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

will mean

$\scriptstyle {(\exists c):\phi x.\equiv _{x}.x=c:\psi c:\supset .p}$,

but

$\scriptstyle {[(}$℩$\scriptstyle {x)(\phi x)]:\psi (}$℩$\scriptstyle {x)(\phi x)(\phi x).\supset .p}$

will mean

$\scriptstyle {(\exists c):\phi x.\equiv _{x}.x=c:\psi c.\supset .p}$.

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

$\scriptstyle {[(}$℩$\scriptstyle {x)(\phi x)].\sim \psi (}$℩$\scriptstyle {x)(\phi x)}$

will mean

$\scriptstyle {(\exists c):\phi x.\equiv _{x}.x=c:\sim \psi c}$

while

$\scriptstyle {\sim \{[(}$℩$\scriptstyle {x)(\phi x)].\sim \psi (}$℩$\scriptstyle {x)(\phi x)\}}$

will mean

$\scriptstyle {\sim \{(\exists c):\phi x.\equiv _{x}.x=c:\psi c\}}$.

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

$\scriptstyle {[(}$℩$\scriptstyle {x)(\phi x)].f(}$℩$\scriptstyle {x)(\phi x).=:(\exists c):\phi x.\equiv _{x}.x=c:fc\quad {\text{Df.}}}$

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

$\scriptstyle {[(}$℩$\scriptstyle {x)(\phi x)].f(}$℩$\scriptstyle {x)(\phi x).\supset .p}$

the scope of $\scriptstyle {(}$℩$\scriptstyle {x)(\phi x)}$ is $\scriptstyle {f(}$℩$\scriptstyle {x)(\phi x)}$; but in

$\scriptstyle {[(}$℩$\scriptstyle {x)(\phi x)]:f(}$℩$\scriptstyle {x)(\phi x).\supset .p}$

the scope is

$\scriptstyle {f(}$℩$\scriptstyle {x)(\phi x).\supset .p}$;

in

$\scriptstyle {\sim \{[(}$℩$\scriptstyle {x)(\phi x)].f(}$℩$\scriptstyle {x)(\phi x)\}}$

the scope is $\scriptstyle {f(}$℩$\scriptstyle {x)(\phi x)}$; but in

$\scriptstyle {[(}$℩$\scriptstyle {x)(\phi x)].\sim f(}$℩$\scriptstyle {x)(\phi x)\}}$

the scope is

$\scriptstyle {\sim f(}$℩$\scriptstyle {x)(\phi x)}$.

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
 (1)

$\scriptstyle {[(}$℩$\scriptstyle {x)(\phi x)]:[(}$℩$\scriptstyle {x)(\psi x)].f\{(}$℩$\scriptstyle {x)(\phi x),(}$℩$\scriptstyle {x)(\psi x)\}}$,

 (2)

$\scriptstyle {[(}$℩$\scriptstyle {x)(\psi x)]:[(}$℩$\scriptstyle {x)(\phi x)].f\{(}$℩$\scriptstyle {x)(\phi x),(}$℩$\scriptstyle {x)(\psi x)\}}$,