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Justin Woods

Russell's Theory of Descriptions in 'On Denoting'


The aim of this essay is to give an exposition of the theory of descriptions as it is first set forth by Russell in his article 'On Denoting' found in Mind 1905.

Each section of this article will be explained in my own words, with the exception of some of the symbolic logic. Russell's own words are indicated by speech marks.

I have tried not to simply re-write what Russell has said, but rather endeavoured to explain, in an original way, each part of Russell's theses, and in the order that they are found in the article.

Firstly, I will outline the article 'On Denoting' giving my own understanding of the theory of descriptions as Russell introduces it. It should be noted that the phrase 'theory of descriptions' is not used in this article, but is coined later in Russell's philosophy.


The theory of descriptions is Russell's solution to the problems caused by the interpretation of denoting phrases.

This solution can be found in his article 'On Denoting' (Mind, 1905). What follows is an outline of the theory as it is proposed in that article.

A denoting phrase is a phrase such as the following: a man, some man, any man, every man, all men, the present king of England, the present king of France.

These denoting phrases can be classed into either one of two groups; those containing definite descriptions and those containing indefinite descriptions.

A denoting phrase is a denoting phrase purely because of its form, not its 'content'. By this is meant, for example, that a denoting phrase need not actually denote anything in order to be a denoting phrase, it simply needs to have the form of a phrase that is denoting something

A phrase can denote a definite object, such as 'the present Queen of England' or it can denote ambiguously 'a man', which is no man in particular, but an undefined man.


To Russell, the interpretation of such phrases is 'a matter of considerable difficulty'.

Russell demonstrates the importance of the subject of denoting in the theory of knowledge. He distinguishes between two types of knowledge:

  1. knowledge by acquaintance; we have knowledge by acquaintance of the objects of our perception, e.g. when I look at my hand.
  2. knowledge about; we have knowledge of the objects of our thought, e.g. when I think of the present Queen of England.

We do not necessarily have knowledge by acquaintance of those objects denoted by some phrases. We may be reliant, therefore, on denoting phrases for the knowledge of those objects that they denote.

A good example from Russell is our knowledge of other minds: '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.'


Russell now begins to state his theory. Hitherto, according to Russell, the interpretation of propositions containing denoting phrases has been based on a 'wrong analysis' of those propositions.

The proper analysis is based on Russell's assumption that denoting phrases have no meaning in isolation but that a meaning is attributed to the propositions in which they are contained.

So, it is not necessary for a denoting phrase to denote anything in order for the verbal expression of the proposition which contains it to have a meaning. Thereby, a denoting phrase is such purely in virtue of its form. This is the principle of the theory Russell wishes to advocate.

The aim of the theory is to provide the tools for showing that this is so. The meaning of any sentence in which denoting phrases occur can be spelled out in such a way that the resulting sentence no longer contains the expression of it or any synonym of it. The theory exposes them by analysis and then eliminates them.

Thus, the theory will give 'a reduction of all propositions in which no such phrases occur.'

We will now show how this is achieved.


Russell formalises his theory into symbolic logic. He takes as fundamental the concept of a propositional function's being always true. In other words, for all the values of the variable the propositional function is true.

He uses 'C(x)' to mean a propositional function, where 'x' is the variable of the proposition 'wholly and essentially undetermined'.

So, we consider then, the notions 'C(x) is always true' and 'C(x) is sometimes true'. The latter being defined in terms of the former when we take it to mean 'It is not true that 'C(x) is false' is always true'.

Then, Russell sets about his interpretation of the phrases referred to as 'the most primitive of denoting phrases', 'everything', 'nothing', and 'something', by formalising them into the above propositional functions, so that:

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'.

So, for the proposition 'everything has the property C' we can say 'C(x) is always true'. Or, for every value of 'x' C(x) is true.

That nothing has the property C means that for every value of 'x' 'C(x) is false' is always true.

Finally, when something has the property C we mean that it is false that ''C(x) is false' is always true, so C(x) is true for at least one value of 'x'.

Now we can apply this method of interpretation to denoting phrases that contain indefinite articles and are therefore indefinite descriptions; such as 'a', 'some', 'any'.

For instance, to say that I have met 'a man' is to say that the propositional function 'x is human, and I have met x' is not always false (or is sometimes true). In other words, true for at least one value of x.

Thus far we have shown how Russell's proper analysis of propositions whose verbal expressions contain denoting phrases has dealt with those phrases that are indefinite descriptions.


Now all that remains for us is to interpret those denoting phrases classed as definite descriptions. In other words, phrases containing 'the x'.

Russell tells us that 'these are by far the most interesting and difficult of denoting phrases.'

As above, we use the theory so far propounded to formalise those propositions whose verbal expressions contain definite descriptions into symbolic logic.

To eliminate denoting phrases from those propositions in which they are contained we simply expand them into existential statements; statements that assert that one thing has the property contained in the description.

In the propositional function, however, we must also stipulate that the function is true for one and only one value of the variable.

Russell achieves this by stipulating that it is always true of any object y that if y satisfies the function, then y is identical with x.

For example, the proposition 'the father of Charles II was executed' contains a 'hidden' existential statement; a statement that asserts that there was an x such that x was the father of Charles II.

It is also made clear to us by Russell that we are using the strict sense of 'the' which, he tells us, 'involves uniqueness', so that we are not only asserting that the father of Charles II had a certain relation with Charles II, but also that nothing else had this relation.

So, we can eliminate the denoting phrase 'the father of Charles II' from the proposition by, firstly, showing that what is being said of a variable x is that 'x was the father of Charles II'.

To eliminate the denoting phrase 'the father' and the uniqueness that it implies we can say that 'x begat Charles II'. Since there is a unique relationship between a father and his son, we need to show this in the new form of the proposition, thus completing our interpretation of definite descriptions. So, we add the amendment that it is always true for y, if y satisfies the function, then y is identical with x.

This leads us to interpret 'the father of Charles II' as:

'x begat Charles II, and 'If y begat Charles II, then y is identical with x' is always true of y'.

So, for one value of x the function is satisfied. The function being any statement made about, or attributed to x. In this case it is the statement that x was executed. This we can interpret as:

'It is not always false of x that x begat Charles II, and that x was executed and that 'if y begat Charles II, then y is identical with x' is always true of y'.

C is the statement that x was executed. However, whatever C may be the proposition will be satisfied for one and only one value of x, x being 'the father of Charles II' in this case.

Consequently, if the function is not satisfied then every proposition that attributes something of 'the father of Charles II' is false.

In other words, every proposition of the form C(the father of Charles II) is false.


This is Russell's account of denoting phrases that are definite descriptions.

His theory has shown a 'reduction of all propositions in which denoting phrases occur to forms in which no such phrases occur'.

They do not occur because the propositions which contain them have been interpreted by Russell's theory into a logical form that takes the meaning of the proposition as a whole, rather than wrongly taking out of context certain phrases and then attempting to attribute meaning to them.

On the contrary, Russell has shown us that these phrases have no meaning by themselves, just as the tooth of a cog wheel is unidentifiable in use until it is seen as a part of the whole.

Any meaning is to be attributed to the verbal expressions of the propositions which contain denoting phrases.


The source of the difficulties hitherto in the interpretation of denoting phrases can be found with the assumption that any grammatically correct denoting phrase can stand for an object.

Russell goes on to criticise one philosopher who has this assumption: Meinong.

Meinong holds that phrases such as 'the present King of France' and 'the round square' stand for genuine objects.

He is not saying that there is a King currently ruling France, or a square that is round in the same sense that there is a Queen of England now, or a three-sided triangle. He is saying that there 'is' an object in some sense of 'being', in some sense of 'is'.

These objects do not subsist as the present Queen of England does now, but they are still objects. Here, subsistence seems to be synonymous with being, rather than existence.

Russell does not elaborate any further than this, but goes on to criticise this theory by proposing that it is apt to admit self-contradictions. For instance, if there is in Meinong's sense an object that answers to the phrase 'the present King of France' yet a glance at the list of present rulers reveals no such ruler, then we are committed to admitting both that there is and there is not a present King of France.

To me, this is a simple semantic problem based on the misuse of these seemingly existential terms. I see no contradiction in admitting that there both is a God and there is not, for the following reason.

God 'is' in the sense that God is the subject of a sentence such as, 'God is merciless'. God 'is' in the sense of 'is' when 'is' is used as an existential term in phrases such as 'I know there is a God'.

Based on the above uses of the ambiguous term 'is' I can now both admit God as a subject of a sentence and dismiss God as existing. In both cases the use of the term 'is' is distinctive, and I have therefore made no contradiction.

It may be interesting to note the following statement Russell gives in defence of philosophy when his mother-in-law assured him that philosophy is only difficult because of the long words that it uses:

'What 'is' means is is and therefore differs from 'is', for "'is' is" would be nonsense.' (My Philosophical Development, 1959)


To continue, Russell tells us that the above infringement of the laws of contradiction can be avoided, while still maintaining the original assumption that grammatically correct denoting phrases can stand for genuine objects, through the efforts of Frege.

According to Russell, Frege distinguishes between two different aspects of a denoting phrase:

  1. its meaning
  2. its denotation

Take the phrase, 'the centre of mass of the solar system at the beginning of the twentieth century'.

Here we can see that the phrase is highly complex in meaning, but its denotation, that which it denotes, is simple; is a certain point.

The centre of mass, the solar system, the beginning of the twentieth century, are all constituents of the phrase's meaning, but the denotation has no constituents, it is simple.

So, Frege avoids self-contradiction by showing that a denoting phrase's meaning is to be found in its constituent meanings, not its denotation.

However, what happens when it is the case that a denotation is absent?

Russell gives the example, 'The King of England is bald.'

This is a statement about the man, the denotation, not the meaning of the phrase. What is predicated by baldness is what the sentence's phrase denotes.

Russell, then asks us to consider the statement 'The King of France is bald.' As above, this too should be about the denotation.

The statement has meaning, i.e; there is a man who currently rules France and is bald. Yet, there is clearly no denotation. Russell's problem with Frege's theory is that a statement 'the king of France is bald' is not about meaning at all, but about the man himself. Therefore, it is a statement about a denotation not a meaning.

Frege's theory shows that denoting phrases may express meaning, but according to Russell they may not denote a denotation.


At this stage, Russell would like to test his theory by subjecting it to three puzzles which a theory of denoting ought to be able to solve.

Puzzle 1 Leibniz's Law.

If a is identical with b, whatever is true of a is true of b. The exchange of a and b in any proposition does not change that propositions truth value.

For instance, if 'Scott was the author of Waverley', then what we are saying is merely that Scott was Scott.

Puzzle 2 The Law of the Excluded Middle.

Either 'A is B' or 'A is not B'. Therefore, either 'the King of France is bald' or 'the king of France is not bald' must be true or false, but not both.

But if we enumerate all things bald and not bald, we will find the king of France on neither list.

Puzzle 3 Difference.

Consider 'A differs from B'. If this is true then there is a difference between A and B. This difference may be expressed as 'the difference between A and B subsists' (He does not mention it, but I take Russell to be avoiding Meinong's use of 'subsist' i.e. that subsistence is synonymous with being, not existence).

If, however, it is false that A differs from B then we say, 'The difference between A and B does not subsist'. So, how can a non-entity be the subject of a proposition?

It seems, therefore, self-contradictory to deny the being of anything.


Russell now postpones the solution to the above puzzles and continues to elaborate on Frege's distinction between a denoting phrase's having both meaning and denotation. This time he looks at the use of inverted commas to distinguish meaning from denotation.

Consider the proposition:

'The first line of Gray's Elegy states a proposition'.

This proposition is about the denotation of the denoting phrase, in this case a definite description.

Now, consider:

'"The first line of Gray's Elegy" does not state a proposition'.

This proposition is about the meaning of what is contained within the inverted commas.

So, C, when it occurs in speech, is about the denotation, and when 'C' occurs it is about the meaning.

Russell states that the relation between C and 'C' is not just linguistic it is a logical relation. It is expressed as 'the meaning denotes the denotation'. The problem is how are we to maintain the connection between meaning and denotation without showing that they are one and the same?

There is the further problem that the meaning cannot be got at except by means of the denoting phrase.

'C' is what we use when we want to talk about the meaning, but it is not the meaning, it is now denoting the meaning.

If we say 'the meaning of C' then we get the denotation. For example, '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''. So, in order to get the meaning rather than the denotation we must avoid saying 'the meaning of C' and instead say 'the meaning of 'C'', which is the same is 'C'.

Similarly, 'the denotation of C' does not mean the denotation we want. This merely denotes what is denoted by the denotation we want. E.g:

C = 'the first line of Gray's Elegy'

The denotation of 'C' = the curfew tolls the knell of parting day.

Yet, what we wanted as the denotation was 'the first line of Gray's Elegy'. Thus, 'we have failed to get what we wanted'.

The problem with speaking of the meaning of a denoting phrase or complex is that the moment the denoting complex is put into a proposition it becomes about the denotation. And if the subject of a proposition is within inverted commas, then it becomes the meaning of the denotation, not the meaning of the 'C'.

In conclusion, some meanings have denotations, but propositions containing denoting phrases about meaning have only meaning and not denotation.

Thus, 'the relation between 'C' and C remains wholly mysterious'.


As we have seen, Russell has shown us that a denoting phrase is part of a sentence. It does not have any significance on its own.

'Scott was a man' has the form of 'x was a man' with 'Scott' as its subject. But, 'the author of Waverley was a man' is not a statement of the form 'x was man' and does not have 'the author of Waverley' as its subject.

'The author of Waverley was a man' can be translated into 'one and only one entity wrote Waverley and that one was a man', which is an entirely different form of statement.

If we wish to attribute a property of the author of Waverley we can then say 'one and only one entity wrote Waverley and that one had the property q'.

So, with every proposition in which 'the author of Waverley' occurs the proposition 'Scott was the author of Waverley' becomes:

'One and only one entity wrote Waverley and that one was identical with Scott'.

Thus, the property had by the phrase 'one and only entity wrote Waverley' is that of identity with Scott.

The entity x is in this case the denotation of the denoting phrase 'C'. So, Scott is the denotation of 'the author of Waverley'.

'C' is then merely the phrase rather than the meaning of it. 'C' has no meaning, per se, because in any proposition in which 'C' occurs, when the proposition is fully expressed, it no longer contains 'C', the denoting phrase, in it. The fully expressed form is of such a nature that it or any implication of it has been eliminated.

The fully expressed form has become:

'It is not always false of x, that x wrote Waverley, that it is always true of y that if y wrote Waverley, then y is identical with x and that Scott is identical with x'.

Solution 1

The solution to the first puzzle has now a simple solution:

'Scott was the author of Waverley' does not contain any constituent 'the author of Waverley' for which we could substitute 'Scott'.

Solution 2

Next, Russell tells us that the truth value of propositions inferred by substituting 'Scott' for 'the author of Waverley' is not affected so long as 'the author of Waverley' has a primary occurrence in the proposition considered.

So now we need to know what he means by this.


When we say 'the so-and-so is true' or 'So-and-so is surprising', 'so-and-so' must be a proposition.

If the proposition contains a denoting phrase, we can eliminate it from the proposition or from the whole sentence in which the proposition occurs.

Different propositions result according to which of these two ways of elimination we choose.

For instance, 'the present King of France is not bald' is a sentence that can be interpreted in two ways, according to the term 'not'.

Consider the following interpretations:

  1. If the scope of 'not' is limited to its adjoining predicate 'bald' we are asserting that there is a present King of France and that he is not bald.
  2. If the scope of 'not' is applied to the whole sentence, then what is asserted is that it is not the case that the present King of France is bald.

With the first interpretation, the denoting phrase 'the present King of France' is said to have a primary occurrence. The denoting phrase of the second interpretation is said to have a secondary occurrence. This is simply an exchange of names for descriptions.

The distinction between primary and secondary occurrences is much more easily seen in symbolic logical form.

Solution 3

How can a non-entity be the subject of a proposition?

As we see from the puzzle, this question appears as a result of the proposition 'If A and B do not differ, then the difference between A and B does not subsist'.

Russell's solution is to again remove the denoting phrase from the proposition, so that:

'If A and B differ, then 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 then there is no such entity.'

So, according to Russell's meaning of denotation, the difference between A and B has a denotation when A and B differ, but has no denotation when they do not.

Thus, 'out of any proposition we can make a denoting phrase which denotes an entity if the proposition is true, but does not denote an entity if it is false.'

So, Russell has shown that all denoted non-entities (denoting phrases that seem to denote non-entities) are denoting phrases that do not denote anything, i.e. that have no denotation./P>


In conclusion, Russell points out one interesting consequence of his theory.

As we know, whenever we use a proposition to talk about an object that we have no knowledge by acquaintance of then we are using denoting phrases to stand for these objects.

It in no way means that the object is an actual constituent of that proposition. The proposition contains merely the constituents of those words contained in the denoting phrase.

Therefore, with every proposition we could possibly apprehend (whether true or not) each of their constituents are real entities with which we do have immediate acquaintance, so long as we can apprehend them.

The minds of others are not known to us by acquaintance but by denoting phrases. We can then know the properties of other minds while not knowing the actual minds themselves, because we can know the constituents of the denoting phrases used to introduce them to us.

Hence, of those propositions that have the actual mind as a constituent we cannot know them.

Lastly, Russell begs the reader not to go against his theory unless at least the reader can himself reveal his own theory on denotation.