Legal philosophers and property scholars sometimes disagree over one or more of the following: the meaning of the word 'property,' the concept of property, and the nature of property. For much of the twentieth century, the work of W.N. Hohfeld and Tony Honoré represented a consensus around property. The consensus often went under the heading of property as bundle of rights, or more accurately as a set of normative relations between persons with respect to things. But by the mid-l 990s, (...) the consensus was under attack. Key figures in the attack were James Penner, a legal philosopher, and Thomas W. Merrill and Henry E. Smith, two highly regarded professors of property law. This article aims to repel the attack and argues for property as a set of normative relations between persons with respect to things. The positive case for this view of property pays special attention to the philosophy of language and the analysis of concepts. The positive case also maintains that the right to use and the power to transfer are as central to property as the right to exclude. It is possible that the virtues of Smith's modular theory of property differ from the virtues of a well-crafted bundle theory. Indeed, it may be the case that these two theories throw light on different features of property law and are not, save at the margin, competitors with each other. The label 'new essentialism' sometimes applied to the work of Penner, Merrill, and Smith seems inapt if property does not have an essence. Of course, they might refuse the label. (shrink)

The term fuzzy logic is used in this paper to describe an imprecise logical system, FL, in which the truth-values are fuzzy subsets of the unit interval with linguistic labels such as true, false, not true, very true, quite true, not very true and not very false, etc. The truth-value set, , of FL is assumed to be generated by a context-free grammar, with a semantic rule providing a means of computing the meaning of each linguistic truth-value in (...) as a fuzzy subset of [0, 1].Since is not closed under the operations of negation, conjunction, disjunction and implication, the result of an operation on truth-values in requires, in general, a linguistic approximation by a truth-value in . As a consequence, the truth tables and the rules of inference in fuzzy logic are (i) inexact and (ii) dependent on the meaning associated with the primary truth-value true as well as the modifiers very, quite, more or less, etc. (shrink)

part 1. Fuzzy logic theory 1 -- part 2. Applications and advanced topics of fuzzy logic.

Medical knowledge as well as clinical practice are characterized by inescapable uncertainty. There are many reasons this is the case, but foremost among them is that almost everything in medicine is inevitably vague, be it something linguistic such as the term “illness”, or something extra-linguistic such as the condition referred to as illness. If we ask ourselves, then, what the term “illness” means exactly, on the one hand; and how we may precisely delimit the condition illness, on the other; we (...) shall realize that to answer these and similar questions requires specific methods that enable us to adequately cope with vagueness. As we shall see below, fuzzy logic provides us with just such methods. (shrink)

This paper deals with the relation with Fuzzy Logic of some of the ideas of Karl Menger published between 1942 and 1966 and concerning what he called “Hazy Sets”, Probabilistic Relations and Statistical Metric Spaces. The author maintains the opinion that if Lofti A. Zadeh is actually the father of Fuzzy Logic, Menger not only was a forerunner of this field but that his ideas were and still are influential on it.

In this introduction to Part 1 of the Common Knowledge symposium, “Fuzzy Studies,” the journal's editor discusses four essays from the 1980s by Richard Rorty, in which Rorty chose to associate himself with various neopragmatists, Continental thinkers, and “left-wing Kuhnians” under the rubric of the “new fuzziness.” The term had been introduced as an insult by a philosopher of science with positivist leanings, but Rorty took it up as an “endearing” compliment, arguing that “to be less fuzzy” was (...) also to be “less genial, tolerant, open-minded, and fallibilist.” He defined the “new fuzziness” as “an attempt to blur just those distinctions between the objective and subjective and between fact and value which the critical conception of rationality has developed.” This introduction also examines W. V. Quine's essay “Speaking of Objects”, which describes objects as fuzzy “half-entities”; Clifford Geertz's essay “Blurred Genres”, which advises social scientists that being “taxonomically upstanding” is futile; and Lofti Zadeh's article “The Concept of a Linguistic Variable and Its Application to Approximate Reasoning”, which abandons “Aristotelian, bivalent logic” in favor of a “fuzzy logic” based on Zadeh's “fuzzy set theory.” This introductory piece relates these theoretical works of the past half-century to the sorites paradox and to classical issues of vagueness raised and still unresolved in Western philosophy. Returning then to Rorty, the author questions how Rorty expected his endorsement of the “new fuzziness” to be applied, as proposed, to theology and politics. Suggesting that such applications are the natural work of historians, the author, having asked the historian Natalie Zemon Davis for comment, then quotes her response—which associates fuzzy studies, “common knowledge,” and peacemaking—at length. (shrink)

In a previous paper the concept of disease was fuzzy-logically analyzed and a sketch was given of a prototype resemblance theory of disease (Sadegh-Zadeh (2000). J. Med. Philos., 25:605–38). This theory is outlined in the present paper. It demonstrates what it means to say that the concept of disease is a nonclassical one and, therefore, not amenable to traditional methods of inquiry. The theory undertakes a reconstruction of disease as a category that in contradistinction to traditional views is (...) not based on a set of common features of its members, that is individual diseases, but on a few best examples of the category, called its prototypes, and a similarity relationship such that a human condition is considered a disease if it resembles a prototype. It enables new approaches to resolving many of the stubborn problems associated with the concept of disease. (shrink)

Complex fuzzy sets are the novel extension of Zadeh’s fuzzy sets. In this paper, we comprise the introduction to the concept of ξ -complex fuzzy sets and proofs of their various set theoretical properties. We define the notion of α, δ -cut sets of ξ -complex fuzzy sets and justify the representation of an ξ -complex fuzzy set as a union of nested intervals of these cut sets. We (...) also apply this newly defined concept to a physical situation in which one may judge the performance of the participants in a given task. In addition, we innovate the phenomena of ξ -complex fuzzy subgroups and investigate some of their fundamental algebraic attributes. Moreover, we utilize this notion to define level subgroups of these groups and prove the necessary and sufficient condition under which an ξ -complex fuzzy set is ξ -complex fuzzy subgroup. Furthermore, we extend the idea of ξ -complex fuzzy normal subgroup to define the quotient group of a group G by this particular ξ -complex fuzzy normal subgroup and establish an isomorphism between this quotient group and a quotient group of G by a specific normal subgroup G A ξ. (shrink)

Two different generalizations of Brouwer–Zadeh posets (BZ posets) are introduced. The former (called pre-BZ poset) arises from topological spaces, whose standard power set orthocomplemented complete atomic lattice can be enriched by another complementation associating with any subset the set theoretical complement of its topological closure. This complementation satisfies only some properties of the algebraic version of an intuitionistic negation, and can be considered as, a generalized form of a Brouwer negation. The latter (called degenerate BZ poset) arises from the (...) so-called special effects on a Hilbert space. It is shown that the standard Brouwer negation for effect operators produces a degenerate BZ poset with respect to the order induced from the partial sum operation. (shrink)

Since the world is full of indeterminacy, the Neutrosophics found their place into contemporary research. We now introduce for the first time the notions of Neutrosophic Crisp Sets and Neutrosophic Topology on Crisp Sets. We develop the 2012 notion of Neutrosophic Topological Spaces and give many practical examples. Neutrosophic Science means development and applications of Neutrosophic Logic, Set, Measure, Integral, Probability etc., and their applications in any field. It is possible to define the neutrosophic measure and consequently the (...) neutrosophic integral and neutrosophic probability in many ways, because there are various types of indeterminacies, depending on the problem we need to solve. Indeterminacy is different from randomness. Indeterminacy can be caused by physical space, materials and type of construction, by items involved in the space, or by other factors. In 1965 [51], Zadeh generalized the concept of crisp set by introducing the concept of fuzzy set, corresponding to the situation in which there is no precisely defined set;there are increasing applications in various fields, including probability, artificial intelligence, control systems, biology and economics. Thus, developments in abstract mathematics using the idea of fuzzy sets possess sound footing. In accordance, fuzzy topological spaces were introduced by Chang [12] and Lowen [33]. After the development of fuzzy sets, much attention has been paid to the generalization of basic concepts of classical topology to fuzzy sets and accordingly developing a theory of fuzzy topology [1-58]. In 1983, the intuitionistic fuzzy set was introduced by K. Atanassov [55, 56, 57] as a generalization of the fuzzy set, beyond the degree of membership and the degree of non-membership of each element. In 1999 and 2002, Smarandache [71, 72, 73, 74] defined the notion of Neutrosophic Sets, which is a generalization of Zadeh’s fuzzy set and Atanassov's intuitionistic fuzzy set. Some neutrosophic concepts have been investigated by Salama et al. [61-70]. Forwarding the study of neutrosophic sets, this book consists of seven chapters, targeting to:  generalize the previous studies in [1-59], and[91-94] so to define the neutrosopic crisp set and neutrosophic set concepts;  discuss their main properties; A. A. Salama & Florentin Smarandache  introduce and study some concepts of neutrosophic crisp and neutrosophic topological spaces and deduce their properties;  deduce many types of functions and give the relationships between different neutrosophic topological spaces, which helps to build new properties of neutrosophic topological spaces;  stress once more the importance of Neutrosophic Ideal as a nontrivial extension of neutrosophic set and neutrosophic logic [71, 72, 73, 74];  propose applications on computer sciences by using neutrosophic sets. (shrink)

This paper examines a novel notion of causality, namely, fuzzy-set-theoretic causality. Over the last decade, a number of conceptual models of causality, in the language of fuzzy-set theory, have appeared in the scientific literature and have been applied to empirical research. They have circulated widely from one scientific discipline to another, weaving a unifying thread through them. However, they have received no philosophical attention. In this paper, I will discuss the value and limitations of this type of model (...) and will read into its application several dimensions of philosophical significance. (shrink)

Smarandache (1995) defined the notion of neutrosophic sets, which is a generalization of Zadeh's fuzzy set and Atanassov's intuitionistic fuzzy set. In this paper, we first develop some similarity measures of neutrosophic sets. We will present a method to calculate the distance between neutrosophic sets (NS) on the basis of the Hausdorff distance. Then we will use this distance to generate a new similarity measure to calculate the degree of similarity between NS. Finally we (...) will prove some properties of the proposed similarity measures. (shrink)

From Introduction: In a 1968 article, ‘Probability Measures of Fuzzy Events’, Lotfi Zadeh pro-posed accounts of absolute and conditional probability for fuzzy sets (Zadeh, 1968).

Today, soft computing is a field that is used a lot in solving real-world problems, such as problems in economics, finance, banking... With the aim to serve for solving the real problem, many new theories and/or tools which were proposed, improved to help soft computing used more efficiently. We can mention some theories as fuzzy sets theory (L. Zadeh, 1965), intuitionistic fuzzy set (K Atanasov, 1986), neutrosophic set (F. Smarandache 1999). In this paper, we introduce a (...) new notion of sup port-neutrosophic set (SNS), which is the combination a neutrosophic set with a fuzzy set. So, SNS set is a direct extension of fuzzy set and neutrosophic sets (F. Smarandache). Then, we define some operators on the support-neutrosophic sets, and investigate some properties of these operators. (shrink)

The notion of fuzzy event is introduced in the theory of measurement in quantum mechanics by indicating in which sense measurements can be considered to yield fuzzy sets. The concept of probability measure on fuzzy events is defined, and its general properties are deduced from the operational meaning assigned to it. It is pointed out that such probabilities can be derived from the formalism of quantum mechanics. Any such probability on a given fuzzy set is (...) related to the frequency of occurrence within that set of points in a random sample, where the sample points are themselves fuzzy sets obtained as outcomes of measurements of, in general, incompatible observables on replicas of the system in the same prepared state. (shrink)

In 1685, in The Art of Discovery, Leibniz set down an extraordinary idea: "The only way to rectify our reasonings is to make them as tangible as those of the Mathematicians, so that we can find our error at a glance, and when there are disputes among persons, we can simply say: Let us calculate [calculemus], without further ado, to see who is right." Calculemus.

The relationship between vague statements and fuzzy sets is examined. It is shown that the probability of vague statements may be defined in a manner analogous to that discussed in Reichenbach's logic of weight.

As a generation of fuzzy set, the notion of complex fuzzy set which is an innovative concept is introduced by Ramot, Milo, Friedman and Kandel. The purpose of this article is to apply complex fuzzy set to BCK/BCI-algebras. The notions of a complex subalgebra and a complex left reduced ideal in a BCK/BCI- algebra are introduced, and related properties are investigated. Characterizations of a complex subalgebra are provided, and the homomorphic image of a complex subalgebra and a (...) complex left reduced ideal. (shrink)

The notions of health, illness, and disease are fuzzy-theoretically analyzed. They present themselves as non-Aristotelian concepts violating basic principles of classical logic. A recursive scheme for defining the controversial notion of disease is proposed that also supports a concept of fuzzy disease. A sketch is given of the prototype resemblance theory of disease.

PurposeDrawing upon the stimulus-organism-response model, this paper aims to investigate the effects of stores’ environmental components on Chinese consumers’ emotions and intentions to purchase luxury brands.Design/Methodology/ApproachData were collected from Chinese consumers who have purchased luxury brands from retail stores. Partial least squares-structural equation modeling and fuzzy-set qualitative comparative analysis were employed for data analysis.FindingsThe results of PLS-SEM indicated that three dimensions of the store environment directly and significantly influenced Chinese consumers’ emotions. However, fsQCA revealed greater heterogeneity among respondents by (...) highlighting stores’ environmental components and Chinese consumers’ emotions.Originality/ValueThis study makes significant contributions to the marketing literature by examining the S-O-R model alongside fsQCA approach to conditionally predict outcomes in a luxury brand context. The present study may be one of the first to examine the effects of stores’ environmental components on Chinese consumers’ emotions and purchase intentions, applying both symmetrical test and asymmetrical test approaches to determine significant components of the store environment as well as component combinations that predict Chinese consumers’ emotions and purchase intentions. (shrink)

A structural (as opposed to Zadeh's quantitative) approach to fuzziness is given, based on the operator "very", which is added to the language of set theory together with some elementary axioms about it. Due to the axiom of foundation and to a lifting axiom, the operator is proved trivial on the cumulative hierarchy of ZF. So we have to drop either foundation or lifting. Since fuzziness concerns complemented predicates rather than sets, a class theory is needed for the (...) very operator. And of them the Kelley-Morse (KM) theory is more appropriate for reasons of class existence. Several definable realizations of the very-operator are presented in KM⁻. In the last section we consider the operator "very" without the lifting axiom on classes of urelements. To each structurally fuzzy set X a traditional quantitative fuzzy set X̄ is assigned -- its quantitative representation. This way we are able partly to recover ordinary fuzzy sets from the structurally fuzzy ones. (shrink)

Many criticisms of prototype theory and/or fuzzy-set theory are based on the assumption that category representativeness (or typicality) is identical with fuzzy membership. These criticisms also assume that conceptual combination and logical rules (all in the Aristotelian sense) are the appropriate criteria for the adequacy of the above “fuzzy typicality”. The present paper discusses these assumptions following the line of their most explicit and most influential expression by Osheron and Smith (1981). Several arguments are made against the (...) above identification, the most important being that representativeness in prototype theory is exclusively based on element-to-element similarity while fuzzy membership is inherently an element-to-category relationship. Also the above criteria for adequacy are criticized from the viewpoint of both prototype theory and fuzzy-set theory as well as from that of both conceptual and logical combination, and also from that of integration. (shrink)

Computability theorists have introduced multiple hierarchies to measure the complexity of sets of natural numbers. The Kleene Hierarchy classifies sets according to the first-order complexity of their defining formulas. The Ershov Hierarchy classifies limit computable sets with respect to the number of mistakes that are needed to approximate them. Biacino and Gerla extended the Kleene Hierarchy to the realm of fuzzy sets, whose membership functions range in a complete lattice. In this paper, we combine the (...) Ershov Hierarchy and fuzzy set theory, by introducing and investigating the Fuzzy Ershov Hierarchy. (shrink)

JSTOR is a not-for-profit organization founded in 1995 to build trusted digital archives for scholarship. We work with the scholarly community to preserve their work and the materials they rely upon, and to build a common research platform that promotes the discovery and use of these resources. For more information about JSTOR, please contact [email protected].

The multiobjective optimization on the basis of ratio analysis method captures diverse features such as the criteria and alternatives of appraising a multiple criteria decision-making problem. At the same time, the multiple criteria problem includes a set of decision makers with diverse expertise and preferences. In fact, the literature lists numerous approaches to aid in this problematic task of choosing the best alternative. Nevertheless, in the MCDM field, there is a challenge regarding intangible information which is commonly involved in multiple (...) criteria decision-making problem; hence, it is substantial in order to advance beyond the research related to this field. Thus, the objective of this paper is to present a fused method between multiobjective optimization on the basis of ratio analysis and Pythagorean fuzzy sets for the choice of an alternative. Besides, multiobjective optimization on the basis of ratio analysis is utilized to choose the best alternatives. Finally, two decision-making problems are applied to illustrate the feasibility and practicality of the proposed method. (shrink)

This book is a collection of contributions written by philosophers and scientists active in different fields, such as mathematics, logics, social sciences, computer sciences and linguistics. They comment on and discuss various parts of and subjects and propositions introduced in the Handbook of Analytical Philosophy of Medicine from Kadem Sadegh-Zadeh, published by Springer in 2012. This volume reports on the fruitful exchange and debate that arose in the fuzzy community upon the publication of the Handbook. This was not (...) only very much appreciated by the community but also seen as a critical starting point for beginning a new discussion. The results of this discussion, which involved many different perspectives from science and the humanities and was warmly encouraged by Kadem Sadegh-Zadeh himself, are accurately reported in this volume, which is intended to be a critical companion to Kadem Sadegh-Zadeh ́s handbook. Rudolf Seising is currently an adjunct researcher at the European Centre for Soft Computing in Mieres, Asturias (Spain) and a college lecturer at the Faculty of History and Arts, at the Ludwig Maximilians University of Munich (Germany). Marco Elio Tabacchi is currently the Scientific Director of the Italian National Research & Survey Organization Demopolis, and a research assistant in the Soft Computing Group at University of Palermo (Italy). (shrink)

This is the first part of a two-part paper in which I conclude the process, initiated elsewhere, of tracking objective conditions of vagueness of representation from language to pictures, from philosophy to imaging science, from vagueness to approximation, from representation to reasoning, with a focus on the application of fuzzy set theory and its challenges.

Conservative agriculture practice adoption literature advocates that adoption is caused by many factors comprising cognitive, social, economic, personal, and CAP-related factors. Evaluating the adoption of CAPs as the outcome is complex and challenging with regression-based models as the systemic interdependencies of the factors offer diverse or varying results. Farmer production and environmental orientations as cognitive stances are notable interpreters of CAP adoption. The appetite level for risk-taking, innovativeness, and trust facilitates the adoption of CAPs. However, a causal-predictive technique should be (...) used to investigate the adoption of CAP. Hence, this study engages in a configuration approach using a fuzzy-set qualitative comparative analysis to analyze the patterns of different types of farmers' orientations, personal level of trust on extension services, and innovativeness risk-taking attitude on the intention to adopt CAPs. The analysis is based on the 155-farmer data collected employing a structured interview from Pakistan. The results suggest that a higher level of environment orientation and innovativeness is sufficient to increase the intention to adopt CAPs. Moreover, a higher intention to adopt CAPs is achieved with a lower production orientation, a higher personal level of innovativeness, and a risk-taking attitude of the farmer. The innovativeness can help to develop the intention to adopt the CAPs among the environment and production-oriented farmers. Causal solutions offer a unique understanding that the farmers' environment and production orientation can combine to suggest inclined to adopt CAPs by having an attitude of innovativeness and risk-taking. The causal solutions achieved significant predictive validity in the holdout samples. Policy and farmer-level suggestions were made to raise the intention to adopt CAPs among the farmers. (shrink)

Based on the expanded theory of planned behavior, this study first explores the configuration relationship between founder management and innovation by using the fuzzy-set qualitative comparative analysis. Based on the theory of planned behavior, this study divides the behavior intention of founders into three categories: Attitude, subjective norm, and perceived behavior control. Using fsQCA, we found that there are two ways to achieve high innovation input of enterprises. In combination with the two ways, the factors such as male and (...) highly educated founder, and large firm size can effectively increase the innovation input of firms, which is consistent with the three aspects of the behavioral intention of the theory of planned behavior, and it proves that the theory of planned behavior can effectively explain the configuration relation between the founder and firm innovation. In addition, this study finds that the innovation output is different from the innovation input, is dependent on the innovation ability of the firm itself, and is less influenced by the external environment. (shrink)

The work on prototypes in ontologies pioneered by Rosch [10] and elaborated by Lakoff [8] and Freund [3] is related to vagueness in the sense that the more remote an instance is from a prototype the fewer people agree that it is an example of that prototype. An intuitive example is the prototypical “mother”, and it is observed that more specific instances like ”single mother”, “adoptive mother”, “surrogate mother”, etc., are less and less likely to be classified as “mothers” by (...) experimental subjects. From a different direction Gärdenfors [4] provided a persuasive account of natural predicates to resolve paradoxes of induction like Goodman’s “Grue” predicate [5]. Gärdenfors proposed that “quality dimensions” arising from human cognition and perception impose topologies on concepts such that the ones that appear “natural” to us are convex in these topologies. We show that these two cognitive principles — prototypes and predicate convexity — are equivalent to unimodal (convex) fuzzy characteristic functions for sets. Then we examine the case when the fuzzy set characteristic function is not convex, in particular when it is multi-modal. We argue that this is an indication that the fuzzy concept should really be regarded as a super concept in which the decomposed components are subconcepts in an ontological taxonomy. (shrink)

It, is shown that Birkhoff –von Neumann quantum logic (i.e., an orthomodular lattice or poset) possessing an ordering set of probability measures S can be isomorphically represented as a family of fuzzy subsets of S or, equivalently, as a family of propositional functions with arguments ranging over S and belonging to the domain of infinite-valued Łukasiewicz logic. This representation endows BvN quantum logic with a new pair of partially defined binary operations, different from the order-theoretic ones: Łukasiewicz intersection and (...) union of fuzzy sets in the first case and Łukasiewicz conjunction and disjunction in the second. Relations between old and new operations are studied and it is shown that although they coincide whenever new operations are defined, they are not identical in general. The hypothesis that quantum-logical conjunction and disjunction should be represented by Łukasiewicz operations, not by order-theoretic join and meet is formulated and some of its possible consequences are considered. (shrink)

To establish the scientific-theoretical and philosophical- methodological foundations for the transition from classical set theory (Cantor set) to fuzzy set theory, the mathematical and ontological structures of the Cantor set are presented in a localized manner. This approach aligns with the research direction, incorporating philosophical reflections on the fundamental concepts of this theory, such as infinity, continuum, and the empty set. Particular emphasis is placed on infinity and infinite sets (countable and continuum), which are crucial to the development (...) of set theory as a scientific field and have been subjects of inquiry in both scientific and philosophical cognitive processes from ancient times to the present. In addition, a relatively concise explanation of the mathematical and ontological structures of both classical set theory and fuzzy set theory is provided. Through certain epistemological and cognitive approaches, the article identifies the scientific-theoretical and philosophical-methodological bases for the emergence of fuzzy set theory, framed within the context of the scientific-philosophical reflections on classical set theory and its core concepts. (shrink)

Selon le concept de Nancy Fraser du triple mouvement de protection sociale, d’émancipation et de marchandisation, les forces d’émancipation peuvent former une alliance avec la protection sociale ou la marchandisation. Un véritable exemple d’émancipation est la transformation des services résidentiels de soins aux personnes handicapées en assistance personnelle. Toutefois, on ne sait pas encore très bien pourquoi certaines réformes se chevauchent davantage avec la marchandisation et d’autres avec la protection sociale, alors que d’autres pays n’ont pas entrepris de réformes généralisées (...) de leurs services de soins aux personnes handicapées. Cet article tente d’éclairer cette question en examinant l’approche morphogénétique pour expliquer l’évolution des services de soins aux personnes handicapées dans dix pays européens. Une fuzzy set ideal type analysis a été utilisée pour délimiter quatre types de services de soins aux personnes handicapées. L’analyse a classé la Grèce, la Slovénie et l’Espagne dans le type domestique-traditionnel, la Belgique, l’Allemagne et la Suisse dans le type bienveillant-paternaliste, la Suède dans le type universel-progressif, et la Lettonie, la République slovaque et le Royaume-Uni dans le type précaire-progressif. (shrink)

The aim of this manuscript is to introduce the \((\alpha,\beta)\)-hesitant fuzzy set and apply it to semigroups. In this paper, as a generalization of the concept of hesitant fuzzy sets to semigroup theory, the concept of \((\alpha,\beta)\)-hesitant fuzzy subsemigroups of semigroups is introduced, and related properties are discussed. Furthermore, we define and study \((\alpha,\beta)\)-hesitant fuzzy ideals on semigroups. In particular, we investigate the structure of \((\alpha,\beta)\)-hesitant fuzzy ideal generated by a hesitant fuzzy ideal (...) in a semigroup. In addition, we also introduce the concepts of \((\alpha,\beta)\)-hesitant fuzzy semiprime sets of semigroups, and characterize regular semigroups in terms of \((\alpha,\beta)\)-hesitant fuzzy left ideals and \((\alpha,\beta)\)-hesitant fuzzy right ideals. Finally, several characterizations of regular and intra-regular semigroups by the properties of \((\alpha,\beta)\)-hesitant ideals are given. (shrink)

The neutrosophic cubic hesitant fuzzy set can efficiently handle the complex information in a decision-making problem because it combines the advantages of the neutrosophic cubic set and the hesitant fuzzy set. The algebraic operations based on Dombi norms and co-norms are more flexible than the usual algebraic operations as they involve an operational parameter. First, this paper establishes Dombi algebraic operational laws, score functions, and similarity measures in neutrosophic cubic hesitant fuzzy sets. Then, we proposed Dombi (...) exponential operational laws in which the exponents are neutrosophic cubic hesitant fuzzy values and bases are positive real numbers. To use neutrosophic cubic hesitant fuzzy sets in decision-making, we are developing Dombi exponential aggregation operators in the current study. In the end, we present applications of exponential aggregation operators in multiattribute decision-making problems. (shrink)

As an introduction to our work, we emphasize the parallel interpretation of abstract tools and the concepts of undetermined and vague information. Imprecision, uncertainty and their relationships are inspected. Suitable interpretations of the fuzzy sets theory are applied to legal phenomena in an attempt to clearly circumscribe the possible applications of the theory. The fundamental notion of reference sets is examined in detail, hence highlighting their importance. A systematic and combinatorial classification of the relevant subsets of the (...) legal field is supplied for practical application. Although the use of the fuzzy sets theory is sometimes suggested as a palliative measure (no competition exists), it can also be complementary (serve as a building block to improve modelisation). An Appendix gives a brief recall of the key-concepts of the axiomatic theory of fuzziness and its developments: fuzzy sets, fuzzy logic, fuzzy control and theory of possibility. (shrink)

Medical practice is practiced morality, and clinical research belongs to normative ethics. The present book elucidates and advances this thesis by: 1. analyzing the structure of medical language, knowledge, and theories; 2. inquiring into the foundations of the clinical encounter; 3. introducing the logic and methodology of clinical decision-making, including artificial intelligence in medicine; 4. suggesting comprehensive theories of organism, life, and psyche; of health, illness, and disease; of etiology, diagnosis, prognosis, prevention, and therapy; and 5. investigating the moral and (...) metaphysical issues central to medical practice and research. Many systems of (classical, modal, non-classical, probability, and fuzzy) logic are introduced and applied. Fuzzy medical deontics, fuzzy medical ontology, fuzzy medical concept formation, fuzzy medical decision-making and biomedicine and many other techniques of fuzzification in medicine are introduced for the first time. -/- . (shrink)

Trapezoidal interval type-2 fuzzy sets are a special kind of type-2 fuzzy sets. TIT2FSs are useful in dealing with fuzziness inherent in decision data and the decision-making process. For multi-attribute group decision-making problems in which the attribute values and attribute weights are TIT2FSs, a new decision-making approach is proposed. On the basis of the concept of barycenters, a new approach to ranking TIT2FSs is given. Four kinds of geometric aggregation operators for TIT2FSs are developed, including the (...) TIT2FS weighted geometric average operator, TIT2FS ordered weighted geometric average operator, TIT2FS hybrid weighted geometric average operator, and extended TIT2FS hybrid weighted geometric average operator. The individual comprehensive values of alternatives are derived through the extended TIT2FS weighted geometric average operators. Using the TIT2FS hybrid weighted geometric average operator and the expert weights, the individual comprehensive values of alternatives are integrated into the collective ones, which are used to rank the alternatives. The practicability and effectiveness of the developed method are illustrated with a teaching quality assessment example. (shrink)

Computability theorists have introduced multiple hierarchies to measure the complexity of sets of natural numbers. The Kleene Hierarchy classifies sets according to the first-order complexity of their defining formulas. The Ershov Hierarchy classifies Δ20\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varDelta ^0_2$$\end{document} sets with respect to the number of mistakes that are needed to approximate them. Biacino and Gerla extended the Kleene Hierarchy to the realm of fuzzy sets, whose membership functions range in (...) a complete lattice L. In this paper, we combine the Ershov Hierarchy and fuzzy set theory, by introducing and investigating the Fuzzy Ershov Hierarchy. In particular, we focus on the fuzzy n-c.e. sets which form the finite levels of this hierarchy. Intuitively, a fuzzy set is n-c.e. if its membership function can be approximated by changing monotonicity at most n-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n-1$$\end{document} times. We prove that the Fuzzy Ershov Hierarchy does not collapse; that, in analogy with the classical case, each fuzzy n-c.e. set can be represented as a Boolean combination of fuzzy c.e. sets; but that, contrary to the classical case, the Fuzzy Ershov Hierarchy does not exhaust the class of all Δ20\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varDelta ^0_2$$\end{document} fuzzy sets. (shrink)

For classical sets one has with the cumulative hierarchy of sets, with axiomatizations like the system ZF, and with the category SET of all sets and mappings standard approaches toward global universes of all sets. We discuss here the corresponding situation for fuzzy set theory.Our emphasis will be on various approaches toward (more or less naively formed)universes of fuzzy sets as well as on axiomatizations, and on categories of fuzzy sets.

For classical sets one has with the cumulative hierarchy of sets, with axiomatizations like the system ZF, and with the category SET of all sets and mappings standard approaches toward global universes of all sets.We discuss here the corresponding situation for fuzzy set theory. Our emphasis will be on various approaches toward (more or less naively formed) universes of fuzzy sets as well as on axiomatizations, and on categories of fuzzy sets.

This paper deals with Gärdenfors’ theory of conceptual spaces. Let \({\mathcal {S}}\) be a conceptual space consisting of 2-type fuzzy sets equipped with several kinds of metrics. Let a finite set of prototypes \(\tilde{P}_1,\ldots,\tilde{P}_n\in \mathcal {S}\) be given. Our main result is the construction of a classification algorithm. That is, given an element \({\tilde{A}}\in \mathcal {S},\) our algorithm classifies it into the conceptual field determined by one of the given prototypes \(\tilde{P}_i.\) The construction of our algorithm uses some (...) physical analogies and the Newton potential plays a significant role here. Importantly, the resulting conceptual fields are not convex in the Euclidean sense, which we believe is a reasonable departure from the assumptions of Gardenfors’ original definition of the conceptual space. A partitioning algorithm of the space \(\mathcal {S}\) is also considered in the paper. In the application section, we test our classification algorithm on real data and obtain very satisfactory results. Moreover, the example we consider is another argument against requiring convexity of conceptual fields. (shrink)