Definite descriptions refer by way of a description only when there is one and only one object satisfying the description; that is, an object satisfying the description must exist and be unique. This is often so for ordinary definite descriptions only when the description is understood to contain an implicit qualification that the object be the most salient one satisfying other parts of the description.

A famous analysis of definite descriptions was first proposed early in the 20^th century by Bertrand Russell. According to Russell’s analysis, a sentence The C is such that (… it …) amounts to Something such that it and only it is a C is such that (… it …). This analysis is equivalent to the conjunction of Some C is such that (… it …) and There is at most one C, so, according to Russell, the effect of using a definite rather than an indefinite article is to imply the latter conjunct. Russell’s analysis treats a definite description as a kind of quantifier phrase and leads to scope ambiguities in negative sentences involving definite descriptions.

An alternative approach avoids this suggestion of ambiguity by treating definite descriptions as individual terms and analyzing them by the use of a description operator, which applies to predicate abstracts to form terms. We use a sans-serif capital I as notation for the description operator, writing Iρ as Ix ρx. A term formed in this way has the sole member of the predicate’s extension as its reference value if that extension has a unique member; otherwise, its reference value is the Nil. We fix a logically constant term ∗ (asterisk operator), the Nil, which always has the Nil as its reference value. The content of … the C … on this analysis can be expressed using a branching conditional as if there is exactly one C, then some C is such that (… it …); otherwise, … the Nil ….

Each of the two approaches to analyzing definite descriptions can be used to exhibit the difference between a restrictive and a non-restrictive relative clause when these modify a common noun governed by the article the. Although both analyses point to differences between such sentences, their accounts of the relations between them differ.