Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/82

second-order functions, or to functions of any order we please. But we must necessarily leave out functions of all but one order. Thus we shall obtain, so to speak, a hierarchy of different degrees of identity. We may say "all the predicates of $\scriptstyle {x}$ belong to $\scriptstyle {y}$," "all second-order properties of $\scriptstyle {x}$ belong to $\scriptstyle {y}$," and so on. Each of these statements implies all its predecessors: for example, if all second-order properties of $\scriptstyle {x}$ belong to $\scriptstyle {y}$, then all predicates of $\scriptstyle {x}$ belong to $\scriptstyle {y}$, for to have all the predicates of $\scriptstyle {x}$ is a second-order property, and this property belongs to $\scriptstyle {x}$. But we cannot, without the help of an axiom, argue conversely that if all the predicates of $\scriptstyle {x}$ belong to $\scriptstyle {y}$, all the second-order properties of $\scriptstyle {x}$ must also belong to $\scriptstyle {y}$. Thus we cannot, without the help of an axiom, be sure that $\scriptstyle {x}$ and $\scriptstyle {y}$ are identical if they have the same predicates. Leibniz's identity of indiscernibles supplied this axiom. It should be observed that by "indiscernibles" he cannot have meant two objects which agree as to all their properties, for one of the properties of $\scriptstyle {x}$ is to be identical with $\scriptstyle {x}$, and therefore this property would necessarily belong to $\scriptstyle {y}$ if $\scriptstyle {x}$ and $\scriptstyle {y}$ agreed in all their properties. Some limitation of the common properties necessary to make things indiscernible is therefore implied by the necessity of an axiom. For purposes of illustration (not of interpreting Leibniz) we may suppose the common properties required for indiscernibility to be limited to predicates. Then the identity of indiscernibles will state that if $\scriptstyle {x}$ and $\scriptstyle {y}$ agree as to all their predicates, they are identical. This can be proved if we assume the axiom of reducibility. For, in that case, every property belongs to the same collection of objects as is defined by some predicate. Hence there is some predicate common and peculiar to the objects which are identical with $\scriptstyle {x}$. This predicate belongs to $\scriptstyle {x}$, since $\scriptstyle {x}$ is identical with itself; hence it belongs to $\scriptstyle {y}$, since $\scriptstyle {y}$ has all the predicates of $\scriptstyle {x}$; hence $\scriptstyle {y}$ is identical with $\scriptstyle {x}$. It follows that we may define $\scriptstyle {x}$ and $\scriptstyle {y}$ as identical when all the predicates of $\scriptstyle {x}$ belong to $\scriptstyle {y}$, i.e. when $\scriptstyle {(\phi ):\phi !x.\supset .\phi x.\phi !y}$. We therefore adopt the following definition of identity^[1]:

$\scriptstyle {x=y.=:(\phi ):\phi !x.\supset .\phi !y\quad {\text{Df.}}}$

But apart from the axiom of reducibility, or some axiom equivalent in this connection, we should be compelled to regard identity as indefinable, and to admit (what seems impossible) that two objects may agree in all their predicates without being identical.

The axiom of reducibility is even more essential in the theory of classes. It should be observed, in the first place, that if we assume the existence of classes, the axiom of reducibility can be proved. For in that case, given any function $\scriptstyle {\phi {\hat {z}}}$ of whatever order, there is a class $\scriptstyle {a}$ consisting of just those objects which satisfy $\scriptstyle {\phi {\hat {z}}}$. Hence "$\scriptstyle {\phi x}$" is equivalent to "$\scriptstyle {x}$ belongs to $\scriptstyle {\alpha }$." But "$\scriptstyle {x}$ belongs to $\scriptstyle {\alpha }$" is a statement containing no apparent variable, and is therefore a predicative function of $\scriptstyle {x}$. Hence if we assume the existence of

1. Note that in this definition the second sign of equality is to be regarded as combining with "Df" to form one symbol; what is defined is the sign of equality not followed by the letters "Df."