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

26

INTRODUCTION

[CHAP.

Here e is chosen as the initial of the word earO. "x e a " may be read " x is
an a." Thus "x e man " will mean "x is a man," and so on. For typographical
convenience we shall put
axme a.= =.(x e a) Df,
x,yea.=.xea.yea Df.
For " class " we shall write " Cls"; thus "a e Cls " means " a is a class."
We have
-: xE~(cz).E-.x,
i.e. "'x is a member of the class determined by ~z' is equivalent to 'x
satisfies f^,' or to 'ox is true."'
A class is wholly determinate when its membership is known, that is,
there cannot be two different classes having the same membership. Thus if
fx, sx are formally equivalent functions, they determine the same class;
for in that case, if x is a member of the class determined by Ox, and therefore
satisfies Ox, it also satisfies fx, and is therefore a member of the class
determined by 4Z. Thus we have:. Z (,Z) = Z (~Z) -: x. -. *X.
The following propositions are obvious and important:
F:. a= z (Oz). =-: x e.-. x,
i.e. a is identical with the class determined by <^ when, and only when,
"x is an a" is formally equivalent to Ox;
F:.a=8. -:xe a.-. xe3,
i.e. two classes a and /3 are identical when, and only when, they have the
same membership;
F. X (x e a) = a,
i.e. the class whose determining function is "x is an a" is a, in other words,
a is the class of objects which are members of a;
F. 2 (z) e Cls,
i.e. the class determined by the function fbz is a class.
It will be seen that, according to the above, any function of one variable
can be replaced by an equivalent function of the form "xea." Hence any
extensional function of functions which holds when its argument is a function
of the form " e a," whatever possible value a may have, will hold also when
its argument is any function OZ. Thus variation of classes can replace
variation of functions of one variable in all the propositions of the sort with
which we are concerned.
In an exactly analogous manner we introduce dual or dyadic relations,
i.e. relations between two terms. Such relations will be called simply
"relations"; relations between more than two terms will be distinguished