Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/74

 since we are concerned to assert all values of "$\scriptstyle{\phi x}$ implies $\scriptstyle{\phi y}$" regarded as a function of $\scriptstyle{\phi}$, we shall be compelled to impose upon $$\scriptstyle{\phi}$$ some limitation which will prevent us from including among values of $$\scriptstyle{\phi}$$ values in which "all possible values of $\scriptstyle{\phi}$" are referred to. Thus for example "$\scriptstyle{x}$ is identical with $\scriptstyle{a}$" is a function of $\scriptstyle{x}$; hence, if it is a legitimate value of $$\scriptstyle{\phi}$$ in "$\scriptstyle{\phi x}$ always implies $\scriptstyle{\phi y}$," we shall be able to infer, by means of the above definition, that if $$\scriptstyle{x}$$ is identical with $\scriptstyle{a}$, and $$\scriptstyle{x}$$ is identical with $\scriptstyle{y}$, then $$\scriptstyle{y}$$ is identical with $\scriptstyle{a}$. Although the conclusion is sound, the reasoning embodies a vicious-circle fallacy, since we have taken "$\scriptstyle{(\phi).\phi x}$ implies $\scriptstyle{\phi a}$" as a possible value of $\scriptstyle{\phi x}$, which it cannot be. If, however, we impose any limitation upon $\scriptstyle{\phi}$, it may happen, so far as appears at present, that with other values of $$\scriptstyle{\phi}$$ we might have $$\scriptstyle{\phi x}$$ true and $$\scriptstyle{\phi y}$$ false, so that our proposed definition of identity would plainly be wrong. This difficulty is avoided by the "axiom of reducibility," to be explained later. For the present, it is only mentioned in order to illustrate the necessity and the relevance of the hierarchy of functions of a given argument.

Let us give the name "$\scriptstyle{a}$-functions" to functions that are significant for a given argument $\scriptstyle{a}$. Then suppose we take any selection of $\scriptstyle{a}$-functions, and consider the proposition "$\scriptstyle{a}$ satisfies all the functions belonging to the selection in question." If we here replace $$\scriptstyle{a}$$ by a variable, we obtain an $\scriptstyle{a}$-function; but by the vicious-circle principle this $\scriptstyle{a}$-function cannot be a member of our selection, since it refers to the whole of the selection. Let the selection consist of all those functions which satisfy $\scriptstyle{f(\phi\hat z)}$. Then our new function is where $$\scriptstyle{x}$$ is the argument. It thus appears that, whatever selection of $\scriptstyle{a}$-functions we may make, there will be other $\scriptstyle{a}$-functions that lie outside our selection. Such $\scriptstyle{a}$-functions, as the above instance illustrates, will always arise through taking a function of two arguments, $$\scriptstyle{\phi\hat z}$$ and $\scriptstyle{x}$, and asserting all or some of the values resulting from varying $\scriptstyle{\phi}$. What is necessary, therefore, in order to avoid vicious-circle fallacies, is to divide our $\scriptstyle{a}$-functions into "types," each of which contains no functions which refer to the whole of that type.

When something is asserted or denied about all possible values or about some (undetermined) possible values of a variable, that variable is called apparent, after Peano. The presence of the words all or some in a proposition indicates the presence of an apparent variable; but often an apparent variable is really present where language does not at once indicate its presence. Thus for example "$\scriptstyle{A}$ is mortal" means "there is a time at which $$\scriptstyle{A}$$ will die." Thus a variable time occurs as apparent variable.

The clearest instances of propositions not containing apparent variables are such as express immediate judgments of perception, such as "this is red" or "this is painful," where "this" is something immediately given. In other