Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/81

 i.e. there is a predicative function which is true when $$\scriptstyle{\phi x}$$ is true and false when $$\scriptstyle{\phi x}$$ is false. In symbols, the axiom is: For two variables, we require a similar axiom, namely: Given any function $\scriptstyle{\phi(\hat x,\hat y)}$, there is a formally equivalent predicative function, i.e.

In order to explain the purposes of the axiom of reducibility, and the nature of the grounds for supposing it true, we shall first illustrate it by applying it to some particular cases.

If we call a predicate of an object a predicative function which is true of that object, then the predicates of an object are only some among its properties. Take for example such a proposition as "Napoleon had all the qualities that make a great general." We may interpret this as meaning "Napoleon had all the predicates that make a great general." Here there is a predicate which is an apparent variable. If we put "$\scriptstyle{f(\phi!\hat z)}$" for "$\scriptstyle{\phi!\hat z}$ is a predicate required in a great general," our proposition is Since this refers to a totality of predicates, it is not itself a predicate of Napoleon. It by no means follows, however, that there is not some one predicate common and peculiar to great generals. In fact, it is certain that there is such a predicate. For the number of great generals is finite, and each of them certainly possessed some predicate not possessed by any other human being—for example, the exact instant of his birth. The disjunction of such predicates will constitute a predicate common and peculiar to great generals. If we call this predicate $\scriptstyle{\psi!\hat z}$, the statement we made about Napoleon was equivalent to $\scriptstyle{\psi!(}$Napoleon$\scriptstyle{)}$. And this equivalence holds equally if we substitute any other individual for Napoleon. Thus we have arrived at a predicate which is always equivalent to the property we ascribed to Napoleon, i.e. it belongs to those objects which have this property, and to no others. The axiom of reducibility states that such a predicate always exists, i.e. that any property of an object belongs to the same collection of object as those that possess some predicate.

We may next illustrate our principle by its application to identity. In this connection, it has a certain affinity with Leibniz's identity of indiscernibles. It is plain that, if $$\scriptstyle{x}$$ and $$\scriptstyle{y}$$ are identical, and $$\scriptstyle{\phi x}$$ is true, then $$\scriptstyle{\phi y}$$ is true. Here it cannot matter what sort of function $$\scriptstyle{\phi\hat x}$$ may be: the statement must bold for any function. But we cannot say, conversely: "If, with all values of $\scriptstyle{\phi}$, $$\scriptstyle{\phi x}$$ implies $\scriptstyle{\phi y}$, then $$\scriptstyle{x}$$ and $$\scriptstyle{y}$$ are identical"; because "all values of $\scriptstyle{\phi}$" is inadmissible. If we wish to speak of "all values of $\scriptstyle{\phi}$," we must confine ourselves to functions of one order. We may confine $$\scriptstyle{\phi}$$ to predicates, or to