Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/110

 but the function "$\scriptstyle{\hat z=}$the author of Waverley" has the property that George IV wished to know whether its value with the argument "Scott" was true, whereas the function "$\scriptstyle{\hat z=}$Scott" has no such property, and therefore the two functions are not identical. Hence there is a propositional function, namely which holds without any exception, and yet does not hold when for $$\scriptstyle{x}$$ we substitute a class, and for $$\scriptstyle{y}$$ and $$\scriptstyle{z}$$ we substitute functions. This is only possible because a class is an incomplete symbol, and therefore "$\scriptstyle{\hat z(\phi z)=\psi!\hat z}$" is not a value of "$\scriptstyle{x=y}$."

It will be observed that "$\scriptstyle{\theta!\hat z=\psi!\hat z}$" is not an extensional function of $\scriptstyle{\psi!\hat z}$. Thus the scope of $$\scriptstyle{\hat z(\phi z)}$$ is relevant in interpreting the product If we take the whole of the product as the scope of $\scriptstyle{\hat z(\phi z)}$, the product is equivalent to

We may say generally that the fact that $$\scriptstyle{\hat z(\phi z)}$$ is an incomplete symbol is not relevant so long as we confine ourselves to extensional functions of functions, but is apt to become relevant for other functions of functions.