Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/108

Just as a class must not be capable of being or not being a member of itself, so a relation must not be referent or relatum with respect to itself. This turns out to be equivalent to the assertion that $\scriptstyle {\phi !({\hat {x}},{\hat {y}})}$ cannot significantly be either of the arguments $\scriptstyle {x}$ or $\scriptstyle {y}$ in $\scriptstyle {\phi !(x,y)}$. This principle, again, results from the limitation to the possible arguments to a function explained at the beginning of Chapter II.

We may sum up this whole discussion on incomplete symbols as follows.

The use of the symbol "$\scriptstyle {(}$℩$\scriptstyle {x)(\phi x)}$" as if in "$\scriptstyle {f(}$℩$\scriptstyle {x)(\phi x)}$" it directly represented an argument to the function $\scriptstyle {f{\hat {z}}}$ is rendered possible by the theorems

$\scriptstyle {\vdash :.\mathbf {E!} (}$℩$\scriptstyle {x)(\phi x).\supset :(x).fx.\supset .f(}$℩$\scriptstyle {x)(\phi x)}$,
$\scriptstyle {\vdash :(}$℩$\scriptstyle {x)(\phi x)=(}$℩$\scriptstyle {x)(\psi x).\supset .f(}$℩$\scriptstyle {x)(\phi x)\equiv f(}$℩$\scriptstyle {x)(\psi x)}$,
$\scriptstyle {\vdash :\mathbf {E!} (}$℩$\scriptstyle {x)(\phi x).\supset .(}$℩$\scriptstyle {x)(\phi x)=(}$℩$\scriptstyle {x)(\phi x)}$,
$\scriptstyle {\vdash :(}$℩$\scriptstyle {x)(\phi x)=(}$℩$\scriptstyle {x)(\psi x).\equiv .(}$℩$\scriptstyle {x)(\psi x)=(}$℩$\scriptstyle {x)(\phi x)}$,
$\scriptstyle {\vdash :(}$℩$\scriptstyle {x)(\phi x)=(}$℩$\scriptstyle {x)(\psi x).(}$℩$\scriptstyle {x)(\psi x)=(}$℩$\scriptstyle {x)(\chi x).\supset .(}$℩$\scriptstyle {x)(\phi x)=(}$℩$\scriptstyle {x)(\chi x)}$.

The use of the symbol "$\scriptstyle {{\hat {x}}(\phi x)}$" (or of a single letter, such as $\scriptstyle {\alpha }$, to represent such a symbol) as if, in "$\scriptstyle {f\{{\hat {x}}(\phi x)\}}$," it directly represented an argument $\scriptstyle {\alpha }$ to a function $\scriptstyle {f{\hat {\alpha }}}$, is rendered possible by the theorems

${\begin{aligned}&\scriptstyle {\vdash :(\alpha ).f\alpha .\supset .f\{{\hat {x}}(\phi x)\},}\\&\scriptstyle {\vdash :{\hat {x}}(\phi x)={\hat {x}}(\psi x).\supset .f\{{\hat {x}}(\phi x)\}\equiv f\{{\hat {x}}(\psi x)\},}\\&\scriptstyle {\vdash .{\hat {x}}(\phi x)={\hat {x}}(\phi x),}\\&\scriptstyle {\vdash :{\hat {x}}(\phi x)={\hat {x}}(\psi x).\equiv .{\hat {x}}(\psi x)={\hat {x}}(\phi x),}\\&\scriptstyle {\vdash :{\hat {x}}(\phi x)={\hat {x}}(\psi x).{\hat {x}}(\psi x)={\hat {x}}(\chi x).\supset .{\hat {x}}(\phi x)={\hat {x}}(\chi x).}\end{aligned}}$

Throughout these propositions the types must be supposed to be properly adjusted, where ambiguity is possible.

The use of the symbol "$\scriptstyle {{\hat {x}}{\hat {y}}\{\phi (x,y)\}}$ (or of a single letter, such as $\scriptstyle {R}$, to represent such a symbol) as if, in "$\scriptstyle {f\{{\hat {x}}{\hat {y}}\phi (x,y)\}}$," it directly represented an argument $\scriptstyle {R}$ to a function $\scriptstyle {f{\hat {R}}}$, is rendered possible by the theorems

${\begin{aligned}&\scriptstyle {\vdash :(R).fR.\supset .f\{{\hat {x}}{\hat {y}}\phi (x,y)\},}\\&\scriptstyle {\vdash :{\hat {x}}{\hat {y}}\phi (x,y)={\hat {x}}{\hat {y}}\phi (x,y)={\hat {x}}{\hat {y}}\psi (x,y).\supset .f\{{\hat {x}}{\hat {y}}\phi (x,y)\}\equiv f\{{\hat {x}}{\hat {y}}\psi (x,y)\},}\\&\scriptstyle {\vdash .{\hat {x}}{\hat {y}}\phi (x,y)={\hat {x}}{\hat {y}}\phi (x,y),}\\&\scriptstyle {\vdash :{\hat {x}}{\hat {y}}\phi (x,y)={\hat {x}}{\hat {y}}\psi (x,y).\equiv .{\hat {x}}{\hat {y}}\psi (x,y)={\hat {x}}{\hat {y}}\phi (x,y),}\\&\scriptstyle {\vdash :{\hat {x}}{\hat {y}}\phi (x,y)={\hat {x}}{\hat {y}}\psi (x,y).{\hat {x}}{\hat {y}}\psi (x,y)={\hat {x}}{\hat {y}}\chi (x,y).}\\&\scriptstyle {\qquad \qquad \qquad \qquad \qquad \quad \supset .{\hat {x}}{\hat {y}}\phi (x,y)={\hat {x}}{\hat {y}}\chi (x,y).}\end{aligned}}$

Throughout these propositions the types must be supposed to be properly adjusted where ambiguity is possible.