Fuzzy Logic with Engineering Applications, 4th Edition

6Historical Methods of Developing Membership Functions

So far as the laws of mathematics refer to reality, they are not certain. And so far as they are certain, they do not refer to reality.

Albert Einstein, theoretical physicist and Nobel Laureate “Geometrie und Erfahrung,”Lecture to Prussian Academy, 1922

Uncertainty is the refuge of hope.

Henri Frederic Amiel, nineteenth‐century Swiss philosopher.

The quest for certainty blocks the search for meaning. Uncertainty is the very condition to impel man to unfold his powers.

Erich Fromm, twentieth‐century psychologist

Not to be absolutely certain is, I think, one of the essential things in rationality.

Bertrand Russell, twentieth‐century British philosopher

The statements from Albert Einstein to Bertrand Russell attest to the fact that few things in real life are certain or can be conveniently reduced to the axioms of mathematical theories and models. A metaphorical expression that represents this idea is known as the Law of Probable Dispersal; to wit, “Whatever it is that hits the fan will not be evenly distributed.” As this enlightened law implies, most things in nature cannot be characterized by simple or convenient shapes or distributions. Membership functions characterize the fuzziness in a fuzzy set—whether the elements in the set are discrete or continuous—in a graphical form for eventual use in the mathematical formalisms of fuzzy set theory. But the shapes used to describe the fuzziness have few restrictions indeed; ...