Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/42

how the argument is to be supplied, and therefore there are objections to omitting an apparent variable where (as in the case before us) this is the argument to the function which is the real variable. This appears more plainly if, instead of a general function $\scriptstyle {\phi {\hat {x}}}$, we take some particular function, say "$\scriptstyle {{\hat {x}}=a}$," and consider the definition of $\scriptstyle {\sim \{(x).x=a\}}$. Our definition gives

$\scriptstyle {\sim \{(x).x=a\}.=.(\exists x).\sim (x=a)\quad {\text{Df}}}$.

But if we had adopted a notation in which the ambiguous value "$\scriptstyle {x=a}$," containing the apparent variable $\scriptstyle {x}$, did not occur in the definiendum, we should have had to construct a notation employing the function itself, namely "$\scriptstyle {{\hat {x}}=a}$." This does not involve an apparent variable, but would be clumsy in practice. In fact we have found it convenient and possible—except in the explanatory portions—to keep the explicit use of symbols of the type "$\scriptstyle {\phi {\hat {x}}}$," either as constants [e.g. $\scriptstyle {{\hat {x}}=a}$] or as real variables, almost entirely out of this work.

Propositions connecting real and apparent variables. The most important propositions connecting real and apparent variables are the following:

(1) "When a propositional function can be asserted, so can the proposition that all values of the function are true." More briefly, if less exactly, "what holds of any, however chosen, holds of all." This translates itself into the rule that when a real variable occurs in an assertion, we may turn it into an apparent variable by putting the letter representing it in brackets immediately after the assertion-sign.

(2) "What holds of all, holds of any," i.e.

$\scriptstyle {\vdash :(x).\phi x.\supset .\phi y}$.

This states "if $\scriptstyle {\phi x}$ is always true, then $\scriptstyle {\phi y}$ is true." (3) " If $\scriptstyle {\phi y}$ is true, then $\scriptstyle {\phi x}$ is sometimes true," i.e.

$\scriptstyle {\vdash :\phi y.\supset .(\exists x).\phi x}$.

An asserted proposition of the form "$\scriptstyle {(\exists x).\phi x}$" expresses an "existence-theorem," namely "there exists an $\scriptstyle {x}$ for which $\scriptstyle {\phi x}$ is true." The above proposition gives what is in practice the only way of proving existence-theorems: we always have to find some particular $\scriptstyle {y}$ for which $\scriptstyle {\phi y}$ holds, and thence to infer "$\scriptstyle {(\exists x).\phi x}$." If we were to assume what is called the multiplicative axiom, or the equivalent axiom enunciated by Zermelo, that would, in an important class of cases, give an existence-theorem where no particular instance of its truth can be found.

In virtue of "$\scriptstyle {\vdash :(x).\phi x.\supset .\phi y}$" and "$\scriptstyle {\vdash :\phi y.\supset .(\exists x).\phi x}$," we have "$\scriptstyle {\vdash :(x).\phi x.\supset .(\exists x).\phi x}$," i.e. "what is always true is sometimes true." This would not be the case if nothing existed; thus our assumptions contain the assumption that there is something. This is involved in the principle