Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/100

philosophically. The technical procedure by which the apparent difficulty is overcome is as follows.

We have seen that an extensional function of a function may be regarded as a function of the class determined by the argument-function, but that an intensional function cannot be so regarded. In order to obviate the necessity of giving different treatment to intensional and extensional functions of functions, we construct an extensional function derived from any function of a predicative function $\scriptstyle {\psi !{\hat {z}}}$, and having the property of being equivalent to the function from which it is derived, provided this function is extensional, as well as the property of being significant (by the help of the systematic ambiguity of equivalence) with any argument $\scriptstyle {\phi {\hat {z}}}$ whose arguments are of the same type as those of $\scriptstyle {\psi !{\hat {z}}}$. The derived function, written "$\scriptstyle {f\{{\hat {z}}(\phi z)\}}$," is defined as follows: Given a function $\scriptstyle {f(\psi !{\hat {z}})}$, our derived function is to be "there is a predicative function which is formally equivalent to $\scriptstyle {\phi {\hat {z}}}$ and satisfies $\scriptstyle {f}$." If $\scriptstyle {\phi {\hat {z}}}$ is a predicative function, our derived function will be true whenever $\scriptstyle {f(\phi {\hat {z}})}$ is true. If $\scriptstyle {f(\phi {\hat {z}})}$ is an extensional function, and $\scriptstyle {\phi {\hat {z}}}$ is a predicative function, our derived function will not be true unless $\scriptstyle {f(\phi {\hat {z}})}$ is true; thus in this case, our derived function is equivalent to $\scriptstyle {f(\phi {\hat {z}})}$. If $\scriptstyle {f(\phi {\hat {z}}}$ is not an extensional function, and if $\scriptstyle {\phi {\hat {z}}}$ is a predicative function, our derived function may sometimes be true when the original function is false. But in any case the derived function is always extensional.

In order that the derived function should be significant for any function $\scriptstyle {\phi {\hat {z}}}$, of whatever order, provided it takes arguments of the right type, it is necessary and sufficient that $\scriptstyle {f(\psi !{\hat {z}})}$) should be significant, where $\scriptstyle {\psi !{\hat {z}}}$ is any predicative function. The reason of this is that we only require, concerning an argument $\scriptstyle {\phi {\hat {z}}}$, the hypothesis that it is formally equivalent to some predicative function, $\scriptstyle {\psi !{\hat {z}}}$, and formal equivalence has the same kind of systematic ambiguity as to type that belongs to truth and falsehood, and can therefore hold between functions of any two different orders, provided the functions take arguments of the same type. Thus by means of our derived function we have not merely provided extensional functions everywhere in place of intensional functions, but we have practically removed the necessity for considering differences of type among functions whose arguments are of the same type. This effects the same kind of simplification in our hierarchy as would result from never considering any but predicative functions.

If $\scriptstyle {f(\psi !{\hat {z}})}$ can be built up by means of the primitive ideas of disjunction, negation, $\scriptstyle {(x).\phi x}$, and $\scriptstyle {(\exists x).\phi x}$, as is the case with all the functions of functions that explicitly occur in the present work, it will be found that, in virtue of the systematic ambiguity of the above primitive ideas, any function $\scriptstyle {\phi {\hat {z}}}$ whose arguments are of the same type as those of $\scriptstyle {\psi !{\hat {z}}}$ can significantly be substituted for $\scriptstyle {\psi !{\hat {z}}}$ in $\scriptstyle {f}$ without any other symbolic change. Thus in