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

*37. PLURAL DESCRIPTIVE FUNCTIONS.

Summary of *37.
In this number, we introduce what may be regarded as the plural of R'y.
" R'y " was defined to mean " the term which has the relation R to y." We
now introduce the notation "R"/3" to mean "the terms which have the
relation R to members of 3." Thus if / is the class of great men, and R is
the relation of wife to husband, R",/ will mean "wives of great men." If
/3 is the class of fractions of the form 1-1/2n for integral values of n, and R
is the relation "less than," R",/ will be the class of fractions each of which
is less than some member of this class of fractions, i.e. R"/, will be the class
of proper fractions. Generally, R"/3 is the class of those referents which
have relata that are members of /.
We require also a notation for the relation of R",/ to /3. This relation
we will call Re. Thus R, is the relation which holds between two classes
a and 8 when a consists of all terms which have the relation R to some
member of /3.
A specially important case arises when R'y always exists if y e /. In this
case, R"/3 is the class of all terms of the form R'y when ye /. We will
denote the hypothesis that R'y always exists if y e, by the notation E!! R"S,
meaning "the R's of f's exist."
The definitions are as follows:
*37 01. R"'/ = {(Ly). y e /3. xRy} Df
*37-02. Re = a (a = R"/3) Df
*37'03. RE = Cnv'(R,) Df
This definition serves merely for the avoidance of brackets. Without it,
" Re" would be ambiguous as between (R), and Cnv'(R,), which are not
equal. In all cases in which a suffix occurs, we shall adopt the same convention, i.e. we shall always put
Rsuffx = Cnv'(Rsuffix).