Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/107

 (3) Relations. With regard to relations, we have a theory strictly analogous to that which we have just explained as regards classes. Relations in extension, like classes, are incomplete symbols. We require a division of functions of two variables into predicative and non-predicative functions, again for reasons which have been explained in Chapter II. We use the notation "$\scriptstyle{\phi!(x,y)}$" for a predicative function of $$\scriptstyle{x}$$ and $\scriptstyle{y}$.

We use "$\scriptstyle{\phi!(\hat x,\hat y)}$" for the function as opposed to its values; and we use "$\scriptstyle{\hat x\hat y\phi(x,y)}$" for the relation (in extension) determined by $\scriptstyle{\phi(x,y)}$. We put Thus even when $$\scriptstyle{f\{\psi!(\hat x,\hat y)\}}$$ is not an extensional function of $\scriptstyle{\psi}$, $$\scriptstyle{f\{\hat x\hat y\phi(x,y)\}}$$ is an extensional function of $\scriptstyle{\phi}$. Hence, just as in the case of classes, we deduce i.e. a relation is determined by its extension, and vice versa.

On the analogy of the definition of "$\scriptstyle{x\in\psi!\hat z}$," we put

This definition, like that of "$\scriptstyle{x\in\psi!\hat z}$," is not introduced for its own sake, but in order to give a meaning to This meaning, in virtue of our definitions, is  and this, in virtue of the axiom of reducibility  Thus we have always:

Whenever the determining function of a relation is not relevant, we may replace $$\scriptstyle{\hat x\hat y\phi(x,y)}$$ by a single capital letter. In virtue of the propositions given above,

Classes of relations, and relations of relations, can be dealt with as classes of classes were dealt with above.