Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/57

Instead of $\scriptstyle {\overset {\rightarrow }{R}}$ and $\scriptstyle {\overset {\leftarrow }{R}}$ we sometimes use $\scriptstyle {{\text{sg}}'R}$, $\scriptstyle {{\text{gs}}'R}$, where "sg" stands for "sagitta," and "gs" is "sg" backwards. Thus we put

${\begin{aligned}\scriptstyle {{\text{sg}}'R={\overset {\rightarrow }{R}}\quad }&\scriptstyle {\text{Df,}}\\\scriptstyle {{\text{gs}}'R={\overset {\leftarrow }{R}}\quad }&\scriptstyle {\text{Df.}}\end{aligned}}$

These notations are sometimes more convenient than an arrow when the relation concerned is represented by a combination of letters, instead of a single letter such as R. Thus e.g. we should write $\scriptstyle {{\text{sg}}'(R{\dot {\cap }}S)}$, rather than put an arrow over the whole length of $\scriptstyle {(R{\dot {\cap }}S)}$. The c1ass of all terms that have the relation R to something or other is called the domain of R. Thus if R is the relation of parent and child, the domain of R will be the class of parents. We represent the domain of R by "D'R." Thus we put

$\scriptstyle {{\text{D}}'R={\hat {x}}\{(\exists y).xRy\}\quad {\text{Df.}}}$

Similarly the class of all terms to which something or other has the relation R is called the converse domain of R; it is the same as the domain of the converse of R. The converse domain of R is represented by "ꓷ'R"; thus

ꓷ$\scriptstyle {'R={\hat {y}}\{(\exists x).xRy\}\quad {\text{Df.}}}$

The sum of the domain and the converse domain is called the field, and is represented by C'R: thus

$\scriptstyle {C'R=D'R\cup }$ꓷ$\scriptstyle {'R\quad {\text{Df.}}}$

The field is chiefly important in connection with series. If R is the ordering relation of a series, C'R will be the class of terms of the series, $\scriptstyle {{\text{D}}'R}$ will be all the terms except the last (if any), and ꓷ$\scriptstyle {'R}$ will be all the terms except the first (if any). The first term, if it exists, is the only member of $\scriptstyle {{\text{D}}'R\cap \lnot }$ꓷ$\scriptstyle {'R}$, since it is the only term which is a predecessor but not a follower. Similarly the last term (if any) is the only member of ꓷ$\scriptstyle {'R\cap \lnot {\text{D}}'R}$. The condition that a series should have no end is ꓷ$\scriptstyle {'R\subset {\text{D}}'R}$, i.e. "every follower is a predecessor"; the condition for no beginning is $\scriptstyle {{\text{D}}'R\subset }$ꓷ$\scriptstyle {'R}$. These conditions are equivalent respectively to $\scriptstyle {D'R=C'R}$ and ꓷ$\scriptstyle {'R=C'R}$.

The relative product of two relations R and S is the relation which holds between x and z when there is an intermediate term y such that x has the relation R to y and y has the relation S to z. The relative product of R and S is represented by $R|S$; thus we put

$\scriptstyle {R|S={\hat {x}}{\hat {z}}\{(\exists y).xRy.ySz\}\quad {\text{Df,}}}$

whence

$\scriptstyle {\vdash :x(R|S)z.\equiv .(\exists y).xRy.ySz}$.

Thus "paternal aunt" is the relative product of sister and father; "paternal grandmother" is the relative product of mother and father; "maternal