Identity in Philosophy of Language

§ 1 Definite Descriptions

Recall that a constant of QL must refer to some member of the UD. This constraint allows us to avoid the problem of non-referring terms. Given a UD that included only actually existing creatures but a constant \(c\) that meant ``chimera'' (a mythical creature), sentences containing \(c\) would become impossible to evaluate.

The most widely influential solution to this problem was introduced by Bertrand Russell in 1905. Russell asked how we should understand this sentence:

$$1. \text{The present king of France is bald.}$$

The phrase 'the present king of France is bald' supposed to pick out an individual by means of a definite description.However, there was no king of France in 1905 and there is none now.Since the description is a non - referring term, we cannot just define a constant to mean ``the present king of France '' and translate the sentence as \(Kf\).

Russell's idea was that sentences that contain definite descriptions have a different logical structure than sentences that contain proper names, even though they share the same grammatical form. What do we mean when we use an unproblematic, referring description, like ``the highest peak in Washington state''? We mean that there is such a peak, because we could not talk about it otherwise. We also mean that it is the only such peak. If there was another peak in Washington state of exactly the same height as Mount Rainier, then Mount Rainier would not be the highest peak.

According to this analysis, the sentence in question is saying three things. First, it makes an existence claim: There is some present king of France. Second, it makes a uniqueness claim: This guy is the only present king of France. Third, it makes a claim of predication: This guy is bald.

In order to symbolize definite descriptions in this way, we need the identity predicate. Without it, we could not translate the uniqueness claim which (according to Russell) is implicit in the definite description.

Let the UD be {x | x is a living person}, let \(Fx\) mean ``\(x\) is the present king of France,'' and let \(Bx\) mean ``\(x\) is bald.'' The sentence can then be translated as \(\exists x[Fx \wedge \neg\exists y(Fy \wedge x\neq y) \wedge Bx]\). This says that there is some guy who is the present king of France, he is the only present king of France, and he is bald.

Understood in this way, the sentence is meaningful but false. It says that this guy exists, but he does not.

The problem of non-referring terms is most vexing when we try to translate negations. So consider this sentence:

$$\text{2. The present king of France is not bald.}$$

According to Russell, this sentence is ambiguous in English. It could mean either of two things: a: It is not the case that the present king of France is bald, or b: The present king of France is non-bald.

Sentence a is called a wide-scope negation, because it negates the entire sentence. It can be translated as \(\neg \exists x [Fx \wedge \wedge \exists y(Fy \wedge x\neq y) \wedge Bx]\). This does not say anything about the present king of France, but rather says that some sentence about the present king of France is false. Since 2 if false, sentence a is true.

Sentence 2 b says something about the present king of France.It says that he lacks the property of baldness.Like sentence 1, it makes an existence claim and a uniqueness claim;it just denies the claim of predication.This is called narrow - scope negation.It can be translated as\(\exists x[Fx\wedge\neg\exists y(Fy\wedge x\neq y)\wedge\neg Bx]\).Since there is no present king of France, this sentence is false.

Russell's theory of definite descriptions resolves the problem of non-referring terms and also explains why it seemed so paradoxical. Before we distinguished between the wide-scope and narrow-scope negations, it seemed that sentences like 2 should be both true and false. By showing that such sentences are ambiguous, Russell showed that they are true understood one way but false understood another way.

§ 2 Quantity Statements Involving Definite Descriptions

Perhaps the most challenging statements to translate are ones that involve both definite descriptions and quantity. But recall that definite descriptions are basically just 'exectly one' statements, so they are not really different. But things can get messy pretty quickly. For instance, consider this monstrosity

$$\text{The friend of Lily,} \text{ who knew everyone she dated,} \text{ liked none of them }$$

To begin, we need to decide on how to understand the English statement itself. For instance, what do 'who' and 'she' refer to? Multiple interpretations, as long as they are reasonably close to the original, are often permissible - we just have to be clear about our decision. Here, the use of 'the' signals the presence of a definite description. This means that we can understand the predicate 'x knew everyone she dated' as a unique description of this friend. From this, we can make the interpretation the friend in question knew everyone Lily dated. In other words: there exists exactly one person x such that x is a friend of Lily and x knows everyone Lily dated, and x liked none of the people Lily dated. It's a mouth full, but its logical structure is more apparent. One possible translation is as follows.

$$\exists x [Fxl \wedge \neg \exists y (Dyl \wedge Kxy \wedge Lxy) \wedge \forall z ((Fzl \wedge \forall w (Dwl \to Kwy) \to x = z)]$$

This sentence can be broken down into 3 small ideas:

$\exists x Fxl$: Someone x is a friend of Lily.

$\neg \exists y (Dyl \wedge Kxy \wedge Lxy)$: There is no one y such that y dates Lily and known to x yet liked by x.

$\forall z (Fzl \wedge \forall w (Dwl \to Kwy)) \to x = z)$: This is the definite description: we are fixing the reference of 'the friend' who knew everyone Lily dated' to x.