Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/45

concerned are all truth-functions and the fact that mathematics is concerned with extensions rather than intensions.

Convenient abbreviation. The following definitions give alternative and often more convenient notations:

${\begin{aligned}\scriptstyle {\phi x.\supset _{x}.\psi x:=:(x):\phi x.\supset .\psi x}&\scriptstyle {\quad {\text{Df,}}}\\\scriptstyle {\phi x.\equiv _{x}.\psi x:=:(x):\phi x.\equiv .\psi x}&\scriptstyle {\quad {\text{Df.}}}\\\end{aligned}}$

This notation "$\scriptstyle {\phi x.\supset _{x}.\psi x}$" is due to Peano, who, however, has no notation for the general idea "$\scriptstyle {(x).\phi x}$." It may be noticed as an exercise in the use of dots as brackets that we might have written

${\begin{aligned}\scriptstyle {\phi x\supset _{x}\psi x.=.(x).\phi x\supset \psi x}&\scriptstyle {\quad {\text{Df,}}}\\\scriptstyle {\phi x\equiv _{x}\psi x.=.(x).\phi x\equiv \psi x}&\scriptstyle {\quad {\text{Df.}}}\\\end{aligned}}$

In practice however, when $\scriptstyle {\phi {\hat {x}}}$ and $\scriptstyle {\psi {\hat {x}}}$ are special functions, it is not possible to employ fewer dots than in the first form, and often more are required. The following definitions give abbreviated notations for functions of two or more variables:

$\scriptstyle {(x,y).\phi (x,y).=:(x):(y).\phi (x,y)\quad {\text{Df,}}}$

and so on for any number of variables;

$\scriptstyle {\phi (x,y).\supset _{x,y}.\psi (x,y):=:(x,y):\phi (x,y).\supset .\psi (x,y)\quad {\text{Df,}}}$

and so on for any number of variables. Identity. The propositional function "$\scriptstyle {x}$ is identical with $\scriptstyle {y}$" is expressed by

$\scriptstyle {x=y.}$

This will be defined (cf. *13·01), but, owing to certain difficult points involved in the definition, we shall here omit it (cf. Chapter II). We have, of course,

${\begin{aligned}&\scriptstyle {\vdash .x=x\quad {\text{(the law of identity),}}}\\&\scriptstyle {\vdash :x=y.\equiv .y=x,}\\&\scriptstyle {\vdash :x=y.y=z.\supset .x=z.}\\\end{aligned}}$

The first of these expresses the reflexive property of identity: a relation is called reflexive when it holds between a term and itself, either universally, or whenever it holds between that term and some term. The second of the above propositions expresses that identity is a symmetrical relation: a relation is called symmetrical if, whenever it holds between $\scriptstyle {x}$ and $\scriptstyle {y}$, it also holds between $\scriptstyle {y}$ and $\scriptstyle {x}$. The third proposition expresses that identity is a transitive relation: a relation is called transitive if, whenever it holds between $\scriptstyle {x}$ and $\scriptstyle {y}$ and between $\scriptstyle {y}$ and $\scriptstyle {z}$, it holds also between $\scriptstyle {x}$ and $\scriptstyle {z}$.

We shall find that no new definition of the sign of equality is required in mathematics: all mathematical equations in which the sign of equality is