Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/64

definite until its values are definite. This is a particular case, but perhaps the most fundamental case, of the vicious-circle principle. A function is what ambiguously denotes some one of a certain totality, namely the values of the function; hence this totality cannot contain any members which involve the function, since, if it did, it would contain members involving the totality, which, by the vicious-circle principle, no totality can do.

It will be seen that, according to the above account, the values of a function are presupposed by the function, not vice versa. It is sufficiently obvious, in any particular case, that a value of a function does not presuppose the function. Thus for example the proposition "Socrates is human" can be perfectly apprehended without regarding it as a value of the function "$\scriptstyle {x}$ is human." It is true that, conversely, a function can be apprehended without its being necessary to apprehend its values severally and individually. If this were not the case, no function could be apprehended at all, since the number of values (true and false) of a function is necessarily infinite and there are necessarily possible arguments with which we are unacquainted. What is necessary is not that the values should be given individually and extensionally, but that the totality of the values should be given intensionally, so that, concerning any assigned object, it is at least theoretically determinate whether or not the said object is a value of the function.

It is necessary practically to distinguish the function itself from an undetermined value of the function. We may regard the function itself as that which ambiguously denotes, while an undetermined value of the function is that which is ambiguously denoted. If the undetermined value is written "$\scriptstyle {\phi x}$," we will write the function itself "$\scriptstyle {\phi {\hat {x}}}$." (Any other letter may be used in place of $\scriptstyle {x}$.) Thus we should say "$\scriptstyle {\phi x}$ is a proposition," but "$\scriptstyle {\phi {\hat {x}}}$ is a propositional function." When we say "$\scriptstyle {\phi x}$ is a proposition," we mean to state something which is true for every possible value of $\scriptstyle {x}$, though we do not decide what value $\scriptstyle {x}$ is to have. We are making an ambiguous statement about any value of the function. But when we say "$\scriptstyle {\phi {\hat {x}}}$ is a function," we are not making an ambiguous statement. It would be more correct to say that we are making a statement about an ambiguity, taking the view that a function is an ambiguity. The function itself, $\scriptstyle {\phi {\hat {x}}}$, is the single thing which ambiguously denotes its many values; while $\scriptstyle {\phi x}$, where $\scriptstyle {x}$ is not specified, is one of the denoted objects, with the ambiguity belonging to the manner of denoting.

We have seen that, in accordance with the vicious-circle principle, the values of a function cannot contain terms only definable in terms of the function. Now given a function $\scriptstyle {\phi {\hat {x}}}$, the values for the function^[1] are all pro-

1. We shall speak in this Chapter of "values for $\scriptstyle {\phi {\hat {x}}}$" and of "values of $\scriptstyle {\phi x}$," meaning in each case the same thing, namely $\scriptstyle {\phi a}$, $\scriptstyle {\phi b}$, $\scriptstyle {\phi c}$, etc. The distinction of phraseology serves to avoid ambiguity where several variables are concerned, especially when one of them is a function.