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

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I] IDENTITY 23
concerned are all truth-functions and the fact that mathematics is concerned
with extensions rather than intensions.
Convenient abbreviation. The following definitions give alternative and
often more convenient notations:
Ox.. *rx =: (x):. ~b. O. 'X Df,
Ofx.-. x: =:(x): X -..x Df.
This notation " Ox. )x. * " is due to Peano, who, however, has no notation
for the general idea " (x). bx." It may be noticed as an exercise in the use
of dots as brackets that we might have written
(fX )a tr. =. (x). fX ~ q$x Df,
fx - xx x. =.(x). fbx x Df.
In practice however, when Ox' and f^r are special functions, it is not possible
to employ fewer dots than in the first form, and often more are required.
The following definitions give abbreviated notations for functions of two
or more variables:
(x, y). (x, y).=: (x): (y). (x, y) Df,
and so on for any number of variables;
b (x, y).;),y. * (x, y): =: (x, y): q (x, y). 2). * (x, y) Df,
and so on for any number of variables.
Identity. The propositional function "x is identical with y" is expressed by
= y.
This will be defined (cf. *13'01), but, owing to certain difficult points involved
in the definition, we shall here omit it (cf. Chapter II). We have, of
course,
F. x = x (the law of identity),
1:x=y..y=,
F:x=y.y= z.. = z.
The first of these expresses the reflexive property of identity: a relation is
called reflexive when it holds between a term and itself, either universally, or
whenever it holds between that term and some term. The second of the
above propositions expresses that identity is a symmetrical relation: a relation
is called symmetrical if, whenever it holds between x and y, it also holds
between y and x. The third proposition expresses that identity is a transitive
relation: a relation is called transitive if, whenever it holds between x and y
and between y and z, it holds also between x and z.
We shall find that no new definition of the sign of equality is required in
mathematics: all mathematical equations in which the sign of equality is