Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/75

judgments, even where at first sight no variable appears to be present, it often happens that there really is one. Take (say) "Socrates is human." To Socrates himself, the word "Socrates" no doubt stood for an object of which he was immediately aware, and the judgment "Socrates is human" contained no apparent variable. But to us, who only know Socrates by description, the word "Socrates" cannot mean what it meant to him; it means rather "the person having such-and-such properties," (say) "the Athenian philosopher who drank the hemlock." Now" in all propositions about "the so-and-so" there is an apparent variable, as will be shown in Chapter III. Thus in what we have in mind when we say "Socrates is human" there is an apparent variable, though there was no apparent variable in the corresponding judgment as made by Socrates, provided we assume that there is such a thing as immediate awareness of oneself.

Whatever may be the instances of propositions not containing apparent variables, it is obvious that propositional functions whose values do not contain apparent variables are the source of propositions containing apparent variables, in the sense in which the function $\scriptstyle {\phi {\hat {x}}}$ is the source of the proposition $\scriptstyle {(x).\phi x}$. For the values for $\scriptstyle {\phi {\hat {x}}}$ do not contain the apparent variable $\scriptstyle {x}$, which appears in $\scriptstyle {(x).\phi x}$; if they contain an apparent variable $\scriptstyle {y}$, this can be similarly eliminated, and so on. This process must come to an end, since no proposition which we can apprehend can contain more than a finite number of apparent variables, on the ground that whatever we can apprehend must be of finite complexity. Thus we must arrive at last at a function of as many variables as there have been stages in reaching it from our original proposition, and this function will be such that its values contain no apparent variables. We may call this function the matrix of our original proposition and of any other propositions and functions to be obtained by turning some of the arguments to the function into apparent variables. Thus for example, if we have a matrix-function whose values are $\scriptstyle {\phi (x,y)}$, we shall derive from it

${\begin{aligned}\scriptstyle {(y)}&\scriptstyle {\cdot \phi (x,y){\text{, which is a function of }}x,}\\\scriptstyle {(x)}&\scriptstyle {\cdot \phi (x,y){\text{, which is a function of }}y,}\\\scriptstyle {(x,y)}&\scriptstyle {\cdot \phi (x,y){\text{, meaning }}^{\prime \prime }\phi (x,y){\text{ is true with all possible values of }}x{\text{ and }}y{\text{.}}^{\backprime \backprime }}\end{aligned}}$

This last is a proposition containing no real variable, i.e. no variable except apparent variables.

It is thus plain that all possible propositions and functions are obtainable from matrices by the process of turning the arguments to the matrices into apparent variables. In order to divide our propositions and functions into types, we shall, therefore, start from matrices, and consider how they are to be divided with a view to the avoidance of vicious-circle fallacies in the definitions of the functions concerned. For this purpose, we will use such letters as $\scriptstyle {a}$, $\scriptstyle {b}$, $\scriptstyle {c}$, $\scriptstyle {x}$, $\scriptstyle {y}$, $\scriptstyle {z}$, $\scriptstyle {w}$, to denote objects which are neither propositions nor functions. Such objects we shall call individuals. Such objects will be