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

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

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 $\scriptstyle {1-2^{-n}}$, 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.

$\scriptstyle {P''\alpha ={\hat {x}}\{(\exists y).y\in \alpha .x=P'y\}}$.

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

$\scriptstyle {\iota 'x={\hat {y}}(y=x)\quad {\text{Df,}}}$

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.

$\scriptstyle {1={\hat {\alpha }}\{(\exists x).\alpha =\iota 'x\}\quad {\text{Df.}}}$

This leads to

$\scriptstyle {\vdash :.\alpha \in 1.\equiv :(\exists x):y\in \alpha .\equiv _{y}.y=x}$.

From this it appears further that

$\scriptstyle {\vdash :\alpha \in 1.\equiv .\mathbf {E!} (}$℩$\scriptstyle {x)(x\in \alpha )}$,

whence

$\scriptstyle {\vdash :{\hat {z}}(\phi z)\in 1.\equiv .\mathbf {E!} (}$℩$\scriptstyle {x)(\phi x)}$,

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