Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/65

positions of the form $\scriptstyle {\phi x}$. It follows that there must be no propositions, of the form $\scriptstyle {\phi x}$, in which $\scriptstyle {x}$ has a value which involves $\scriptstyle {\phi {\hat {x}}}$. (If this were the case, the values of the function would not all be determinate until the function was determinate, whereas we found that the function is not determinate unless its values are previously determinate.) Hence there must be no such thing as the value for $\scriptstyle {\phi {\hat {x}}}$ with the argument $\scriptstyle {\phi {\hat {x}}}$, or with any argument which involves $\scriptstyle {\phi {\hat {x}}}$. That is to say, the symbol "$\scriptstyle {\phi (\phi {\hat {x}})}$" must not express a proposition, as "$\scriptstyle {\phi a}$" does if $\scriptstyle {\phi a}$ is a value for $\scriptstyle {\phi {\hat {x}}}$. In fact "$\scriptstyle {\phi (\phi {\hat {x}})}$" must be a symbol which does not express anything: we may therefore say that it is not significant. Thus given any function $\scriptstyle {\phi {\hat {x}}}$, there are arguments with which the function has no value, as well as arguments with which it has a value. Ve will call the arguments with which $\scriptstyle {\phi {\hat {x}}}$ has a value "possible values of $\scriptstyle {x}$." We will say that $\scriptstyle {\phi {\hat {x}}}$ is "significant with the argument $\scriptstyle {x}$" when $\scriptstyle {\phi {\hat {x}}}$ has a value with the argument $\scriptstyle {x}$.

When it is said that e.g. "$\scriptstyle {\phi (\phi {\hat {z}})}$" is meaningless, and therefore neither true nor false, it is necessary to avoid a misunderstanding. If "$\scriptstyle {\phi (\phi {\hat {z}})}$" were interpreted as meaning "the value for $\scriptstyle {\phi {\hat {z}}}$ with the argument $\scriptstyle {\phi {\hat {z}}}$ is true," that would be not meaningless, but false. It is false for the same reason for which "the King of France is bald" is false, namely because there is no such thing as "the value for $\scriptstyle {\phi {\hat {z}}}$ with the argument $\scriptstyle {\phi {\hat {z}}}$." But when, with some argument $\scriptstyle {a}$, we assert $\scriptstyle {\phi a}$, we are not meaning to assert "the value for $\scriptstyle {\phi {\hat {x}}}$ with the argument $\scriptstyle {a}$ is true"; we are meaning to assert the actual proposition which is the value for $\scriptstyle {\phi {\hat {x}}}$ with the argument $\scriptstyle {a}$. Thus for example if $\scriptstyle {\phi {\hat {x}}}$ is "$\scriptstyle {\hat {x}}$ is a man," $\scriptstyle {\phi ({\text{Socrates}})}$ will be "Socrates is a man," not "the value for the function '$\scriptstyle {\hat {x}}$ is a man,' with the argument Socrates, is true." Thus in accordance with our principle that "$\scriptstyle {\phi (\phi {\hat {z}})}$" is meaningless, we cannot legitimately deny "the function '$\scriptstyle {\hat {x}}$ is a man' is a man," because this is nonsense, but we can legitimately deny "the value for the function '$\scriptstyle {\hat {x}}$ is a man' with the argument, '$\scriptstyle {\hat {x}}$ is a man' is true," not on the ground that the value in question is false, but on the ground that there is no such value for the function.

We will denote by the symbol "$\scriptstyle {(x).\phi x}$" the proposition "$\scriptstyle {\phi x}$ always^[1]," i.e. the proposition which asserts all the values for $\scriptstyle {\phi {\hat {x}}}$. This proposition involves the function $\scriptstyle {\phi {\hat {x}}}$, not merely an ambiguous value of the function. The assertion of $\scriptstyle {\phi x}$, where $\scriptstyle {x}$ is unspecified, is a different assertion from the one which asserts all values for $\scriptstyle {\phi {\hat {x}}}$, for the former is an ambiguous assertion, whereas the latter is in no sense ambiguous. It will be observed that "$\scriptstyle {(x).\phi x}$" does not assert "$\scriptstyle {\phi x}$ with all values of $\scriptstyle {x}$," because, as we have seen, there must be values of $\scriptstyle {x}$ with which "$\scriptstyle {\phi x}$" is meaningless. What is asserted by "$\scriptstyle {(x).\phi x}$" is all propositions which are values for $\scriptstyle {\phi {\hat {x}}}$; hence it is

1. We use "always" as meaning "in all cases," not "at all times." Similarly "sometimes" will mean "in some cases."