Principia mathematica, by Alfred North Whitehead ... and Bertrand Russell.

12

CARDINAL ARITH~METIC

[PART III

This notation would enable us to deal with descending relational cardinals.
But it is not required in the present work, and is therefore not introduced
among the numbered propositions.
When a typically ambiguous symbol, such as "sm" or "Nc," occurs more
than once in a given context, it must not be assumed, unless required by the
conditions of significance, that it is to receive the same typical determination
in each case. Thus e.g. we shall write "asm 3. ).sm a," although, if
a and /3 are of different types, the two symbols "sin" must receive different
typical determinations.
Formulae which are typically ambiguous, or only partially definite as to
type, must not be admitted unless every significant interpretation is true.
Thus for example we may admit
"F. a e Nc'a"
because here "Nc" must mean "Nc(as)," so that the only ambiguity remaining
is as to the type of a, and the formula holds whatever type a may belong to,
provided "Nc'a" is significant, i.e. provided a is a class. But we must not,
from "aeNc'a," allow ourselves to infer
"a! Nc'a."
For here the conditions of significance no longer demand that "Nc" should
mean "Nc(aa)": it might just as well mean "Nc(/a3)." And as we saw, if
B3 is a lower type than a, and a is sufficiently large of its type, we may have
Nc (a)'a= A,
so that "a! Nc'a" is not admissible without qualification. Nevertheless, as
we shall see in *100, there are a certain number of propositions to be made
about a wholly ambiguous Nc or NC.