On Denoting
Bertrand Russell
1905
By a `denoting 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 presenting King of
France, the center 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 considerably
difficulty; 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 very great importance, not only in logic and mathematics,
but also in the theory of knowledge. For example, we know that the center of mass of the
solar system at a definite instant is some definite point, and we can affirm a number of
propositions about 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 center of mass. In perception we have acquaintance
with 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 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 shall then discuss the theories of Frege and Meinong, showing why neither
of them satisfies me; then I shall give the grounds in favor 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 proposition 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'. 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 proposition in whose
verbal expression they occur has a meaning. The difficulties concerning denoting are, I
believe, all the result of 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.
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:
`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 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:
`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)', 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 t`he 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 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 II'. To get an equivalent of `x was the father of Charles II', we must add `If y is
other than x, y did not 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' becomes: `x begat Charles II; and
``If y begat Charles II, y is identical with x '' is always true of y'.
Thus `the father of Charles II was executed' becomes: `It is not always false of x that x
begat Charles II and that x was executed and that ``if y begat Charles II, y is identical
with x '' is always true of y'.
This may 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, according to
the above interpretation, whatever statement 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 condition fails, every proposition of the form `C(the father of
Charles II)' is false. Thus e.g. every proposition of the form `C(the present King of
France)' is false. This is a great advantage to the present theory. I shall show later that it
is not contrary to the law of contradiction, as might be at first supposed.
The above gives a reduction of all propositions in which denoting phrases occur to forms
in which no such phrases occur. Why it is imperative to effect such a reduction, the
subsequent discussion will endeavor 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 possible theories which
admit such constituents the simplest is that of Meinong. This theory regards any
grammatically correct denoting phrase as standing for an object. Thus `the present King
of France', `the round square', etc., are supposed to be genuine objects. It is admitted that
such objects do not subsist, 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
infringe the law of contradiction. It is contended, for example, that the 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 meaning and the
denotation. Thus `the center of mass of the solar system at the beginning 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. 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 favor 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, 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 ought to be about the denotation of the phrase `the King of
France'. But this phrase, though it has a meaning provided `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 is u is not a unit class,
`the u' seems to denote nothing; hence our proposition would seem to become nonsense
as soon as u 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 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 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, 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
Waverley; and in fact Scott was the author of Waverley. Hence we may substitute Scott
for the author of `Waverley', 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 the 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, not 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
difficulties must be wrong.
When we wish to speak about the meaning of a denoting phrase, as opposed to its
denotation, the natural mode of doing so is by inverted commas. Thus we say:
The center of mass of the solar system is a point, not a denoting complex;
`The center 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 denotation is not
merely linguistic through the phrase: there must be a logical relation involved, which we
express by saying 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
`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 to
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 the complex in a proposition, the proposition is about the denotation; 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
that 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 center 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 not be the meaning, but must be
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
meaning, 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 t
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