Formal theories of fuzzy logic

1. Introduction to the problem

Mathematical fuzzy logic is a special formal theory of many-valued logic generalizing classical mathematical logic that focuses on the development of tools for modeling vagueness phenomenon. We distinguish fuzzy logic in narrow (FLn) and in broader sense (FLb). Fuzzy logic has been initiated at the end of sixties and beginning of seventies by L. A. Zadeh and J. A. Goguen. Nowadays, it is a well developed theory thanks to the results of an international group of mathematicians (e.g., P. Hájek, D. Mundici, S. Gottwald, F. Esteva, L. Godo, A. diNola, F. Montagna, V. Novák, P. Cintula, L. Běhounek, and many others).

FLn provide fundamental formalism while FLb is its extension aiming at modeling human way of reasoning, important feature of which is the use of natural language. Hence, the necessary constituent of FLb is formalization of approximate reasoning – the reasoning realized by people on the basis of imprecise information formulated in natural language. Fuzzy logic puts forth special questions either not raised it classical logic, or having no sense in it. Typical problems of fuzzy logic is formalization of approximate equality, formalization of semantics of evaluative linguistic expressions, finding a conclusion on the basis of the assumption fulfilled only approximately, etc.

As usual for any mathematical logic, in fuzzy logic we clearly distinguish syntax from semantics. While semantics is always many-valued, based on a certain algebraic structure of truth values, syntax can be either traditional or evaluated. In comparison with classical logic, fuzzy logic has different and wider repertoire of connectives and logical axioms but similar inference rules. All known kinds of fuzzy logic with traditional syntax including first-order are complete that is, a formula is provable if and only if it is true in the degree one in all models.

A more radical departure from classical logic is fuzzy logic with evaluated syntax (J. Pavelka, V. Novák). Its fundamental concept is evaluated formula a/A where A is a formula and a is its syntactic evaluation that is, a lower bound for the truth of A in any model. This makes it possible to consider axioms that need not be fully convincing. The concepts such as evaluated proof, provability degree and some other ones are introduced. Important result is a generalization of the Gödel completeness theorem: the provability degree of an arbitrary formula in arbitrary fuzzy theory is equal to its truth degree. It has been proved that such generalization is possible only in case that the structure of truth values is isomorphic with the standard or finite Łukasiewicz algebra.

2. Focus of our research

IRAFM has focused on the following topics:

• Formal systems of fuzzy logic in narrow sense:
• fuzzy logic with evaluated syntax,
• higher order fuzzy logic (fuzzy type theory),
• Model theory for fuzzy logic including models of higher order fuzzy logic,
• Model theory of fuzzy logic in categories,
• Theory of generalized quantifiers, nonstandard extensions of fuzzy logic,
• Category of fuzzy sets and sets with generalized equality as a basis for models of fuzzy logic.

3. Description of the main results

1. We have developed in detail calculus of predicate first-order fuzzy logic with evaluated syntax. The results are summarized in the book [47]. Further results including model theory of this logic are presented in [3], [48], [28], [32], [34], [36], [37], [38], [49], [39], [45]. In model theory we have generalized Robinson-Craig theorem on union of fuzzy theories [29] and omitting types theorem [23], [24], [26]. Results concerning automation of provability in this logic are contained in [5], [6], [7], [8].
2. We have developed full calculus of fuzzy type theory that is, a higher-order fuzzy logic of Henkin type. The fundamental paper is [33] where the considered structure of truth values is IMTL-algebra. Further extension is contained in [35] (extension by description operator). Fuzzy type theory for Łukasiewicz, BL and ŁΠ algebras of truth values is presented in [1], [41], [42]. The completeness theorem for all the above mentioned kinds of fuzzy type theory with respect to general models has been proved.
3. We have described special categories, namely categories of fuzzy sets with values in MV-algebra, studied their morphisms and subobjects with respect to interpretation of fuzzy logic in such categories [14], [15], [16], [17], [19], [20]. The other special categories are categories of fuzzy sets with generalized equality.
4. We have extended fuzzy logic by the theory of generalized quantifiers as a direct generalization of classical theory of generalized quantifiers [9], [10] , [11].

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