Principia mathematica, by Alfred North Whitehead ... and Bertrand Russell.

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INTRODUCTION

[CHtAP. III

but the function "2=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 " Z= Scott" has no such property, and therefore the two
functions are not identical. Hence there is a propositional function, namely
a=y.=z.). y=z,
which holds without any exception, and yet does not hold when for x we
substitute a class, and for y and z we substitute functions. This is only
possible because a class is an incomplete symbol, and therefore "^z(qz)=J!z"
is not a value of " x = y."
It will be observed that " 9!0 = -! ~" is not an extensional function of! z. Thus the scope of Z (4z) is relevant in interpreting the product
(z) =!. = (z()=!.
If we take the whole of the product as the scope of Z (4z), the product is
equivalent to
(g0): kx= Z 9 x.! z = qJ!. 0!z = X! z,
and this does imply! =*! Z.
We may say generally that the fact that Z (Oz) 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.