Definite descriptions are the logical constructions that Russell has in mind when when he describes them as “incomplete symbols”.
And that's not just because "the" sounds incomplete -- cfr. "The The").
The notion of a “logical fiction”, on the other hand, applies most straightforwardly to classes.
Other constructions, such as the notions of the domain and range of a relation, and of one to one mappings that are crucial to the development of arithmetic, are only “incomplete” in an indirect sense due to their being defined as classes of a certain sort, which are in turn constructions.
Russell's theory of descriptions was introduced in his paper “On Denoting” published in the journal Mind.
Russell's theory provides an analysis of sentences of the form
is called a definite description in contrast with
which is an indefinite description.
as in Bradley, "A king of Utopia" (1883) -- "A King of Utopia died yesterday".
The analysis proposes that
is equivalent to
There is one and only one
Given this analysis, the logical properties of descriptions can be deduced using just the logic of quantifiers and identity.
Among the theorems in *14 of Principia Mathematica are those showing that,
(1) if there is just one
(2) if the
These theorems show that proper (uniquely referring) descriptions behave like proper names, the “singular terms” of logic.
Some of these results have been controversial — Strawson (1950), who was Grice's student at St. John's, claimed that an utterance of
The present King of France is bald.
should be truth valueless since there is no present king of France, rather than “plainly” false, as Russell's theory predicts.
Russell's reply to Strawson in (Russell 1959, 239–45) is helpful for understanding Russell's philosophical methodology of which logical construction is just a part.
It is, however, by assessing the logical consequences of a construction that it is to be judged, and so Strawson challenged Russell in an appropriate way.
The theory of descriptions introduces Russell's notion of incomplete symbol.
This arises because no definitional equivalent of
appears in the formal analysis of sentences in which the description occurs.
of which no subformula, or even a contiguous segment, can be identified as the analysis of
Similarly, talk about “the average family” as in
The average family has 2.2 children.
becomes “The number of children in families divided by the number of families = 2.2”.
There is no segment of that analysis that corresponds to “the average family”.
Instead we are given a procedure for eliminating such expressions from contexts in which they occur, hence this is another example of an “incomplete symbol” and the definition of an average is an example of a “contextual definition.”
It is arguable that Russell's definition of definite descriptions was the most prominent early example of the philosophical distinction between surface grammatical form and logical form, and thus marks the beginnings of linguistic analysis as a method in philosophy.
Linguistic analysis begins by looking past superficial linguistic form to see an underlying philosophical analysis.
Frank Plumpton Ramsey (whom Grice quotes in "Method in philosophical psychology" -- Ramsified naming, Ramsified describing) described the theory of descriptions as a “paradigm of philosophy” (Ramsey 1929, 1).
While in itself surely not a model for all philosophy, it was at least a paradigm for the other examples of logical constructions that Russell listed when looking back on the development of his philosophy in 1924.
The theory of descriptions has been criticized by some philosophers who see descriptions and other noun phrases as full-fledged linguistic constituents of sentences, and who see the sharp distinction between grammatical and logical form as a mistake.
Following Gilbert Ryle's (1931) influential criticisms of Meinong's theory of non-existent objects, the theory of descriptions has been taken as a model for avoiding ontological commitment to objects, and so logical constructions in general are often seen as being chiefly used to eliminate purported entities. In fact, that goal is at most peripheral to many constructions.
The principal goal of these constructions is to allow the proof of propositions that would otherwise have to be assumed as axioms or hypotheses.
Nor need the introduction of constructions always result in the elimination of problematic entities.
Yet other constructions should be seen more as reductions of one class of entity to another, or replacements of one notion by a more precise, mathematical, substitute.