Fuzzy Logic with Engineering Applications, 4th Edition

13Monotone Measures: Belief, Plausibility, Probability, and Possibility

Only those who attempt the absurd can achieve the impossible.

Albert Einstein, theoretical physicist philosopher and Nobel Prize Winner

A bag contains 2 counters, as to which nothing is known except that each is either black or white. Ascertain their colours without taking them out of the bag.

Lewis Carroll, author and mathematician, Pillow Problems, 1893

I still believe in the possibility of a model of reality, that is to say, of a theory, which represents things themselves and not merely the probability of their occurrence.

Albert Einstein, commenting on quantum physics after his Herbert Spencer Lecture, 1933

Most of this text has dealt with the quantification of various forms of nonnumeric uncertainty. Two prevalent forms of uncertainty are those arising from vagueness and from imprecision. How do vagueness and imprecision differ as forms of uncertainty? Often, vagueness and imprecision are used synonymously, but they can differ in the following sense. Vagueness can be used to describe certain kinds of uncertainty associated with linguistic information or intuitive information. Examples of vague information are that the image quality is “good,” or that the transparency of an optical element is “acceptable.” Imprecision can be associated with quantitative or countable data as well as uncountable data. As an example of the latter, one might say the length of a bridge span is “long.” An example ...