Mathematical Fuzzy Logic In Narrow And Broader Sense — A Unified Concept

The main idea motivating the development of fuzzy logic (FL) can be formulated as follows: Fuzzy logic is a special many-valued logic addressing the vagueness phenomenon and developing tools for its modeling via truth degrees taken from an ordered scale. It is expected to preserve as many properties of classical logic as possible. During the past 15 years, the mathematical FL has been developed as fuzzy logic in narrow sense (FLn). Various theories based on it are often gathered under a common name fuzzy logic in broad sense. While the former denotes special kinds of many-valued logics, the latter has been coined by L. A. Zadeh to denote all kinds of applications that use fuzzy sets. Since this is too extensive, I have proposed in [9] (and elsewhere) the concept of fuzzy logic in broader sense (FLb) as an extension of FLn. The goal of FLb is to become a formal theory of human way of reasoning that would include a mathematical model of the meaning of some expressions of natural language (evaluating linguistic expressions), the theory of generalized quantifiers and their use in human reasoning. Claims made on mathematical fuzzy logic (both FLn as well as FLb) can be summarized as follows: (i) It must be a well established sound formal system to make its applications well justified. (ii) It should be an open system allowing extension by new connectives and by generalized quantifiers. Moreover, some specific phenomena of natural language semantics should also be expressible in it, such as non-commutativity of conjunction and disjunction. (iii) It has a specific agenda, special technique and concepts. Among them we can rank evaluating linguistic expressions, linguistic variable, fuzzy IF-THEN rules, fuzzy quantification, defuzzification, fuzzy equality, etc. (iv) It must accomplish special inference schemes including sophisticated inference schemes of human reasoning (e.g., compositional rule of inference, reasoning based natural language expressions, nonmonotonic reasoning, abduction, etc.). It follows from the above claims that fuzzy logic must be well established formal system capable at fulfilling specific agenda stemming from the goal to capture and model the vagueness phenomenon. The state of the art of mathematical fuzzy logic is now very promising, especially thanks to P. Hájek and his book [5], but also thanks many other researchers (M. Baaz, P. Cintula, A. DiNola, F. Esteva, L. Godo, S. Gottwald, E. P. Klement, R. Mesiar, F. Montagna, D. Mundici, J. Pavelka, I. Perfilieva, E. Walker, and others†)). Among many systems of FLn, the following classification emerged: FLn with traditional and evaluated syntax. The systems of FLn themselves differ primarily in the structure of truth values that, of course, determines their properties. There are a lot of fuzzy logics with traditional syntax, such as MTL, BL, IMTL, product, à Lukasiewicz, ΠMTL, nilpotent-minimum, etc., but only one FL with evaluated syntax, namely that based on à Lukasiewicz algebra of truth values (Evà L). The most distinguished properties important for the agenda of fuzzy logic have the systems based on IMTL-, BL-, à Land à LΠ-algebras. It †)It is hardly possible to name all important contributors. Omitting any of them from this list is unintentional. should be stressed that these systems are extended to higher order fuzzy logic — the fuzzy type theory. The latter plays a crucial role in the development of FLb since it seems to have the biggest potential for the agenda of fuzzy logic, namely, for the formal theory of human reasoning. A feature of any good science is its inner beauty coming out of the its harmony and balance. This feature is not always explicitly formulated but is generally accepted and must not be neglected. Fuzzy logic is indubitably a beautiful formal mathematical theory.