Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/50

These propositions are analogous to those previously given for classes. It results from them that any function of two variables is formally equivalent to some function of the form $\scriptstyle {xRy}$; hence, in extensional functions of two variables, variation of relations can replace variation of functions of two variables.

Both classes and relations have properties analogous to most of those of propositions that result from negation and the logical sum. The logical product of two classes $\scriptstyle {\alpha }$ and $\scriptstyle {\beta }$ is their common part, i.e. the class of terms which are members of both. This is represented by $\scriptstyle {\alpha \cap \beta }$. Thus we put

$\scriptstyle {\alpha \cap \beta ={\hat {x}}(x\in \alpha .x\in \beta )\quad {\text{Df.}}}$

This gives us

$\scriptstyle {\vdash :x\in \alpha \cap \beta .\equiv .x\in \alpha .x\in \beta }$,

i.e. "$\scriptstyle {x}$ is a member of the logical product of $\scriptstyle {\alpha }$ and $\scriptstyle {\beta }$" is equivalent to the logical product of "$\scriptstyle {x}$ is a member of $\scriptstyle {\alpha }$" and "$\scriptstyle {x}$ is a member of $\scriptstyle {\beta }$." Similarly the logical sum of two classes $\scriptstyle {\alpha }$ and $\scriptstyle {\beta }$ is the class of terms which are members of either; we denote it by $\scriptstyle {\alpha \cup \beta }$. The definition is

$\scriptstyle {\alpha \cup \beta ={\hat {x}}(x\in \alpha .\lor .x\in \beta )\quad {\text{Df,}}}$

and the connection with the logical sum of propositions is given by

$\scriptstyle {\vdash :.x\in \alpha \cup \beta .\equiv :x\in \alpha .\lor .x\in \beta }$.

The negation of a class $\scriptstyle {\alpha }$ consists of those terms $\scriptstyle {x}$ for which "$\scriptstyle {x\in \alpha }$" can be significantly and truly denied. We shall find that there are terms of other types for which "$\scriptstyle {x\in \alpha }$" is neither true nor false, but nonsense. These terms are not members of the negation of $\scriptstyle {\alpha }$.

Thus the negation of a class $\scriptstyle {\alpha }$ is the class of terms of suitable type which are not members of it, i.e. the class $\scriptstyle {{\hat {x}}(x\sim \in \alpha )}$. We call this class "$\scriptstyle {\lnot \alpha }$" (read "$\scriptstyle {{\text{not}}-\alpha }$"); thus the definition is

$\scriptstyle {\lnot \alpha ={\hat {x}}(x\sim \in \alpha )\quad {\text{Df,}}}$

and the connection with the negation of propositions is given by

$\scriptstyle {\vdash :x\in \lnot \alpha .\equiv .x\sim \in \alpha }$.

In place of implication we have the relation of inclusion. A class $\scriptstyle {\alpha }$ is said to be included or contained in a class $\scriptstyle {\beta }$ if all members of $\scriptstyle {\alpha }$ are members of $\scriptstyle {\beta }$, i.e. if $\scriptstyle {x\in \alpha .\supset _{x}.x\in \beta }$. We write "$\scriptstyle {\alpha \subset \beta }$" for "$\scriptstyle {\alpha }$ is contained in $\scriptstyle {\beta }$." Thus we put

$\scriptstyle {\alpha \subset \beta .=:x\in \alpha .\supset _{x}.x\in \beta \quad {\text{Df.}}}$

Most of the formulae concerning $\scriptstyle {p\cdot q}$, $\scriptstyle {p\lor q}$, $\scriptstyle {p\supset q}$ remain true if we substitute $\scriptstyle {\alpha \cap \beta }$, $\scriptstyle {\alpha \cup \beta }$, $\scriptstyle {\lnot \alpha }$, $\scriptstyle {\alpha \subset \beta }$. In place of equivalence, we substitute identity; for "$\scriptstyle {p\equiv q}$" was defined as "$\scriptstyle {p\supset q.q\supset p}$," but "$\scriptstyle {\alpha \subset \beta .\beta \subset \alpha }$" gives "$\scriptstyle {x\in \alpha .\equiv _{x}.x\in \beta }$," whence $\scriptstyle {\alpha =\beta }$.