The Sigma-Max System Induced from Randomness and Fuzziness

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.