Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/48

Here $\scriptstyle {\in }$ is chosen as the initial of the word éorí. "$\scriptstyle {x\in \alpha }$" may be read "$\scriptstyle {x}$ is an $\scriptstyle {\alpha }$." Thus "$\scriptstyle {x\in }$ man" will mean "$\scriptstyle {x}$ is a man," and so on. For typographical convenience we shall put

${\begin{aligned}\scriptstyle {x\sim \in \alpha .=.\sim (x\in \alpha )\quad }&\scriptstyle {\text{Df,}}\\\scriptstyle {x,y\in \alpha .=.x\in \alpha .y\in \alpha \quad }&\scriptstyle {\text{Df.}}\end{aligned}}$

For "class" we shall write "$\scriptstyle {\text{Cls}}$"; thus "$\scriptstyle {\alpha \in {\text{Cls}}}$" means "$\scriptstyle {\alpha }$ is a class."

We have

$\scriptstyle {\vdash :x\in {\hat {z}}(\phi z).\equiv .\phi x}$,

i.e. "'$\scriptstyle {x}$ is a member of the class determined by $\scriptstyle {\phi {\hat {z}}}$' is equivalent to '$\scriptstyle {x}$ satisfies $\scriptstyle {\phi {\hat {z}}}$,' or to '$\scriptstyle {\phi x}$ 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 $\scriptstyle {\phi x}$, $\scriptstyle {\psi x}$ are formally equivalent functions, they determine the same class; for in that case, if $\scriptstyle {x}$ is a member of the class determined by $\scriptstyle {\phi {\hat {x}}}$, and therefore satisfies $\scriptstyle {\phi x}$, it also satisfies $\scriptstyle {\psi x}$, and is therefore a member of the class determined by $\scriptstyle {\psi {\hat {x}}}$. Thus we have

$\scriptstyle {\vdash :.{\hat {z}}(\phi z)={\hat {z}}(\psi z).\equiv :\phi x.\equiv _{x}.\psi x}$.

The following propositions are obvious and important:

$\scriptstyle {\vdash :.\alpha ={\hat {z}}(\phi z).\equiv :x\in \alpha .\equiv _{x}.\phi x}$,

i.e. $\scriptstyle {\alpha }$ is identical with the class determined by $\scriptstyle {\phi {\hat {z}}}$ when, and only when, "$\scriptstyle {x}$ is an $\scriptstyle {\alpha }$" is formally equivalent to $\scriptstyle {\phi x}$;

$\scriptstyle {\vdash :.\alpha =\beta .\equiv :x\in \alpha .\equiv _{x}.x\in \beta }$,

i.e. two classes $\scriptstyle {\alpha }$ and $\scriptstyle {\beta }$ are identical when, and only when, they have the same membership;

$\scriptstyle {\vdash .{\hat {x}}(x\in \alpha )=\alpha }$,

i.e. the class whose determining function is "$\scriptstyle {x}$ is an $\scriptstyle {\alpha }$" is $\scriptstyle {\alpha }$, in other words, $\scriptstyle {\alpha }$ is the class of objects which are members of $\scriptstyle {\alpha }$;

$\scriptstyle {\vdash .{\hat {z}}(\phi z)\in {\text{Cls}}}$,

i.e. the class determined by the function $\scriptstyle {\phi {\hat {z}}}$ 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 "$\scriptstyle {x\in \alpha }$." Hence any extensional function of functions which holds when its argument is a function of the form "$\scriptstyle {{\hat {z}}\in \alpha }$," whatever possible value $\scriptstyle {\alpha }$ may have, will hold also when its argument is any function $\scriptstyle {\phi {\hat {z}}}$. 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