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

In order to allow for the analysis of data sets including numerical attributes, several generalizations of association rule mining based on fuzzy sets have been proposed in the literature. While the formal specification of fuzzy associations is more or less straightforward, the assessment of such rules by means of appropriate quality measures is less obvious. Particularly, it assumes an understanding of the semantic meaning of a fuzzy rule. This aspect has been ignored by most existing proposals, which must therefore be considered as ad-hoc to some extent. In this paper, we develop a systematic approach to the assessment of fuzzy association rules. To this end, we proceed from the idea of partitioning the data stored in a database into examples of a given rule, counterexamples, and irrelevant data. Evaluation measures are then derived from the cardinalities of the corresponding subsets. The problem of finding a proper partition has a rather obvious solution for standard association rules but becomes less trivial in the fuzzy case. Our results not only provide a sound justification for commonly used measures but also suggest a means for constructing meaningful alternatives.

As a natural generalization of a measure space, Butnariu and Klement introduced T -tribes of fuzzy sets with T -measures. They gave a complete characterization of T -measures for a Frank triangular norm T . Here we characterize T-measures with respect to non-Frank strict triangular norms. We show the specific roles of Frank triangular norms and a newly introduced family, nearly Frank triangular norms. 1 The notion of T-measure We start with the basic definitions from [4]. Let X be a set and B a oe-algebra of subsets of X . The B-generated tribe is the collection T of all functions A: X ! [0; 1] (fuzzy subsets of X) which are B-measurable. In order to define measures on T , we fix a t-norm T (fuzzy conjunction), i.e., a binary operation T : [0; 1] 2 ! [0; 1] which is commutative, associative, nondecreasing, and satisfies the boundary condition T (a; 1) = a for all a 2 [0; 1] (see [15]). For the other necessary fuzzy logical operations, we take the standard fuzzy negation 0 : [0; 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. 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 ...

In this paper, we derive new neuro-fuzzy structures called flexible neuro-fuzzy inference systems or FLEXNFIS. Based on the input–output data, we learn not only the parameters of the membership functions but also the type of the systems (Mamdani or logical). Moreover, we introduce: 1) softness to fuzzy implication operators, to aggregation of rules and to connectives of antecedents; 2) certainty weights to aggregation of rules and to connectives of antecedents; and 3) parameterized families of T-norms and S-norms to fuzzy implication operators, to aggregation of rules and to connectives of antecedents. Our approach introduces more flexibility to the structure and design of neuro-fuzzy systems. Through computer simulations, we show that Mamdani-type systems are more suitable to approximation problems, whereas logical-type systems may be preferred for classification problems.

This article discusses a range of regression techniques specifically tailored to building aggregation operators from empirical data. These techniques identify optimal parameters of aggregation operators from various classes (triangular norms, uninorms, copulas, ordered weighted aggregation (OWA), generalized means, and compensatory and general aggregation operators), while allowing one to preserve specific properties such as commutativity or associativity. 2003 Wiley Periodicals, Inc

In this paper, a new method for merging multiple inconsistent knowledge bases in the framework of possibilistic logic is presented. We divide the fusion process into two steps: one is called the splitting step and the other is called the combination step. Given several inconsistent possibilistic knowledge bases (i.e. the union of these possibilistic bases is inconsistent) , we split each of them into two subbases according to the upper free degree of their union, such that one subbase contains formulas whose necessity degrees are less than the upper free degree and the other contains formulas whose necessity degrees are greater than the upper free degree.

A neural approach to propositional multi-adjoint logic programming was recently introduced. In this paper we extend the neural approach to multi-adjoint deduction and, furthermore, modify it to cope with abductive multi-adjoint reasoning, where adaptations of the uncertainty factor in a knowledge base are carried out automatically so that anumber of given observations can be adequately explained.

Aggregation operators generated by means of additive generators are discussed. Depending on the properties of additive generators, some classes of generated aggregation operators are derived. Several examples are included.

It is well accepted that inconsistency may exist in a database system or an intelligent information system (Benferhat et al. 1993a; 1993b; 1997b; 1998; Benferhat & Kaci 2003; Elvang-Gransson & Hunter 1995; Gabbay & Hunter 1991; Lin 1994; Priest et al. 1989; Priest 2001). Inconsistency can either appear in the given knowledge bases or as a result of combination or revision. In this paper, we will propose two different methods to combine individually inconsistent possibilistic knowledge bases. The first method, called an argument-based method, is a generalization of the merging method introduced in (Benferhat & Kaci 2003). When the knowledge bases to be merged are self-consistent, this method coincides with the original one. The second method, called a multiple-operator based method, combines the consistent and the conflict information using different operators. This method