Suppose an approximate concept represented by a predicate *p*(*x*) has a
sufficient condition *suff*(*x*) and a necessary condition *nec*(*x*).
Thus we have

In general the sufficient and the necessary conditions will not coincide, i.e. we will not have

However, they made coincide with some restriction on *x*,
i.e. we may have

Another way an approximate concept may become definite is by a mapping from the space in which it is first formalized into more restricted space. We'll combine specialization with mapping in

where the function *f* maps a subset of the original domain into a
specialized domain in which the concept *p*(*x*) becomes definite.

Wed Feb 2 15:59:04 PST 2000