Florentin Smarandache

This edited volume presents recent theoretical advances, methodological innovations, and interdisciplinary applications in the rapidly evolving field of neutrosophic theories and applications, with particular emphasis on uncertainty modelling, indeterminacy representation, and multi-criteria decision-making (MCDM). Building upon the foundational principles of neutrosophic set theory (NST) introduced by Florentin Smarandache, the volume explores the integration of neutrosophic approaches with Pythagorean fuzzy sets, Fermatean fuzzy sets, rough set theory, pentapartitioned neutrosophic structures, and continuum uncertainty frameworks to address complex real-world problems characterized by vagueness, incompleteness, inconsistency, and contradictory information. The book encompasses diverse theoretical contributions, including neutrosophic number–based machine learning, learning analytics, information geometry, graph theory, topological structures, self-reference stability, algebraic systems, and Gödel-type incompleteness extensions, alongside novel aggregation operators and advanced decision-making methodologies. Several chapters develop innovative MCDM and MADM frameworks in rough neutrosophic, interval rough neutrosophic, trapezoidal neutrosophic, Fermatean neutrosophic, and pentapartitioned neutrosophic environments, demonstrating applications in engineering, biomedical sciences, environmental sustainability, healthcare, artificial intelligence, education, manufacturing systems, and sustainable waste management. By integrating mathematical rigor with practical applicability, this volume advances the theoretical foundations of neutrosophic science while offering robust analytical tools for reasoning under uncertainty across multidisciplinary domains. The book serves as a valuable resource for researchers, academicians, practitioners, and graduate students working in fuzzy systems, artificial intelligence, computational intelligence, decision sciences, mathematical modelling, and uncertainty-aware systems.

This volume presents an Assamese translation of selected one-line poems by Florentin Smarandache, a Romanian-American writer, mathematician, and professor of science and mathematics at the University of New Mexico. The collection introduces Assamese readers to the literary form of one-line poetry (eshoriya kobita), a concise poetic genre that has gained recognition in Western literature but remains relatively unexplored in Assamese literary traditions. The poems, translated from English into Assamese by Dr. Ranjit Bhattacharyya, preserve the philosophical depth, irony, existential reflection, emotional resonance, and minimalist aesthetics characteristic of Smarandache’s poetic style. Through linguistic adaptation and cultural mediation, this work creates a bridge between global experimental poetry and Assamese literary expression, expanding the scope of contemporary Assamese poetic discourse. The translation also contributes to intercultural literary exchange by making internationally recognized minimalist poetry accessible to Assamese-speaking readers.

Este libro desarrolla un análisis comparativo integral de la plitogenia como un marco metafilosófico capaz de interpretar las relaciones entre diversas tradiciones filosóficas. La plitogenia, formulada originalmente en el ámbito de la lógica matemática, describe la formación y evolución de entidades mediante la interacción de múltiples elementos que pueden ser compatibles, contradictorios o neutrales. Al extender este concepto formal al campo de la filosofía, la obra examina cómo los sistemas filosóficos pueden entenderse no como doctrinas aisladas o mutuamente excluyentes, sino como estructuras conceptuales en interacción dentro de un campo más amplio de pluralidad intelectual. El estudio explora sistemáticamente las conexiones entre la plitogenia y los principales movimientos filosóficos, incluyendo el empirismo, el racionalismo, el idealismo, el realismo, la fenomenología y el epifenomenalismo, así como teorías éticas como la deontología, el utilitarismo y el hedonismo. Asimismo, aborda el posmodernismo, el pragmatismo y dominios contemporáneos como la ética digital, la bioética, la filosofía ambiental y la teoría de la justicia. Una contribución distintiva de esta edición ampliada reside en la integración de tradiciones filosóficas latinoamericanas e indígenas, incluyendo conceptos como yanantin, ch’ixi, pachakuti, ayni, Sumak Kawsay (Buen Vivir) y teotl, destacando sus afinidades estructurales con el razonamiento plitogénico. Mediante el empleo de una metodología comparativa consistente —centrada en similitudes, diferencias y conexiones conceptuales— el libro demuestra que la pluralidad filosófica puede modelarse como un sistema dinámico de perspectivas en interacción. La plitogenia se presenta así no como un reemplazo de las filosofías existentes, sino como una lente analítica unificadora que posibilita el diálogo entre tradiciones preservando al mismo tiempo sus diferencias. La obra contribuye a la filosofía contemporánea al ofrecer un marco adecuado para abordar la complejidad, la interdisciplinariedad y la coexistencia de epistemologías diversas tanto en contextos teóricos como aplicados.

The Honest Classroom explores how education must evolve in response to the widespread adoption of artificial intelligence and the emergence of highly fluent but epistemically unreliable AI systems. The book argues that traditional educational approaches are insufficient for preparing students to critically evaluate AI-generated content because they lack a formal framework for representing uncertainty, contradiction, and ignorance. Drawing on neutrosophic logic, the authors introduce the Truth–Indeterminacy–Falsity (T, I, F) framework as a practical and theoretical tool for navigating complex knowledge environments where confidence does not guarantee accuracy. The work examines the cognitive biases that influence human reasoning, demonstrates how AI systems can amplify these biases, and proposes pedagogical strategies that cultivate epistemic responsibility, critical thinking, and intellectual honesty. Through educational models, assessment instruments, AI tutoring frameworks, and classroom applications, the book develops the concept of the “Honest Classroom,” an educational environment that treats indeterminacy as a legitimate and valuable component of learning. By integrating neutrosophic logic into teaching, assessment, and AI-assisted learning, the book provides educators, researchers, and policymakers with a comprehensive framework for fostering informed judgment, responsible AI use, and resilient epistemic practices in the age of confident machines.

This book presents a comprehensive study of topological indices in the framework of single-valued neutrosophic graphs, extending classical graph-theoretical concepts to environments characterized by uncertainty, indeterminacy, and inconsistency. Building upon the foundations of graph theory and neutrosophic set theory, the authors define and analyze several important neutrosophic topological indices, including the connectivity index, Wiener index, and Sombor index. The work develops formal definitions, computational methods, theoretical properties, bounds, and proofs related to these indices within neutrosophic graph structures. Special attention is given to connected neutrosophic graphs, neutrosophic trees, cycles, stars, and super hyper power graphs. Through detailed examples and mathematical demonstrations, the book illustrates how neutrosophic representations provide a more realistic modeling framework for complex systems affected by ambiguous or conflicting information. The volume also explores applications of neutrosophic graph indices in network analysis, decision-making systems, and complex relational structures. Designed for researchers and graduate students in mathematics, computer science, engineering, and data science, this work contributes to the advancement of neutrosophic graph theory by establishing a rigorous foundation for future research on uncertainty-aware topological analysis and generalized graph invariants.

This book introduces Type-k Neutrosophic Sets, a recursive generalization of classical neutrosophic sets designed to represent nested and higher-order uncertainty. Unlike traditional neutrosophic frameworks, where truth, indeterminacy, and falsity are represented as independent scalar values, the Type-k model recursively assigns a complete neutrosophic triplet to each component, producing increasingly expressive structures capable of modeling meta-uncertainty and epistemic self-assessment. The work establishes the formal mathematical foundations of Type-k neutrosophic representations, proves their strict expressive hierarchy, and develops recursive aggregation operators that preserve compatibility with classical single-valued neutrosophic systems. The book further extends neutrosophic theory through tensorial frameworks, including single-valued neutrosophic tensors and plithogenic tensors, enabling advanced multi-criteria decision-making and multilinear analysis. A neutrosophic paraconsistent logic is introduced to address contradiction and indeterminacy within formal reasoning systems. The theoretical developments are applied to the epistemic auditing of large language models, where extensive empirical studies demonstrate the limitations of conventional probabilistic approaches and reveal the necessity of extended neutrosophic representations for capturing hallucinations, paradoxes, contradictions, and structural uncertainty. The volume concludes with open problems and future research directions connecting neutrosophic mathematics, artificial intelligence auditing, decision sciences, and paraconsistent reasoning.

This research monograph introduces the theory of Neutrosophic Super Hyper Sets (NSHS), a novel mathematical framework that integrates neutrosophic set theory with the hierarchical structure of iterated power sets to model uncertainty in multi-level and nested systems. Building upon the evolution from classical, fuzzy, intuitionistic fuzzy, and neutrosophic sets, the authors address situations in which information is associated not only with individual elements but also with collections, collections of collections, and higher-order hierarchical structures. The work develops the foundational theory of NSHS, including definitions, set-theoretic operations, algebraic properties, lattice structures, topology, relations, mappings, similarity measures, entropy, and distance metrics. Advanced extensions such as interval-valued, bipolar, plithogenic, and multi-polar neutrosophic super hyper sets are also formulated. The monograph establishes rigorous mathematical results concerning partial orders, distributive lattices, aggregation operators, score and accuracy functions, and topological structures, while demonstrating practical applications in decision-making, medical diagnosis, environmental risk assessment, data mining, machine learning, knowledge graphs, network science, and complex systems analysis. By combining hierarchical modeling with the expressive representation of truth, indeterminacy, and falsity, the NSHS framework provides a powerful tool for analyzing uncertainty in increasingly interconnected and multi-layered scientific, technological, and social systems.

This volume proposes a radical reconceptualization of reality, knowledge, and existence through the development of Meta-Garde theory, a structural framework designed to articulate systems governed not by binary exclusion, but by the coexistence of multiple, non-exclusive states. Moving beyond its origins in avant-garde aesthetics and paradoxism, Meta-Garde is extended here into the domains of epistemology, ontology, and decolonial philosophy to address the limitations of universalist, binary logic in capturing the complexity of Indigenous cosmologies and Latin American thought. The central thesis posits that many Indigenous worldviews—from Mesoamerican triadic ontologies and Andean yanantin (complementary duality) to Amazonian distributed agency and perspectivism—already operate through pluriversal structures where contradiction, indeterminacy, and relational multiplicity are constitutive rather than anomalous. To formally articulate these coexistential dynamics, the book integrates neutrosophic logic (the triadic system of Truth, Indeterminacy, and Falsity) and n-alectics (a logic of coexistence beyond dialectical synthesis) to model realities where affirmation and negation persist simultaneously without requiring resolution. Through a five-part interdisciplinary inquiry, the text traverses the foundations of pluriversal thought, analyzes specific Indigenous cosmological structures as proto-Meta-Garde systems, and explores the role of language as a site of epistemic resistance against colonial erasure. It further develops a Meta-Garde Complexity Model to address recursive contradictions and nonlinear epistemologies, culminating in the formulation of the Pluriversal Meta-Garde Space—a formal field where heterogeneous ontologies interact without collapsing into a singular universal order. The book concludes by outlining the ethical, political, and anthropological implications of this framework, proposing new models for environmental ethics (such as the Meta-Garde Sustainability Index) and a Meta-Garde Anthropology that studies contradiction as a structural principle of social life. By bridging formal logic, decolonial theory, and Indigenous epistemologies, Meta-Garde and the Pluriversal Condition offers a transformative philosophical orientation capable of engaging the dynamic, unresolved, and profoundly interconnected realities of the twenty-first century.

This book develops a formal, quantitative theory of anti-art by integrating multidimensional modeling, neutrosophic logic, and geometric analysis. Moving beyond traditional binary aesthetics, it reconceptualizes art as a structured configuration space in which affirmation, negation, and indeterminacy coexist. The core contribution is the construction of the Meta-Garde framework, which includes: (1) a high-dimensional state space representing artworks as vectors of structural variables, (2) the Meta-Garde Intensity Index (MGII) as a composite functional measuring anti-art intensity, and (3) a geometric and taxonomic system enabling clustering, comparison, and classification of artistic movements. Through this framework, diverse practices—ranging from Dada and Fluxus to Conceptual Art, Glitch Art, and AI Art—are mapped as distinct regions within a unified analytical space. Anti-art is thus shown to be multidimensional, structured, and measurable, rather than merely disruptive or anomalous. A central result of the study is the identification of Outer Art as a high-coherence meta-garde configuration, characterized by simultaneous elevation across key variables such as definitional destabilization, paradox density, reflexivity, and logical explicitness. While not maximizing intensity in any single dimension, Outer Art uniquely achieves multi-variable structural coherence, positioning it as a calibration case and representative of a distinct mathematical class within the state space. The framework culminates in a general theory of neutrosophic aesthetics, in which artworks are modeled through triadic logic ⟨T, I, F⟩ and embedded within a multidimensional geometric structure. This synthesis establishes the foundations for a formal science of art, capable of analyzing aesthetic complexity, contradiction, and evolution across both historical and emerging artistic practices.

This book develops a comprehensive theoretical framework for understanding pArAdOXisM as a central formation within contemporary avant-garde literature and introduces meta-garde as a broader structural condition of artistic and linguistic expression. Challenging traditional binary aesthetics—such as art versus non-art or meaning versus meaninglessness—the work argues that literary and artistic objects must be analyzed through the simultaneous coexistence of affirmation, negation, and indeterminacy. Through historical reconstruction, conceptual analysis, and comparative mapping, the study demonstrates that pArAdOXisM is not merely a marginal or derivative movement but a system in which contradiction functions as a generative principle rather than a stylistic device. The book formalizes paradoxist poetics by identifying key operations such as semantic inversion, anti-language, non-poems, and structural contradiction, and situates these within a multidimensional analytical model derived from neutrosophic logic. By comparing pArAdOXisM with other avant-garde movements—including Dada, Surrealism, Lettrism, Oulipo, and conceptual writing—the work establishes its distinctiveness as a meta-avant-garde system characterized by high paradox density, reflexivity, and definitional instability. Ultimately, the book proposes meta-garde as a generalizable framework extending beyond aesthetics into a broader philosophical logic of coexistence, where contradiction and indeterminacy are understood as fundamental structural conditions rather than anomalies.

This volume develops meta-garde as a structural condition of art in which aesthetic states are no longer governed by binary distinctions, but coexist in multivalent configurations. Moving beyond the traditional oppositions that have shaped aesthetic theory—such as art and non-art, meaning and non-meaning—the study argues that contemporary artistic practices reveal a condition in which affirmation, negation, and indeterminacy operate simultaneously within the same object. The book proceeds by first examining the historical development of aesthetics, from classical frameworks grounded in binary logic to the fragmentation associated with postmodern theory. While postmodernism recognizes the instability of aesthetic categories, it remains largely descriptive. In response, this work introduces a formal articulation of coexistence through a triadic structure that allows contradictory states to be represented without reduction. This framework is then applied to a range of aesthetic phenomena, including anti-art, paradoxism, conceptual practices, and hybrid and processual forms. Particular attention is given to the role of indeterminacy, reflexivity, and contradiction as constitutive features rather than anomalies. Through the development of a minimal axiomatic system and the concept of an aesthetic state space, the study provides tools for analyzing objects that resist classification within traditional models. Finally, the volume situates meta-garde within the history of aesthetic thought and argues for its extension beyond aesthetics. Rather than a movement or style, meta-garde is proposed as a general condition that may also be observed in language, culture, and non-binary philosophical frameworks. In this sense, the book advances a shift from exclusion-based models of understanding toward a logic of coexistence, offering a foundation for further theoretical and interdisciplinary development.

This book develops a comprehensive theoretical framework for the study of Hierarchical SuperHyperGraphs and Recursive SuperHyperGraphs, extending classical graph and hypergraph theory toward more expressive higher-order and multi-level relational structures. SuperHyperGraphs generalize hypergraphs through iterated powerset constructions, allowing vertices to be nested, set-valued, and hierarchically organized. The work introduces recursive SuperHyperGraphs by integrating n-level supervertices with recursive superhyperedges of bounded depth, where edges may contain both supervertices and lower-level recursive edges under typed incidence and well-foundedness constraints. Furthermore, the book formulates hierarchical SuperHyperGraphs as mixed-level incidence systems governed by coherence conditions to ensure structural consistency across powerset levels. Several important extensions are explored, including fuzzy, neutrosophic, uncertain, soft, rough, weighted, directed, and bidirected variants. Theoretical comparisons are also provided with related graph-theoretic models such as Meta-Graphs, Iterated Meta-Graphs, Filtrated Graphs, Iterated MultiGraphs, and Neural Graphs. Emphasizing definitions, formal properties, examples, and structural comparisons, the book establishes a foundational basis for future computational developments and interdisciplinary applications in complex systems, artificial intelligence, network science, and higher-order modeling.

This book develops a systematic and unified framework for the study of planar graph theory under uncertainty, integrating classical graph-theoretic concepts with fuzzy, intuitionistic fuzzy, neutrosophic, plithogenic, and uncertain graph models. The work revisits foundational graph classes—including planar, outerplanar, apex, quasi-planar, and related graph structures—and extends them into uncertainty-aware environments where vertices and edges may carry graded, indeterminate, or attribute-dependent information. Special attention is given to fuzzy planar graphs, intuitionistic fuzzy planar graphs, neutrosophic planar graphs, plithogenic planar graphs, and uncertain planar graph formulations, emphasizing both structural consistency and model-independent interpretations of planarity. In addition, the book explores related extensions to hypergraphs and SuperHyperGraphs, thereby broadening the conceptual landscape of planar structures in higher-order settings. By combining rigorous definitions, examples, comparative perspectives, and theoretical analysis, the book establishes a coherent foundation for future investigations in uncertain network modeling, graph theory, artificial intelligence, graph neural networks, computer vision, VLSI design, and complex systems analysis.

Artificial Intelligence (AI) is rapidly transforming teacher education through intelligent tutoring systems, automated assessment, learning analytics, virtual teaching simulations, and AI-assisted curriculum design. While these technologies offer substantial benefits in terms of efficiency, personalization, scalability, and data-informed decision-making, they also introduce significant challenges related to ethical responsibility, professional autonomy, algorithmic bias, privacy, and pedagogical integrity. Conventional evaluative approaches often rely on binary or probabilistic reasoning, limiting their ability to capture the complexity and uncertainty inherent in AI-supported educational environments. This book introduces a neutrosophic framework for analyzing AI-driven teacher education, grounded in the concepts of truth, indeterminacy, and falsity. Through this multidimensional perspective, AI technologies are examined as educational phenomena that simultaneously generate positive outcomes, unresolved uncertainties, and potential limitations. The work develops a comprehensive theoretical foundation, presents a neutrosophic assessment framework, explores empirical applications and case studies, and discusses ethical, methodological, and policy implications of AI integration in teacher preparation. By embracing uncertainty as an essential dimension of educational reality, the book advocates a balanced and human-centered approach to technological innovation. It argues that AI should function as an augmentative partner that enhances reflective practice, pedagogical creativity, and professional growth while preserving the relational and ethical foundations of teaching. The neutrosophic perspective offers researchers, educators, and policymakers a robust analytical tool for understanding and guiding the future development of AI-enabled teacher education systems.

This book presents a collection of recent advances in neutrosophic mathematics, highlighting the growing role of neutrosophic structures in addressing uncertainty, indeterminacy, and inconsistency within modern mathematical systems. The volume integrates developments from neutrosophic graph theory, algebra, metric spaces, and functional analysis, demonstrating both theoretical depth and practical applicability. The first part of the book investigates novel classes of neutrosophic graphs, including Min-Max neutrosophic graphs and neutrosophic bidirected graphs. New combinatorial constructions, structural properties, graph operations, and network modeling applications are developed, providing mathematical tools for the analysis of uncertain and complex networks. The second part focuses on neutrosophic algebraic structures through the study of neutrosophic Lie subalgebras and ideals. Fundamental results concerning homomorphisms, images, inverse images, quotient structures, and preservation properties are established, extending classical Lie algebra theory to neutrosophic environments. The third part explores analytical aspects of neutrosophic mathematics. New common fixed-point results in neutrosophic metric spaces are developed under integral-type contractive conditions and applied to dynamic programming problems. Furthermore, the theory of rough I-statistical convergence of order α′ in neutrosophic normed spaces is introduced and characterized, with applications to dynamical systems, iterative processes, and sensor-network data fusion. The book concludes with a comprehensive discussion of emerging research directions, including neutrosophic artificial intelligence, machine learning, quantum neutrosophic structures, network science, algebraic topology under uncertainty, and sustainability applications. By bringing together diverse mathematical perspectives, this volume provides a unified overview of contemporary developments in neutrosophic mathematics and serves as a valuable resource for researchers, graduate students, and professionals working in uncertainty modeling and advanced mathematical sciences.

his book introduces Type-k Neutrosophic Sets as a recursive extension of classical neutrosophic logic. While standard neutrosophic sets represent each proposition through three independent components—truth, indeterminacy, and falsity—Type-k Neutrosophic Sets allow each of these components to be recursively characterized by further neutrosophic triplets. This creates a hierarchical epistemic structure capable of representing nested uncertainty, meta-uncertainty, contradiction, ignorance, and paraconsistent states that cannot be adequately captured by classical probability, fuzzy logic, or intuitionistic fuzzy models. The book develops the formal foundations of this recursive framework, proves its expressive hierarchy, and extends it through single-valued neutrosophic tensors and plithogenic tensors for multi-criteria decision-making. It also explores connections with neutrosophic paraconsistent logic and applies the framework to the epistemic auditing of Large Language Models. Particular attention is given to hallucination detection, the limitations of softmax-based confidence, and the need for triadic and recursive models of truth, falsity, and indeterminacy in AI evaluation. By combining non-classical logic, tensorial structures, decision theory, and artificial intelligence auditing, the work proposes a new formal and computational framework for understanding uncertainty in complex epistemic systems. It is intended for researchers in neutrosophic logic, philosophy of logic, formal epistemology, artificial intelligence, paraconsistent reasoning, and decision sciences.

When a newspaper systematically omits a relevant causal explanation, it does not merely fail to inform — it performs an illocutionary act. Drawing on Austin's speech act theory and Grice's cooperative principle, this paper argues that the systematic press silence on structural drivers of urban violence in Guayaquil, Ecuador (2025) constitutes an act of structural erasure: a communicative action that produces a distorted public understanding of violence causation with real perlocutionary effects on policy, investment, and citizen perception. We provide the first empirical, quantitative evidence for this claim at scale. Using four Large Language Models (LLMs) to score 31 press sources (16 local Spanish-language, 15 international English-language) across eight structural causal conditions, and applying Neutrosophic fsQCA (N-fsQCA) to compute illocutionary consistency and indeterminacy, we find that five of eight structural drivers are systematically under-represented in the press relative to a validated structural baseline: weapons trafficking (gap=-0.444), extortion (gap=-0.345), prison linkages (gap=-0.340), drug routes (gap=-0.290), and economic pressure (gap=-0.214). Crucially, this silence is not random: it is patterned, stable across 31 sources and 4 LLMs, and exhibits a local/international asymmetry that reveals two distinct illocutionary communities enacting different acts of erasure about the same referential reality. We term this empirical pragmatics of violent silence: the systematic, measurable breach of Gricean maxims in press accounts of urban violence.

This book provides a comprehensive and systematic survey of modern set-theoretic frameworks developed to model uncertainty, vagueness, indeterminacy, and partial truth in real-world phenomena. It synthesizes a wide spectrum of classical and advanced theories, including fuzzy sets, intuitionistic fuzzy sets, neutrosophic sets, plithogenic sets, and numerous related extensions. The work organizes these concepts across multiple structural and conceptual dimensions—such as interval, granular, probabilistic, and complex extensions—highlighting their mathematical foundations, interrelations, and distinguishing characteristics. By unifying diverse approaches under a coherent analytical perspective, the book clarifies how different uncertainty models complement or generalize one another. It also explores emerging directions in set theory, including hybrid and dynamic frameworks, offering insights into their applicability in fields such as decision-making, control systems, artificial intelligence, and data analysis. Serving both as a reference and a conceptual guide, the book contributes to advancing the theoretical understanding and practical deployment of uncertainty-oriented mathematical models.

This book presents a comprehensive neutrosophic evaluation of the Mid-Day Meal Scheme (MDMS) in India, a large-scale public welfare initiative aimed at improving child nutrition, educational participation, and social equity. Recognizing the inherent complexity, uncertainty, and contradictions in policy implementation, the authors propose a neutrosophic framework that extends beyond classical and fuzzy evaluation models by incorporating degrees of truth, indeterminacy, and falsity. Using a micro-level case study approach, the research systematically analyzes key dimensions of the scheme, including nutritional adequacy, enrolment and attendance, food quality and hygiene, administrative efficiency, and social inclusion. The methodology integrates primary and secondary data, expert judgment, and neutrosophic aggregation techniques to construct a multidimensional performance assessment. Special emphasis is placed on identifying zones of indeterminacy and contradictory evidence, which are often overlooked in traditional evaluations. The study further develops neutrosophic indices, applies sensitivity and robustness analyses, and explores the implications of uncertainty-aware decision-making in public policy. The findings demonstrate that neutrosophic logic provides a more nuanced and realistic representation of policy outcomes, offering valuable insights for policymakers, researchers, and stakeholders. Ultimately, the book contributes both methodologically and practically to the application of neutrosophic theory in socio-economic policy analysis.

Real-world systems frequently involve vagueness, partial truth, incomplete information, and various forms of indeterminacy. To address these challenges, numerous generalized set-theoretic frameworks have been developed, including fuzzy sets, intuitionistic fuzzy sets, neutrosophic sets, plithogenic sets, and related extensions. These approaches provide powerful tools for modeling uncertainty in a mathematically rigorous and flexible manner. This book presents a comprehensive and systematic survey of fuzzy and uncertainty-oriented concepts in applied mathematics. It brings together a wide range of theoretical frameworks and applications, spanning discrete structures, algebraic systems, analytical models, and quantitative methods. By organizing these diverse topics within a unified perspective, the work clarifies their underlying mathematical structures, highlights their interrelationships, and compares their expressive capabilities. The book aims to serve both as a reference and as a conceptual guide, offering readers a coherent overview of the landscape of uncertainty modeling. It emphasizes the role of advanced frameworks—particularly neutrosophic and plithogenic systems—in extending classical fuzzy theory and enabling more refined representations of complex, inconsistent, and multi-attribute information in applied mathematics.

This book introduces and develops neutrosophy as a unifying philosophical and mathematical framework for the study of neutralities and their interactions with truth and falsity. Extending beyond classical, fuzzy, and intuitionistic logics, the work formalizes neutrosophic logic, in which every proposition is characterized by three independent components: truth (T), indeterminacy (I), and falsity (F). These components are defined over a non-standard unit interval, allowing values beyond conventional probabilistic constraints and enabling the modeling of incomplete, inconsistent, and paradoxical information. The volume systematically generalizes several foundational domains, including logic, set theory, probability, and statistics, by introducing corresponding neutrosophic structures such as neutrosophic sets, neutrosophic probability, and neutrosophic statistics. The framework accommodates contradictions, uncertainty, and vagueness in a unified mathematical structure, overcoming limitations of existing theories that cannot adequately represent indeterminacy or simultaneous truth and falsity. The book also establishes formal definitions based on non-standard analysis and hyper-real numbers, providing a rigorous mathematical foundation for neutrosophic components and operations. By integrating philosophical insight with mathematical formalism, the work opens new directions for research across disciplines, including artificial intelligence, decision theory, physics, and social sciences, where complex and uncertain phenomena require more expressive representational models.

Filosofia greacă antică reinterpretată: în dialog cu neutrosofia și plitogenia propune o reinterpretare sistematică a gândirii filozofice grecești timpurii prin prisma unor cadre logice contemporane – neutrosofia și plitogenia. Cartea analizează principalele concepte dezvoltate de filozofii presocratici, sofiști și reprezentanți ai Academiei platonice, evidențiind tensiunile logice dintre concepte precum ființă și devenire, unitate și pluralitate, adevăr și relativism. Autorul argumentează că logica clasică, bazată pe binaritate, este insuficientă pentru a surprinde complexitatea acestor idei și introduce neutrosofia ca un sistem care integrează simultan grade de adevăr, falsitate și indeterminare. Plitogenia extinde acest cadru prin modelarea realității ca sistem multi-atribut, caracterizat de grade de apartenență și contradicție. Printr-o metodă comparativă, fiecare filozof este reinterpretat în acest nou cadru conceptual, fără a-i atribui anacronic idei moderne, ci evidențiind convergențele posibile. Lucrarea contribuie astfel la dezvoltarea unei filozofii a cunoașterii extinse, în care indeterminarea, contradicția și complexitatea devin elemente fundamentale ale analizei filozofice.

Many real-world systems involve uncertainty not only in individual attributes but also in the relationships between entities. To address this, a variety of graph-theoretic frameworks have been developed that incorporate uncertainty directly into vertices, edges, and higher-level structural properties. Among these, fuzzy graphs, intuitionistic fuzzy graphs, neutrosophic graphs, and plithogenic graphs represent key approaches to modeling uncertainty in network structures. This book presents a comprehensive and systematic survey of graph theory under uncertainty, with particular emphasis on the unifying role of the uncertain graph framework. It reviews fundamental concepts, structural properties, graph classes, and graph parameters within fuzzy, neutrosophic, and related models, while also introducing a wide range of extensions such as uncertain digraphs, hypergraphs, superhypergraphs, and dynamic graphs. In addition to theoretical developments, the book explores practical applications, including uncertain molecular graphs, decision-making systems, graph neural networks, knowledge graphs, and cognitive maps. By organizing diverse uncertainty-aware graph models within a common perspective, this work provides a coherent framework for understanding their relationships, capabilities, and applications in complex systems.

Mategrams represents an original artistic expression situated at the intersection of literature, mathematics, and visual art. The book introduces the concept of “mategrams”—condensed thoughts and reflections that naturally organize themselves into geometric and algebraic structures, becoming mathematical-poetic figures akin to anti-calligrams. Rather than following a linear textual format, the content is embedded within visual compositions that exploit space, form, and symbolic arrangement, creating a nontraditional reading experience. Each mategram functions simultaneously as idea, image, and structure, inviting reflection on human condition, society, and the relationship between logic and creativity. By merging poetic language with graphical representations and mathematical concepts, the volume proposes an innovative interdisciplinary aesthetic in which meaning emerges from the interaction between text and form.

Paradoxica propune o reconstrucție radicală a limbajului prin integrarea paradoxului ca principiu fundamental de generare a sensului. Pornind de la critica logicii clasice, care privilegiază stabilitatea, identitatea și coerența, autorul dezvoltă un sistem alternativ în care contradicția nu este exclusă, ci devine condiție esențială a semnificației. Cartea introduce „Paradoxica” ca limbă formală și poetică în care fiecare enunț conține o tensiune semantică internă, bazată pe opoziție, inversiune sau autoreferențialitate. Sensul nu mai este un rezultat fix, ci un proces emergent generat din conflictul dintre concepte și anticoncepte. Lucrarea dezvoltă, de asemenea, un sistem formal de reguli, structuri și axiome, precum și o reprezentare vizuală (Scriptul Paradoxica), unde spațiul, simbolurile și relațiile dintre ele devin purtători de sens. În plan poetic, sunt prezentate multiple forme paradoxiste (distihuri duale, oxymorfoze, anacoluturi, structuri recursive), fiecare ilustrând mecanisme distincte de producere a sensului prin instabilitate și tensiune. În ansamblu, Paradoxica nu este doar o colecție de texte, ci o propunere epistemologică și lingvistică, în care adevărul nu este afirmat, ci generat prin contradicție, iar limbajul devine un sistem dinamic, deschis și autoreflexiv.

This book develops a comprehensive comparative analysis of plithogeny as a meta-philosophical framework capable of interpreting the relationships among diverse philosophical traditions. Plithogeny, originally formulated within the domain of mathematical logic, describes the formation and evolution of entities through the interaction of multiple elements that may be compatible, contradictory, or neutral. Extending this formal concept into philosophy, the work examines how philosophical systems can be understood not as isolated or mutually exclusive doctrines, but as interacting conceptual structures within a broader field of intellectual plurality. The study systematically explores the connections between plithogeny and major philosophical movements, including empiricism, rationalism, idealism, realism, phenomenology, and epiphenomenalism, as well as ethical theories such as deontology, utilitarianism, and hedonism. It further engages with postmodernism, pragmatism, and contemporary domains such as digital ethics, bioethics, environmental philosophy, and justice theory. A distinctive contribution of the expanded edition lies in its integration of Latin American and Indigenous philosophical traditions, including concepts such as yanantin, ch’ixi, pachakuti, ayni, Sumak Kawsay (Buen Vivir), and teotl, highlighting their structural affinities with plithogenic reasoning. By employing a consistent comparative methodology—focusing on similarities, distinctions, and conceptual connections—the book demonstrates that philosophical plurality can be modeled as a dynamic system of interacting perspectives. Plithogeny is thus presented not as a replacement for existing philosophies, but as a unifying analytical lens that enables dialogue across traditions while preserving their differences. The work contributes to contemporary philosophy by offering a framework suited to complexity, interdisciplinarity, and the coexistence of diverse epistemologies in both theoretical and applied contexts.

This third edition of Representing Higher-Order Networks: A Survey of Graph-Based Frameworks presents a substantially expanded and conceptually enriched treatment of higher-order network models. While retaining the foundational survey character of earlier editions, the present version introduces not only editorial refinements but also significant conceptual and structural developments that deepen and broaden the scope of the work. Beyond corrections and improved exposition, this edition incorporates a range of newly developed frameworks, including advanced tensor-based representations, recursive and iterated graph constructions, and extended superhypergraph formalisms. These additions reflect a shift from a primarily descriptive catalog of models toward a more systematic and integrative perspective, emphasizing relationships, hierarchies, and interoperability among diverse higher-order structures. The book further strengthens its unifying framework by organizing higher-order graph models into coherent families—combinatorial, topological, tensorial, and semantic—while highlighting their underlying principles and modeling capabilities. In doing so, it advances a more comprehensive theoretical landscape that supports both comparative analysis and practical application across disciplines such as network science, data modeling, and knowledge representation. Overall, this edition represents a qualitative evolution of the work, transitioning from a survey of existing approaches to a more cohesive and forward-looking synthesis of higher-order network theory.

SuperHyperScapes: An Ontological Atlas is a visual–conceptual exploration situated within the theoretical framework of oUTER-aRT, extending its principles into a structured yet open-ended system of image production. Composed of 77 generative visual fields, the album organizes reality into phased domains—ranging from perception and cognition to matter, society, and abstraction—each articulated through a distinct visual grammar. Grounded in neutrosophic logic, the project rejects binary representation and instead constructs images as spaces of simultaneous affirmation, negation, and indeterminacy. These “scapes” are not representations of reality, but ontological propositions: dynamic environments where contradictions coexist, structures destabilize, and meaning emerges through tension rather than resolution. By integrating superhyperstructural thinking with recursive and self-referential visual systems, SuperHyperScapes proposes an expanded ontology of art—one that transcends medium, dissolves disciplinary boundaries, and situates artistic practice within a continuum between system and anomaly, order and breakdown. As both atlas and anti-atlas, the work maps not what reality is, but how it can be simultaneously constructed, deconstructed, and reimagined.