always
false of x that x wrote Waverley, that it is always true of y
that if y wrote Waverley y is identical with x, and that Scott
is identical with x ". Thus if " C " is a denoting phrase, it

may happen that there is one entity x (there cannot be more
than one) for which the proposition "x is identical with C"
is true, this proposition being interpreted as above. We
may then say that the entity x is the denotation of the
phrase "C ". Thus Scott is the denotation of " the author
of Waverley". The "-C " in inverted commas will be merely
the phrase, not anything-that can be called the meaning. The
phrase per se has no meaning, because in any proposition in
which it occurs the proposition, fully expressed, does not
contain the phrase, which has been broken up.
The puzzle about George IV.'s curiosity is now seen to
have a very simple solution. The proposition " Scott was

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ON DENOTING. 489

the author of Waverley," which was written out in its unabbreviated form in the preceding paragraph, does not contain any constituent "the author of Waverley " for which
we could substitute " Scott ". This does not interfere with
the truth of inferences resulting from making what is verbally
the substitution of " Scott " for " the author of Waverley," so
long as " the author of Waaverley" has what I call a primary
occurrence in the proposition considered. The difference of
primary and secondary occurrences of denoting phrases is
as follows:When we say: " George IV. wished to know whether soand-so," or when we say " So-and-so is surprising" or " So-

and-so is true," etc., the " so-and-so " must be a proposition.
Suppose now that " so-and-so " contains a denoting phrase.
We may either eliminate this denoting phrase from the
;subordinate proposition " so-and-so," or from the whole proposition in which " so-and-so " is a mere constituent. Different propositions result according to which we do. I have
heard of a touchy owner of a yacht to whom a guest, on first
seeing it, remarked, " I thought your yacht was larger than
it is"; and the owner replied, " No, my yacht is not larger
than it is ". What the guest meant was, " The size that I
thought your yacht was is greater than the size your yacht
is "; the meaning attributed to him is, " I thought the size
of your yacht was greater than the size of your yacht". To
return to George IV. and Waverley, when we say, "George
IV. wished to know whether Scott was the author of
Woaverley," we normally mean " George IV. wished to know
whether one- and only one man wrote Waverley and Scott
was that man"; but we may also mean: "One and only
one -man wrote Waverley, and George IV. wished to know
whether Scott was that man ". In the latter, " the author
of Waverley" has a primary occurrence; in the former, a
secondary. The latter might be expressed by "George IV.
wished to know, concerning the man who in fact wrote
Waverley, whether he was Scott ". This would be true, for
example, if George IV. had seen Scott at a distance, and
had asked "Is that Scott?" A secondary occurrence of a
denoting phrase may be defined as one in which the phrase
occurs in a proposition p which is a mere counstituent of the
proposition we are considering, and the substitution for the
denoting phrase is to be effected inmp, not in the whole proposition concerned. The ambiguity as between primary and
secondary occurrences is hard to avoid in language; but it

does no harm if we are on our guard against it. In symbolic
logic it is of course easily avoided.
33

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490 BERTRAND -RUSSELL:

The distinction of primary and secondary occurrences also
enables us to deal with the question whether the present
King of France is bald or not bald, and generally with the
logical status of denoting phrases that denote nothing. If
"C" is a denoting phrase, say " the term having the property F," then
" C has the property b" means "one and only one term
has the property F, and that one has the property b".6
If now the property F belongs to no terms, or to several, it
follows that " C has the property sb" is false for all values
of '. Thus " the present King of France is bald " is certainly
false; and " the present King of France is not bald" is false
if it means
"There is an entity which is now King of France and is not
bald,
but is true if it means
"It is false that there is an entity which is now King of
France and is bald ".
That is, "the King of France is not bald " is false if the
occurrence of " the King of France " is primary, and true if
it is secondary. Thus all propositions in which " the King of
France" has a primary occurrence are false; the denials of
such propositions are true, but in them " the King of France "
has a secondary occurrence. Thus we escape the conclusion
that the King of France has a wig.
We can now see also how to deny that there is such an
object as the difference between A and B in the case when A
and B do not differ. If A and B do differ, there is one and
only one entity x such that " x is the difference between A and
B " is a true proposition; if A and B do not differ, there is
no such entity x. Thus according to the meaning of denotation lately explained, " the difference between A and B " has
a denotation when A and B differ, but not otherwise. This
difference applies to true and false propositions generally. If
"'a R b " stands for " a has the relation R to b," then when
aR b is true, there is such an entity as the relation R between
a and b; when a R b is false, there is no such entity. Thus

out of any7 proposition we can make a denoting phrase, which

denotes an entity if the proposition is true, but does not denote an entity if the proposition is false. E.g., it is true (at
least we will suppose so) that the earth revolves round the
sun, and false that the sun revolves round the earth; hence
"the revolution of the earth round the sun" denotes an
I This is the abbreviated, not the stricter, interpretation.

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ON DENOTING. 491

entity, while " the revolution of the sun round the earth"
does not denote an entity.'

The whole realm of non-entities, such as " the round
square," "the even prime other than 2," " Apollo," " Hamlet," etc., can now be satisfactorily dealt with. All these are

denoting phrases which do not denote anything. A pro-

position about Apollo means what we get by substituting
what the classical dictionary tells us is meant by Apollo,
say " the sun-god ". All propositions in which Apollo occurs
are to be interpreted by the above rules for denoting phrases.
If " Apollo " has a primary occurrence, the proposition containing the occurrence is false; if the occurrence is secondary,

the proposition may be true. So again " the round square is

round " means " there is one and only one entity x which is
round and square, and that entity is round," which is a
false proposition, not, as Meinong maintains, a true one.

" The most perfect Being has all perfections; existence is

a perfection; therefore the most perfect Being exists" becomes:

" There is one and only one entity x which is most perfect;

that one has all perfections; existence is a perfection; there-

fore that one exists". As a proof, this fails for want of a
proof of the premiss "there is one and only one entity x
which is most perfect "'.2
Mr. MacColl (MIND, N.S., No. 54, and again No. 55, p. 401)
regards individuals as of two sorts, real and unreal; hence
he defines the null-class as the class consisting of all unreal
individuals. This assumes that such phrases as " the

present King of France," which do not denote a real in-

dividual, do, nevertheless, denote an individual, but an unreal one. This is essentially Meinong's theory, which we
have seen reason to reject because it conflicts with the law
of contradiction. With our theory of denoting, we are able
to hold that there are no unreal individuals; so that the
null-class is the class containing no members, not the class
containing as members all unreal individuals.
It is important to observe the effect of our theory on the

interpretation of definitions which proceed by means of de1 The propositions fromh which such entities are derived are not identical either with these entities or with the propositions that these entities
have being.

2The argument can be made to prove validly that all members of the
class of most perfect Beings exist; it can also be proved formally that
this class cannot have more than one member; but, taking the definition
of perfection as possession of all positive predicates, it can be proved
almost equally formally that the class does not have even one member.

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492 BERTRAND RUSSELL:

noting phrases. Most mathematical definitions are of this
sort: for example, " n-n means the number which, added to
n, gives n ". Thus n - n is defined as meaning the same as
a certain 'denoting phrase; but we agreed that denoting
phrases have no meaning in isolation. Thus what the definition really ought to be is: " Any proposition containing rn-n
is to mean the proposition which results from substituting
for 'rm-n' 'the number which, added to n, gives rn' ". The
resulting proposition is interpreted according to the rules
already given for interpreting propositions whose verbal expression contains a denoting phrase. In the case where m
and n are such that there is one and only one number x
which, added to n, gives rn, there is a number x which can
be substituted for n - n in any proposition containing m - n
without altering the truth or falsehood of the proposition.

But in other cases, all propositions in which "in - n" has a

primary occurrence are false.

The usefulness of identity is explained by the above theory.
No one outside a logic-book ever wishes' to say "x is x," and
yet assertions of identity are often made in such forms as
" Scott was the author of Waverley " or " thou art the man ".
The meaning of such propositions cannot be stated without
the notion of identity, although they are not simply statements that Scott is identical with another term, the author
of Waverley, or that thou art identical with another term,
the man. The shortest statement of " Scott is the author
of Waverley" seems to be: "Scott, wrote Waverley; and it
is always true of y that if y wrote Waverley, y is identical with

Scott ". It is in this way that identity enters into " Scott is
the author of Waverley"; and it is owing to such uses that

identity is worth affirming.

One interesting result of the above theory of denoting is
this: when there is anything with which we do not have
immediate acquaintance, but only definition by denoting
phrases, then the propositions in which this thing is introduced by means of a denoting phrase do not really contain
this thing as a constituent, but contain instead the constituents expressed by the several words of the denoting phrase.
Thus in every proposition that we can apprehend (i.e. not
only in those whose truth or falsehood we can judge of, but
in all that we can think about), all the constituents are really
entities with which we have immediate acquaintance. Now
such things as matter (in the sense in which matter occurs
in physics) and the minds of other people are known to us
only by denoting phrases, i.e., we are not acquainted with
them, but we know them as what has such and such proper-

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ON DENOTING. 493

ties. Hence, although we can form propositional functions
C (x) which must hold of such and such a material particle,
or of So-and-so's mind, yet we are not acquainted with the
propositions which affirm these things that we know must
be true, because we cannot apprehend the aetual entities
concerned. What we know is " So-and-so has a mind which
has such and such properties " but we do not know " A has
such and such properties," where A is the mind in question.
In such a case, we know the properties of a thing without
having acquaintance with the thing itself, and without, consequently, knowing any single proposition of which the thing
itself is a constituent.
Of the many other consequences of the view I have been
advocating, I will say nothing. I will only beg the reader
not to make up his mind against the view-as he might be
tempted to do, on account of its apparently excessive complication-until he has attempted to construct a theory of
his own on the subject of denotation. This attempt, I believe, will convince him that, whatever the true theory may

be, it cannot have such a simplicity as one might have expected beforehand.

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