  Based on the in-depth analysis of the nature and features of vague
phenomenon, this paper focuses on establishing the axiomatical foundation of
the membership degree theory for vague phenomenon, presents an axiomatic system
of governing membership degrees and their interconnections. Some important
basic notions, such as vague variable, vague partition etc. are defined, their
useful properties are characterized. Moreover, the notion of fuzzy set is also
redefined by the notion of vague partition on the basis of the axiomatic
system. Hence, this work can serve as a mathematical model of dealing with the
phenomena of vagueness by axiomatical approach from the many-valued point of
view, as well as the axiomatical foundation of fuzzy sets and its applications.
The thesis defended in this paper is that the difference among vague attributes
is the key point to recognize and model vague phenomena, membership degrees
should be considered in a vague membership space. In other words, vagueness
should be treated from a global and overall (instead of local) of view.
