Unit III: Russell’s theory of descriptions


January 22, 2004

1 Propositions and propositional functions
2 Descriptions and names
 2.1 Problems with indefinite descriptions
 2.2 Three puzzles caused by the assimilation of definite descriptions to names
  2.2.1 The problem of negative existentials
  2.2.2 Substitution failures
 2.3 The distinction between primary and secondary occurrence
3 Russell’s theory of descriptions
 3.1 Indefinite descriptions
 3.2 Definite descriptions
 3.3 ‘Everyone’, ‘someone’
  3.3.1 How Russell’s theory solves the three puzzles
  3.3.2 How it explains the difference between primary and secondary occurrence
4 Objections to Russell’s theory
 4.1 Incomplete definite descriptions
 4.2 The view that sentences containing descriptions say something about propositional functions
5 Russell’s view of names
6 The metaphysical importance of Russell’s theory

1 Propositions and propositional functions

Russell (but not contemporary theorists) means by ‘proposition’, as he puts it, “primarily a form of words which expresses what is either true or false.” Roughly, then, ‘propositions’ in Russell refers to ‘declarative sentences.’

Propositional functions are something else:

“A ‘propositional function,’ in fact, is an expression containing one or more undetermined constituents, such that, when values are assigned to these constituents, the expression becomes a proposition. ...Examples of propositional functions are easy to give: “x is human” is a propositional function; so long as x remains undetermined, it is neither true nor false, but when a value is assigned to x it becomes a true or false proposition.” (An Introduction to Mathematical Philosophy, pp. 155-156)

What does it mean for ‘x’ to ‘remain undetermined’, or to have a ‘value’? To understand what Russell is talking about we will need to explain some of the background of this passage in the theory of reference.

To a first approximation, the basic idea of the theory of reference is that we can assign values, or references, to expressions which will explain the conditions under which sentences involving those expressions are true. You can think of reference as ‘power to affect truth value.’

What kind of value we assign to an expression will depend on what kind of expression it is. Focus for now on simple sentences of the form ‘n is F’, where ‘n’ is a name, like ‘Bob,’ and ’is F’ is a predicate, like ‘is male’. Intuitively, a sentence of this form is true just in case the object picked out by the name is a member of the class of things which ‘are F’. In the case of this example, the sentence is true just in case Bob, who is picked out by the name ‘Bob’, is a member of the class of males -- the class of things which satisfy the predicate ‘is male.’

This suggests a natural idea for the references of names and predicates. We should let the reference of a name (if it has one) be an object (as the reference of ‘Bob’ is Bob), and the reference of a predicate be a class of things (as the reference of ‘is male’ will be the class of things which are male). We will then have an explanation of what it takes for a simple sentence to be true: the sentence will be true if and only if the reference of the name is a member of the class which is the reference of the predicate.

Now return to Russell’s definition of a propositional function. To say that a propositional function is an expression containing one or more undetermined constituents is to say that it is an expression containing one or more expressions which have not been assigned a reference. This is the case with Russell’s example:

x is human.

Here, ‘x’ has no reference; we might as well have written

___ is human.

Once those undetermined constituents are ‘filled in’ by assigning them a reference (or by replacing them with words which have a reference), we will have a proposition: a form of words which expresses something which can be either true or false.

2 Descriptions and names

A first natural thought is that what goes for names like ‘Bob’ should also go for descriptions -- both indefinite descriptions, like ‘a man’, and definite descriptions, like ‘the tallest man in this room.’ After all, they seem to play the same grammatical role as proper names; just as we can say

Bob is happy.

we can say

A man is happy.

The tallest man in this room is happy.

On this view, we should let the references of such expressions be the objects which those expressions pick out.

Russell, however, argues that this does not capture the real nature of either indefinite or definite descriptions.

2.1 Problems with indefinite descriptions

The first problem Russell notes is that there is a clear sense in which indefinite descriptions do not stand for objects at all:

‘Our question is: What do I really assert when I assert “I met a man”? Let us assume, for the moment, that my assertion is true, and that in fact I met Jones. It is clear that what I assert is not “I met Jones.” I may say “I met a man, but it was not Jones”; in that case, though I lie, I do not contradict myself, as I should do if when I say I met a man I really mean that I met Jones. ...not only Jones, but no actual man, enters into my statement.’ (167-8)

This is puzzling; if the value assigned to ‘a man’ is not an object, what could it be?

2.2 Three puzzles caused by the assimilation of definite descriptions to names

Different, but equally pressing, problems arise from assimilating definite descriptions to names. (Russell discusses these and other problems in his article, ‘On Denoting.’)

2.2.1 The problem of negative existentials

Russell asks us to consider sentences like:

The round square is unreal.

The round square is nonexistent.

These sentences are called ‘negative existentials’ because they can be understood as the negation of an existence claim.

If it were the case that definite descriptions were to be understood as a kind of name, then we could give an account of these sentences using the elementary theory of reference sketched above: that is, the sentences would be true just in case there was some object referred to by ‘the round square’ which was, respectively, among the unreal things or the nonexistent things.

Russell does not think that this is plausible; there is, after all, no object -- the round square -- which could be the referent of the ‘the round square.’ He says:

2.2.2 Substitution failures

2.3 The distinction between primary and secondary occurrence

There is another difference between names and descriptions which Russell notes only in passing toward the end of the article, but which can also be used as an argument that descriptions function quite differently than names. Consider the following three sentences:

3 Russell’s theory of descriptions

3.1 Indefinite descriptions

3.2 Definite descriptions

3.3 ‘Everyone’, ‘someone’

3.3.1 How Russell’s theory solves the three puzzles

3.3.2 How it explains the difference between primary and secondary occurrence

4 Objections to Russell’s theory

4.1 Incomplete definite descriptions

4.2 The view that sentences containing descriptions say something about propositional functions

5 Russell’s view of names

Genuine names vs. disguised descriptions; why some names must be regarded as disguised descriptions. The characteristics of genuine, or ‘logically’ proper names.

6 The metaphysical importance of Russell’s theory