The notion of validation set of a formula in a fuzzy logic was introduced by Butnariu, Klement and Zafrany. It is the set of all evaluations of the formula for all possible evaluations of its atomic symbols. We generalize this notion to sets of formulas. This enables us to formulate and prove generalized theorems on satisfiability and compactness of various fuzzy logics. We also propose and study new types of satisfiability and consistency degree of a set of formulas. Key words: Fuzzy logic, many-valued logic, triangular norm, compactness of a logic. 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. There are different ways of interpretation of the implication; this, together with the choice of the triangular norm, leads to a large collection of fuzzy logics with different semantics. Here we wor...

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

Abstract Algebraic Logic is a general theory of the algebraization of deductive systems arising as an abstraction of the well-known Lindenbaum-Tarski process. The notions of logical matrix and of Leibniz congruence are among its main building blocks. Its most successful part has been developed mainly by BLOK, PIGOZZI and CZELAKOWSKI, and obtains a deep theory and very nice and powerful results for the so-called protoalgebraic logics. I will show how the idea (already explored by WÓJCKICI and NOWAK) ofdeÞning logics using a scheme of “preservation of degrees of truth ” (as opposed to the more usual one of “preservation of truth”) characterizes a wide class of logics which are not necessarily protoalgebraic and provide another fairly general framework where recent methods in Abstract Algebraic Logic (developed mainly by JANSANA and myself) can give some interesting results. After the general theory is explained, I apply it to an inÞnite family of logics deÞned in this way from subalgebras of the real unit interval taken as an MV-algebra. The general theory determines the algebraic counterpart of each of these logics without having to perform any computations for each particular case, and proves some interesting properties common to all of them. Moreover, in the Þnite case the logics so obtained are protoalgebraic, which implies they have a “strong version ” deÞned from their Leibniz Þlters; again, the general theory helps in showing that it is the logic deÞned from the same subalgebra by the truth-preserving scheme, that is, the corresponding Þnite-valued logic in the most usual sense. However, for inÞnite subalgebras the obtained logic turns out to be the same for all such subalgebras and is not protoalgebraic, thus the ordinary methods do not apply. After introducing some (new) more general abstract notions for non-protoalgebraic logics I can Þnally show that this logic too has a strong version, and that it coincides with the ordinary inÞnite-valued logic of Łukasiewicz. 1 1

This contribution is concerned with a detailed investigation of linearity axioms for fuzzy orderings. Different existing concepts are evaluated with respect to three fundamental correspondences from the classical case—linearizability of partial orderings, intersection representation, and one-to-one correspondence between linearity and maximality. As a main result, we obtain that it is virtually impossible to simultaneously preserve all these three properties in the fuzzy case. If we do not require a one-to-one correspondence between linearity and maximality, however, we obtain that an implication-based definition appears to constitute a sound compromise, in particular, if Łukasiewicz-type logics are considered. Key words: completeness, fuzzy ordering, fuzzy preference modeling, fuzzy relation,

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.

Abstract. The monadic fragments of first-order Gödel logics are investigated. It is shown that all finite-valued monadic Gödel logics are decidable; whereas, with the possible exception of one (G↑), all infinitevalued monadic Gödel logics are undecidable. For the missing case G↑ the decidability of an important sub-case, that is well motivated also from an application oriented point of view, is proven. A tight bound for the cardinality of finite models that have to be checked to guarantee validity is extracted from the proof. Moreover, monadic G↑, like all other infinite-valued logics, is shown to be undecidable if the projection operator △ is added, while all finite-valued monadic Gödel logics remain decidable with △. 1

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Fuzzy logic is now one of the leading and most successful methodologies for the treatment of the vagueness phenomenon. It is a well-established sound formal system with numerous applications. In recent years, several significant books have been published where fuzzy logic is investigated deeply and so, its

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