Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/54

We now proceed to define "$\scriptstyle {\mathbf {E!} (}$℩$\scriptstyle {x)(\phi x)}$" so that it can be read "the $\scriptstyle {x}$ satisfying $\scriptstyle {\phi x}$ exists." (It will be observed that this is a different meaning of existence from' that which we express by "$\scriptstyle {\exists }$.") Its definition is

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

i.e. "the $\scriptstyle {x}$ satisfying $\scriptstyle {\phi {\hat {x}}}$ exists" is to mean "there is an object $\scriptstyle {c}$ such that $\scriptstyle {\phi x}$ is true when $\scriptstyle {x}$ is $\scriptstyle {c}$ but not otherwise."

The following are equivalent forms:

$\scriptstyle {\vdash :.\mathbf {E!} (}$℩$\scriptstyle {x).\equiv :(\exists c):\phi c:\phi x.\supset _{x}.x=c}$,
$\scriptstyle {\vdash :.\mathbf {E!} (}$℩$\scriptstyle {x).\equiv :(\exists c):\phi c:\phi x.\phi y.\supset _{x,y}.x=y}$,
$\scriptstyle {\vdash :.\mathbf {E!} (}$℩$\scriptstyle {x).\equiv :(\exists c):\phi c:x\neq c.\supset _{x}.\sim \phi x}$.

The last of these states that "the $\scriptstyle {x}$ satisfying $\scriptstyle {\phi {\hat {x}}}$ exists" is equivalent to "there is an object $\scriptstyle {c}$ satisfying $\scriptstyle {\phi {\hat {x}}}$, and every object other than $\scriptstyle {c}$ does not satisfy $\scriptstyle {\phi {\hat {x}}}$."

The kind of existence just defined covers a great many cases. Thus for example "the most perfect Being exists" will mean:

$\scriptstyle {(\exists c):x{\text{ is most perfect}}.\equiv _{x}.x=c}$,

which, taking the last of the above equivalences, is equivalent to

$\scriptstyle {(\exists c):c{\text{ is most perfect}}:x\neq c.\supset _{x}.x{\text{ is not most perfect.}}}$

A proposition such as "Apollo exists" is really of the same logical form, although it does not explicitly contain the word the. For "Apollo" means really "the object having such-and-such properties," say "the object having the properties enumerated in the Classical Dictionary^[1]." If these properties make up the propositional function $\scriptstyle {\phi x}$, then "Apollo" means "$\scriptstyle {(}$℩$\scriptstyle {x)(\phi x)}$," and "Apollo exists" means "$\scriptstyle {\mathbf {E!} (}$℩$\scriptstyle {x)(\phi x)}$." To take another illustration, "the author of Waverley" means "the man who (or rather, the object which) wrote Waverley." Thus "Scott is the author of Waverley" is

$\scriptstyle {{\text{Scott}}=(}$℩$\scriptstyle {x)(x{\text{ wrote Waverley}})}$.

Here (as we observed before) the importance of identity in connection with descriptions plainly appears.

The notation "$\scriptstyle {(}$℩$\scriptstyle {x)(\phi x)}$," which is long and inconvenient, is seldom used, being chiefly required to lead up to another notation, namely "$\scriptstyle {R'y}$," meaning "the object having the relation $\scriptstyle {R}$ to $\scriptstyle {y}$." That is, we put

$\scriptstyle {R'y=(}$℩$\scriptstyle {x)(xRy)\quad {\text{Df.}}}$

The inverted comma may be read "of." Thus "$\scriptstyle {R'y}$" is read "the $\scriptstyle {R}$ of $\scriptstyle {y}$." Thus if $\scriptstyle {R}$ is the relation of father to son, "$\scriptstyle {R'y}$" means "the father of $\scriptstyle {y}$"; if $\scriptstyle {R}$ is the relation of son to father, "$\scriptstyle {R'y}$" means "the the son of $\scriptstyle {y}$," which will

1. The same principle applies to many uses of the proper names of existent objects, e.g. to all uses of proper names for objects known to the speaker only by report, and not by personal acquaintance.