Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/55

only "exist" if $\scriptstyle {y}$ has one son and no more. $\scriptstyle {R'y}$ is a function of $\scriptstyle {y}$, but not a propositional function; we shall call it a descriptive function. All the ordinary functions of mathematics are of this kind, as will appear more fully in the sequel. Thus in our notation, "$\scriptstyle {\sin y}$" would be written "$\scriptstyle {\sin 'y}$," and "$\scriptstyle {\sin }$" would stand for the relation which $\scriptstyle {\sin 'y}$ has to $\scriptstyle {y}$. Instead of a variable descriptive function $\scriptstyle {fy}$, we put $\scriptstyle {R'y}$, where the variable relation $\scriptstyle {R}$ takes the place of the variable function $\scriptstyle {f}$. A descriptive function will in general exist while $\scriptstyle {y}$ belongs to a certain domain, but not outside that domain; thus if we are dealing with positive rationals, $\scriptstyle {^{\sqrt {}}y}$ will be significant if $\scriptstyle {y}$ is a perfect square, but not otherwise; if we are dealing with real numbers, and agree that "$\scriptstyle {^{\sqrt {}}y}$" is to mean the positive square root (or, is to mean the negative square root), $\scriptstyle {^{\sqrt {}}y}$ will be significant provided $\scriptstyle {y}$ is positive, but not otherwise; and so on. Thus every descriptive function has what we may call a "domain of definition" or a "domain of existence," which may be thus defined: If the function in question is $\scriptstyle {R'y}$, its domain of definition or of existence will be the class of those arguments $\scriptstyle {y}$ for which we have $\scriptstyle {\mathbf {E!} R'y}$, i.e. for which $\scriptstyle {\mathbf {E!} (}$℩$\scriptstyle {x)(xRy)}$, i.e. for which there is one $\scriptstyle {x}$, and no more, having the relation $\scriptstyle {R}$ to $\scriptstyle {y}$.

If $\scriptstyle {R}$ is any relation, we will speak of $\scriptstyle {R'y}$ as the "associated descriptive function." A great many of the constant relations which we shall have occasion to introduce are only or chiefly important on account of their associated descriptive functions. In such cases, it is easier (though less correct) to begin by assigning the meaning of the descriptive function, and to deduce the meaning of the relation from that of the descriptive function. This will be done in the following explanations of notation.

Various descriptive functions of relations. If $\scriptstyle {R}$ is any relation, the converse of $\scriptstyle {R}$ is the relation which holds between $\scriptstyle {y}$ and $\scriptstyle {x}$ whenever $\scriptstyle {R}$ holds between $\scriptstyle {x}$ and $\scriptstyle {y}$. Thus greater is the converse of less, before of after, cause of effect, husband of wife, etc. The converse of $\scriptstyle {R}$ is written^[1] $\scriptstyle {{\text{Cnv}}'R}$ or $\scriptstyle {\breve {R}}$. The definition is

${\begin{aligned}\scriptstyle {{\breve {R}}={\hat {x}}{\hat {y}}(yRx)\quad }&\scriptstyle {\text{Df,}}\\\scriptstyle {{\text{Cnv}}'R={\breve {R}}\qquad }&\scriptstyle {\text{Df.}}\end{aligned}}$

The second of these is not a formally correct definition, since we ought to define "$\scriptstyle {\text{Cnv}}$" and deduce the meaning of $\scriptstyle {{\text{Cnv}}'R}$. But it is not worth while to adopt this plan in our present introductory account, which aims at simplicity rather than formal correctness. A relation is called symmetrical if $\scriptstyle {R={\breve {R}}}$, i.e. if it holds between $\scriptstyle {y}$ and $\scriptstyle {x}$ whenever it holds between $\scriptstyle {x}$ and $\scriptstyle {y}$ (and therefore vice versa). Identity,

1. The second of these notations is taken from Schröder's Algebra und Logik der Relative.