Bertrand Russell:

On Denoting

How can a sentence describe the world?

What does 'is' mean?

Background

Bertrand Russell (1872 - 1970) was a British philosopher, logician, mathematician, historian, writer, social critic, political activist and Nobel laureate. He was one of the foremost philosophers of the 20th century and made significant contributions to logic, epistemology and the philosophy of mathematics. Russell was also known for his activism for peace and his popular writing on social, political and moral subjects.

Denoting phrases

Firstly, Russell gives examples of ‘denoting phrases’:

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:

A phrase may be denoting, and yet not denote anything; e.g., 'the present King of France'.
A phrase may denote one definite object; e.g., 'the present King of England' denotes a certain man.
A phrase may denote ambiguously; e.g. 'a man' denotes not many men, but an ambiguous man.

—Bertrand Russell, On Denoting

Knowledge by description vs knowledge by acquaintance

He points out that often we have knowledge only of descriptions and not of acquaintance. For example, I understand what ‘the centre of mass of the Solar System’ means, but I have no direct acquaintance of it - I only know it through the description.

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

—Bertrand Russell, On Denoting

Russell's theory of definite descriptions

Then he explains his theory. Russell gives the following way of thinking about the words all, none, and some:

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.'

—Bertrand Russell, On Denoting

Perhaps the central insight of Russell's paper is that it shows how logical form may differ radically from grammatical form. For example, the sentence ‘I met a man’ can be interpreted in Russell’s theory as

“‘I met x, and x is a man’ is not always false”.

Or ‘All men are mortal’ can be interpreted as

“‘If x is human, x is mortal’ is always true”

Russell gives us the means to analyse the logical form of the sentence by recognising three different uses of the word ‘is’:

The ‘is’ of existence: Cicero is
The ‘is’ of identity: Cicero is Tully
The ‘is’ of predication: Cicero is wise

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 that nothing else had this relation. The relation in question, without the assumption of uniqueness, and without any denotingphrases, 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.

Russell believes that this can solve three logical problems that we come across when analysing sentences:

Problem 1: the law of the excluded middle

The law of the excluded middle says that for every proposition, either the proposition or its negation is true. So either

The mug is on the table.

The mug is not on the table.

Russell criticises (what he thinks of as) Alexius Meinong’s theory - that any grammatically correct denoting phrase stands for an object. Therefore, ‘the present king of France’ stands for an object, even if that object doesn’t ‘subsist’ (exist).

However, statements like, ‘the present King of France is not bald’, seem to cause a problem for this rule. It is not true that ‘the present King of France is not bald’ (because there is no present king of France) but neither is it true that ‘the present King of France is bald’. We seem to have to invent a mysterious entity called ‘the King of France’ in order to keep the law of the excluded middle intact.

Therefore, we can break down the troublesome sentence into:

An existential claim: there exists an x who is the present King of France
A claim of identity: any y, if that y is the present King of France, then y=x
A predicate: this x is bald

It is clear from this analysis that it is the first, existential claim that is false. And so we can clearly say, ‘it is not the case that there exists an x who is the present King of France (that is bald).

Problem 2: the law of identity

According to Frege, a denoting phrase has two elements: a meaning (or sense) and a denotation (or reference). Thus, ‘the centre of mass of the Solar System at the beginning of the twentieth century’ is a highly complex is meaning but very simple in denotation.

Russell rejects this view because it results in confusions:

Russell uses the following example. A man called Walter Scott wrote a book called Waverley. So presumably, ‘Scott’ denotes the same thing as ‘the author of Waverley’. And if that’s true, then, according to the law of identity (which states that each thing is identical with itself) we can say that:

Scott = the author of Waverley

So these two terms are entirely synonymous. It seems sensible enough. But Russell comes up with the following example:

George IV wanted to know whether Scott was the author of Waverley.

We can’t substitute one for the other here. That would result in the nonsense sentence:

George IV wanted to know whether Scott was Scott.

However, Russell can use his form of analysis to make it work by showing that the ‘is’ of predication is not the same as the ‘is’ of identity.

there exists an x who is the author of Waverley
any y, if that y is the author of Waverley, then y=x
this x is Walter Scott

Problem 3: true negative existential claims

The question here is how can someone believe that the subject of a sentence does not have a reference whilst at the same time believing that it does not exist. The question is ‘what is it that doesn’t exist?’ If I say, ‘The god Apollo does not exist’, what does ‘the god Apollo’ refer to?

With Russell’s method, we can show that we are saying the following:

There is no x such that x is a god & x is Apollo. (where ‘is a god’ and ‘is Apollo’ are predicates).

Tasks

Explain the three different uses of the word 'is'.
Explain how this solved the problem of the excluded middle.
Explain how this solved the problem of the law of identity.
Explain how this solved the problem of negative existential claims.
What do you think of Russell's method of analysis?

#logic

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