The paper considers the fundamental notions of many- valued logic together with some of the main trends of the recent development of infinite valued systems, often called mathematical fuzzy logics. Besides this logical approach also a more algebraic approach is discussed. And the paper ends with some hints toward applications which are based upon actual theoretical considerations about infinite valued logics. Key words: mathematical fuzzy logic, algebraic semantics, continuous t-norms, left-continuous t-norms, Pavelka-style fuzzy logic, fuzzy set theory, non-monotonic fuzzy reasoning 1 Basic ideas 1.1 From classical to many-valued logic Logical systems in general are based on some formalized language which includes a notion of well formed formula, and then are determined either semantically or syntactically. That a logical system is semantically determined means that one has a notion of interpretation or model 1 in the sense that w.r.t. each such interpretation every well formed formula has some (truth) value or represents a function into

Among various approaches to fuzzy logics, we have chosen two of them, which are built up in a similar way. Although starting from different basic logical connectives, they both use interpretations based on Frank t-norms. Different interpretations of the implication lead to different axiomatizations, but most logics studied here are complete. We compare the properties, advantages and disadvantages of the two approaches. Key words: Fuzzy logic, many-valued logic, Frank t-norm 1 Introduction A many-valued propositional logic with a continuum of truth values modelled by the unit interval [0; 1] is quite often called a fuzzy logic. In such a logic, the conjunction is usually interpreted by a triangular norm. In this context, a (propositional) fuzzy logic is considered as an ordered pair P = (L; Q) of a language (syntax ) L and a structure (semantics) Q described as follows: (i) The language of P is a pair L = (A; C), where A is an at most countable set of atomic symbols and C is ...

The paper considers some of the main trends of the recent development of mathematical fuzzy logic as an important tool in the toolbox of approximate reasoning techniques. Particularly the focus is on fuzzy logics as systems of formal logic constituted by a formalized language, by a semantics, and by a calculus for the derivation of formulas. Besides this logical approach also a more algebraic approach is discussed. And the paper ends with some hints toward applications which are based upon these theoretical considerations. Key words: mathematical fuzzy logic, algebraic semantics, continuous t-norms, left-continuous t-norms, Pavelka-style fuzzy logic, fuzzy set theory, non-monotonic fuzzy reasoning 1

Different languages that are offered to model vague preferences are reviewed and an interval-valued language is proposed to resolve a particular difficulty encountered with other languages. It is shown that interval-valued languages are well defined for DeMorgan triples constructed by continuous triangular norms, conorms and a strong negation function. A new transitivity condition for vague preferences is suggestedand its relationships to known transitivity conditions are established. A complete characterization of intervalvalued preference structures is also provided.

ABSTRACT: There is a programme in the formal foundations of fuzzy mathematics proposed by the authors, the goal of which is to encompass a large part of existing fuzzy mathematics within a general logical formalism. This paper presents the methodology behind this programme and reviews the technical aspects of a particular apparatus for this enterprise.

s with the corresponding consequent, i.e., 8i 2 f1; : : : ; ng : Int \Theta (X i ) = Y i ; [Int2] for each normal observation X 2 F(X), the conclusion, Int \Theta (X ), is not contained in all consequents, i.e., there is an index i 2 f1; : : : ; ng with Int \Theta (X ) 6 Y i . [Int3] the conclusion Int \Theta (X ) belongs to the convex hull of (Y i ) i2F , where F = fi j Supp X i " Supp X 6= ;g. Axiom [Int1] states that the "typical inputs" used in the rules produce the corresponding outputs

This contribution presents a comprehensive view on problems of approximate reasoning with imprecise knowledge in the form of a collection of fuzzy IF-THEN rules formalized by approximating formulas of a special type. Two alternatives that follow from the dual character of approximating formulas are developed in parallel. The link to the theory of fuzzy control systems is also explained.

The core point of fuzzy control approaches are finite lists of linguistic control rules. For computerbased automatic control these lists have to be transformed into control algorithms which can be realized on a computer. The main general idea of this fuzzy control approach is that such an algorithm should yield a fuzzy subset of the output space of the control problem if confronted with a fuzzy subset of the input space. This paper surveys mathematical problems which are connected with, and arose out of these basic ideas. The main formal tools used in these mathematical considerations are fuzzy sets and fuzzy relations together with some generalized, viz. many-valued logic which underlies these considerations. And the essential way of understanding the mathematical context of fuzzy control is to look at it as an interpolation problem: one has to determine a fuzzy control function out of a finite list of interpolation nodes.

The paper discusses some open problems in the field of mathematical fuzzy logic which may have a decisive influence for the future development of fuzzy logic within the next decade. 1

Among various approaches to fuzzy logics, we have chosen two of them, which are built up in a similar way. Although starting from different basic logical connectives, they both use interpretations based on Frank t-norms. Different interpretations of the implication lead to different axiomatizations, but most logics studied here are complete. We compare the properties, advantages and disadvantages of the two approaches. We deal also with logics containing infinitary conjunctions, and we show that they are semantically "stronger" than all the other logics studied in this paper. Key words: Fuzzy logic, many-valued logic, Frank t-norm 1 Frank t-norms Triangular norms were introduced in the framework of probabilistic metric spaces [33, 32, 34], based on ideas first presented in [22], and they are applied in several fields, e.g., in fuzzy sets [35], fuzzy logics [2, 14, 28] and their applications, but also in the theory of generalized measures [1, 19] and nonlinear differential an...