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

24

INTRODUCTION

[CHAP.

used in the ordinary way express some identity, and thus use the sign of
equality in the above sense.
If x and y are identical, either can replace the other in any proposition
without altering the truth-value of the proposition; thus we have: x = y. D. Ofx_ by.
This is a fundamental property of identity, from which the remaining properties
mostly follow.
It might be thought that identity would not have much importance, since
it can only hold between x and y if x and y are different symbols for the
same object. This view, however, does not apply to what we shall call
"descriptive phrases," i.e. "the so-and-so." It is in regard to such phrases
that identity is important, as we shall shortly explain. A proposition such
as " Scott was the author of Waverley" expresses an identity in which there
is a descriptive phrase (namely " the author of Waverley"); this illustrates
how, in such cases, the assertion of identity may be important. It is
essentially the same case when the newspapers say "the identity of the
criminal has not transpired." In such a case, the criminal is known by a
descriptive phrase, namely "the man who did the deed," and we wish to
find an x of whom it is true that "x = the man who did the deed." When
such an x has been found, the identity of the criminal has transpired.
Classes and relations. A class (which is the same as a manifold or
aggregate) is all the objects satisfying some propositional function. If a is
the class composed of the objects satisfying Ox, we shall say that a is the class
determined by Ox. Every propositional function thus determines a class,
though if the propositional function is one which is always false, the class
will be null, i.e. will have no members. The class determined by the function
Ox will be represented by z (pz)*. Thus for example if Ox is an equation,
z (Oz) will be the class of its roots; if Ox is " x has two legs and no feathers,"
z (4z) will be the class of men; if Ox is " 0 < x < 1," 2 (bz) will be the class
of proper fractions, and so on.
It is obvious that the same class of objects will have many determining
functions. When it is not necessary to specify a determining function of a
class, the class may be conveniently represented by a single Greek letter.
Thus Greek letters, other than those to which some constant meaning is
assigned, will be exclusively used for classes.
There are two kinds of difficulties which arise in formal logic; one kind
arises in connection with classes and relations and the other in connection
with descriptive functions. The point of the difficulty for classes and
relations, so far as it concerns classes, is that a class cannot be an object
suitable as an argument to any of its determining functions. If a represents
* Any other letter may be used instead of z.