Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/58

grandfather" is the relative product of father and mother. The relative product is not commutative, but it obeys the associative law, i.e.

$\scriptstyle {\vdash .(P|Q)|R=P|(Q|R)}$.

It also obeys the distributive law with regard to the logical addition of relations, i.e. we have

${\begin{aligned}&\scriptstyle {\vdash .P|(Q~\cdot \!\!\!\cup \,R)=(P|Q)~\cdot \!\!\!\cup \,(P|R),}\\&\scriptstyle {\vdash .(Q~\cdot \!\!\!\cup \,R)|P=(Q|P)~\cdot \!\!\!\cup \,(Q|R).}\end{aligned}}$

But with regard to the logical product, we have only

${\begin{aligned}&\scriptstyle {\vdash .P|(Q{\dot {\cap }}R)~\cdot \!\!\!\!\subset \,(P|Q){\dot {\cap }}(P|R),}\\&\scriptstyle {\vdash .(Q{\dot {\cap }}R)|P~\cdot \!\!\!\!\subset \,(Q|P){\dot {\cap }}(Q|R).}\end{aligned}}$

The relative product does not obey the law of tautology, i.e. we do not have in general $\scriptstyle {R|R=R}$. We put

$\scriptstyle {R^{2}=R|R\quad {\text{Df.}}}$

${\begin{aligned}\scriptstyle {\text{Thus paternal grandfather}}&\scriptstyle {=({\text{father}})^{2},}\\\scriptstyle {\text{maternal grandmother}}&\scriptstyle {=({\text{mother}})^{2}.}\end{aligned}}$ A relation is called transitive when $\scriptstyle {R^{2}~\cdot \!\!\!\!\subset \,R}$, i.e. when, if $\scriptstyle {xRy}$ and $\scriptstyle {yRz}$, we always have $\scriptstyle {xRz}$, i.e. when

$\scriptstyle {xRy.yRz.\supset _{x,y,z}.xRz}$.

Relations which generate series are always transitive; thus e.g.

$\scriptstyle {x>y.y>z.\supset _{x,y,z}.x>z}$.

If $\scriptstyle {P}$ is a relation which generates a series, $\scriptstyle {P}$ may conveniently be read "precedes"; thus "$\scriptstyle {xPy.yPz.\supset _{x,y,z}.xPz}$" becomes "if $\scriptstyle {x}$ precedes $\scriptstyle {y}$ and $\scriptstyle {y}$ precedes $\scriptstyle {z}$, then $\scriptstyle {x}$ always precedes $\scriptstyle {z}$." The class of relations which generate series are partially characterized by the fact that they are transitive and asymmetrical, and never relate a term to itself. If $\scriptstyle {P}$ is a relation which generates a series, and if we have not merely $\scriptstyle {P^{2}~\cdot \!\!\!\!\subset \,P}$, but $\scriptstyle {P^{2}=P}$, then $\scriptstyle {P}$ generates a series which is compact (überall dicht), i.e. such that there are terms between any two. For in this case we have

$\scriptstyle {xPz.\supset .(\exists y).xPy.yPz}$,

i.e. if $\scriptstyle {x}$ precedes $\scriptstyle {z}$, there is a term $\scriptstyle {y}$ such that $\scriptstyle {x}$ precedes $\scriptstyle {y}$ and $\scriptstyle {y}$ precedes $\scriptstyle {z}$, i.e. there is a term between $\scriptstyle {x}$ and $\scriptstyle {z}$. Thus among relations which generate series, those which generate compact series are those for which $\scriptstyle {P^{2}=P}$.

Many relations which do not generate series are transitive, for example, identity, or the relation of inclusion between classes. Such cases arise when the relations are not asymmetrical. Relations which are transitive and symmetrical are an important class: they may be regarded as consisting in the possession of some common property.