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



Summary of *9.

In the present number, we introduce two new primitive ideas, which may be expressed as "$\scriptstyle{\phi x}$ is always true" and "$\scriptstyle{\phi x}$ is sometimes 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 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 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 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