The new notions of super column FRM model, super row FRM model and mixed super FRM model are introduced in this book. These three models are introduced specially to analyze the biased role of the print media on 27 percent reservation for the Other Backward Classes (OBCs) in educational institutions run by the Indian Central Government.

This book represents the fourth volume in the series Collected Papers on Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond. This volume specifically delves into the concept of the HyperUncertain Set, building on the foundational advancements introduced in previous volumes. The series aims to explore the ongoing evolution of uncertain combinatorics through innovative methodologies such as graphization, hyperization, and uncertainization. These approaches integrate and extend core concepts from fuzzy, neutrosophic, soft, and rough (...) set theories, providing robust frameworks to model and analyze the inherent complexity of real-world uncertainties. At the heart of this series lies combinatorics and set theory—cornerstones of mathematics that address the study of counting, arrangements, and the relationships between collections under defined rules. Traditionally, combinatorics has excelled in solving problems involving uncertainty, while advancements in set theory have expanded its scope to include powerful constructs like fuzzy and neutrosophic sets. These advanced sets bring new dimensions to uncertainty modeling by capturing not just binary truth but also indeterminacy and falsity. In this fourth volume, the integration of set theory with graph theory takes center stage, culminating in "graphized" structures such as hypergraphs and superhypergraphs. These structures, paired with innovations like Neutrosophic Oversets, Undersets, Offsets, and the Nonstandard Real Set, extend the boundaries of mathematical abstraction. This fusion of combinatorics, graph theory, and uncertain set theory creates a rich foundation for addressing the multidimensional and hierarchical uncertainties prevalent in both theoretical and applied domains. The book is structured into thirteen chapters, each contributing unique perspectives and advancements in the realm of HyperUncertain Sets and their related frameworks. -/- The first chapter (Advancing Traditional Set Theory with Hyperfuzzy, Hyperneutrosophic, and Hyperplithogenic Sets) explores the evolution of classical set theory to better address the complexity and ambiguity of real-world phenomena. By introducing hierarchical structures like hyperstructures and superhyperstructures—created through iterative applications of power sets—it lays the groundwork for more abstract and adaptable mathematical tools. The focus is on extending three foundational frameworks: Fuzzy Sets, Neutrosophic Sets, and Plithogenic Sets into their hyperforms: Hyperfuzzy Sets, Hyperneutrosophic Sets, and Hyperplithogenic Sets. These advanced concepts are applied across diverse fields such as statistics, clustering, evolutionary theory, topology, decision-making, probability, and language theory. The goal is to provide a robust platform for future research in this expanding area of study. The second chapter (Applications and Mathematical Properties of Hyperneutrosophic and SuperHyperneutrosophic Sets) extends the work on Hyperfuzzy, Hyperneutrosophic, and Hyperplithogenic Sets by delving into their advanced applications and mathematical foundations. Building on prior research, it specifically examines Hyperneutrosophic and SuperHyperneutrosophic Sets, exploring their integration into: Neutrosophic Logic, Cognitive Maps,Graph Neural Networks, Classifiers, and Triplet Groups. The chapter also investigates their mathematical properties and applicability in addressing uncertainties and complexities inherent in various domains. These insights aim to inspire innovative uses of hypergeneralized sets in modern theoretical and applied research. The third chapter (New Extensions of Hyperneutrosophic Sets – Bipolar, Pythagorean, Double-Valued, and Interval-Valued Sets) studies advanced variations of Neutrosophic Sets, a mathematical framework defined by three membership functions: truth (T), indeterminacy (I), and falsity (F). By leveraging the concepts of Hyperneutrosophic and SuperHyperneutrosophic Sets, the study extends: Bipolar Neutrosophic Sets, Interval-Valued Neutrosophic Sets, Pythagorean Neutrosophic Sets, and Double-Valued Neutrosophic Sets. These extensions address increasingly complex scenarios, and a brief analysis is provided to explore their potential applications and mathematical underpinnings. Building on prior research, the fourth chapter (Hyperneutrosophic Extensions of Complex, Single-Valued Triangular, Fermatean, and Linguistic Sets) expands on Neutrosophic Set theory by incorporating recent advancements in Hyperneutrosophic and SuperHyperneutrosophic Sets. The study focuses on extending: Complex Neutrosophic Sets, Single-Valued Triangular Neutrosophic Sets, Fermatean Neutrosophic Sets, and Linguistic Neutrosophic Sets. The analysis highlights the mathematical structures of these hyperextensions and explores their connections with existing set-theoretic concepts, offering new insights into managing uncertainty in multidimensional challenges. The fifth chapter (Advanced Extensions of Hyperneutrosophic Sets – Dynamic, Quadripartitioned, Pentapartitioned, Heptapartitioned, and m-Polar) delves deeper into the evolution of Neutrosophic Sets by exploring advanced frameworks designed for even more intricate applications. New extensions include: Dynamic Neutrosophic Sets, Quadripartitioned Neutrosophic Sets, Pentapartitioned Neutrosophic Sets, Heptapartitioned Neutrosophic Sets, and m-Polar Neutrosophic Sets. These developments build upon foundational research and aim to provide robust tools for addressing multidimensional and highly nuanced problems. The sixth chapter (Advanced Extensions of Hyperneutrosophic Sets – Cubic, Trapezoidal, q-Rung Orthopair, Overset, Underset, and Offset) builds upon the Neutrosophic framework, which employs truth (T), indeterminacy (I), and falsity (F) to address uncertainty. Leveraging advancements in Hyperneutrosophic and SuperHyperneutrosophic Sets, the study extends: Cubic Neutrosophic Sets, Trapezoidal Neutrosophic Sets, q-Rung Orthopair Neutrosophic Sets, Neutrosophic Oversets, Neutrosophic Undersets, and Neutrosophic Offsets. The chapter provides a brief analysis of these new set types, exploring their properties and potential applications in solving multidimensional problems. The seventh chapter (Specialized Classes of Hyperneutrosophic Sets – Support, Paraconsistent, and Faillibilist Sets) delves into unique classes of Neutrosophic Sets extended through Hyperneutrosophic and SuperHyperneutrosophic frameworks to tackle advanced theoretical challenges. The study introduces and extends: Support Neutrosophic Sets, Neutrosophic Intuitionistic Sets, Neutrosophic Paraconsistent Sets, Neutrosophic Faillibilist Sets, Neutrosophic Paradoxist and Pseudo-Paradoxist Sets, Neutrosophic Tautological and Nihilist Sets, Neutrosophic Dialetheist Sets, and Neutrosophic Trivialist Sets. These extensions address highly nuanced aspects of uncertainty, further advancing the theoretical foundation of Neutrosophic mathematics. The eight chapter (MultiNeutrosophic Sets and Refined Neutrosophic Sets) focuses on two advanced Neutrosophic frameworks: MultiNeutrosophic Sets, and Refined Neutrosophic Sets. Using Hyperneutrosophic and nn-SuperHyperneutrosophic Sets, these extensions are analyzed in detail, highlighting their adaptability to multidimensional and complex scenarios. Examples and mathematical properties are provided to showcase their practical relevance and theoretical depth. The ninth chapter (Advanced Hyperneutrosophic Set Types – Type-m, Nonstationary, Subset-Valued, and Complex Refined) explores extensions of the Neutrosophic framework, focusing on: Type-m Neutrosophic Sets, Nonstationary Neutrosophic Sets, Subset-Valued Neutrosophic Sets, and Complex Refined Neutrosophic Sets. These extensions utilize the Hyperneutrosophic and SuperHyperneutrosophic frameworks to address advanced challenges in uncertainty management, expanding their mathematical scope and practical applications. The tenth chapter (Hyperfuzzy Hypersoft Sets and Hyperneutrosophic Hypersoft Sets) integrates the principles of Fuzzy, Neutrosophic, and Soft Sets with hyperstructures to introduce: Hyperfuzzy Hypersoft Sets, and Hyperneutrosophic Hypersoft Sets. These frameworks are designed to manage complex uncertainty through hierarchical structures based on power sets, with detailed analysis of their properties and theoretical potential. The eleventh chapter (A Review of SuperFuzzy, SuperNeutrosophic, and SuperPlithogenic Sets) revisits and extends the study of advanced set concepts such as: SuperFuzzy Sets, Super-Intuitionistic Fuzzy Sets,Super-Neutrosophic Sets, and SuperPlithogenic Sets, including their specialized variants like quadripartitioned, pentapartitioned, and heptapartitioned forms. The work serves as a consolidation of existing studies while highlighting potential directions for future research in hierarchical uncertainty modeling. Focusing on decision-making under uncertainty, the tweve chapter (Advanced SuperHypersoft and TreeSoft Sets) introduces six novel concepts: SuperHypersoft Rough Sets,SuperHypersoft Expert Sets, Bipolar SuperHypersoft Sets, TreeSoft Rough Sets, TreeSoft Expert Sets, and Bipolar TreeSoft Sets. Definitions, properties, and potential applications of these frameworks are explored to enhance the flexibility of soft set-based models. The final chapter (Hierarchical Uncertainty in Fuzzy, Neutrosophic, and Plithogenic Sets) provides a comprehensive survey of hierarchical uncertainty frameworks, with a focus on Plithogenic Sets and their advanced extensions: Hyperplithogenic Sets, SuperHyperplithogenic Sets. It examines relationships with other major concepts such as Intuitionistic Fuzzy Sets, Vague Sets, Picture Fuzzy Sets, Hesitant Fuzzy Sets, and multi-partitioned Neutrosophic Sets, consolidating their theoretical interconnections for modeling complex systems. This volume not only reflects the dynamic interplay between theoretical rigor and practical application but also serves as a beacon for future research in uncertainty modeling, offering advanced tools to tackle the intricacies of modern challenges. (shrink)

An algebra with fuzzy equality is a set with operations on it that is equipped with similarity ≈, i.e. a fuzzy equivalence relation, such that each operation f is compatible with ≈. Described verbally, compatibility says that each f yields similar results if applied to pairwise similar arguments. On the one hand, algebras with fuzzy equalities are structures for the equational fragment of fuzzy logic. On the other hand, they are the formal counterpart to the intuitive (...) idea of having functions that are not allowed to map similar objects to dissimilar ones. In this paper, we present a generalization of the well-known Birkhoff’s variety theorem: a class of algebras with fuzzy equality is the class of all models of a fuzzy set of identities iff it is closed under suitably defined morphisms, substructures, and direct products. (shrink)

Following the idea of L -fuzzy generalized neighborhood systems as introduced by Zhao et al., we will give the join-complete lattice structures of lower and upper approximation operators based on L -fuzzy generalized neighborhood systems. In particular, as special approximation operators based on L -fuzzy generalized neighborhood systems, we will give the complete lattice structures of lower and upper approximation operators based on L -fuzzy relations. Furthermore, if L satisfies the double negative law, then there exists (...) an order isomorphic mapping between upper and lower approximation operators based on L -fuzzy generalized neighborhood systems; when L -fuzzy generalized neighborhood system is serial, reflexive, and transitive, there still exists an order isomorphic mapping between upper and lower approximation operators, respectively, and both lower and upper approximation operators based on L -fuzzy relations are complete lattice isomorphism. (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)

This book is the sixth volume in the series of Collected Papers on Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond. Building upon the foundational contributions of previous volumes, this edition focuses on the exploration and development of Various New Uncertain Concepts, further enriching the study of uncertainty and complexity through innovative theoretical advancements and practical applications. The volume is meticulously organized into 15 chapters, each presenting unique perspectives and contributions to the field. (...) From theoretical explorations to real-world applications, these chapters provide a cohesive and comprehensive overview of the state of the art in uncertain combinatorics, emphasizing the versatility and power of the newly introduced concepts and methodologies. The first chapter (SuperHypertree-depth – Structural Analysis in SuperHyperGraphs) explores the concept of SuperHypertree-depth, an extension of the classical graph parameter Tree-depth and its hypergraph counterpart Hypertree-depth. By introducing hierarchical nesting within SuperHyperGraphs, where both vertices and edges can represent recursive subsets, this study investigates the mathematical properties and structural implications of these extended parameters. The findings highlight the relationships between SuperHypertree-depth and its traditional graph-theoretic equivalents, providing a deeper understanding of their applicability to hierarchical and complex systems. The second chapter (Obstructions for Hypertree-width and SuperHypertree-width) examines the role of ultrafilters as obstructions in determining Hypertree-width and extends the concept to SuperHypertree-width. Building on hypergraph theory, which abstracts traditional graph frameworks into more complex domains, the study investigates how recursive structures within SuperHyperGraphs redefine the computational and structural properties of these parameters. Ultrafilters, with their broad mathematical significance, serve as critical tools for understanding the limitations and potentials of these advanced graph metrics. The third chapter (SuperHypertree-Length and SuperHypertree-Breadth in SuperHyperGraphs) investigates the extension of the graph-theoretic parameters Tree-length and Tree-breadth to the realms of hypergraphs and SuperHyperGraphs. By leveraging the hierarchical nesting of SuperHyperGraphs, the study explores how these parameters adapt to increasingly complex and multi-level structures. Comparative analyses between these extended parameters and their classical counterparts reveal new insights into their relevance and utility in advanced graph and hypergraph theory. Plithogenic Sets, which generalize Fuzzy and Neutrosophic Sets, are extended in the fourth chapter (Extended HyperPlithogenic Sets and Generalized Plithogenic Graphs) to Extended Plithogenic Sets, HyperPlithogenic Sets, and SuperHyperPlithogenic Sets. This study further investigates their application to graph theory through the concepts of Extended Plithogenic Graphs and Generalized Extended Plithogenic Graphs. The chapter provides a concise exploration of these frameworks, offering insights into their potential for addressing uncertainty and complexity in graph structures. Soft Sets provide an effective framework for decision-making by mapping parameters to subsets of a universal set, addressing uncertainty and vagueness. The fifth chapter (Double-Framed Superhypersoft Set and Double-Framed Treesoft Set) introduces the Double-Framed SuperHypersoft Set and the Double-Framed Treesoft Set as extensions of traditional and advanced soft set frameworks, such as Hypersoft and SuperHypersoft Sets. The chapter explores their relationships with existing concepts, offering new tools to handle complex decision-making scenarios with enhanced structural flexibility. The sixth paper (HyperPlithogenic Cubic Set and SuperHyperPlithogenic Cubic Set) introduces the concepts of the HyperPlithogenic Cubic Set and SuperHyperPlithogenic Cubic Set, which extend the Plithogenic Cubic Set by integrating both interval-valued and single-valued fuzzy memberships. These sets leverage multi-attribute aggregation techniques inherent to plithogenic structures, allowing for nuanced representations of uncertainty. Additionally, related constructs such as the HyperPlithogenic Fuzzy Cubic Set, HyperPlithogenic Intuitionistic Fuzzy Cubic Set, and HyperPlithogenic Neutrosophic Cubic Set are explored, further enriching the theoretical and practical applications of this framework. The seventh chapter (L-Neutrosophic Sets and Nonstationary Neutrosophic Sets) extends the foundational concepts of fuzzy sets by integrating Neutrosophic and Plithogenic frameworks. By introducing L-Neutrosophic Sets and Nonstationary Neutrosophic Sets, the study enhances the representation of uncertainty through independent membership components: truth, indeterminacy, and falsity. These advanced constructs also incorporate multi-dimensional and contradictory attributes, providing a robust means of modeling complex decision-making and uncertain data. Plithogenic and Rough Sets, known for generalizing uncertainty modeling and classification, are extended in the eight chapter (Forest HyperPlithogenic and Forest HyperRough Sets) to Forest HyperPlithogenic Sets, Forest SuperHyperPlithogenic Sets, Forest HyperRough Sets, and Forest SuperHyperRough Sets. These frameworks incorporate hierarchical and recursive structures to advance existing set-theoretic paradigms. The chapter explores their applications in multi-level data analysis and uncertainty classification, demonstrating their adaptability to complex systems. Building on Fuzzy, Neutrosophic, and Plithogenic Sets, the tenth chapter (Symbolic HyperPlithogenic Sets) introduces Symbolic HyperPlithogenic Sets and Symbolic n-SuperHyperPlithogenic Sets. These sets incorporate symbolic components and algebraic coefficients, enabling flexible operations within a defined prevalence order. By extending symbolic representation into hyperplithogenic and superhyperplithogenic domains, the chapter opens new pathways for addressing uncertainty and hierarchical complexity in mathematical modeling. Soft Sets, designed to manage uncertainty and imprecision, have evolved through various extensions like Hypersoft Sets and SuperHypersoft Sets. The eleventh chapter (N-SuperHypersoft and Bijective SuperHypersoft Sets) introduces N-SuperHypersoft Sets, N-Treesoft Sets, Bijective SuperHypersoft Sets, and Bijective Treesoft Sets. These new constructs enhance decision-making frameworks by incorporating advanced hierarchical and bijective relationships, building on existing theories and expanding their applications. Plithogenic Sets, known for integrating multi-valued attributes and contradictions, and Rough Sets, which partition data into definable approximations, are combined in the twelfth chapter (Plithogenic Rough Sets) to form Plithogenic Rough Sets. This fusion provides a powerful framework for addressing uncertainty in dynamic and complex decision-making scenarios, offering a novel approach to uncertainty modeling. Expanding on Neutrosophic Sets, which represent truth, indeterminacy, and falsehood, this chapter introduces Plithogenic Duplets and Plithogenic Triplets. These constructs leverage the Plithogenic framework to incorporate attributes, values, and contradiction measures. The thirteenth chapter (Plithogenic Duplets and Triplets) examines their relationships with Neutrosophic Duplets and Triplets, offering new tools for multi-dimensional data representation and decision-making. Building on foundational concepts like Rough Sets and Vague Sets, the fourteenth chapter (SuperRough and SuperVague Sets) introduces SuperRough Sets and SuperVague Sets. These generalized frameworks extend uncertainty modeling by incorporating hierarchical structures. The study also demonstrates that SuperRough Sets can evolve into SuperHyperRough Sets, providing further generalizations for advanced data classification and analysis. The fifteenth chapter (Neutrosophic TreeSoft Expert and ForestSoft Sets) revisits the Neutrosophic TreeSoft Set, which combines the hierarchical structure of TreeSoft Sets with the Neutrosophic framework for uncertainty representation. Additionally, it introduces the Neutrosophic TreeSoft Expert Set, incorporating expert knowledge into the model. The chapter also explores the ForestSoft Set and its extension, the Neutrosophic ForestSoft Set, to provide multi-level, tree-structured approaches for complex data representation and analysis. (shrink)

Although machine intelligence is increasingly employed in healthcare, the realm of decision-making in medical ethics remains largely unexplored from a technical perspective. We propose an approach based on fuzzy cognitive maps (FCMs), which builds on Beauchamp and Childress’ prima-facie principles. The FCM’s weights are optimized using a genetic algorithm to provide recommendations regarding the initiation, continuation, or withdrawal of medical treatment. The resulting model approximates the answers provided by our team of medical ethicists fairly well and offers a (...) high degree of interpretability. Possible applications of such a system include informal guidance on medical ethics dilemmas as well as educational purposes. (shrink)

Influencers belong to the daily media diet of many adolescents. As role models, they have the potential to play a crucial role in the identity construction of their viewers. In the age of social media, such role models may now be found more local – from the same city – and perhaps with more diverse backgrounds. This may be particularly valuable to adolescents growing up in super-diverse cities, as they are surrounded by a multitude of groups and identities during (...) a life phase in which they have to make sense of who they are and where they belong. Despite the heterogeneity of these identities, there is one thing all have in common: the city they live in. With the city as a common framework, local influencers may be important role models for these adolescents, particularly in negotiating their urban identity. This paper aims toward mapping the ways in which social media can play a role in the negotiation of urban identity among youngsters by investigating how YouTube influencers from a super-diverse city are related to each other online, and how their content relates to the (super-diverse) city of Rotterdam. Findings show that in their videos and on their channel pages, influencers mainly affiliate themselves with the city through having the city as the background and context of the videos, through their involvement with cultural trends (e.g. soccer, hip-hop) that link to the city, and through their affiliation with other local influencers. We argue that influencers may therefore provide their viewers with content that may potentially help their local viewers strengthen their urban identity. (shrink)

AbstarctBoth the superhero and the supervillain operate outside the law. The former replaces law with a form of substantive justice while the latter seeks to invert or overturn the law in favour of a new grundnorm that best serves their vision for how society should operate. In this paper I consider what this prefix “super” really means in relation to these two classes, drawing on Nietzsche’s original definition of the ubermensch and its relationship to legal concepts such as the (...) state of exception, sovereign power and substantive justice. These relationships will be mapped across a number of examples from the Marvel Studios films—paying particular attention to the antagonists in these movies, both the state and the supervillains, arguing that it is these antagonists who are most illustrative of these concepts in play. (shrink)

The developing academic field of machine ethics seeks to make artificial agents safer as they become more pervasive throughout society. In contrast to computer ethics, machine ethics is concerned with the behavior of machines toward human users and other machines. This study aims to use an action-based ethical theory founded on the combinational aspects of deontological and teleological theories of ethics in the construction of an artificial moral agent (AMA).,The decision results derived by the AMA are acquired via fuzzy (...) logic interpretation of the relative values of the steady-state simulations of the corresponding rule-based fuzzy cognitive map (RBFCM).,Through the use of RBFCMs, the following paper illustrates the possibility of incorporating ethical components into machines, where latent semantic analysis (LSA) and RBFCMs can be used to model dynamic and complex situations, and to provide abilities in acquiring causal knowledge.,This approach is especially appropriate for data-poor and uncertain situations common in ethics. Nonetheless, to ensure that a machine with an ethical component can function autonomously in the world, research in artificial intelligence will need to further investigate the representation and determination of ethical principles, the incorporation of these ethical principles into a system’s decision procedure, ethical decision-making with incomplete and uncertain knowledge, the explanation for decisions made using ethical principles and the evaluation of systems that act based upon ethical principles.,To date, the conducted research has contributed to a theoretical foundation for machine ethics through exploration of the rationale and the feasibility of adding an ethical dimension to machines. Further, the constructed AMA illustrates the possibility of utilizing an action-based ethical theory that provides guidance in ethical decision-making according to the precepts of its respective duties. The use of LSA illustrates their powerful capabilities in understanding text and their potential application as information retrieval systems in AMAs. The use of cognitive maps provides an approach and a decision procedure for resolving conflicts between different duties.,This paper suggests that cognitive maps could be used in AMAs as tools for meta-analysis, where comparisons regarding multiple ethical principles and duties can be examined and considered. With cognitive mapping, complex and abstract variables that cannot easily be measured but are important to decision-making can be modeled. This approach is especially appropriate for data-poor and uncertain situations common in ethics. (shrink)

Farmers are key actors in the transition towards sustainable agricultural practices. Therefore, it is important to understand farmers’ motivations to encourage lasting change. This study investigated how farmers from the two different social contexts of the northeast US and Denmark perceive sustainability. Twenty farmers constructed Fuzzy Cognitive Maps to model their practices and perceived outcomes. The maps were analyzed using social capital as an analytical framework. The results showed that sustainability perceptions differed between US and Danish farmers. (...) Specifically, Danish farmers focused mainly on environmental sustainability while US farmers distributed their focus more evenly across environmental, social, and economic factors. Further, US and Danish farmers had different notions of community engagement. US farmers emphasized the importance of interdependence with their non-farmer community members, where farmers contributed to the wellbeing and livelihoods of non-farmers and vice versa. For Danish farmers, the importance of community engagement lay mainly in localizing food systems, the public image of farming, and the cultural value of farms. Differences in the dominant types of social capital (farmer-to-government vs. farmer-to-non-farmer-community) can explain this pattern. We argue that understanding social capital and primary level of influence can lead to more effective and efficient policies. (shrink)

In this book we study the concepts of Fuzzy Cognitive Maps and their Neutrosophic analogue, the Neutrosophic Cognitive Maps.

Fuzzy cognitive maps is a system modeling methodology which applies mostly in complex dynamic systems by describing causal relationships that exist between its parameters called concepts. Fuzzy cognitive map theories have been used in many applications but they present several drawbacks and deficiencies. These limitations are addressed and analyzed fuzzy cognitive map theories are readdressed. A new novel approach in modelling fuzzy cognitive maps is proposed to increase the knowledge of the system and overcome (...) some of its limitations. The state space approach is used for the new model to disaggregate the concepts into different categories. The disaggregation of the concepts into state concepts, input concepts and output concepts is mathematically formulated. The proposed method and the new model is used for the calculation of a building’s energy consumption and the management of its load. Simulations are performed as a case study testing the new proposed method. The problem of the high energy consumption of the building sector is studied using the new fuzzy cognitive map model. Discussions of the obtained results along with future research directions are provided. (shrink)

The concept of supermatrix for social scientists was first introduced by Paul Horst. The main purpose of his book was to introduce this concept to social scientists, students, teachers and research workers who lacked mathematical training. This book introduces the concept of fuzzy super matrices and operations on them. The author has provided only those operations on fuzzy supermatrices that are essential for developing super fuzzy multi expert models. This book will be highly useful to (...) social scientists who wish to work with multi expert models. (shrink)

If X is set and L a lattice, then an L-subset or fuzzy subset of X is any map from X to L, [11]. In this paper we extend some notions of recursivity theory to fuzzy set theory, in particular we define and examine the concept of almost decidability for L-subsets. Moreover, we examine the relationship between imprecision and decidability. Namely, we prove that there exist infinitely indeterminate L-subsets with no more precise decidable versions and classical subsets whose (...) unique shaded decidable versions are the L-subsets almost-everywhere indeterminate. (shrink)

. With an overall objective of establishing association between air pollutants and incidence of respiratory diseases, the environmental professionals and medical practitioners have made significant contribution, using statistical mechanics in modelling epidemiological data, population characteristics, and pollution parameters. Broadly speaking, the studies have shown that the increase in vehicular traffic has been one of the causes of respiratory diseases. However, the WHO Centre for Environment and Health, Europe in its 2005 document states: “There is little evidence for a causal relationship (...) between asthma prevalence/incidence and air pollution in general, though the evidence is suggestive of a causal association between the prevalence/incidence of asthma symptoms and living in close proximity to traffic”. Decision making process in a real world is invariably based on perceptions which are expressed in words or may be in numeric terms and not in probability terms. In the paper, we made an attempt to model the perceptions of experienced pulmonologists in arriving at their combined degree of belief/plausibility/ignorance for all the possible combinations of identified respiratory diseases, using evidence theory and fuzzy relational calculus without collecting sizeable parametric data accumulated over a period of years. Tightening pollution norms by the regulatory authorities is an overall objective of the global efforts on greenhouse gases reduction in general, and air pollution mitigation in particular. The concept of solar battery operated electric vehicles for road transport is advocated, initially for two/three wheelers, and extending it to four wheelers, especially in the developing countries. (shrink)

Goguen categories extend the relational calculus and its categorical formalization to the fuzzy world. Starting from the fundamental concepts of sets, binary relations and lattices, this book introduces several categorical formulations of an abstract theory of relations such as allegories, Dedekind categories and related structures. It is shown that neither theory is sufficiently rich to describe basic operations on fuzzy relations.

In a supply chain system, the prices with which the suppliers supply its local commodity to the retailers should satisfy the requirements of the retailers and the consumers. The supply and demand scheme satisfying these requirements is reduced into fuzzy relation inequalities with min-product composition. Due to the difference between the min-product composition and the classical max-t-norm one, we first study the resolution of such min-product FRI system. For optimization management in the supply chain system, we further investigate a (...) maximin programming problem subject to the min-product FRIs. An algorithm is proposed to obtain the optimal solution based on the quasi-maximal matrix and corresponding index set. To illustrate the efficiency of our proposed algorithm, we provide a simple numerical example. The obtained optimal solution reflects an optimal pricing scheme, which maximizes the minimum prices of the commodity from the suppliers. (shrink)

Over a decade after publication of Thinking Again: Education After Postmodernism (1998) contention still emerges among Foucaultians over whether discursively made‐up things really exist, and whether removal of the constituent subject leaves room for agency within techniques of caring for the self. That these questions are kept alive shows that some readers have not rethought Foucault, finding what possibly comes after postmodernism. Using Wittgenstein to ‘reciprocally illuminate’ Foucault (after Tully and Marshall), I open teacher inspection and reforms to problematization, as (...) relations to bedrock rules governing games of truth. ‘How, upon entering classrooms, do inspectors know “teaching” is taking place and not crazy and fuzzy things in its name?’ Taking up Hirst's vexing question, I move beyond liberal‐analytic concept‐mapping and neo‐liberal individualism to more fully assay the political ground for judging teaching practices through genealogy. Epistemological, political and ethical concerns intersect as we approach the problem through Foucault's three axes of an historical‐ontology of the present: knowledge(s), power relations, and arts of the self. Drawing on recent Governmentality Studies in Education (Peters., 2009), we aver the impasses of postmodern relativism while finding limited ranges of agency along each axis, as teachers practice freedoms by critiquing and renegotiating rules. (shrink)

This paper addresses a set-theoretic completeness based on a relational semantics for fuzzy extensions of two versions Rt and R T of R (Relevance logic). To this end, two fuzzy logics FRt and FRT as extensions of Rt and R T, respectively, and the relational semantics, so called Routley-Meyer semantics, for them are first recalled. Next, on the semantics completeness results are provided for them using a set-theoretic way.

Modelling with Words is an emerging modelling methodology closely related to the paradigm of Computing with Words introduced by Lotfi Zadeh. This book is an authoritative collection of key contributions to the new concept of Modelling with Words. A wide range of issues in systems modelling and analysis is presented, extending from conceptual graphs and fuzzy quantifiers to humanist computing and self-organizing maps. Among the core issues investigated are - balancing predictive accuracy and high level transparency in learning (...) - scaling linguistic algorithms to high-dimensional data problems - integrating linguistic expert knowledge with knowledge derived from data - identifying sound and useful inference rules - integrating fuzzy and probabilistic uncertainty in data modelling. (shrink)

Since the 1990s the so-called transparency of experience has played a crucial role in core debates in philosophy of mind. However, recent developments in the literature have made transparency itself quite opaque. The very idea of transparent experience has become quite fuzzy, due to the articulation of many different notions of transparency and transparency theses. Absent a unified logical space where these notions and theses can be mapped and confronted, we are left with an overall impression of conceptual chaos. (...) This is a problem, given the constant and ubiquitous references to transparency in the literature and its prominent position in the contemporary philosophy of mind. My goal in this paper is to restore clarity through proper analysis of the mutual relations between the different transparency theses. This allows me to uncover a unitary multidimensional logical space where existing (as well as possible) views can be properly singled out and located. (shrink)

Clark & Thornton have demonstrated the paradox between the opacity of the transformations that underlie relational mappings and the ease with which people learn such mappings. However, C&T's trading-spaces proposal resolves the paradox only in the broadest outline. The general-purpose algorithm promised by C&T remains to be developed. The strategy of doing so is to analyze and formulate computational mechanisms for known cases of recoding.

This paper proposes a novel adaptive fuzzy super-twisting sliding mode control scheme for microgyroscopes with unknown model uncertainties and external disturbances. Firstly, an adaptive algorithm is used to estimate the unknown parameters and angular velocity of microgyroscopes. Secondly, in order to improve the performance of the system and the superiority of the super-twisting algorithm, this paper utilizes the universal approximation characteristic of the fuzzy system to approach the gain of the super-twisting sliding mode controller and (...) identify the gain of the controller online, realizing the adaptive adjustment of the controller parameters. Simulation results verify the superiority and the effectiveness of the proposed approach, compared with adaptive super-twisting sliding mode control without fuzzy approximation; the proposed method is more effective. (shrink)

It is proposed to analyze fuzzy concepts and relations between them in the frame of triplet concept modeling. Fuzzy concepts are introduced by means of the so-called fuzzification of dichotomous concepts. The cognitive and psychological aspects of concept possession are separated and studied.

The paper explicates the stages of the author’s philosophical evolution in the light of Kopnin’s ideas and heritage. Starting from Kopnin’s understanding of dialectical materialism, the author has stated that category transformations of physics has opened from conceptualization of immutability to mutability and then to interaction, evolvement and emergence. He has connected the problem of physical cognition universals with an elaboration of the specific system of tools and methods of identifying, individuating and distinguishing objects from a scientific theory domain. The (...) role of vacuum conception and the idea of existence (actual and potential, observable and nonobservable, virtual and hidden) types were analyzed. In collaboration with S.Crymski heuristic and regulative functions of categories of substance, world as a whole as well as postulates of relativity and absoluteness, and anthropic and self-development principles were singled out. Elaborating Kopnin’s view of scientific theories as a practically effective and relatively true mapping of their domains, the author in collaboration with M. Burgin have originated the unified structure-nominative reconstruction (model) of scientific theory as a knowledge system. According to it, every scientific knowledge system includes hierarchically organized and complex subsystems that partially and separately have been studied by standard, structuralist, operationalist, problem-solving, axiological and other directions of the current philosophy of science. 1) The logico-linguistic subsystem represents and normalizes by means of different, including mathematical, languages and normalizes and logical calculi the knowledge available on objects under study. 2) The model-representing subsystem comprises peculiar to the knowledge system ways of their modeling and understanding. 3) The pragmatic-procedural subsystem contains general and unique to the knowledge system operations, methods, procedures, algorithms and programs. 4) From the viewpoint of the problem-heuristic subsystem, the knowledge system is a unique way of setting and resolving questions, problems, puzzles and tasks of cognition of objects into question. It also includes various heuristics and estimations (truth, consistency, beauty, efficacy, adequacy, heuristicity etc) of components and structures of the knowledge system. 5) The subsystem of links fixes interrelations between above-mentioned components, structures and subsystems of the knowledge system. The structure-nominative reconstruction has been used in the philosophical and comparative case-studies of mathematical, physical, economic, legal, political, pedagogical, social, and sociological theories. It has enlarged the collection of knowledge structures, connected, for instance, with a multitude of theoreticity levels and with an application of numerous mathematical languages. It has deepened the comprehension of relations between the main directions of current philosophy of science. They are interpreted as dealing mainly with isolated subsystems of scientific theory. This reconstruction has disclosed a variety of undetected knowledge structures, associated also, for instance, with principles of symmetry and supersymmetry and with laws of various levels and degrees. In cooperation with the physicist Olexander Gabovich the modified structure-nominative reconstruction is in the processes of development and justification. Ideas and concepts were also in the center of Kopnin’s cognitive activity. The author has suggested and elaborated the triplet model of concepts. According to it, any scientific concept is a dependent on cognitive situation, dynamical, multifunctional state of scientist’s thinking, and available knowledge system. A concept is modeled as being consisted from three interrelated structures. 1) The concept base characterizes objects falling under a concept as well as their properties and relations. In terms of volume and content the logical modeling reveals partially only the concept base. 2) The concept representing part includes structures and means (names, statements, abstract properties, quantitative values of object properties and relations, mathematical equations and their systems, theoretical models etc.) of object representation in the appropriate knowledge system. 3) The linkage unites a structures and procedures that connect components from the abovementioned structures. The partial cases of the triplet model are logical, information, two-tired, standard, exemplar, prototype, knowledge-dependent and other concept models. It has introduced the triplet classification that comprises several hundreds of concept types. Different kinds of fuzziness are distinguished. Even the most precise and exact concepts are fuzzy in some triplet aspect. The notions of relations between real scientific concepts are essentially extended. For example, the definition and strict analysis of such relations between concepts as formalization, quantification, mathematization, generalization, fuzzification, and various kinds of identity are proposed. The concepts «PLANET» and «ELEMENTARY PARTICLE» and some of their metamorphoses were analyzed in triplet terms. The Kopnin’s methodology and epistemology of cognition was being used for creating conception of the philosophy of law as elaborating of understanding, justification, estimating and criticizing legal system. The basic information on the major directions in current Western philosophy of law (legal realism, feminism, criticism, postmodernism, economical analysis of law etc.) is firstly introduced to the Ukrainian audience. The classification of more than fifty directions in modern legal philosophy is suggested. Some results of historical, linguistic, scientometric and philosophic-legal studies of the present state of Ukrainian academic science are given. (shrink)

While the past century of neuroscientific research has brought considerable progress in defining the boundaries of the human cerebral cortex, there are cases in which the demarcation of one area from another remains fuzzy. Despite the existence of clearly demarcated areas, examples of gradual transitions between areas are known since early cytoarchitectonic studies. Since multi-modal anatomical approaches and functional connectivity studies brought renewed attention to the topic, a better understanding of the theoretical and methodological implications of fuzzy boundaries (...) in brain science can be conceptually useful. This article provides a preliminary conceptual framework to understand this problem by applying philosophical theories of vagueness to three levels of neuroanatomical research. For the first two levels (cytoarchitectonics and fMRI studies), vagueness will be distinguished from other forms of uncertainty, such as imprecise measurement or ambiguous causal sources of activation. The article proceeds to discuss the implications of these levels for the anatomical study of connectivity between cortical areas. There, vagueness gets imported into connectivity studies since the network structure is dependent on the parcellation scheme and thresholds have to be used to delineate functional boundaries. Functional connectivity may introduce an additional form of vagueness, as it is an organizational principle of the brain. The article concludes by discussing what steps are appropriate to define areal boundaries more precisely. (shrink)

ABSTRACT In this work the connections between the fuzzy closure operators and the graded consequence relations are examined Namely, as it is well known, in the crisp case there is a complete equivalence between the notion of closure operator and the one of consequence relation. We extend this result by proving that the graded consequence relations are related to a particular class of fuzzy closure operators, namely the class of fuzzy closure operators that can be obtained by (...) a chain of classical closure operators. (shrink)

The traditional soft set is a mapping from a parameter set to family of all crisp subsets of a universe. Molodtsov introduced the soft set as a generalized tool for modelling complex systems involving uncertain or not clearly defined objects. In this paper, the notion of neutrosophic soft set is reanalysed. The novel theory is a combination of neutrosophic set theory and soft set theory. The complement, “and”, “or”, intersection and union operations are defined on the neutrosophic soft sets. The (...) neutrosophic soft relations accompanied with their compositions are also defined. The basic properties of the neutrosophic soft sets, neutrosophic soft relations and neutrosophic soft compositions are also discussed. (shrink)

Actions can be physically executed, observed, imagined, or simply thought about. Unifying mental processes, such as simulation, emulation, or predictive processing, are thought to underlie different action types, whether they are mental states, as in the case of motor imagery and action observation, or involve physical execution. While overlapping brain activity is typically observed across different actions which indicates commonalities, research interest is also concerned with investigating the distinct functional components of these action types. Unfortunately, untangling subtleties associated with the (...) neurocognitive bases of different action types is a complex endeavour due to the high dimensional nature of their neural substrate (e.g., any action process is likely to activate multiple brain regions thereby having multiple dimensions to consider when comparing across them). This has impeded progress in action-related theorising and application. The present study addresses this challenge by using the novel approach of multidimensional modeling to reduce the high-dimensional neural substrate of four action-related behaviours (motor imagery, action observation, action-related language, and action execution), find the least number of dimensions that distinguish or relate these action types, and characterise their neurocognitive relational links. Data for the model comprised brain activations for action types from whole-brain analyses reported in 53 published articles. Eighty-two dimensions (i.e., 82 brain regions) for the action types were reduced to a three-dimensional model, that mapped action types in ordination space where the greater the distance between the action types, the more dissimilar they are. A series of one-way ANOVAs and post-hoc comparisons performed on the mean coordinates for each action type in the model showed that across all action types, action execution and concurrent action observation (AO)-motor imagery (MI) were most neurocognitively similar, while action execution and AO were most dissimilar. Most action types were similar on at least one neurocognitive dimension, the exception to this being action-related language. The import of the findings are discussed in terms of future research and implications for application. (shrink)

We consider countable so‐called rich subsemigroups of ; each such semigroup T gives a variety CPEAT that is axiomatizable by a finite schema of equations taken in a countable subsignature of that of ω‐dimensional cylindric‐polyadic algebras with equality where substitutions are restricted to maps in T. It is shown that for any such T, if and only if is representable as a concrete set algebra of ω‐ary relations. The operations in the signature are set‐theoretically interpreted like in polyadic equality (...) set algebras, but such operations are relativized to a union of cartesian spaces that are not necessarily disjoint. This is a form of guarding semantics. We show that CPEAT is canonical and atom‐canonical. Imposing an extra condition on T, we prove that atomic algebras in CPEAT are completely representable and that CPEAT has the super amalgamation property. If T is rich and finitely represented, it is shown that CPEAT is term definitionally equivalent to a finitely axiomatizable Sahlqvist variety. Such semigroups exist. This can be regarded as a solution to the central finitizability problem in algebraic logic for first order logic with equality if we do not insist on full fledged commutativity of quantifiers. The finite dimensional case is approached from the view point of guarded and clique guarded (relativized) semantics of fragments of first order logic using finitely many variables. Both positive and negative results are presented. (shrink)

Three experiments investigated whether the similarity of relational structures influences the interpretation of spatial representations. Adults were shown diagrams of hand gestures paired with simple statements and asked to judge the meaning of new gestures. In Experiment 1 the gestures were paired with active declarative statements. In Experiment 2, the gestures were paired with conjunctive and disjunctive relations. Experiment 3 used statements similar to those used in Experiment 1, but eliminated the initial object‐to‐object mapping provided in Experiments 1 and (...) 2. In all three experiments, most participants chose an interpretation that set up a parallel relational structure between the gesture and its meaning: spatial elements were paired with conceptual elements and spatial relations were paired with conceptual relations. These results are consistent with the hypothesis that similarity of relational structures influences spatial reasoning, and have implications for analogical reasoning, diagrammatic reasoning, and language processing. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Age and Ageing in Contemporary Speculative and Science Fiction by Sarah Falcus and Maricel Oró-PiquerasMariana Batista da CruzSarah Falcus and Maricel Oró-Piqueras, eds. Age and Ageing in Contemporary Speculative and Science Fiction. London: Bloomsbury, 2023. 248 pp., hardcover, $115.00. ISBN 9781350230668.The pervasiveness of questions of temporality, futurity, and immortality in science and speculative fiction opens new perspectives on aging and generationality. However, despite the potential of these genres (...) to illuminate alternative ways of thinking about the human being in time, there has been a clear tendency within the field of aging studies to favor the analysis of realist narratives. To be sure, Age and Ageing in Contemporary Speculative and Science Fiction does not venture into completely uncharted territory. Teresa Mangum's study of "rejuvenation narratives," Andreu Domingo's conceptualization of "demodystopias," and Sarah Falcus's analysis of generational anachronism in dystopian novels have attempted to fill related gaps in aging studies.1 Yet Sarah Falcus and Maricel Oró-Piqueras's edited collection of essays is the first extensive study of speculative and science fiction (SF) as cultural productions that encode prevalent concerns about age and aging societies.Focusing on texts from Europe, North America, and South Asia, with particular emphasis on the anglophone world, the scope of the collection is limited but its contributions are rich in the range of themes that they address. This broad variety, touching on questions of genre, immortality, bio-power, demographic change, temporalities, and transhumanism, renders the book relevant not only to those working within the field of aging studies, in [End Page 266] particular scholars of SF and those more generally interested in representations of futurities.The collection is divided into two parts: The first part consists of six essays, each of which analyzes a range of SF narratives and identifies "genre-based trends in the representation of age and ageing" (4); the second part presents five detailed case studies that consider specific SF texts. This structure suits the book's goal of mapping the intersections between studies of SF and aging studies. The first part illuminates recurrent patterns in the representation of aging in works of SF, while the cases analyzed in the second part demonstrate the range of creative and aesthetic possibilities that SF affords to reimagine human temporality. The two parts contain chapters on literary as well as audiovisual texts, including essays on film, TV, and a play. The latter are a valuable addition to the collection, not only, as the editors point out, due to "the importance of the visual medium to SF/speculative writing" (4), but also given the popularity and reach of these textual modes, which establish SF as particularly influential forms of cultural representation.The chapters in the first half of the book center around the topic of longevity, exploring narratives that capture hopes and anxieties connected to the possibility of prolonging human life. Teresa Botelho's essay on the opportunities that science fiction provides to reimagine humanity's "biological destiny" (10) opens the collection well, as it introduces topics that are further developed in subsequent chapters. Focusing on texts that construct transhumanist scenarios where enhancement technologies have made rejuvenation, immortality, or life extension tangible realities, this chapter considers how SF narratives have questioned the desirability and political implications of these changes to the human life span. Peter Goggin and Ulla Kriebernegg study age regression and progression as narrative devices to show how popular SF films and TV series have assigned meanings to old age and given expression to dominant concerns regarding aging. The narratives that they analyze have done little to subvert dominant cultural assumptions about old age, and older women in particular continue to be underrepresented in SF.Generationalism is the focus of Falcus and Oró-Piqueras's discussion of anglophone longevity narratives, one of the highlights of the first part of the book. This chapter examines literary and audiovisual texts that project future worlds where the aging process has been successfully halted or delayed: the novels Super Sad True Love Story by Gary Shteyngart (2010), The Postmortal by Drew Magary (2011), and Everything Belongs to the Future by Laurie Penny (2016), as well as the film... (shrink)

A theory of analogy must describe how the meaning of an analogy is derived from the meanings of its parts. In the structure‐mapping theory, the interpretation rules are characterized as implicit rules for mapping knowledge about a base domain into a target domain. Two important features of the theory are (a) the rules depend only on syntactic properties of the knowledge representation, and not on the specific content of the domains; and (b) the theoretical framework allows analogies to be distinguished (...) cleanly from literal similarity statements, applications of abstractions, and other kinds of comparisons.Two mapping principles are described: (a) Relations between objects, rather than attributes of objects, are mapped from base to target; and (b) The particular relations mapped are determined by systematicity, as defined by the existence of higher‐order relations. (shrink)

The concept of Galois connection between power sets is generalized from the point of view of fuzzy logic. Studied is the case where the structure of truth values forms a complete residuated lattice. It is proved that fuzzy Galois connections are in one-to-one correspondence with binary fuzzy relations. A representation of fuzzy Galois connections by Galois connections is provided.

This study examined the metaphorical expressions of death in Chinese quadrisyllabic idioms. Specifically, the research investigated the cultural connotations and implications conveyed through death-related idioms by analyzing metaphorical examples. Adopting the ‘Great Chain of Being’ framework (Lakoff and Turner, 1989), a total of 579 death-related idioms with metaphorical and euphemistic meanings were classified and examined. These idioms were further categorized based on the gender of the deceased as well as the initial metaphors and expressions employed. Our research findings highlight cultural (...) parallels evident in death-related idiomatic expressions. Both English and Chinese languages employ metaphors like “DEATH IS SLEEP” and “DEATH IS DEPARTURE.” These pervasive metaphors can be attributed to shared aspects of the human experience. Meanwhile, our findings also revealed a strong intertwining of metaphors and culture within death-related idioms, such that the metaphor “TO DIE IS TO FALL” is frequently used in Mandarin but is not as common in English. The metaphorical mapping mechanisms employed in these idioms reflect the Chinese way of projecting thoughts onto objects and highlight the unique features of the Chinese language. Chinese culture places great emphasis on social classes, relationships, and hierarchical structures. These notions are not only reflected in social etiquette and how one handles interactions, but are also evident in euphemisms and idioms associated with death. (shrink)

I discuss the attitude of Jewish law sources from the 2nd–:5th centuries to the imprecision of measurement. I review a problem that the Talmud refers to, somewhat obscurely, as impossible reduction. This problem arises when a legal rule specifies an object by referring to a maximized measurement function, e.g., when a rule applies to the largest part of a divided whole, or to the first incidence that occurs, etc. A problem that is often mentioned is whether there might be hypothetical (...) situations involving more than one maximal value of the relevant measurement and, given such situations, what is the pertinent legal rule. Presumption of simultaneous occurrences or equally measured values are also a source of embarrassment to modern legal systems, in situations exemplified in the paper, where law determines a preference based on measured values. I contend that the Talmudic sources discussing the problem of impossible reduction were guided by primitive insights compatible with fuzzy logic presentation of the inevitable uncertainty involved in measurement. I maintain that fuzzy models of data are compatible with a positivistic epistemology, which refuses to assume any precision in the extra-conscious world that may not be captured by observation and measurement. I therefore propose this view as the preferred interpretation of the Talmudic notion of impossible reduction. Attributing a fuzzy world view to the Talmudic authorities is meant not only to increase our understanding of the Talmud but, in so doing, also to demonstrate that fuzzy notions are entrenched in our practical reasoning. If Talmudic sages did indeed conceive the results of measurements in terms of fuzzy numbers, then equality between the results of measurements had to be more complicated than crisp equations. The problem of impossible reduction could lie in fuzzy sets with an empty core or whose membership functions were only partly congruent. Reduction is impossible may thus be reconstructed as there is no core to the intersection of two measures. I describe Dirichlet maps for fuzzy measurements of distance as a rough partition of the universe, where for any region A there may be a non-empty set of - _A, where the problem of impossible reduction applies. This model may easily be combined with probabilistic extention. The possibility of adopting practical decision standards based on -cuts is discussed in this context. I propose to characterize the uncertainty that was presumably capped by the old sages as U- uncertainty, defined, for a non-empty fuzzy set A on the set of real numbers, whose -cuts are intervals of real numbers, as U = 1/h 0 h log [1+]d, where h is the largest membership value obtained by any element of A and is the measure of the -cut of A defined by the Lebesge integral of its characteristic function. (shrink)

In a legal expert system based on CBR (Case-Based Reasoning), legal statute rules are interpreted on the basis of precedents. This interpretation, because of its vagueness and uncertainty of the interpretation cannot be handled with the means used for crisp cases. In our legal expert system, on the basis of the facts of precedents, the statute rule is interpreted as a form of case rule, the application of which involves the concepts of membership and vagueness. The case rule is stored (...) in a data base by means of fuzzy frames. The inference based on a case rule is made by fuzzy YES and fuzzy NO, and the degree of similarity of cases. The system proposed here will be used for legal education; its main area of application is contract, especially in relation to the United Nations Convention on Contracts for the International Sale of Goods (CISG). (shrink)