Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/158

(The definitions *9.07-08 are to apply also when $\scriptstyle {\phi }$ and $\scriptstyle {\psi }$ are not both elementary functions.)

In virtue of these definitions, the true scope of an apparent variable is always the whole of the asserted proposition in which it occurs, even when, typographically, its scope appears to be only part of the asserted proposition. Thus when $\scriptstyle {(\exists x).\phi x}$ or $\scriptstyle {(x).\phi x}$ appears as part of an asserted proposition, it does not really occur, since the scope of the apparent variable really extends to the whole asserted proposition. It will be shown, however, that, so far as the theory of deduction is concerned, $\scriptstyle {(\exists x).\phi x}$ and $\scriptstyle {(x).\phi x}$ behave like propositions not containing apparent variables.

The definitions of implications, the logical product, and equivalence are to be transferred unchaged to $\scriptstyle {(x).\phi x}$ and $\scriptstyle {(\exists x).\phi x}$.

The above definitions can be repeated for successive types, and thus reach propositions of any type.

Primitive Propositions. The primitive propositions required are six in number, and may be divided into three sets of two. We have first two propositions which effect the passage from elementary to first-order proposition, namely

*9.1. $\scriptstyle {\vdash :\phi x.\supset .(\exists z).\phi z}$ Pp

*9.11. $\scriptstyle {\vdash :\phi x\lor \phi y.\supset .(\exists z).\phi z}$ Pp

Of these, the first states that if $\scriptstyle {\phi x}$ is true, then there is a value of $\scriptstyle {\phi {\hat {z}}}$ which is true; i.e. if we can find an instance of a function which is true, then the function is "sometime true." (When we speak of a function as "sometimes" true, we do not mean to assert that there is more than one argument for which it is true, but only that there is at least one.) Practically, the above primitive proposition gives the only method of proving "existence-theorems": in order to prove such theorems, it is necessary (and sufficient) to find some instance in which an object possesses the property in question. If we were to assume what may be called "existence-axioms," i.e. axioms stating $\scriptstyle {(\exists z).\phi z}$ for some particular $\scriptstyle {\phi }$, these axioms would give other methods of proving existence. Instances of such axioms are the multiplicative axiom (*88) and the axiom of infinity (defined in *120.03). But we have not assumed any such axioms in the present work.

The second of the above primitive propositions is only used once, in proving $\scriptstyle {(\exists z).\phi z.\lor .(\exists z).\phi z:\supset .(\exists z).\phi z}$, which is the analogue of *1.2 (namely $\scriptstyle {p\lor p.\supset .p}$ when $\scriptstyle {p}$ is replaced by $\scriptstyle {(\exists z).\phi z}$. The effect of this primitive proposition is to emphasize the ambiguity of the $\scriptstyle {z}$ required in order to secure $\scriptstyle {(\exists z).\phi z}$. We have, of course, in virtue of *9.1,

$\scriptstyle {\phi x.\supset .(\exists z).\phi z{\text{ and }}\phi y.\supset .(\exists z).\phi z}$.

But is we try to infer from these that $\scriptstyle {\phi x\lor \phi y.\supset .(\exists z).\phi z}$, we must use the