therefore, be introduced first for one combination and later reintroduced
for another. For example, once negation has been introduced, we must
understand it both in propositions of the form 'Pp' and in propositions
like 'P(p C q)', '(dx) . Pfx', etc. We must not introduce it first for the
one class of cases and then for the other, since it would then be left in
doubt whether its meaning were the same in both cases, and no reason would
have been given for combining the signs in the same way in both cases. (In
short, Frege's remarks about introducing signs by means of definitions (in
The Fundamental Laws of Arithmetic ) also apply, mutatis mutandis, to the
introduction of primitive signs.)


5.452 The introduction of any new device into the symbolism of logic is
necessarily a momentous event. In logic a new device should not be
introduced in brackets or in a footnote with what one might call a
completely innocent air. (Thus in Russell and Whitehead's Principia
Mathematica there occur definitions and primitive propositions expressed in
words. Why this sudden appearance of words? It would require a
justification, but none is given, or could be given, since the procedure is
in fact illicit.) But if the introduction of a new device has proved
necessary at a certain point, we must immediately ask ourselves, 'At what
points is the employment of this device now unavoidable ?' and its place in
logic must be made clear.


5.453 All numbers in logic stand in need of justification. Or rather, it
must become evident that there are no numbers in logic. There are no pre-
eminent numbers.