ry rule, nor one that
is justified by its success in practice: its point is that unnecessary
units in a sign-language mean nothing. Signs that serve one purpose
are logically equivalent, and signs that serve none are logically
meaningless.

5.4733 Frege says that any legitimately constructed proposition must
have a sense. And I say that any possible proposition is legitimately
constructed, and, if it has no sense, that can only be because we have
failed to give a meaning to some of its constituents. (Even if we think
that we have done so.) Thus the reason why 'Socrates is identical' says
nothing is that we have not given any adjectival meaning to the word
'identical'. For when it appears as a sign for identity, it symbolizes
in an entirely different way--the signifying relation is a different
one--therefore the symbols also are entirely different in the two cases:
the two symbols have only the sign in common, and that is an accident.

5.474 The number of fundamental operations that are necessary depends
solely on our notation.

5.475 All that is required is that we should construct a system of signs
with a particular number of dimensions--with a particular mathematical
multiplicity.

5.476 It is clear that this is not a question of a number of primitive
ideas that have to be signified, but rather of the expression of a rule.

5.5 Every truth-function is a result of successive applications to
elementary propositions of the operation '(-----T)(E,....)'. This
operation negates all the propositions in the right-hand pair of
brackets, and I call it the negation of those propositions.

5.501 When a bracketed expression has propositions as its terms--and the
order of the terms inside the brackets is indifferent--then I indicate
it by a sign of the form '(E)'. '(E)' is a variable whose values
are terms of the bracketed expression and the bar over the variable
indicates that it is the representative of all its values in the
brackets. (E.g. if E has the three values P,Q, R, then (E) = (P, Q, R).
) What the values of the variable are is something that is stipulated.
The stipulation is a description of the propositions that have the
variable as their representative. How the description of the terms
of the bracketed expression is produced is not essential. We can
distinguish three kinds of description: 1. Direct enumeration, in which
case we can simply substitute for the variable the constants that are
its values;