Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/109

It follows from these three groups of theorems that these incomplete symbols are obedient to the same formal rules of identity as symbols which directly represent objects, so long as we only consider the equivalence of the resulting variable (or constant) values of propositional functions and not their identity. This consideration of the identity of propositions never enters into our formal reasoning.

Similarly the limitations to the use of these symbols can be summed up as follows. In the. case of $\scriptstyle {(}$℩$\scriptstyle {x)(\phi x)}$, the chief way in which its incompleteness is relevant is that we do not always have

$\scriptstyle {(x).fx.\supset .f(}$℩$\scriptstyle {x)(\phi x)}$,

i.e. a function which is always true may nevertheless not be true of $\scriptstyle {(}$℩$\scriptstyle {x)(\phi x)}$. This is possible because $\scriptstyle {f(}$℩$\scriptstyle {x)(\phi x)}$ is not a value of $\scriptstyle {f{\hat {x}}}$, so that even when all values of $\scriptstyle {f{\hat {x}}}$ are true, $\scriptstyle {f(}$℩$\scriptstyle {x)(\phi x)}$ may not be true. This happens when $\scriptstyle {(}$℩$\scriptstyle {x)(\phi x)}$ does not exist. Thus for example we have $\scriptstyle {(x).x=x}$, but we do not have

the round square=the round square.

The inference

$\scriptstyle {(x).fx.\supset .f(}$℩$\scriptstyle {x)(\phi x)}$

is only valid when $\scriptstyle {\mathbf {E!} (}$℩$\scriptstyle {x)(\phi x)}$. As soon as we know $\scriptstyle {\mathbf {E!} (}$℩$\scriptstyle {x)(\phi x)}$, the fact that $\scriptstyle {(}$℩$\scriptstyle {x)(\phi x)}$ is an incomplete symbol becomes irrelevant so long as we confine ourselves to truth-functions^[1] of whatever proposition is its scope. But even when $\scriptstyle {\mathbf {E!} (}$℩$\scriptstyle {x)(\phi x)}$, the incompleteness of $\scriptstyle {(}$℩$\scriptstyle {x)(\phi x)}$ may be relevant when we pass outside truth-functions. For example, George IV wished to know whether Scott was the author of Waverley, i.e. he wished to know whether a proposition of the form "$\scriptstyle {c=(}$℩$\scriptstyle {x)(\phi x)}$" was true. But there was no proposition of the form "$\scriptstyle {c=y}$" concerning which he wished to know if it was true.

In regard to classes, the relevance of their incompleteness is somewhat different. It may he illustrated by the fact that we may have

$\scriptstyle {{\hat {z}}(\phi z)=\psi !{\hat {z}}(\phi z)=\chi !{\hat {z}}}$

without having

$\scriptstyle {\psi !{\hat {z}}=\chi !{\hat {z}}}$.

For, by a direct application of the definitions, we find that

$\scriptstyle {\vdash :{\hat {z}}(\phi z)=\psi !{\hat {z}}.\equiv .\phi x\equiv _{x}\psi !x}$.

Thus we shall have

$\scriptstyle {\vdash :\phi x\equiv _{x}\psi !x.\phi x,]phix\equiv _{x}\chi !x.\supset .{\hat {z}}(\phi z)=\psi !{\hat {z}}.{\hat {z}}(\phi z)=\chi !{\hat {z}}}$,

but we shall not necessarily have $\scriptstyle {\psi !{\hat {z}}=\chi !{\hat {z}}}$ under these circumstances, for two functions may well be formally equivalent without being identical; for example,

$\scriptstyle {x=}$Scott$\scriptstyle {.\equiv _{x}.x=}$the author of Waverley,

1. Cf. p. 8.