Concluding Remarks

Abstract

Let me start with a recollection of my first acquaintance with Prof. Lotfi Zadeh. It was in 2001 in Villa Real, Portugal, where Prof. Pedro Melo-Pinto organized a school on fuzzy sets. Prof. Zadeh was invited for a 3-hour lecture, which he concluded with presentation of slides with articles by Samuel Kleene, Kurt Gödel and other luminaries of mathematical logic, who have written against the fuzzy sets. The fact that the sublime mathematician and logician Gödel had sometimes made slips in his judgments can be confirmed by the cosmologists, yet I was astonished by his opinion. Of course, nowadays, when we are aware of the enormous number of publications in the field of fuzzy sets, as well as of the various impressive applications of these, it is easy to say that Gödel had mistaken. However, I have been long tormented by the question why these mathematicians had opposed the fuzzy sets while they did not have anything against the three- and multi-valued logics of Jan Lukasiewicz. Thus I reached the conclusion that the reason for the then negative attitude towards fuzzy sets was hidden in the presence of the [0, 1] interval as the set of the fuzzy sets’ membership function (see, e.g, [301, 592, 593]).