Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/96

The first of these, eliminating $\scriptstyle {(}$℩$\scriptstyle {x)(\phi x)}$, becomes
 (3)

$\scriptstyle {(\exists c):\phi x.\equiv _{x}.x=c:[(}$℩$\scriptstyle {x)(\psi x)].f\{c,(}$℩$\scriptstyle {x)(\psi x)\}}$,

which, eliminating $\scriptstyle {(}$℩$\scriptstyle {x)(\psi x)}$, becomes
 (4)

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

and the same proposition results if, in (1), we eliminate first $\scriptstyle {(}$℩$\scriptstyle {x)(\psi x)}$ and then $\scriptstyle {(}$℩$\scriptstyle {x)(\phi x)}$. Similarly (2) becomes, when $\scriptstyle {(}$℩$\scriptstyle {x)(\phi x)}$ and $\scriptstyle {(}$℩$\scriptstyle {x)(\psi x)}$ are eliminated,
 (5)

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

(4) and (5) are equivalent, so that the truth-value of a proposition containing two descriptions is independent of the question which has the larger scope.

It will be found that, in most cases in which descriptions occur, their scope is, in practice, the smallest proposition enclosed in dots or other brackets in which they are contained. Thus for example

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

will occur much more frequently than

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

For this reason it is convenient to decide that, when the scope of an occurrence of $\scriptstyle {(}$℩$\scriptstyle {x)(\phi x)}$ is the smallest proposition, enclosed in dots or other brackets, in which the occurrence in question is contained, the scope need not be indicated by "$\scriptstyle {[(}$℩$\scriptstyle {x)(\phi x)]}$." Thus e.g.

$\scriptstyle {p.\supset .a=(}$℩$\scriptstyle {x)(\phi x)}$

will mean

$\scriptstyle {p.\supset .[(}$℩$\scriptstyle {x)(\phi x)].a=(}$℩$\scriptstyle {x)(\phi x)}$;

and

$\scriptstyle {p.\supset .(\exists a).a=(}$℩$\scriptstyle {x)(\phi x)}$

will mean

$\scriptstyle {p.\supset .(\exists a).[(}$℩$\scriptstyle {x)(\phi x)].a=(}$℩$\scriptstyle {x)(\phi x)}$;

and

$\scriptstyle {p.\supset .a\neq (}$℩$\scriptstyle {x)(\phi x)}$

will mean

$\scriptstyle {p.\supset .[(}$℩$\scriptstyle {x)(\phi x)].\sim \{a=(}$℩$\scriptstyle {x)(\phi x)\}}$;

but

$\scriptstyle {p.\supset .\sim \{a=(}$℩$\scriptstyle {x)(\phi x)\}}$

will mean

$\scriptstyle {p.\supset .\sim \{[(}$℩$\scriptstyle {x)(\phi x)].a=(}$℩$\scriptstyle {x)(\phi x)\}}$.

This convention enables us, in the vast majority of cases that actually occur, to dispense with the explicit indication of the scope of a descriptive symbol; and it will be found that the convention agrees very closely with the tacit conventions of ordinary language on this subject. Thus for example, if "$\scriptstyle {(}$℩$\scriptstyle {x)(\phi x)}$" is "the so-and-so," "$\scriptstyle {a\neq (}$℩$\scriptstyle {x)(\phi x)}$" is to be read "$\scriptstyle {a}$ is not the so-and-so," which would ordinarily be regarded as implying that "the so-and-so" exists; but "$\scriptstyle {\sim \{a=(}$℩$\scriptstyle {x)(\phi x)\}}$" is to be read "it is not true that $\scriptstyle {a}$ is the so-and-so," which would generally be allowed to hold if "the so-and-so" does not exist. Ordinary language is, of course, rather loose and fluctuating in its implications on this matter; but subject to the requirement of definiteness, our convention seems to keep as near to ordinary language as possible.