Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/92

 or some equivalent. Here the apparent grammatical subject $$\scriptstyle{(}$$$$\scriptstyle{x)(\phi x)}$$ has completely disappeared; thus in "$\scriptstyle{\sim\mathbf{E!}(}$$\scriptstyle{x)(\phi x)}$," $$\scriptstyle{(}$$$$\scriptstyle{x)(\phi x)}$$ is an incomplete symbol.

By an extension of the above argument, it can easily be shown that $$\scriptstyle{(}$$$$\scriptstyle{x)(\phi x)}$$ is always an incomplete symbol. Take, for example, the following proposition: "Scott is the author of Waverley." [Here "the author of Waverley" is $$\scriptstyle{(}$$$$\scriptstyle{x)(x}$$ wrote Waverley$\scriptstyle{)}$.] This proposition expresses an identity; thus if "the author of Waverley" could be taken as a proper name, and supposed to stand for some object $\scriptstyle{c}$, the proposition would be "Scott is $\scriptstyle{c}$." But if $$\scriptstyle{c}$$ is anyone except Scott, this proposition is false; while if $$\scriptstyle{c}$$ is Scott, the proposition is "Scott is Scott," which is trivial, and plainly different from "Scott is the author of Waverley." Generalizing, we see that the proposition is one which may be true or may be false, but is never merely trivial, like $\scriptstyle{a=a}$; whereas, if $$\scriptstyle{(}$$$$\scriptstyle{x)(\phi x)}$$ were a proper name, $$\scriptstyle{a=(}$$$$\scriptstyle{x)(\phi x)}$$ would necessarily be either false or the same as the trivial proposition $\scriptstyle{a=a}$. We may express, this by saying that $$\scriptstyle{a=(}$$$$\scriptstyle{x)(\phi x)}$$ is not a value of the propositional function $\scriptstyle{a=y}$, from which it follows that $$\scriptstyle{(}$$$$\scriptstyle{x)(\phi x)}$$ is not a value of $\scriptstyle{y}$. But since $$\scriptstyle{y}$$ may be anything, it follows that $$\scriptstyle{(}$$$$\scriptstyle{x)(\phi x)}$$ is nothing. Hence, since in use it has meaning, it must be an incomplete symbol.

It might be suggested that "Scott is the author of Waverley" asserts that "Scott" and "the author of Waverley" are two names for the same object. But a little reflection will show that this would be a mistake. For if that were the meaning of "Scott is the author of Waverley," what would be required for its truth would be that Scott should have been called the author of Waverley: if he had been so called, the proposition would be true, even if some one else had written Waverley; while if no one called him so, the proposition would be false, even if he had written Waverley. But in fact he was the author of Waverley at a time when no one called him so, and he would not have been the author if everyone had called him so but some one else had written Waverley. Thus the proposition "Scott is the author of Waverley" is not a proposition about names, like "Napoleon is Bonaparte"; and this illustrates the sense in which "the author of Waverley" differs from a true proper name.

Thus all phrases (other than propositions) containing the word the (in the singular) are incomplete symbols: they have a meaning in use, but not in isolation. For "the author of Waverley" cannot mean the same as "Scott," or "Scott is the author of Waverley" would mean the same as "Scott is Scott," which it plainly does not; nor can "the author of Waverley" mean anything other than "Scott," or "Scott is the author of Waverley" would be false. Hence "the author of Waverley" means nothing.