Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/71

up to $\scriptstyle {n}$. But no such judgment can include itself in its own scope, since such a judgment is always of higher order than the judgments to which it refers.

Let us consider next what is meant by the negation of a proposition of the form "$\scriptstyle {(x).\phi x}$." We observe, to begin with, that "$\scriptstyle {\phi x}$ in some cases," or "$\scriptstyle {\phi x}$ sometimes," is a judgment which is on a par with "$\scriptstyle {\phi x}$ in all cases," or "$\scriptstyle {\phi x}$ always." The judgment "$\scriptstyle {\phi x}$ sometimes" is true if one or more values of $\scriptstyle {x}$ exist for which $\scriptstyle {\phi x}$ is true. We will express the proposition "$\scriptstyle {\phi x}$ sometimes" by the notation "$\scriptstyle {(\exists x).\phi x}$," where "$\scriptstyle {\exists }$" stands for "there exists," and the whole symbol may be read "there exists an $\scriptstyle {x}$ such that $\scriptstyle {\phi x}$." We take the two kinds of judgment expressed by "$\scriptstyle {(x).\phi x}$" and "$\scriptstyle {(\exists x).\phi x}$" as primitive ideas. We also take as a primitive idea the negation of an elementary proposition. We can then define the negations of $\scriptstyle {(x).\phi x}$ and $\scriptstyle {(\exists x).\phi x}$. The negation of any proposition $\scriptstyle {p}$ will be denoted by the symbol "$\scriptstyle {\sim p}$." Then the negation of $\scriptstyle {(x).\phi x}$ will be defined as meaning

"$\scriptstyle {(\exists x).\sim \phi x}$,"

and the negation of $\scriptstyle {(\exists x).\phi x}$ will be defined as meaning "$\scriptstyle {(x).\sim \phi x}$." Thus, in the traditional language of formal logic, the negation of a universal affirmative is to be defined as the particular negative, and the negation of the particular affirmative is to be defined as the universal negative. Hence the meaning of negation for such propositions is different from the meaning of negation for elementary propositions. An analogous explanation will apply to disjunction. Consider the statement "either $\scriptstyle {p}$, or $\scriptstyle {\phi x}$ always." We will denote the disjunction of two propositions $\scriptstyle {p,~q}$ by "$\scriptstyle {p\lor q}$." Then our statement is "$\scriptstyle {p.\lor .(x).\phi x}$." We will suppose that $\scriptstyle {p}$ is an elementary proposition, and that $\scriptstyle {\phi x}$ is always an elementary proposition. We take the disjunction of two elementary propositions as a primitive idea, and we wish to define the disjunction

"$\scriptstyle {p.\lor .(x).\phi x}$."

This may be defined as "$\scriptstyle {(x).p\lor \phi x}$," i.e. "either $\scriptstyle {p}$ is true, or $\scriptstyle {\phi x}$ is always true" is to mean "'$\scriptstyle {p}$ or $\scriptstyle {\phi x}$' is always true." Similarly we will define

"$\scriptstyle {p.\lor .(\exists x).\phi x}$"

as meaning "$\scriptstyle {(\exists x).p\lor \phi x}$," i.e. we define "either $\scriptstyle {p}$ is true or there is an $\scriptstyle {x}$ for which $\scriptstyle {\phi x}$ is true" as meaning "there is an $\scriptstyle {x}$ for which either $\scriptstyle {p}$ or $\scriptstyle {\phi x}$ is true." Similarly we can define a disjunction of two universal propositions: "$\scriptstyle {(x).\phi x.\lor .(y).\psi y}$" will be defined as meaning "$\scriptstyle {(x,y).\phi x\lor \psi y}$," i.e. "either $\scriptstyle {\phi x}$ is always true or $\scriptstyle {\psi y}$ is always true" is to mean "'$\scriptstyle {\phi x}$ or $\scriptstyle {\psi y}$' is always true." By this method we obtain definitions of disjunctions containing propositions of the form $\scriptstyle {(x).\phi x}$ or $\scriptstyle {(\exists x).\phi x}$ in terms of disjunctions of elementary propositions; but the meaning of "disjunction" is not the