Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/39

i.e. "either $\scriptstyle {\phi x}$ is always true, or $\scriptstyle {p}$ is true" is to mean "'$\scriptstyle {\phi x}$ or $\scriptstyle {p}$' is always true," with similar definitions in other cases. This subject is resumed in Chapter II, and in *9 in the body of the work.

Apparent variables. The symbol "$\scriptstyle {(x).\phi x}$" denotes one definite proposition, and there is no distinction in meaning between "$\scriptstyle {(x).\phi x}$" and "$\scriptstyle {(y).\phi y}$" when they occur in the same context. Thus the "$\scriptstyle {x}$" in "$\scriptstyle {(x).\phi x}$" is not an ambiguous constituent of any expression in which "$\scriptstyle {(x).\phi x}$" occurs; and such an expression does not cease to convey a determinate meaning by reason of the ambiguity of the $\scriptstyle {x}$ in the "$\scriptstyle {\phi x}$." The symbol "$\scriptstyle {(x).\phi x}$" has some analogy to the symbol

"$\scriptstyle {\int _{a}^{b}\phi (x)\,dx}$"

for definite integration, since in neither case is the expression a function of $\scriptstyle {x}$.

The range of $\scriptstyle {x}$ in "$\scriptstyle {(x).\phi x}$" or "$\scriptstyle {(\exists x).\phi x}$" extends over the complete field of the values of $\scriptstyle {x}$ for which "$\scriptstyle {\phi x}$" has meaning, and accordingly the meaning of "$\scriptstyle {(x).\phi x}$" or "$\scriptstyle {(\exists x).\phi x}$" involves the supposition that such a field is determinate. The $\scriptstyle {x}$ which occurs in "$\scriptstyle {(x).\phi x}$" or "$\scriptstyle {(\exists x).\phi x}$" is called (following Peano) an "apparent variable." It follows from the meaning of "$\scriptstyle {(\exists x).\phi x}$" that the $\scriptstyle {x}$ in this expression is also an apparent variable. A proposition in which $\scriptstyle {x}$ occurs as an apparent variable is not a function of $\scriptstyle {x}$. Thus e.g. "$\scriptstyle {(x).x=x}$" will mean "everything is equal to itself." This is an absolute constant, not a function of a variable $\scriptstyle {x}$. This is why the $\scriptstyle {x}$ is called an apparent variable in such cases.

Besides the "range" of $\scriptstyle {x}$ in "$\scriptstyle {(x).\phi x}$" or "$\scriptstyle {(\exists x).\phi x}$," which is the field of the values that $\scriptstyle {x}$ may have, we shall speak of the "scope" of $\scriptstyle {x}$, meaning the function of which all values or some value are being affirmed. If we are asserting all values (or some value) of "$\scriptstyle {\phi x}$," "$\scriptstyle {\phi x}$" is the scope of $\scriptstyle {x}$; if we are asserting all values (or some value) of "$\scriptstyle {\phi x\supset p}$," "$\scriptstyle {\phi x\supset p}$" is the scope of $\scriptstyle {x}$; if we are asserting all values (or some value) of "$\scriptstyle {\phi x\supset \psi x}$," "$\scriptstyle {\phi x\supset \psi x}$" will be the scope of $\scriptstyle {x}$, and so on. The scope of $\scriptstyle {x}$ is indicated by the number of dots after the "$\scriptstyle {(x)}$" or "$\scriptstyle {(\exists x)}$"; that is to say, the scope extends forwards until we reach an equal number of dots not indicating a logical product, or a greater number indicating a logical product, or the end of the asserted proposition in which the "$\scriptstyle {(x)}$" or "$\scriptstyle {(\exists x)}$" occurs, whichever of these happens first^[1]. Thus e.g.

"$\scriptstyle {(x):\phi x.\supset .\psi x}$"

will mean "$\scriptstyle {\phi x}$ always implies $\scriptstyle {\psi x}$," but

"$\scriptstyle {(x).\phi x.\supset .\psi x}$"

will mean "if $\scriptstyle {\phi x}$ is always true, then $\scriptstyle {\psi x}$ is true for the argument $\scriptstyle {x}$." Note that in the proposition

$\scriptstyle {(x).\phi x.\supset .\psi x}$

1. This agrees with the rules for the occurrences of dots of the type of Group II as explained above, pp. 9 and 10.