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This paper managed to induce probability theory (sigma system) andpossibility theory (max system) respectively from randomness and fuzziness,through which the premature theory of possibility is expected to be wellfounded. Such an objective is achieved by addressing three open key issues: a)the lack of clear mathematical definitions of randomness and fuzziness; b) thelack of intuitive mathematical definition of possibility; c) the lack ofabstraction procedure of the axiomatic definitions of probability/possibilityfrom their intuitive definitions. Especially, the last issue involves thequestion 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-strictdisjunctive operator that is applicable across the fuzzy event space, and is anexact operator for fuzzy feature extraction that assures the max inference isan exact mechanism. It is fair to claim that the long-standing problem of lackof consensus to the foundation of possibility theory is well resolved, whichwould facilitate wider adoption of possibility theory in practice and promotecross prosperity of the two uncertainty theories of probability andpossibility.

The Sigma-Max System Induced from Randomness and Fuzziness