Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/95

 It is important to distinguish these two, for if $\scriptstyle{(}$$\scriptstyle{x)(\phi x)}$ does not exist, the first is true and the second false. Again Here again, when $\scriptstyle{(}$$\scriptstyle{x)(\phi x)}$ does not exist, the first is false and the second true.

In order to avoid this ambiguity in propositions containing $\scriptstyle{(}$$\scriptstyle{x)(\phi x)}$, we amend our definition, or rather our notation, putting By means of this definition, we avoid any doubt as to the portion of our whole asserted proposition which is to be treated as the "$\scriptstyle{f(}$$\scriptstyle{x)(\phi x)}$" of the definition. This portion will be called the scope of $\scriptstyle{f(}$$\scriptstyle{x)(\phi x)}$. Thus in the scope of $\scriptstyle{(}$$\scriptstyle{x)(\phi x)}$ is $\scriptstyle{f(}$$\scriptstyle{x)(\phi x)}$; but in the scope is $\scriptstyle{f(}$$\scriptstyle{x)(\phi x)}$; but in {{centre|$$\scriptstyle{[(}$$$$\scriptstyle{x)(\phi x)].\sim f(}$$$$\scriptstyle{x)(\phi x)\}}$$}}

It will be seen that when $\scriptstyle{(}$$\scriptstyle{x)(\phi x)}$ has the whole of the proposition concerned for its scope, the proposition concerned cannot be true unless $\scriptstyle{\mathbf{E!}(}$$\scriptstyle{x)(\phi x)}$; but when $\scriptstyle{(}$$\scriptstyle{x)(\phi x)}$ has only part of the proposition concerned for its scope, it may often be true even when $\scriptstyle{(}$$\scriptstyle{x)(\phi x)}$ does not exist. It will be seen further that when $\scriptstyle{\mathbf{E!}(}$$\scriptstyle{x)(\phi x)}$, we may enlarge or diminish the scope of $\scriptstyle{(}$$\scriptstyle{x)(\phi x)}$ as much as we please without altering the truth-value of any proposition in which it occurs.

If a proposition contains two descriptions, say $\scriptstyle{(}$$\scriptstyle{x)(\phi x)}$ and $\scriptstyle{(}$$\scriptstyle{x)(\psi x)}$, we have to distinguish which of them has the larger scope, i.e. we have to distinguish {{centre|$$\scriptstyle{[(}$$$$\scriptstyle{x)(\psi x)]:[(}$$$$\scriptstyle{x)(\phi x)].f\{(}$${{unicode|&#8489;}}$$\scriptstyle{x)(\phi x),(}$${{unicode|&#8489;}}$$\scriptstyle{x)(\psi x)\}}$$,}}