Supersound many-valued logics and Dedekind-MacNeille completions

In Hájek et al. (J Symb Logic 65(2):669–682, 2000) the authors introduce the concept of supersound logic, proving that first-order Gödel logic enjoys this property, whilst first-order ?ukasiewicz and product logics do not; in Hájek and Shepherdson (Ann Pure Appl Logic 109(1–2):65–69, 2001) this result is improved showing that, among the logics given by continuous t-norms, Gödel logic is the only one that is supersound. In this paper we will generalize the previous results. Two conditions will be presented: the first one implies the supersoundness and the second one non-supersoundness. To develop these results we will use, between the other machineries, the techniques of completions of MTL-chains developed in Labuschagne and van Alten (Proceedings of the ninth international conference on intelligent technologies, 2008) and van Alten (2009). We list some of the main results. The first-order versions of MTL, SMTL, IMTL, WNM, NM, RDP are supersound; the first-order version of an axiomatic extension of BL is supersound if and only it is n-potent (i.e. it proves the formula \({\varphi^{n}\,\to\,\varphi^{n\,{+}\,1}}\) for some \({n\,\in\,\mathbb{N}^+}\)). Concerning the negative results, we have that the first-order versions of ΠMTL, WCMTL and of each non-n-potent axiomatic extension of BL are not supersound.     

Cosine similarity measures for intuitionistic fuzzy sets and their applications

In this work, considering the information carried by the membership degree and the non-membership degree in Atanassov’s intuitionistic fuzzy sets (IFSs) as a vector representation with the two elements, a cosine similarity measure and a weighted cosine similarity measure between IFSs are proposed based on the concept of the cosine similarity measure for fuzzy sets. To demonstrate the efficiency of the proposed cosine similarity measures, the existing similarity measures between IFSs are compared with the cosine similarity measure between IFSs by numerical examples. Finally, the cosine similarity measures are applied to pattern recognition and medical diagnosis.     

Partitionable sets,almost partitionable sets,and their applications

This paper introduces almost partitionable sets (APSs) to generalize the known concept of partitionable sets. These notions provide a unified frame to construct $\mathbb{Z}$‐cyclic patterned starter whist tournaments and cyclic balanced sampling plans excluding contiguous units. The existences of partitionable sets and APSs are investigated. As an application, a large number of optical orthogonal codes achieving the Johnson bound or the Johnson bound minus one are constructed.     

Upper semicontinuous fuzzy sets and applications

Let (X, τ) be a generalized topological space of type $\text{V}$_α (see A. Appert and K. Fan, “Espaces topologiques intermédiares,” Herman, Paris, 1951) and (L, ?) be a complete Brouwerian lattice such that the dual lattice of (L, ?) is also Brouwerian. We prove that every upper semicontinuous L-fuzzy subset of X can be represented by a τ-closed random set. As an important application we obtain a fuzzification of measurable spaces as well as of topological spaces. In particular a concept of measurable (open) L-fuzzy sets is developed.     

L-equivalence of systems of functions in many-valued logics

Translated from Algebra i Logika, Vol. 27, No. 1, pp. 37–47, January–February, 1988.     

Correlation coefficients of hesitant fuzzy sets and their applications to clustering analysis

Hesitant fuzzy sets (HFSs), which allow the membership degree of an element to a set represented by several possible values, can be considered as a powerful tool to express uncertain information in the process of group decision making. We derive some correlation coefficient formulas for HFSs and apply them to clustering analysis under hesitant fuzzy environments. Two real world examples, i.e. software evaluation and classification as well as the assessment of business failure risk, are employed to illustrate the actual need of the clustering algorithm based on HFSs, which can incorporate the difference of evaluation information provided by different experts in clustering processes. In order to extend the application domain of the clustering algorithm in the framework of HFSs, we develop the interval-valued HFSs and the corresponding correlation coefficient formulas, and then demonstrate their application in clustering with interval-valued hesitant fuzzy information through a specific numerical example.     

On generalized fuzzy relational equations and their applications

The paper provides an idea of generalization of fuzzy relational equations where t- and s-norms are introduced. The first part contains an extensive presentation of the resolution of fuzzy relational equations; next the solutions are specified for a list of several triangular norms. Moreover the dual equations are considered. The second part deals with the applicational aspects of these equations in systems analysis, decision-making, and arithmetic of fuzzy numbers.     

Sufficient triangular norms in many-valued logics with standard negation

In many-valued logics with the unit interval as the set of truth values, from the standard negation and the product (or, more generally, from any strict Frank t-norm) all measurable logical functions can be derived, provided that also operations with countable arity are allowed. The question remained open whether there are other t-norms with this property or whether all strict t-norms possess this property. We give a full solution to this problem (in the case of strict t-norms), together with convenient sufficient conditions. We list several families of strict t-norms having this property and provide also counterexamples (the Hamacher product is one of them). Finally, we discuss the consequences of these results for the characterization of tribes based on strict t-norms.     

Some formal relationships among soft sets,fuzzy sets,and their extensions

We prove that every hesitant fuzzy set on a set E can be considered either a soft set over the universe $[0,1]$ or a soft set over the universe E. Concerning converse relationships, for denumerable universes we prove that any soft set can be considered even a fuzzy set. Relatedly, we demonstrate that every hesitant fuzzy soft set can be identified with a soft set, thus a formal coincidence of both notions is brought to light. Coupled with known relationships, our results prove that interval type-2 fuzzy sets and interval-valued fuzzy sets can be considered as soft sets over the universe $[0,1]$. Altogether we contribute to a more complete understanding of the relationships among various theories that capture vagueness and imprecision.     

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     

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.     

On ranking fuzzy sets

A mathematical model describing a fuzzy ordering of fuzzy sets is suggested. In this model an induced fuzzy ordering is defined as the inverse image of the natural linear ordering on real numbers under natural fuzzy correspondence between fuzzy sets and real numbers. The transitivity property of the strict fuzzy ordering associated with the induced fuzzy ordering is established.     

Some similarity measures of intuitionistic fuzzy sets and their applications to multiple attribute decision making

Atanassov (1986) defined the notion of intuitionistic fuzzy set, which is a generalization of the notion of Zadeh’ fuzzy set. In this paper, we first develop some similarity measures of intuitionistic fuzzy sets. Then, we define the notions of positive ideal intuitionistic fuzzy set and negative ideal intuitionistic fuzzy set. Finally, we apply the similarity measures to multiple attribute decision making under intuitionistic fuzzy environment.     

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)     

On D-posets of fuzzy sets

D-posets of fuzzy sets constitute a natural simple mathematical structure in which relevant notions of generalized probability theory can be formalized. We present a classification of D-posets leading to a hierarchy of distinguished subcategories of D-posets related to probability and study their relationships. This contributes to a better understanding of the transition from classical probability theory to fuzzy probability theory. In particular, we describe the transition from the Boolean cogenerator {0, 1} to the fuzzy cogenerator [0, 1] and prove that the generated ?ukasiewicz tribes form an epireflective subcategory of the bold algebras.     

On intutionistic fuzzy gpr-closed sets

In this paper, a new class of intuitionistic fuzzy closed sets called intuitionistic fuzzy generalized preregular closed sets (briefly intuitionistic fuzzy gpr-closed sets) and intuitionistic fuzzy generalized preregular open sets (briefly intuitionistic fuzzy gpr-open sets) are introduced and their properties are studied. Further the notion of intuitionistic fuzzy preregular T _1/2-spaces and intuitionistic fuzzy generalized preregular continuity (briefly intuitionistic fuzzy gpr-continuity) are introduced and studied.