This contains the decisive point. We have said that some
things are arbitrary in the symbols that we use and that some things are
not. In logic it is only the latter that express: but that means that logic
is not a field in which we express what we wish with the help of signs, but
rather one in which the nature of the absolutely necessary signs speaks for
itself. If we know the logical syntax of any sign-language, then we have
already been given all the propositions of logic.
6.125 It is possible--indeed possible even according to the old conception
of logic--to give in advance a description of all 'true' logical
propositions.
6.1251 Hence there can never be surprises in logic.
6.126 One can calculate whether a proposition belongs to logic, by
calculating the logical properties of the symbol. And this is what we do
when we 'prove' a logical proposition. For, without bothering about sense
or meaning, we construct the logical proposition out of others using only
rules that deal with signs . The proof of logical propositions consists in
the following process: we produce them out of other logical propositions by
successively applying certain operations that always generate further
tautologies out of the initial ones. (And in fact only tautologies follow
from a tautology.) Of course this way of showing that the propositions of
logic are tautologies is not at all essential to logic, if only because the
propositions from which the proof starts must show without any proof that
they are tautologies.
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