Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/72

 same for propositions of the forms $\scriptstyle{(x).\phi x}$, $\scriptstyle{(\exists x).\phi x}$, as it was for elementary propositions.

Similar explanations could be given for implication and conjunction, but this is unnecessary, since these can be defined in terms of negation and disjunction.

The considerations so far adduced in favour of the view that a function cannot significantly have as argument anything defined in terms of the function itself have been more or less indirect. But a direct consideration of the kinds of functions which have functions as arguments and the kinds of functions which have arguments other than functions will show, if we are not mistaken, that not only is it impossible for a function $$\scriptstyle{\phi\hat z}$$ to have itself or anything derived from it as argument, but that, if $$\scriptstyle{\psi\hat z}$$ is another function such that there are arguments $$\scriptstyle{a}$$ with which both "$\scriptstyle{\phi a}$" and "$\scriptstyle{\psi a}$" are significant, then $$\scriptstyle{\psi\hat z}$$ and anything derived from it cannot significantly be argument to $\scriptstyle{\phi\hat z}$. This arises from the fact that a function is essentially an ambiguity, and that, if it is to occur in a definite proposition, it must occur in such a way that the ambiguity has disappeared, and a wholly unambiguous statement has resulted. A few illustrations will make this clear. Thus "$\scriptstyle{(x).\phi x}$," which we have already considered, is a function of $\scriptstyle{\phi\hat x}$; as soon as $$\scriptstyle{\phi\hat x}$$ is assigned, we have a definite proposition, wholly free from ambiguity. But it is obvious that we cannot substitute for the function something which is not a function: "$\scriptstyle{(x).\phi x}$" means "$\scriptstyle{\phi x}$ in all cases," and depends for its significance upon the fact that there are "cases" of $\scriptstyle{\phi x}$, i.e. upon the ambiguity which is characteristic of a function. This instance illustrates the fact that, when a function can occur significantly as argument, something which is not a function cannot occur significantly as argument. But conversely, when something which is not a function can occur significantly as argument, a function cannot occur significantly. Take, e.g. "$\scriptstyle{x}$ is a man," and consider "$\scriptstyle{\phi\hat x}$ is a man." Here there is nothing to eliminate the ambiguity which constitutes $\scriptstyle{\phi\hat x}$; there is thus nothing definite which is said to be a man. A function, in fact, is not a definite object, which could be or not be a man; it is a mere ambiguity awaiting determination, and in order that it may occur significantly it must receive the necessary determination, which it obviously does not receive if it is merely substituted for something determinate in a proposition. This argument does not, however, apply directly as against such a statement as "$\scriptstyle{\{(x).\phi x\}}$|undefined is a man." Common sense would pronounce such a statement to be meaningless, but it cannot be condemned on the ground of ambiguity in its subject. We need