Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/59

 Plural descriptive functions. The class of terms $$\scriptstyle{x}$$ which have the relation R to some member of a class $$\scriptstyle{\alpha}$$ is denoted by $$\scriptstyle{R''\alpha}$$ or $\scriptstyle{R_\in'\alpha}$. The definition is

Thus for example let $$\scriptstyle{R}$$ be the relation of inhabiting, and $$\scriptstyle{\alpha}$$ the class of towns; then $\scriptstyle{R''\alpha=}$inhabitants of towns. Let $$\scriptstyle{R}$$ be the relation "less than" among rationals, and $$\scriptstyle{\alpha}$$ the class of those rationals which are of the form $\scriptstyle{1-2^{-n}}$,|undefined for integral values of $\scriptstyle{n}$; then $$\scriptstyle{R\alpha}$$ will be all rationals less than some member of $\scriptstyle{\alpha}$, i.e.'' all rationals less than 1. If $$\scriptstyle{P}$$ is the generating relation of a series, and $$\scriptstyle{\alpha}$$ is any class of members of $\scriptstyle{P}$, $$\scriptstyle{P\alpha}$$ will be predecessors of $\scriptstyle{\alpha}$'s, i.e.'' the segment defined by $\scriptstyle{\alpha}$. If $$\scriptstyle{P}$$ is a relation such that $$\scriptstyle{P'y}$$ always exists when $\scriptstyle{y\in\alpha}$, $$\scriptstyle{P\alpha}$$ will be the class of all terms of the form $$\scriptstyle{P'y}$$ for values of $$\scriptstyle{y}$$ which are members of $\scriptstyle{\alpha}$; i.e.''

Thus a member of the class "fathers of great men" will be the father of $\scriptstyle{y}$, where $$\scriptstyle{y}$$ is some great man. In other cases, this will not hold; for instance, let $$\scriptstyle{P}$$ be the relation of a number to any number of which it is a factor; then $$\scriptstyle{P}$$ (even numbers)=factors of even numbers, but this class is not composed of terms of the form "the'' factor of $\scriptstyle{x}$," where $$\scriptstyle{x}$$ is an even number, because numbers do not have only one factor apiece.

Unit classes. The class whose only member is $$\scriptstyle{x}$$ might be thought to be identical with $\scriptstyle{x}$, but Peano and Frege have shown that this is not the case. (The reasons why this is not the case will be explained in a preliminary way in Chapter II of the Introduction.) We denote by "$\scriptstyle{\iota'x}$" the class whose only member is $\scriptstyle{x}$: thus

i.e. "$\scriptstyle{\iota'x}$" means "the class of objects which are identical with $\scriptstyle{x}$."

The class consisting of $$\scriptstyle{x}$$ and $$\scriptstyle{y}$$ will be $\scriptstyle{\iota'x\cup\iota'y}$; the class got by adding $$\scriptstyle{x}$$ to a class $$\scriptstyle{\alpha}$$ will be $\scriptstyle{\alpha\cup\iota'x}$; the class got by taking away $$\scriptstyle{x}$$ from a class $$\scriptstyle{\alpha}$$ will be $\scriptstyle{\alpha-\iota'x}$. (We write $$\scriptstyle{\alpha-\beta}$$ as an abbreviation for $\scriptstyle{\alpha\cap-\beta}$.)

It will be observed that unit classes have been defined without reference to the number 1; in fact, we use unit classes to define the number 1. This number is defined as the class of unit classes, i.e.

This leads to

From this it appears further that

i.e. "$\scriptstyle{\hat z(\phi z)}$ is a unit class" is equivalent to "the $$\scriptstyle{x}$$ satisfying $$\scriptstyle{\phi\hat x}$$ exists."