Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/56

diversity, agreement or disagreement in any respect, are symmetrical relations. A relation is called asymmetrical when it is incompatible with its converse, i.e. when $\scriptstyle {R{\dot {\cap }}{\breve {R}}={\dot {\Lambda }}}$, or, what is equivalent,

$\scriptstyle {xRy.\supset _{x,y}.\sim (yRx)}$.

Before and after, greater and less, ancestor and descendant, are asymmetrical, as are all other relations of the sort that lead to series. But there are many asymmetrical relations which do not lead to series, for instance, that of wife's brother^[1]. A relation may be neither symmetrical nor asymmetrical; for example, this holds of the relation of inclusion between classes: $\scriptstyle {\alpha \subset \beta }$ and $\scriptstyle {\beta \subset \alpha }$ will both be true if $\scriptstyle {\alpha =\beta }$, but otherwise only one of them, at most, will be true. The relation brother is neither symmetrical nor asymmetrical, for if $\scriptstyle {x}$ is the brother of $\scriptstyle {y}$, $\scriptstyle {y}$ may be either the brother or the sister of $\scriptstyle {x}$.

In the propositional function $\scriptstyle {xRy}$, we call $\scriptstyle {x}$ the referent and $\scriptstyle {y}$ the relatum. The class $\scriptstyle {{\hat {x}}(xRy)}$, consisting of all the $\scriptstyle {x}$'s which have the relation $\scriptstyle {R}$ to $\scriptstyle {y}$, is called the class of referents of $\scriptstyle {y}$ with respect to $\scriptstyle {x}$; the class $\scriptstyle {{\hat {y}}(xRy)}$, consisting of all the $\scriptstyle {y}$'s to which $\scriptstyle {x}$ has the relation $\scriptstyle {R}$, is called the class of relata of $\scriptstyle {x}$ with respect to $\scriptstyle {R}$. These two classes are denoted respectively by $\scriptstyle {{\overset {\rightarrow }{R}}'y}$ and $\scriptstyle {{\overset {\leftarrow }{R}}'x}$. Thus

${\begin{aligned}\scriptstyle {{\overset {\rightarrow }{R}}'y={\hat {x}}(xRy)\quad {\text{Df,}}}\\\scriptstyle {{\overset {\leftarrow }{R}}'x={\hat {y}}(yRx)\quad {\text{Df.}}}\end{aligned}}$

The arrow runs towards $\scriptstyle {y}$ in the first case, to show that we are concerned with things having the relation $\scriptstyle {R}$ to $\scriptstyle {y}$; it runs away from $\scriptstyle {x}$ in the second case to show that the relation $\scriptstyle {R}$ goes from $\scriptstyle {x}$ to the members of $\scriptstyle {{\overset {\leftarrow }{R}}'x}$. It runs in fact from a referent and towards a relatum. The notations $\scriptstyle {{\overset {\rightarrow }{R}}'y}$, $\scriptstyle {{\overset {\leftarrow }{R}}'x}$ are very important, and are used constantly. If $\scriptstyle {R}$ is the relation of parent to child, $\scriptstyle {{\overset {\rightarrow }{R}}'y=}$the parents of $\scriptstyle {y}$, $\scriptstyle {{\overset {\leftarrow }{R}}'x=}$the children of $\scriptstyle {x}$. We have

$\scriptstyle {\vdash :x\in {\overset {\rightarrow }{R}}'y.\equiv .xRy}$

and

$\scriptstyle {\vdash :y\in {\overset {\leftarrow }{R}}'x.\equiv .xRy}$.

These equivalences are often embodied in common language. For example, we say indiscriminately "$\scriptstyle {x}$ is an inhabitant of London" or "$\scriptstyle {x}$ inhabits London." If we put "$\scriptstyle {R}$" for "inhabits," "$\scriptstyle {x}$ inhabits London" is "$\scriptstyle {xR}$ London," while "$\scriptstyle {x}$ is an inhabitant of London" is $\scriptstyle {x\in {\overset {\rightarrow }{R}}'}$ London."

1. This relation is not strictly asymmetrical, but is so except when the wife's brother is also the sister's husband. In the Greek Church the relation is strictly asymmetrical.