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 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. Thus e.g. will mean "$\scriptstyle{\phi x}$ always implies $\scriptstyle{\psi x}$," but  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