Main trends of research in the social and human ... - unesdoc - Unesco

562 Raymond Boudon
On the other hand, although the structure of type (I) sentences may seem
more elementary than that of type (2) sentences, it can be demonstrated that it
is impossible to generate type (I) sentences by means of context-free grammars.31
In order to generate sentences of this type, a context-sensitive grammar like the
following must be used:
rule I S+uS
rule 2 S+ bS
rule 3 # kS # -+ # kk #
where k represents any arbitrary string of terminal vocabulary. With this
grammar it is now possible, by applying rule I n times in succession, to generate
the rewritten expression:
S -+ anS,
then by applying rule 2 m times, obtain:
S -+ u"b"'S
Finally, by applying rule 3 the following sentence is obtained:
u"bmu"bm
which is in fact a type (I) sentence. As can be seen, rule 3 is incompatible with
the definition of the context-free grammars considered above, since the expres-
sion which appears to the left of the arrow is not 'a single symbol of the non-
terminal vocabulary'.
These examples should be sufficient to give an idea of the type of results
achieved in the 'formal theory of grammars'. Although such research derives
from considerations relating to the analysis of natural languages, it is obvious
that the theorems which it yields are of a purely formal nature The sentences
considered by the formal theory of grammars are constructs of the mind:
certainly, they are inspired by natural languages; but they are for the most part
independent of such languages.
The theory of grammars, therefore, is one of a set of purely mathematical
theories, in which the r61e of human phenomena is essentially to trigger a line
of thought. Other, more widely known examples of such theories are the mathe-
matical theories connected with the problem of the transmission of informa-
tion: information theory or code theory.
The three types of 'theory' which we have examined more or less exhaust the
types of general theoretical model discoverable in the social sciences. More
frequently the theoretical models met with in the literature of the subject are
specific models expressing specific situations and forming units in themselves.
It is these specific models which we shall now consider.
A typical instance of this state of affairs is provided by the mathematical
researches associated with group dynamics. These research items do not consti-
tute a general theory comparable to game theory, or even an assemblage of
models presenting logical affinities as with theories of economic cycles. There
have been many attempts at formalization in this domain, but the models which
can be instanced are each concerned with some separate aspect of the func-