Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/93

It follows from the above that we must not attempt to define $\scriptstyle {(}$℩$\scriptstyle {x)(\phi x)}$," but must define the uses of this symbol, i.e. the propositions in whose symbolic expression it occurs. Now in seeking to define the uses of this symbol, it is important to observe the import of propositions in which it occurs. Take as an illustration: "The author of Waverley was a poet." This implies (1) that Waverley was written, (2) that it was written by one man, and not in collaboration, (3) that the one man who wrote it was a poet. If any one of these fails, the proposition is false. Thus "the author of 'Slawkenburgius on Noses' was a poet" is false, because no such book was ever written; "the author of 'The Maid's Tragedy' was a poet" is false, because this play was written by Beaumont and Fletcher jointly. These two possibilities of falsehood do not arise if we say "Scott was a poet." Thus our interpretation of the uses of $\scriptstyle {(}$℩$\scriptstyle {x)(\phi x)}$ must be such as to allow for them. Now taking $\scriptstyle {\phi x}$ to replace "$\scriptstyle {x}$ wrote Waverley," it is plain that any statement apparently about $\scriptstyle {(}$℩$\scriptstyle {x)(\phi x)}$ requires (1) $\scriptstyle {(\exists x).(\phi x)}$ and (2) $\scriptstyle {\phi x.\phi y.\supset _{x,y}.x=y}$; here (1) states that at least one object satisfies $\scriptstyle {\phi x}$, while (2) states that at most one object satisfies $\scriptstyle {\phi x}$. The two together are equivalent to

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

which we defined as

$\scriptstyle {\mathbf {E!} (}$℩$\scriptstyle {x)(\phi x)}$.

Thus "$\scriptstyle {\mathbf {E!} (}$℩$\scriptstyle {x)(\phi x)}$" must be part of what is affirmed by any proposition about $\scriptstyle {(}$℩$\scriptstyle {x)(\phi x)}$. If our proposition is $\scriptstyle {f\{(}$℩$\scriptstyle {x)(\phi x)\}}$, what is further affirmed is $\scriptstyle {fc}$, if $\scriptstyle {\phi x.\equiv _{x}.x=c}$. Thus we have

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

i.e. "the $\scriptstyle {x}$ satisfying $\scriptstyle {\phi x}$ satisfies $\scriptstyle {fx}$" is to mean: "There is an object $\scriptstyle {c}$ such that $\scriptstyle {\phi x}$ is true when, and only when, $\scriptstyle {x}$ is $\scriptstyle {c}$, and $\scriptstyle {fc}$ is true," or, more exactly: "There is a $\scriptstyle {c}$ such that '$\scriptstyle {\phi x}$' is always equivalent to '$\scriptstyle {x}$ is $\scriptstyle {c}$,' and $\scriptstyle {fc}$." In this, "$\scriptstyle {(}$℩$\scriptstyle {x)(\phi x)}$" has completely disappeared; thus "$\scriptstyle {(}$℩$\scriptstyle {x)(\phi x)}$" is merely symbolic, and does not directly represent an object, as single small Latin letters are assumed to do^[1]. The proposition "$\scriptstyle {a=(}$℩$\scriptstyle {x)(\phi x)}$" is easily shown to be equivalent to "$\scriptstyle {\phi x.\equiv _{x}.x=a}$." For, by the definition, it is

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

i.e. "there is a $\scriptstyle {c}$ for which $\scriptstyle {\phi x.\equiv _{x}.x=c}$, and this $\scriptstyle {c}$ is $\scriptstyle {a}$," which is equivalent to "$\scriptstyle {\phi x.\equiv _{x}.x=a}$." Thus "Scott is the author of Waverley" is equivalent to:

"'$\scriptstyle {x}$ wrote Waverley' is always equivalent to '$\scriptstyle {x}$ is Scott,'"

i.e. "$\scriptstyle {x}$ wrote Waverley" is true when $\scriptstyle {x}$ is Scott and false when $\scriptstyle {x}$ is not Scott.

Thus although "$\scriptstyle {(}$℩$\scriptstyle {x)(\phi x)}$" has no meaning by itself, it may be substituted for $\scriptstyle {y}$ in any propositional function $\scriptstyle {fy}$, and we get a significant proposition, though not a value of $\scriptstyle {fy}$.

1. We shall generally write "$\scriptstyle {f(}$℩$\scriptstyle {x)(\phi x)}$" rather than "$\scriptstyle {f\{(}$℩$\scriptstyle {x)(\phi x)\}}$" in future.