Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/77

(like all matrices) they contain no apparent variables. Any such matrix, if it contains more than one variable, gives rise to new functions of one variable by turning all its arguments except one into apparent variables. Thus we obtain the functions

$\scriptstyle {(\phi ).g(\phi !{\hat {z}},\psi !{\hat {z}})}$, which is a function of $\scriptstyle {\psi !{\hat {z}}}$.
$\scriptstyle {(x).F(\phi !{\hat {z}},x)}$, which is a function of $\scriptstyle {\phi !{\hat {z}}}$.
$\scriptstyle {(\phi ).F(\phi !{\hat {z}},x)}$, which is a function of $\scriptstyle {x}$.

We will give the name of second-order matrices to such matrices as have first-order functions among their arguments, and have no arguments except first-order functions and individuals. (It is not necessary that they should have individuals among their arguments.) We will give the name of second-order functions to such as either are second-order matrices or are derived from such matrices by turning some of the arguments into apparent variables. It will be seen that either an individual or a first-order function may appear as argument to a second-order function. Second-order functions are such as contain variables which are first-order functions, but contain no other variables except (possibly) individuals.

We now have various new classes of functions at our command. In the first place, we have second-order functions which have one argument which is a first-order function. We will denote a variable function of this kind by the notation $\scriptstyle {f!({\hat {\phi }}!{\hat {z}})}$, and any value of such a function by $\scriptstyle {f!(\phi !{\hat {z}})}$. Like $\scriptstyle {\phi !x}$, $\scriptstyle {f!(\phi !{\hat {z}})}$ is a function of two variables, namely $\scriptstyle {f!({\hat {\phi }}!{\hat {z}})}$ and $\scriptstyle {\phi !{\hat {z}}}$. Among possible values of $\scriptstyle {f!(\phi !{\hat {z}})}$ will be $\scriptstyle {\phi !a}$ (where $\scriptstyle {a}$ is constant), $\scriptstyle {(x).\phi !x}$, $\scriptstyle {(\exists x).\phi !x}$, and so on. (These result from assigning a value to $\scriptstyle {f}$, leaving $\scriptstyle {\phi }$ to be assigned.) We will call such functions "predicative functions of first-order functions."

In the second place, we have second-order functions of two arguments, one of which is a first-order function while the other is an individual. Let us denote undetermined values of such functions by the notation

$\scriptstyle {f!(\phi !{\hat {z}},x)}$.

As soon as $\scriptstyle {x}$ is assigned, we shall have a predicative function of $\scriptstyle {\phi !{\hat {z}}}$. If our function contains no first-order function as apparent variable, we shall obtain a predicative function of $\scriptstyle {x}$ if we assign a value to $\scriptstyle {\phi !{\hat {z}}}$. Thus, to take the simplest possible case, if $\scriptstyle {f!(\phi !{\hat {z}},x)}$ is $\scriptstyle {\phi !x}$, the assignment of a value to $\scriptstyle {\phi }$ gives us a predicative function of $\scriptstyle {x}$, in virtue of the definition of "$\scriptstyle {\phi !x}$." But if $\scriptstyle {f!(\phi !{\hat {z}},x)}$ contains a first-order function as apparent variable, the assignment of a value to $\scriptstyle {\phi !{\hat {z}}}$ gives us a second-order function of $\scriptstyle {x}$.

In the third place, we have second-order functions of individuals. These will all be derived from functions of the form $\scriptstyle {f!(\phi !{\hat {z}},x)}$ by turning $\scriptstyle {\phi }$ into an apparent variable. We do not, therefore, need a new notation for them.

We have also second-order functions of two first-order functions, or of two such functions and an individual, and so on.