Fuzzy, Possibility, Probability, and Generalized Uncertainty Theory in Mathematical Analysis

This presentation outlines from a quantitative point of view, the relationships between probability theory, possibility theory, and generalized uncertainty theory, and the role that fuzzy set theory plays in the context of these theories. Fuzzy sets, possibility, and probability entities are de ned in terms of a function. In each case, these three functions map the real numbers to the interval [0,1]. However, each of these functions are de ned with di¤erent properties. There are generalizations associated with these three theories that lead to intervals (sets of connected real numbers bounded by two points) and interval functions (sets of functions that are bounded by known upper and lower functions). An interval or interval function encodes the fact that it is unknown which of the points or functions is the point or function in questions, that is, the numerical value or realvalued function is unknown, it is uncertain. For generalizations given by pairs of numbers or functions, a case is made for a particular type of generalized uncertainty theory, interval-valued probability measures, as a way to unify the generalizations of probability, possibility theory, as well as other generalized probability theories via fuzzy intervals and fuzzy interval functions. This presentation brings a new understanding of quantitative fuzzy set theory, possibility theory, probability theory, and generalized uncertainty and gleans from existing research with the intent to organize and further clarify existing approaches.