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 mustmust neither be nor [sic] 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

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 "$\scriptstyle{f\{\hat x(\phi x)\}}$,"|undefined it directly represented an argument $$\scriptstyle{\alpha}$$ to a function $\scriptstyle{f\hat\alpha}$, is rendered possible by the theorems

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

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

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