The Parable of Zoltan

Abstract

I learned about fuzzy sets in 1969 when I was a graduate student in Applied Mathematics at Cornell University. Subsequently, I based my PhD thesis on Fuzzy Clustering. The notion of fuzzy sets was not only novel then, but controversial. And its basic premise - that there is a type of imprecision which cannot be adequately accounted for with probability - continues to bother many engineers and scientists today. This note is about the turbulence that is still created by this division of beliefs about mathematical models of uncertainty.

The growth of the theory and applications of fuzzy sets in the 1970s-1980s created a demand at conferences for tutorials about fuzzy models, and I sometimes gave such lectures on the use of fuzzy sets in pattern recognition. A common question then that persists to the present day was “can you give us an example that shows a real difference between fuzzy and probabilistic uncertainty?”. My response to that question led me to propose an example called the “potable drinks” example. I often used this example in the late 1980s, and finally published it, first in Bezdek and Pal [1], and then again, in my introduction to fuzzy models that served as a preamble to the inaugural issue of the IEEE Transactions on Fuzzy Systems [2].