On many-valued logics,fuzzy sets,fuzzy logics and their applications

This paper gives a survey of some aspects of many-valued logics and the theory of fuzzy sets and fuzzy reasoning, as advocated in particular by Zadeh. It starts with a short discussion of the development of many-valued logics and its philosophical background. In particular, the systems of Lukasiewicz and their algebraic models are presented. In connection with the famous Arrow paradoxon, Boolean valued and fuzzy social orderings are discussed. After some remarks on inference, fuzzy sets are introduced and it is shown that their definition is sound if some acceptable rationality requirements are demanded. Deformable prototypes are suggested in order to obtain the numerical values of the membership function for some applications. Finally, a recent paper of Bellman and Zadeh on a fuzzy logic, where the truth values themselves are fuzzy, is reviewed.     

Commutative basic algebras and non-associative fuzzy logics

Several investigations in probability theory and the theory of expert systems show that it is important to search for some reasonable generalizations of fuzzy logics (e.g. ?ukasiewicz, Gödel or product logic) having a non-associative conjunction. In the present paper, we offer a non-associative fuzzy logic L _CBA having as an equivalent algebraic semantics lattices with section antitone involutions satisfying the contraposition law, so-called commutative basic algebras. The class (variety) CBA of commutative basic algebras was intensively studied in several recent papers and includes the class of MV-algebras. We show that the logic L _CBA is very close to the ?ukasiewicz one, both having the same finite models, and can be understood as its non-associative generalization.     

Residuated fuzzy logics with an involutive negation

Residuated fuzzy logic calculi are related to continuous t-norms, which are used as truth functions for conjunction, and their residua as truth functions for implication. In these logics, a negation is also definable from the implication and the truth constant , namely is $\varphi \to \overline{0}$. However, this negation behaves quite differently depending on the t-norm. For a nilpotent t-norm (a t-norm which is isomorphic to Łukasiewicz t-norm), it turns out that is an involutive negation. However, for t-norms without non-trivial zero divisors, is G?del negation. In this paper we investigate the residuated fuzzy logics arising from continuous t-norms without non-trivial zero divisors and extended with an involutive negation. Received: 14 April 1998     

On elementary equivalence in fuzzy predicate logics

Our work is a contribution to the model theory of fuzzy predicate logics. In this paper we characterize elementary equivalence between models of fuzzy predicate logic using elementary mappings. Refining the method of diagrams we give a solution to an open problem of Hájek and Cintula (J Symb Log 71(3):863–880, 2006, Conjectures 1 and 2). We investigate also the properties of elementary extensions in witnessed and quasi-witnessed theories, generalizing some results of Section 7 of Hájek and Cintula (J Symb Log 71(3):863–880, 2006) and of Section 4 of Cerami and Esteva (Arch Math Log 50(5/6):625–641, 2011) to non-exhaustive models.     

Fuzzy events and fuzzy logics in classical information systems

Classical information systems are introduced in the framework of measure and integration theory. The measurable characteristic functions are identified with the exact events while the fuzzy events are the real measurable functions whose range is contained in the unit interval. Two orthogonality relations are introduced on fuzzy events, the first linked to the fuzzy logic and the second to the fuzzy structure of partial a Baer¹-ring. The fuzzy logic is then compared with the “empirical” fuzzy logic induced by the classical information system. In this context, quantum logics could be considered as those empirical fuzzy logics in which it is not possible to have preparation procedures which provide physical systems whose “microstate” is always exactly defined.     

On n ‐contractive fuzzy logics

It is well known that MTL satisfies the finite embeddability property. Thus MTL is complete w. r. t. the class of all finite MTL‐chains. In order to reach a deeper understanding of the structure of this class, we consider the extensions of MTL by adding the generalized contraction since each finite MTL‐chain satisfies a form of this generalized contraction. Simultaneously, we also consider extensions of MTL by the generalized excluded middle laws introduced in [9] and the axiom of weak cancellation defined in [31]. The algebraic counterpart of these logics is studied characterizing the subdirectly irreducible, the semisimple, and the simple algebras. Finally, some important algebraic and logical properties of the considered logics are discussed: local finiteness, finite embeddability property, finite model property, decidability, and standard completeness. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Normal forms for fuzzy logics: a proof-theoretic approach

A method is described for obtaining conjunctive normal forms for logics using Gentzen-style rules possessing a special kind of strong invertibility. This method is then applied to a number of prominent fuzzy logics using hypersequent rules adapted from calculi defined in the literature. In particular, a normal form with simple McNaughton functions as literals is generated for ?ukasiewicz logic, and normal forms with simple implicational formulas as literals are obtained for Gödel logic, Product logic, and Cancellative hoop logic.     

On interplay of quantifiers in Gödel-Dummett fuzzy logics

Axiomatization of Gödel-Dummett predicate logics S2G, S3G, and PG, where PG is the weakest logic in which all prenex operations are sound, and the relationships of these logics to logics known from the literature are discussed. Examples of non-prenexable formulas are given for those logics where some prenex operation is not available. Inter-expressibility of quantifiers is explored for each of the considered logics.     

The fuzzy core in games with fuzzy coalitions

In this paper, the fuzzy core of games with fuzzy coalition is proposed, which can be regarded as the generalization of crisp core. The fuzzy core is based on the assumption that the total worth of a fuzzy coalition will be allocated to the players whose participation rate is larger than zero. The nonempty condition of the fuzzy core is given based on the fuzzy convexity. Three kinds of special fuzzy cores in games with fuzzy coalition are studied, and the explicit fuzzy core represented by the crisp core is also given. Because the fuzzy Shapley value had been proposed as a kind of solution for the fuzzy games, the relationship between fuzzy core and the fuzzy Shapley function is also shown. Surprisingly, the relationship between fuzzy core and the fuzzy Shapley value does coincide, as in the classical case.     

On the algebraic structure of linear, relevance, and fuzzy logics

 Substructural logics are obtained from the sequent calculi for classical or intuitionistic logic by suitably restricting or deleting some or all of the structural rules (Restall, 2000; Ono, 1998). Recently, this field of research has come to encompass a number of logics - e.g. many fuzzy or paraconsistent logics - which had been originally introduced out of different, possibly semantical, motivations. A finer proof-theoretical analysis of such logics, in fact, revealed that it was possible to subsume them under the previous definition (see e.g. Aguzzoli and Ciabattoni, 2000). Although proof systems for substructural logics are currently being investigated with remarkable success, their algebraic models do not seem equally satisfactory. In fact: (i) such structures are often very weak, i.e. they do not possess many interesting algebraic properties; (ii) as a consequence, their theories of ideals, congruences, and representation are as a rule scarcely developed, or even lacking. In this paper, we address these difficulties. Received: 18 February 2000 / Published online: 12 December 2001     

Arithmetical complexity of fuzzy predicate logics — A survey II

Results on arithmetical complexity of important sets of formulas of several fuzzy predicate logics (tautologies, satisfiable formulas, …) are surveyed and some new results are proven.     

First-order t-norm based fuzzy logics with truth-constants: Distinguished semantics and completeness properties

This paper aims at being a systematic investigation of different completeness properties of first-order predicate logics with truth-constants based on a large class of left-continuous t-norms (mainly continuous and weak nilpotent minimum t-norms). We consider standard semantics over the real unit interval but also we explore alternative semantics based on the rational unit interval and on finite chains. We prove that expansions with truth-constants are conservative and we study their real, rational and finite chain completeness properties. Particularly interesting is the case of considering canonical real and rational semantics provided by the algebras where the truth-constants are interpreted as the numbers they actually name. Finally, we study completeness properties restricted to evaluated formulae of the kind , where φ has no additional truth-constants.     

The fuzzy core and Shapley function for dynamic fuzzy games on matroids

In this paper, the generalized forms of the fuzzy core and the Shapley function for dynamic fuzzy games on matroids are given. An equivalent form of the fuzzy core is researched. In order to better understand the fuzzy core and the Shapley function for dynamic fuzzy games on matroids, we pay more attention to study three kinds of dynamic fuzzy games on matroids, which are named as fuzzy games with multilinear extension form, with proportional value and with Choquet integral form, respectively. Meantime, the relationship between the fuzzy core and the Shapley function for dynamic fuzzy games on matroids is researched, which coincides with the crisp case.     

Random fuzzy shock models and bivariate random fuzzy exponential distribution

In this paper, a random fuzzy shock model and a random fuzzy fatal shock model are proposed. Then bivariate random fuzzy exponential distribution is derived from the random fuzzy fatal shock model. Furthermore, some properties of the bivariate random fuzzy exponential distribution are proposed. Finally, an example is given to show the application of the bivariate random fuzzy exponential distribution.     

Spatial reasoning under imprecision using fuzzy set theory,formal logics and mathematical morphology

In spatial reasoning, in particular for applications in image understanding, structure recognition and computer vision, a lot of attention has to be paid to spatial relationships and to the imprecision attached to information and knowledge to be handled. Two main components are knowledge representation and reasoning. We show in this paper that the fuzzy set framework associated to the formalism provided by mathematical morphology and formal logics allows us to derive appropriate representations and reasoning tools.