Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/94

When $\scriptstyle {f\{(}$℩$\scriptstyle {x)(\phi x)\}}$, as above defined, forms part of some other proposition, we shall say that $\scriptstyle {(}$℩$\scriptstyle {x)(\phi x)}$ has a secondary occurrence. When $\scriptstyle {(}$℩$\scriptstyle {x)(\phi x)}$ has a secondary occurrence, a proposition in which it occurs may be true even when $\scriptstyle {(}$℩$\scriptstyle {x)(\phi x)}$ does not exist. This applies, e.g. to the proposition: "There is no such person as the King of France." We may interpret this as

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

or as

$\scriptstyle {\sim \{(\exists c).c=(}$℩$\scriptstyle {x)(\phi x)\}}$,

if "$\scriptstyle {\phi x}$" stands for "$\scriptstyle {x}$ is King of France." In either case, what is asserted is that a proposition $\scriptstyle {p}$ in which $\scriptstyle {(}$℩$\scriptstyle {x)(\phi x)}$ occurs is false, and this proposition $\scriptstyle {p}$ is thus part of a larger proposition. The same applies to such a proposition as the following: "If France were a monarchy, the King of France would be of the House of Orleans." It should be observed that such a proposition as

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

is ambiguous; it may deny $\scriptstyle {f\{(}$℩$\scriptstyle {x)(\phi x)\}}$, in which case it will be true if $\scriptstyle {(}$℩$\scriptstyle {x)(\phi x)}$ does not exist, or it may mean

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

in which case it can only be true if $\scriptstyle {(}$℩$\scriptstyle {x)(\phi x)}$ exists. In ordinary language, the latter interpretation would usually be adopted. For example, the proposition "the King of France is not bald" would usually be rejected as false, being held to mean "the King of France exists and is not bald," rather than "it is false that the King of France exists and is bald." When $\scriptstyle {(}$℩$\scriptstyle {x)(\phi x)}$ exists, the two interpretations of the ambiguity give equivalent results; but when $\scriptstyle {(}$℩$\scriptstyle {x)(\phi x)}$ does not exist, one interpretation is true and one is false. It is necessary to be able to distinguish these in our notation; and generally, if we have such propositions as

$\scriptstyle {\psi (}$℩$\scriptstyle {x)(\phi x).\supset .p}$,
$\scriptstyle {p.\supset .\psi (}$℩$\scriptstyle {x)(\phi x)}$,
$\scriptstyle {\psi (}$℩$\scriptstyle {x)(\phi x).\supset .\chi (}$℩$\scriptstyle {x)(\phi x)}$,

and so on, we must be able by our notation to distinguish whether the whole or only part of the proposition concerned is to be treated as the "$\scriptstyle {f(}$℩$\scriptstyle {x)(\phi x)}$" of our definition. For this purpose, we will put "$\scriptstyle {[(}$℩$\scriptstyle {x)(\phi x)]}$" followed by dots at the beginning of the part (or whole) which is to be taken as $\scriptstyle {f(}$℩$\scriptstyle {x)(\phi x)}$, the dots being sufficiently numerous to bracket off the $\scriptstyle {f(}$℩$\scriptstyle {x)(\phi x)}$; i.e. $\scriptstyle {f(}$℩$\scriptstyle {x)(\phi x)}$ is to be everything following the dots until we reach an equal number of dots not signifying a logical product, or a greater number signifying a logical product, or the end of the sentence, or the end of a bracket enclosing "$\scriptstyle {[(}$℩$\scriptstyle {x)(\phi x)]}$." Thus

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