Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/154

Section B.

Theory of Apparent Variables.

*9. Extension of the Theory of Deduction from Lower to Higher Types of Propositions

Summary of *9.

In the present number, we introduce two new primitive ideas, which may be expressed as "$\scriptstyle {\phi x}$ is always^[1] true" and "$\scriptstyle {\phi x}$ is sometimes^[1] true," or, more correctly, as "$\scriptstyle {\phi x}$ always" and "$\scriptstyle {\phi x}$ sometimes." When we assert "$\scriptstyle {\phi x}$ always," we are asserting all values of $\scriptstyle {\phi {\hat {x}}}$, where "$\scriptstyle {\phi {\hat {x}}}$" means the function itself, as opposed to an ambiguous value of the function (cf. pp. 15, 42); we are not asserting that $\scriptstyle {\phi x}$ is true for all values of $\scriptstyle {x}$, because, in accordance with the theory of types, there are values of $\scriptstyle {x}$ for which "$\scriptstyle {\phi x}$" is meaningless; for example, the function $\scriptstyle {\phi {\hat {x}}}$ itself must be such a value. We shall denote "$\scriptstyle {\phi x}$ always" by the notation

$\scriptstyle {(x).\phi x}$,

where the "$\scriptstyle {(x)}$" will be followed by a sufficiently large number of dots to cover the function of which "all values" are concerned. The form in which such propositions most frequently occur is the "formal implication," i.e. such a proposition as

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

i.e. "$\scriptstyle {\phi x}$ always implies $\scriptstyle {\psi x}$." This is the form in which we express the universal affirmative "all objects having the property $\scriptstyle {\phi }$ have the property $\scriptstyle {\psi }$." We shall denote "$\scriptstyle {\phi x}$ sometimes" by the notation

$\scriptstyle {(\exists x).\phi x}$.

Here "$\scriptstyle {\exists }$" stands for "there exists," and the whole symbol may be read "there exists an $\scriptstyle {x}$ such that $\scriptstyle {\phi x}$."

In a proposition of either of the two forms $\scriptstyle {(x).\phi x}$, $\scriptstyle {(\exists x).\phi x}$, the $\scriptstyle {x}$ is called an apparent variable. A proposition which contains no apparent variables is called "elementary," and a function, all whose values are

1. We use "always" as meaning "in all cases," not "at all times." A similar remark applies to "sometimes."