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

198

MATHEMATICAL LOGIC

[PART I

symbol of the form z (G)! z), where 4 is properly the apparent variable concerned. The use of single letters in place of such symbols as z(<z) or z (! z)
is practically almost indispensable, since otherwise the notation rapidly
becomes intolerably cumbrous. Thus "x e a " will mean " x is a member of
the class a," and may be used wherever no special defining function of the
class a is in question.
The following definition defines what is meant by a class.
*20-03. Cls = a 1(g>). a = (4! z)} Df
Note that the expression (" a{(a<4). a = (4>! z)}" has no meaning in
isolation: we have merely defined (in *20*01) certain uses of such expressions.
What the above definition decides is that the symbol " Cls " may replace the
symbol "a {(g^). a = z (! z)}," wherever the latter occurs, and that the
meaning of the combination of symbols concerned is to be unchanged thereby.
Thus " Cls," also, has no meaning in isolation, but merely in certain uses.
The above definition, like many future definitions, is ambiguous as to
type. The Latin letter z, according to our conventions, is to represent the
lowest type concerned; thus 4 is of the type next above this. It is convenient to speak of a class as being of the same type as its defining function;
thus a is of the type next above that of z, and "Cls" is of the type next above
that of a. Thus the type of "CIs" is fixed relatively to the lowest type concerned; but if, in two different contexts, different types are the lowest concerned,
the meaning of " Cls " will be different in these two contexts. The meaning
of " CIs " only becomes definite when the lowest type concerned is specified.
Equality between classes is defined by applying *13 01, symbolically unchanged, to their defining functions, and then using *2001.
The propositions of the present number may be divided into three sets.
First, we have those that deal with the fundamental properties of classes;
these end with *2043. Then we have a set of propositions dealing with
both classes and descriptions; these extend from *205 to *20'59 (with the
exception of *20 53 54). Lastly, we have a set of propositions designed to
prove that classes of classes have all the same formal properties as classes of
individuals.
In the first set, the principal propositions are the following.
*2015. h:. w --. % Z: E. (fz)= 2 (z)
I.e. two classes are identical when, and only when, their defining functions
are formally equivalent. This is the principal property of classes.
*2031.. H (4z)=z(%z).: X ez(z). E.xez(Xz)
I.e. two classes are identical when, and only when, they have the same
members.