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

$\scriptstyle {x=y.x=z.\supset .y=z}$,

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

$\scriptstyle {{\hat {z}}(\phi z)=\psi !{\hat {z}}.{\hat {z}}(\phi z)=\chi !{\hat {z}}}$.

If we take the whole of the product as the scope of $\scriptstyle {{\hat {z}}(\phi z)}$, the product is equivalent to

$\scriptstyle {(\exists \theta ):\phi x\equiv _{x}\theta !x.\theta !{\hat {z}}=\psi !{\hat {z}}.\theta !{\hat {z}}=\chi !{\hat {z}}}$,

and this does imply

$\scriptstyle {\psi !{\hat {z}}=\chi !{\hat {z}}}$.

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.