# Definite description

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Definite description

A definite description is a denoting phrase in the form of "the X" where X is a noun-phrase or a singular common noun. The definite description is proper if X applies to a unique individual or object. For example: "the first person in space" and "the 42nd President of the United States of America", are proper. The definite descriptions "the person in space" and "the Senator from Ohio" are improper because the noun phrase X applies to more than one thing, and the definite descriptions "the first man on Mars" and "the Senator from Washington D.C." are improper because X applies to nothing. Improper descriptions raise some difficult questions about the law of excluded middle, denotation, modality, and mental content.

## Russell's analysis

France is presently a republic, and has no king. Bertrand Russell pointed out that this raises a puzzle about the truth value of the sentence "The present King of France is bald."

The sentence does not seem to be true: if we consider all the bald things, the present King of France isn't among them, since there is no present King of France. But if it is false, then one would expect that the negation of this statement, that is, "It is not the case that the present King of France is bald," or its logical equivalent, "The present King of France is not bald," is true. But this sentence doesn't seem to be true either: the present King of France is no more among the things that fail to be bald than among the things that are bald. We therefore seem to have a violation of the Law of Excluded Middle.

Is it meaningless, then? One might suppose so (and some philosophers have; see below) since "the present King of France" certainly does fail to refer. But on the other hand, the sentence "The present King of France is bald" (as well as its negation) seem perfectly intelligible, suggesting that "the Present King of France" can't be meaningless.

Russell proposed to resolve this puzzle via his theory of descriptions. A definite description like "the present King of France", he suggested, isn't a referring expression, as we might naively suppose, but rather an "incomplete symbol" that introduces quantificational structure into sentences in which it occurs. The sentence "the present King of France is bald", for example, is analyzed as a conjunction of the following three quantified statements:

1. there is an x such that x is presently King of France: ∃x[PKoF(x)] (using 'PKoF' for 'presently King of France')
2. for any x and y, if x is presently King of France and y is presently King of France, then x=y (i.e. there is at most one thing which is presently King of France): ∀x∀y[[PKoF(x) & PKoF(y)] → y=x]
3. for every x that is presently King of France, x is bald: ∀x[PKoF(x) → B(x)]

More briefly put, the claim is that "The present King of France is bald" says that some x is such that x is presently King of France, and that any y is presently King of France only if y = x, and that x is bald:

∃x[PKoF(x) & ∀y[PKoF(y) → y=x] & B(x)]

This is false, since it is not the case that some x is presently King of France.

The negation of this sentence, i.e. "The present King of France is not bald", is ambiguous. It could mean one of two things, depending on where we place the negation 'not'. On one reading, it could mean that there is no one who is presently King of France and bald:

~∃x[PKoF(x) & ∀y[PKoF(y) → y=x] & B(x)]

On this disambiguation, the sentence is true (since there is indeed no x that is presently King of France).

On a second, reading, the negation could be construed as attaching directly to 'bald', so that the sentence means that there is presently a King of France, but that this King fails to be bald:

∃x[PKoF(x) & ∀y[PKoF(y) → y=x] & ~B(x)]

On this disambiguation, the sentence is false (since there is no x that is presently King of France).

Thus, whether "the present King of France is not bald" is true or false depends on how it is interpreted at the level of logical form: if the negation is construed as taking wide scope (as in ~∃x[PKoF(x) & ∀y[PKoF(y) → y=x] & B(x)]), it is true, whereas if the negation is construed as taking narrow scope (with the existential quantifier taking wide scope, as in ∃x[PKoF(x) & ∀y[PKoF(y) → y=x] & ~B(x)]), it is false. In neither case does it lack a truth value.

So we do not have a failure of the Law of Excluded Middle: "the present King of France is bald" (i.e. ∃x[PKoF(x) & ∀y[PKoF(y) → y=x] & B(x)]) is false, because there is no present King of France. The negation of this statement is the one in which 'not' takes wide scope: ~∃x[PKoF(x) & ∀y[PKoF(y) → y=x] & B(x)]. This statement is true because there does not exist anything which is presently King of France.

## Generalized Quantifier Analysis

Stephen Neale, among others, has defended Russell's theory, and incorporated it into the theory of generalized quantifiers. On this view, 'the' is a quantificational determiner like 'some', 'every', 'most' etc. The definite description 'the' has the following denotation (using lambda notation):

λf.λg.[∃x(f(x)=1 & ∀y(f(y)=1 → y=x)) & g(x)=1].

Given the denotation of the predicates 'present King of France' (again PKoF for short) and 'bald (B for short)'

λx.[PKoF(x)]
λx.[B(x)]

we then get the Russellian truth conditions: 'The present King of France is bald' is true just in case ∃x[PKoF(x) & ∀y[PKoF(y) → y=x] & B(x)]. On this view, definite descriptions like 'the present King of France' do have a denotation (specifically, definite descriptions denote a function from properties to truth values -- they are in that sense not syncategorematic, or "incomplete symbols"); but the view retains the essentials of the Russellian analysis, yielding exactly the truth conditions Russell argued for.

## Symbolic form

When using the definite descriptor in a formal logic context, it can be symbolized by $\scriptstyle\iota x$, so that

ιxx)

means "the $\scriptstyle x$ such that $\scriptstyle\phi x$", and

ψ(ιxx))

is equivalent to "There is exactly one $\scriptstyle\phi$ and it has the property $\scriptstyle\psi$":

$\exists x\forall y (\phi(y) \iff y=x \and \psi(y))$

## References

• Donnellan, Keith, "Reference and Definite Descriptions," in Philosophical Review 75 (1966): 281-304.
• Neale, Stephen, Descriptions, MIT Press, 1990.
• Ostertag, Gary (ed.). (1998) Definite Descriptions: A Reader Bradford, MIT Press. (Includes Donnellan (1966), Chapter 3 of Neale (1990), Russell (1905), and Strawson (1950).)
• Reimer, Marga and Bezuidenhout, Anne (eds.) (2004), Descriptions and Beyond, Clarendon Press, Oxford
• Russell, Bertrand, "On Denoting," in Mind 14 (1905): 479-493. Online text
• Strawson, P. F., "On Referring," in Mind 59 (1950): 320-344.

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