Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/76

constituents of propositions or functions, and will be genuine constituents, in the sense that they do not disappear on analysis, as (for example) classes do, or phrases of the form "the so-and-so."

The first matrices that occur are those whose values are of the forms

$\scriptstyle {\phi x,~\psi (x,y),~\chi (x,y,z\ldots ),}$

i.e. where the arguments, however many there may be, are all individuals. The functions $\scriptstyle {\phi }$, $\scriptstyle {\psi }$, $\scriptstyle {\chi \ldots }$, since (by definition) they contain no apparent variables, and have no arguments except individuals, do not presuppose any totality of functions. From the functions $\scriptstyle {\psi ,~\chi \ldots }$ we may proceed to form other functions of $\scriptstyle {x}$, such as $\scriptstyle {(y).\psi (x,y)}$, $\scriptstyle {(\exists y).\psi (x,y)}$, $\scriptstyle {(y,z).\chi (x,y,z)}$, $\scriptstyle {(y):(\exists z).\chi (x,y,z)}$, and so on. All these presuppose no totality except that of individuals. We thus arrive at a certain collection of functions of $\scriptstyle {x}$, characterized by the fact that they involve no variables except individuals. Such functions we will call "first-order functions." We may now introduce a notation to express "any first-order function." We will denote any first-order function by "$\scriptstyle {\phi !{\hat {x}}}$." and any value for such a function by "$\scriptstyle {\phi !x}$." Thus "$\scriptstyle {\phi !x}$" stands for any value for any function which involves no variables except individuals. It will be seen that "$\scriptstyle {\phi !x}$" is itself a function of two variables, namely $\scriptstyle {\phi !{\hat {z}}}$ and $\scriptstyle {x}$. Thus $\scriptstyle {\phi !x}$ involves a variable which is not an individual, namely $\scriptstyle {\phi !{\hat {z}}}$. Similarly "$\scriptstyle {(x).\phi !x}$" is a function of the variable $\scriptstyle {\phi !{\hat {z}}}$, and thus involves a variable other than an individual. Again, if $\scriptstyle {a}$ is a given individual,

"$\scriptstyle {\phi !x}$ implies $\scriptstyle {\phi !a}$ with all possible values of $\scriptstyle {\phi }$"

is a function of $\scriptstyle {x}$, but it is not a function of the form $\scriptstyle {\phi !x}$, because it involves an (apparent) variable $\scriptstyle {\phi }$ which is not an individual. Let us give the name "predicate" to any first-order function $\scriptstyle {\phi !{\hat {x}}}$. (This use of the word "predicate" is only proposed for the purposes of the present discussion.) Then the statement "$\scriptstyle {\phi !x}$ implies $\scriptstyle {\phi !a}$ with all possible values of $\scriptstyle {\phi }$" may be read "all the predicates of $\scriptstyle {x}$ are predicates of $\scriptstyle {a}$." This makes a statement about $\scriptstyle {x}$, but does not attribute to $\scriptstyle {x}$ a predicate in the special sense just defined. Owing to the introduction of the variable first-order function $\scriptstyle {\phi !{\hat {z}}}$, we now have a new set of matrices. Thus "$\scriptstyle {\phi !x}$" is a function which contains no apparent variables, but contains the two real variables $\scriptstyle {\phi !{\hat {z}}}$ and $\scriptstyle {x}$. (It should be observed that when $\scriptstyle {\phi }$ is assigned, we may obtain a function whose values do involve individuals as apparent variables, for example if $\scriptstyle {\phi !x}$ is $\scriptstyle {(y).\psi (x,y)}$. But so long as $\scriptstyle {\phi }$ is variable, $\scriptstyle {\phi !x}$ contains no apparent variables.) Again, if $\scriptstyle {a}$ is a definite individual, $\scriptstyle {\phi !a}$ is a function of the one variable $\scriptstyle {\phi !{\hat {z}}}$. If $\scriptstyle {a}$ and $\scriptstyle {b}$ are definite individuals, "$\scriptstyle {\phi !a}$ implies $\scriptstyle {\psi !b}$" is a function of the two variables $\scriptstyle {\phi !{\hat {z}}}$, $\scriptstyle {\psi !{\hat {z}}}$, and so on. We are thus led to a whole set of new matrices,

$\scriptstyle {f(\phi !{\hat {z}}),~g(\phi !{\hat {z}}),~F(\phi !{\hat {z}})}$, and so on.

These matrices contain individuals and first-order functions as arguments, but