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When langage is used for comparatively narrow professsional purposes
there is a tendency to limit the number of terms used and to give
them more precise and constant meaning. We say that the language
is being formalized. If this process is carried through
to its logical conclusion, the language will be completely
formalized, or formal. A language is formal if
its usage relies only on the `form' of linguistic objects, and not
their intuitive meanings.
To make this definition precise, we must specify a set of perceptions
(that is abstractions) and
actions which are registered and performed in the same way by
all members of the society whom the languages serves.
We shall refere to these perceptions and actions
as universally defined. A language is formal if all the processes
involved in its usage, namely the representation function R(w),
the modelling function M(r), and (for command languages) the set of
possible actions are expressed in terms of universally defined perceptions
We usually assume that universally defined perceptions and actions
can be relegated to the machine. The question is still open
whether this is a realistic assumption. We accept it with a qualification
that if there is a doubt about a specific abstraction or action,
it nust be excluded from the universally defined set. The a formal
language is a language usable by a properly constructed machine.
A machine of that kind becomes an objective model of reality,
independent from the human brain which created it. This makes possible
a series of consecutive metasystem transitions, where each next level
deals with a well-defined, objective reality of the previous levels.
Thus the language becomes an ultrametasystem, exhibiting the
stair-case effect (see [Tur77]) and an explosive growth in volume
and power. Just as mastering the general principle of using tools
to make better tools gives rise to consecutive metasystem transitions
and the creation of industrial system, so mastering the principle of
describing (modelling) by means of a formalized language gives rise to
the creation of the hierarchical system of formal languages on which
the modern science is based.