Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/49

as multiple relations, or (when the number of their terms is specified) as triple, quadruple,…relations, or as triadic, tetradic,…relations. Such relations will not concern us until we come to Geometry. For the present, the only relations we are concerned with are dual relations.

Relations, like classes, are to be taken in extension, i.e. if $\scriptstyle {R}$ and $\scriptstyle {S}$ are relations which hold between the same pairs of terms, $\scriptstyle {R}$ and $\scriptstyle {S}$ are to be identical. We may regard a relation, in the sense in which it is required for our purposes, as a class of couples; i.e. the couple $\scriptstyle {(x,y)}$ is to be one of the class of couples constituting the relation $\scriptstyle {R}$ if $\scriptstyle {x}$ has the relation $\scriptstyle {R}$ to $\scriptstyle {y}$^[1]. This view of relations as classes of couples will not, however, be introduced into our symbolic treatment, and is only mentioned in order to show that it is possible so to understand the meaning of the word relation that a relation shall be determined by its extension.

Any function $\scriptstyle {\phi (x,y)}$ determines a relation $\scriptstyle {R}$ between $\scriptstyle {x}$ and $\scriptstyle {y}$. If we regard a relation as a class of couples, the relation determined by $\scriptstyle {\phi (x,y)}$ is the class of couples $\scriptstyle {(x,y)}$ for which $\scriptstyle {\phi (x,y)}$ is true. The relation determined by the function $\scriptstyle {\phi (x,y)}$ will be denoted by

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

We shall use a capital letter for a relation when it is not necessary to specify the determining function. Thus whenever a capital letter occurs, it is to be understood that it stands for a relation. The propositional function "$\scriptstyle {x}$ has the relation $\scriptstyle {R}$ to $\scriptstyle {y}$" will be expressed by the notation

$\scriptstyle {xRy}$.

This notation is designed to keep as near as possible to common language, which, when it has to express a relation, generally mentions it between its terms, as in "$\scriptstyle {x}$ loves $\scriptstyle {y}$," "$\scriptstyle {x}$ equals $\scriptstyle {y}$," "$\scriptstyle {x}$ is greater than $\scriptstyle {y}$," and so on. For "relation" we shall write "$\scriptstyle {\text{Rel}}$"; thus "$\scriptstyle {R\in {\text{Rel}}}$" means "$\scriptstyle {R}$ is a relation." Owing to our taking relations in extension, we shall have

$\scriptstyle {\vdash :.{\hat {x}}{\hat {y}}\phi (x,y)={\hat {x}}{\hat {y}}\psi (x,y).\equiv :\phi (x,y).\equiv _{x,y}.\psi (x,y)}$,

i.e. two functions of two variables determine the same relation when, and only when, the two functions are formally equivalent. We have

$\scriptstyle {\vdash .z\{{\hat {x}}{\hat {y}}\phi (x,y)\}w.\equiv .\phi (z,w)}$,

i.e. "$\scriptstyle {z}$ has to $\scriptstyle {w}$ the relation determined by the function $\scriptstyle {\phi (x,y)}$" is equivalent to $\scriptstyle {\phi (z,w)}$;

${\begin{aligned}&\scriptstyle {\vdash :.R={\hat {x}}{\hat {y}}\phi (x,y).\equiv :xRy.\equiv _{x,y}.\phi (x,y),}\\&\scriptstyle {\vdash :.R=S.\equiv :xRy.\equiv _{x,y}.xSy,}\\&\scriptstyle {\vdash .{\hat {x}}{\hat {y}}(xRy)=R,}\\&\scriptstyle {\vdash .\{{\hat {x}}{\hat {y}}\phi (x,y)\}\in {\text{Rel}}.}\end{aligned}}$

1. Such a couple has a sense, i.e. the couple $\scriptstyle {(x,y)}$ is different from the couple $\scriptstyle {(y,x)}$, unless $\scriptstyle {x=y}$. We shall call it a "couple with sense," to distinguish it from the class consisting of $\scriptstyle {x}$ and $\scriptstyle {y}$. It may also be called an ordered couple.