  This paper managed to induce probability theory (sigma system) and
possibility theory (max system) respectively from randomness and fuzziness,
through which the premature theory of possibility is expected to be well
founded. Such an objective is achieved by addressing three open key issues: a)
the lack of clear mathematical definitions of randomness and fuzziness; b) the
lack of intuitive mathematical definition of possibility; c) the lack of
abstraction procedure of the axiomatic definitions of probability/possibility
from their intuitive definitions. Especially, the last issue involves the
question why the key axiom of "maxitivity" is adopted for possibility measure.
By taking advantage of properties of the well-defined randomness and fuzziness,
we derived the important conclusion that "max" is the only but un-strict
disjunctive operator that is applicable across the fuzzy event space, and is an
exact operator for fuzzy feature extraction that assures the max inference is
an exact mechanism. It is fair to claim that the long-standing problem of lack
of consensus to the foundation of possibility theory is well resolved, which
would facilitate wider adoption of possibility theory in practice and promote
cross prosperity of the two uncertainty theories of probability and
possibility.
