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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 largest prime number" are improper because X applies to nothing. Improper descriptions raise some difficult questions about the law of excluded middle, denotation, modality, and mental content.

Main article: Theory of descriptions

France is a republic, and has no king. Consider the statement "The present King of France is bald." Is this statement true? Is it false? Is it meaningless?

It does not seem to be true, for there is no present King of France. But if it is false, then one would suppose that the negation of the statement is true, that is, "The present King of France has hair (is not bald)." But that seems no more true than the original statement.

Is it meaningless, then? One might suppose so (and some philosophers have; see below), because it certainly does fail to denote in a sense, but on the other hand it seems to mean something that we can quite clearly understand.

Bertrand Russell, extending the work of Gottlob Frege, who had similar thoughts, proposed according to his 'theory of descriptions' that when we say "the present King of France is bald", we are making three separate assertions:

1. there is an x such that x is the King of France (∃x(Fx))
2. for every x that is the King of France and every y that is the King of France, x equals y (i.e. there is at most one King of France) (∀x(Fx → ∀y(Fy → y=x)))
3. x is bald. (∀x(Fx → Bx))

Since assertion 1 is plainly false, and our statement is the conjunction of all three assertions, our statement is false.

Similarly, for "the present King of France is not bald", we have the identical assertions 1 and 2 plus

4. x is not bald

so "the present King of France is not bald", because it consists of a conjunction, one of whose terms is assertion 1, is also false.

The law of the excluded middle is not violated because by denying both "the King of France is bald" and "the King of France is not bald," we are not asserting the existence of some x which is neither bald nor not bald, but denying the existence of some x which is the King of France.

There is a second way of stating "the present King of France is not bald". Instead of substituting x in the sentence "x is not bald" as we have done above, we may negate (1) yielding "it is not the case that there exists an x and x is bald" (alternatively "it is not the case that there exists an x, therefore x is neither bald nor not bald". This sentence is true as opposed to the statement obtained by the previous method. Second, it is easier to see that it does not violate the law of excluded middle. Russell's analysis has been attacked by P.F. Strawson, Keith Donnellan and others, and it has been defended and refined by Stephen Neale.

When using the definite descriptor in a formal logical context, it is common to denote it by the symbol iota:

ψ(ιx(φx)) ≔ ∃x(∀y(φy ↔ y = x) ∧ ψx).

• 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).)
• Russell, Bertrand, "On Denoting," in Mind 14 (1905): 479-493. Online text
• Strawson, P. F., "On Referring," in Mind 59 (1950): 320-344.

• Stanford Encyclopedia of Philosophy