Ulf G. Indahl MATFORSK Osloveien 1 N-1430 As NORWAY Fax: +47-64 97 03 33 Tel.: +47-64 97 01 11 E-mail: ulf.indahl@matforsk.nlh.no Extending the laws of probability to Fuzzy sets Ulf G. Indahl MATFORSK, Osloveien 1, N-1430 As, NORWAY June 12, 1996 SUMMARY Although taken seriously by several other areas of mathematical modelling, Fuzzy sets and Fuzzy Logic seem to have been missing the probabilistic basis necessary for a breakthrough in statistics and applied probability. By improving the ideas of Zadeh we demonstrate how Fuzzy sets can be given a straightforward treatment within the framework of traditional probability theory. The Fuzzy sets defined over a universe respects the structure of a De Morgan Algebra. By establishing an isomorphism to a De Morgan Algebra of ordinary sub- sets of an extended universe, we can define the probability of a Fuzzy set once a probability distribution on the extended space is given. Problems involving stochastic optimization of utility- and loss-functions can in the framework of Fuzzy sets be given a direct probabilistic interpretation. Keywords: PROBABILITY; FUZZY SETS; FUZZY LOGIC; POSSIBILITY; BOOLEAN ALGEBRA; DE MORGAN ALGEBRA 1 INTRODUCTION During the last couple of decades modelling based on Fuzzy Logic and Fuzzy sets has been demonstrated powerful for solving practical problems in disciplines like process con- trol, expert systems, computer vision, image analysis and processing of natural language, see [32] and [35]. Fuzzy methods are considered to be serious alternatives to traditional mathematical and statistical modelling in these and several other areas. 1