Russell and Definate Descriptions

University Degree Historical and Philosophical studies

Was Russell right to think that definite descriptions are quantifying expressions?

Russell believed that our language is too ambiguous and misleading for properly representing the truth and that a purer and more rigorous language would be necessary for minimal mistakes and assumptions in philosophy. In “On Denoting” Russell introduces one of his most significant contributions to linguistic theory known as the theory of descriptions. The theory of descriptions encompasses a formal and idealized language which is based on mathematical logic, and looks more like a string of math equations rather than what the ordinary man may recognize as “language”.

In his work, Russell focuses on finding a method for comprehending propositions that include definite descriptions. A definite description is defined as any word, name, or phrase that denotes a particular object. “The present queen of England”, and “The cat on the mat” are both examples of definite descriptions. Russell created the theory of descriptions with the intention to deal with sentences such as “The King of America is bald”, where the object referred by the definite description is either ambiguous or nonexistent. These are known as incomplete symbols. Russell demonstrates how they can actually be broken down into logical atoms, which allow us to judge the truth-value or validity of statements more efficiently by exposing their underlying assumptions.

For example, take the statement “The author of Waverley was Scotch”. This appears to be a simple sentence, but in Russell’s view the sentence can be broken down into three logical atoms in the form of quantifying expressions:

At least one person authored Waverley
At most one person authored Waverley
Whoever authored Waverley was Scotch.

The three logical atoms are all intuitively necessary for the truth of the original sentence. If the first logical atom was false, then the original statement is rendered untrue. If the second logical atom was false (that there were in fact 2 people who authored Waverley), then the usage of “the” in the sentence would have been inappropriate. Then it logically follows at 3 that whoever authored Waverley was Scotch. The mathematical notation used by Russell to represent the above conditions would appear as follows:

*”W” is the predicate “authored Waverley”

**”S” is “…was Scotch”

(∃x) Wx
(x)(Wx→(y)(Wy→y=x)
(x)(Wx→Sx)

The 3 logical atoms ultimately sum up to become the logical form of the original sentence “The author of Waverley was Scotch”:

4. (∃x)(Wx & ((y)(Wy→y=x) & Sx)).

This is a preview of the whole essay

*”W” is the predicate “authored Waverley”

**”S” is “…was Scotch”

(∃x) Wx
(x)(Wx→(y)(Wy→y=x)
(x)(Wx→Sx)

The 3 logical atoms ultimately sum up to become the logical form of the original sentence “The author of Waverley was Scotch”:

4. (∃x)(Wx & ((y)(Wy→y=x) & Sx)).

As a result 4 is the pure, referentially transparent form of the original sentence that is not obscured by the grammatical form. This shows that any sentence involving definite descriptions are, in actual terms merely a shorthand notations for a series of claims. Russell believes that through exposing the logical structures that underlie ordinary language, the application of the theory may allow linguists and philosophers to avoid ambiguity and paradox when making claims of their own. The same can be applied to sentences such as “The present King of France is bald” except in such case certain glitches in the theory have lured attack by other philosophers. We shall now examine Russell’s approach to apparent reference to non-existents.

We all know that France, as a republic, does not have a king and that the statement “The present King of France” is bald appears to be false as there exists no present King of France. However, it is uncertain as to whether the statement can properly be deemed as false as one may then logically conclude that the negation of the statement (“The present King of France has hair”) is true. However, in this case the negated statement appears just as untrue as the original sentence. What then, can we make of a sentence that is neither true nor false? We shall begin by following Russell’s method of analyzing sentences by breaking it down to logical atoms which can also be quantified mathematically:

There exists a King of France: (∃x(Fx)) – there is an x such that x is the King of France

There is only one King of France: (∀x(Fx→∀y(Fy→y=x) – for every y that is the King of France, x equals y; there is at most one King of France

The King of France has no hair. : (∀x(Fx→Bx)) – x is bald

Russell proposes that because the logical atom 1 is false, and our sentence is the conjunction of all three logical atoms, the sentence is false. Therefore, the negation of the sentence is similarly false as it consists of the same logical atoms 1 & 2 but simply with a different 3rd atom (x is not bald). This means that we do not violate the “law of the excluded middle” as by denying both “the King of France is bald” and its negation we completely discard the assertion of some existence of x which is the King of France. Hence we do not contradictingly assert the existence of an x, which is neither bald nor not bald. An easier way of showing a conformation to the law of the excluded middle in this case would be to negate logical atom 1 instead of substituting x in the sentence “x is bald” (or its negation), which leaves us with “it is not the case that there exists an x and x is bald”. This, however, may appear to be an absolutely pointless claim and philosophers have argued over whether the sentence is in fact meaningful at all. Nonetheless, it becomes clear that applying concepts of logical atomism to language reveals the complexity of truth and validity in concepts expressed by language.

Strawson for one, argues against Russell’s analysis. Let us examine the following argument which according to Russell would be invalid :

If “The present King of France is bald” is meaningful, it is either true or false
“The present King of France is bald” is true if the PKoF is bald, and false if the PKoF is not bald
But both the statement “The present King of France is bald” and its negation may only be true if there is something which IS the “present King of France”
Hence, there must (at least in some possible world) exist the present King of France.

Russell does not accept the above argument because it necessitates the existence of a present King of France, who does not exist. As he says himself in On Denoting :

“Logic is concerned with the real world just as truly as zoology, though with its more abstract and general features…There is only one world, one “real” world…A robust sense of reality is very necessary in framing a correct analysis of propositions about unicorms….and other such pseudo-objects (this includes the present King of France)”

Strawson, despite agreeing with the fact that the aforementioned argument is invalid, draws attention to Russell’s mistake in rejecting the meaningfulness of the sentence. He claims that Russell is wrong in two ways: 1. Russell confuses meaning with denotation 2. As in the case of sentences like “The present King of France is bald” where the subject of the subject-predicate sentence is something that acts as a logically proper name and the sentence is meaningful, it is wrong for Russell to conclude that there must exist something to which the subject term refers. Strawson points out that Russell’s mistake is based on the wrong conflation of what he introduces as the three-way distinction for both sentences (such as “The present King of France is bald”) and expressions (definite descriptions such as “The present King of France”). The three-way distinction is :

The sentence or expression
A use of the sentence of expression
An utterance of the sentence or expression

According to Strawson, Russell considers things that are true of uses as if they were true of sentences or expressions themselves. For example suppose we are now in a world where France actually had a bald King 2 years ago, and a King with hair right now. If person A and B both say that “The present King of France is bald” 2 years ago, and person C says the same sentence “The present King of France is bald” right now, then

The sentence used remains the same for all people A,B,C
The USE, however is same for persons A and B, but not C as C is referring to someone different.
There are 3 different utterances (but we shall ignore this bit because it does not affect the rest of the argument)

According to Russell, “The present King of France is bald” is itself a sentence which must either be true or false about the King of France. However, Strawson states that it does not make sense to talk of a sentence as being true or false. This is because sentences are “used” to make true or false assertions, and are not supposed to have any truth-value themselves. Only statements we make when we “use” sentences can refer to anything or have any truth value. But because in reality, the “present King of France” does not exist, the sentence is neither true nor false.

Keith Donnellan, however, holds an objection against both Russell and Strawson in that they appear to be trying to analyse sentences without thinking about the context in which they are used. For example, there are contexts where it may be possible that the speaker is referring to a person who is thought by the general public to be the King of France, making the truth-value of the statement dependent on the existence of hairs on the head of the impostor. (It can however be argued that in such cases, it is unsuitable for the impostor to be labeled the King of France at all, making all statements about him meaningless altogether…but to solve that problem will take another complete essay so I will stop here)

Ultimately, it is difficult to say whether Russell was “right” to think that definite descriptions are quantifying expressions, as rightness is always subjective. What we may like to think about is how Russell’s theory benefits us in clearing up potential confusions and ambiguities in language. We may, however, remain skeptical of whether the theory of descriptions really help us understand how our words gain meaning or how they are used to refer to the world. Russell may have made a very good attempt at showing the logical relation between definite descriptions and the world, but I remain uncertain as to whether this is necessary and whether quantificational logic is in fact a most appropriate way for uncovering the underlying nature of our thoughts.