Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/52

Similarly

${\begin{aligned}\scriptstyle {R~\cdot \!\!\!\cup \,S={\hat {x}}{\hat {y}}(xRy.\lor .xSy)\quad }&\scriptstyle {\text{Df,}}\\\scriptstyle {{\dot {\lnot }}R={\hat {x}}{\hat {y}}\{\sim (xRy)\}\qquad }&\scriptstyle {\text{Df,}}\\\scriptstyle {R~\cdot \!\!\!\!\subset \,S.=.xRy.\supset _{x,y}.xSy\quad }&\scriptstyle {\text{Df.}}\end{aligned}}$

Generally, when we require analogous but different symbols for relations and for classes, we shall choose for relations the symbol obtained by adding a dot, in some convenient position, to the corresponding symbol for classes. (The dot must not be put on the line, since that would cause confusion with the use of dots as brackets.) But such symbols require and receive a special definition in each case.

A class is said to exist when it has at least one member: "$\scriptstyle {\alpha }$ exists" is denoted by "$\scriptstyle {\exists !\alpha }$." Thus we put

$\scriptstyle {\exists !\alpha .=.(\exists x).x\in \alpha \quad {\text{Df.}}}$

The class which has no members is called the "null-class," and is denoted by "$\scriptstyle {\Lambda }$." Any propositional function which is always false determines the null-class. One such function is known to us already, namely "$\scriptstyle {x}$ is not identical with $\scriptstyle {x}$," which we denote by "$\scriptstyle {x\neq x}$." Thus we may use this function for defining $\scriptstyle {\Lambda }$, and put

$\scriptstyle {\Lambda ={\hat {x}}(x\neq x)\quad {\text{Df.}}}$

The class determined by a function which is always true is called the universal class, and is represented by $\scriptstyle {\text{V}}$; thus

$\scriptstyle {{\text{V}}={\hat {x}}(x=x)\quad {\text{Df.}}}$

Thus $\scriptstyle {\Lambda }$ is the negation of $\scriptstyle {\text{V}}$. We have

$\scriptstyle {\vdash .(x).x\in {\text{V}}}$,

i.e. "'$\scriptstyle {x}$ is a member of $\scriptstyle {\text{V}}$' is always true"; and

$\scriptstyle {\vdash .(x).x\sim \in \Lambda }$,

i.e. "'$\scriptstyle {x}$ is a member of $\scriptstyle {\Lambda }$' is always false." Also

$\scriptstyle {\vdash :\alpha =\Lambda .\equiv .\sim \exists !\alpha }$,

i.e. "$\scriptstyle {\alpha }$ is the null-class" is equivalent to "$\scriptstyle {\alpha }$ does not exist." For relations we use similar notations. We put

$\scriptstyle {{\dot {\exists }}!R.=.(\exists x,y).xRy}$,

i.e. "$\scriptstyle {{\dot {\exists }}!R}$" means that there is at least one couple $\scriptstyle {x,y}$ between which the relation $\scriptstyle {R}$ holds. $\scriptstyle {\dot {\Lambda }}$ will be the relation which never holds, and $\scriptstyle {\dot {\text{V}}}$ the relation which always holds. $\scriptstyle {\dot {\text{V}}}$ is practically never required; $\scriptstyle {\dot {\Lambda }}$ will be the relation $\scriptstyle {{\hat {x}}{\hat {y}}(x\neq x.y\neq y)}$. We have

$\scriptstyle {\vdash .(x,y).\sim (x{\dot {\Lambda }}y)}$,

and

$\scriptstyle {\vdash :R={\dot {\Lambda }}.\equiv .\sim {\dot {\exists }}!R}$.