Fuzzy implication can be arbitrarily complicated: A theorem

| In fuzzy logic, there are several methods of representing implication in terms of &, _, and :; in particular, explicit representations deene a class of S implications, implicit representations deene a class of R implications. Some reasonable implication operations have been proposed, such as Yager's a b , that are diicult to represent as S or R implications. For such operations, a new class of representations has been recently proposed called A implications, for which the relationship between implications and the basic operations &, _, and : is even more complicated. A natural question is: is this complexity really necessary? In other words, is it true that A operations cannot be described as S or R ones, or they can, but we simply have not found these representations? In this paper, we show that yes, the complexity is necessary, because there are operations that cannot be represented in a simpler form. For every problem, be it medical, engineering, etc., we would like to use the best experts. Best experts are few, and it is impossible to use them for all practical problems. So, it is desirable to design computer-based systems that will use the knowledge of the best experts to make decisions comparable in quality with the decisions of these experts. Designing such expert systems is, therefore, an extremely important area of computer science. To develop an expert system, we must describe the expert knowledge in a way that a computer can use it. Expert knowledge, in addition to well-formulated precise statements about which experts are 100% sure, usually contains statements that are not 100% veriied, that are somehow more intuitive than precise. In other words, experts may have different degrees of belief in the statements that comprise their knowledge. Hence, when we describe expert statements, then instead of simply classifying them as \true" or \false" (i.e., instead of using the traditional two-valued logic), we must describe the expert's degree of belief in these statements; this degree of belief may range from \absolutely sure" (that correspond to \true") to \absolutely wrong" (which corresponds to \false"), but in addition to these two extreme cases, there may be many intermediate degrees of belief that an expert expresses by words like \possi-ble", \most probable", etc. In other words, instead of a two-valued logic, we have a multi-valued logic, in which instead of a truth value that can take only two possible values, …