Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/63

II. The Nature of Propositional Functions.

By a "propositional function" we mean something which contains a variable $\scriptstyle {x}$, and expresses a proposition as soon as a value is assigned to $\scriptstyle {x}$. That is to say, it differs from a proposition solely by the fact that it is ambiguous: it contains a variable of which the value is unassigned. It agrees with the ordinary functions of mathematics in the fact of containing an unassigned variable: where it differs is in the fact that the values of the function are propositions. Thus e.g. "$\scriptstyle {x}$ is a man" or "$\scriptstyle {\sin x=1}$" is a propositional function. We shall find that it is possible to incur a vicious-circle fallacy at the very outset, by admitting as possible arguments to a propositional function terms which presuppose the function. This form of the fallacy is very instructive, and its avoidance leads, as we shall see, to the hierarchy of types.

The question as to the nature of a function^[1] is by no means an easy one. It would seem, however, that the essential characteristic of a function is ambiguity. Take, for example, the law of identity in the form "$\scriptstyle {A}$ is $\scriptstyle {A}$," which is the form in which it is usually enunciated. It is plain that, regarded psychologically, we have here a single judgment. But what are we to say of the object of the judgment? We are not judging that Socrates is Socrates, nor that Plato is Plato, nor any other of the definite judgments that are instances of the law of identity. Yet each of these judgments is, in a sense, within the scope of our judgment. We are in fact judging an ambiguous instance of the propositional function "$\scriptstyle {A}$ is $\scriptstyle {A}$." We appear to have a single thought which does not have a definite object, but has as its object an undetermined one of the values of the function "$\scriptstyle {A}$ is $\scriptstyle {A}$." It is this kind of ambiguity that constitutes the essence of a function. When we speak of "$\scriptstyle {\phi x}$," where $\scriptstyle {x}$ is not specified, we mean one value of the function, but not a definite one. We may express this by saying that "$\scriptstyle {\phi x}$" ambiguously denotes $\scriptstyle {\phi a}$, $\scriptstyle {\phi b}$, $\scriptstyle {\phi c}$, etc., where $\scriptstyle {\phi a}$, $\scriptstyle {\phi b}$, $\scriptstyle {\phi c}$, etc., are the various values of "$\scriptstyle {\phi x}$."

When we say that "$\scriptstyle {\phi x}$" ambiguously denotes $\scriptstyle {\phi a}$, $\scriptstyle {\phi b}$, $\scriptstyle {\phi c}$, etc., we mean that "$\scriptstyle {\phi x}$" means one of the objects $\scriptstyle {\phi a}$, $\scriptstyle {\phi b}$, $\scriptstyle {\phi c}$, etc., though not a definite one, but an undetermined one. It follows that "$\scriptstyle {\phi x}$" only has a well-defined meaning (well-defined, that is to say, except in so far as it is of its essence to be ambiguous) if the objects $\scriptstyle {\phi a}$, $\scriptstyle {\phi b}$, $\scriptstyle {\phi c}$, etc., are well-defined. That is to say, a function is not a well-defined function unless all its values are already well-defined. It follows from this that no function can have among its values anything which presupposes the function, for if it had, we could not regard the objects ambiguously denoted by the function as definite until the function was definite, while conversely, as we have just seen, the function cannot be

1. When the word "function" is used in the sequel, "propositional function" is always meant. Other functions will not be in question in the present Chapter.