Principia mathematica, by Alfred North Whitehead ... and Bertrand Russell.

DESCRIPTIONS

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There are no classes which contain objects of more than one type.
Accordingly there is a universal class and a null-class proper to each type
of object. But these symbols need not be distinguished, since it will be
found that there is no possibility of confusion. Similar remarks apply to
relations.
Descriptions. By a "description" we mean a phrase of the form "the
so-and-so" or of some equivalent form. For the present, we confine our
attention to the in the singular. We shall use this word strictly, so as to
imply uniqueness; e.g. we should not say " A is the son of B " if B had other
sons besides A. Thus a description of the form "the so-and-so" will only
have an application in the event of there being one so-and-so and no more.
Hence a description requires some propositional function Ox which is satisfied
by one value of x and by no other values; then "the x which satisfies ^ "
is a description which definitely describes a certain object, though we may
not know what object it describes. For example, if y is a man, "x is the
father of y" must be true for one, and only one, value of x. Hence "the
father of y" is a description of a certain man, though we may not know what
man it describes. A phrase containing "the" always presupposes some
initial propositional function not containing "the"; thus instead of "x is the
father of y" we ought to take as our initial function " x begot y"; then "the
father of y" means the one value of x which satisfies this propositional
function.
If O' is a propositional function, the symbol "(?x)((x)" is used in our
symbolism in such a way that it can always be read as "the x which
satisfies Pb." But we do not define "(?xz)(x)" as standing for "the x
which satisfies Ox," thus treating this last phrase as embodying a primitive
idea. Every use of "(?x)((x)," where it apparently occurs as a constituent
of a proposition in the place of an object, is defined in terms of the primitive
ideas already on hand. An example of this definition in use is given by
the proposition "E! (x) )(x)" which is considered immediately. The whole
subject is treated more fully in Chapter III.
The symbol should be compared and contrasted with "x^(Ox)" which in
use can always be read as "the x's which satisfy fx." Both symbols are
incomplete symbols defined only in use, and as such are discussed in
Chapter III. The symbol "2(qbx)" always has an application, namely to
the class determined by 4x; but " (x) (ox)" only has an application when
f$S is only satisfied by one value of x, neither more nor less. It should also
be observed that the meaning given to the symbol by the definition, given
immediately below, of E! (7x)(ox) does not presuppose that we know the
meaning of "one." This is also characteristic of the definition of any other
use of (?x) (x).