Maikel Leyva

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.

This article proposes The Paraconsistent as a philosophical-political category to name the subject who exists constitutively in ontological contradiction — not due to epistemic deficiency but by structural condition — and who from that position produces thought, knowledge and action without collapsing into either of its poles. Drawing on a critical revision of two dominant traditions for handling contradiction — presented here as interpretive models called the First Logos and the Second Logos — the article proposes the Third Logos as a form of rationality especially adequate for contexts where contradiction is structural, persistent and irreducible. Logical colonialism — the imposition of coherence and non-contradiction criteria as conditions for recognizing knowledge, identity or decisions as valid — is identified as the object of liberation this philosophy proposes. The praxis operates in three simultaneous registers: pedagogical, technical and epistemic.

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.

Building on Nunes Filho's recent positioning of paraconsistent logic as a constitutive element of Latin American philosophy (RUDN Journal of Philosophy, 2025), this paper traces a continuation of that tradition into the contemporary problem of artificial intelligence alignment. We argue that the Latin American paraconsistent project — initiated by Miro Quesada's coining of the term in 1976 and formalized by Newton da Costa's C-systems — finds its natural twenty-first century extension in Florentin Smarandache's neutrosophic logic (1995), which generalizes paraconsistency by introducing three independent dimensions of evidence: truth (T), indeterminacy (I), and falsity (F), with no sum constraint. We further contextualize this development within the broader Eastern European tradition of non-classical logic, from Vasiliev's non-Aristotelian logic (1911) and the Romanian school of Lukasiewicz–Moisil multi-valued algebras to Smarandache's emigration from Romania in 1988 and his subsequent founding of neutrosophy. We then show that contemporary AI alignment frameworks — Reinforcement Learning from Human Feedback, Constitutional AI, Direct Preference Optimization — operate under an implicit epistemological monism that mirrors precisely the classical-logic universalism the Latin American tradition critiqued. Anthropic's own documentation of Collective Constitutional AI confirms that contradictions among diverse constitutional principles are resolved by majority vote — what we term enforced convergence rather than alignment. We propose Pluriversal Alignment: a research program that operationalizes paraconsistent neutrosophic triplets as reward signals, constitutional evaluations, and policy outputs, allowing T+F>1 as a structural feature rather than an error. We close by arguing that the Latin American paraconsistent tradition, in convergence with its Russian and Eastern European counterparts, is uniquely positioned to contribute the formal apparatus that the decolonial critique of AI has named philosophically but not yet formalized.

"Neutrosophic Computing and Machine Learning" (NCML) es una revista académica dedicada a la publicación de estudios avanzados en neutrosofía, conjuntos neutrosóficos, lógica neutrosófica, probabilidad neutrosófica, estadística neutrosófica y enfoques neutrosóficos para el aprendizaje automático y sus aplicaciones en diversos campos. La revista da la bienvenida a trabajos profesionales en inglés o español que incluyan un breve resumen de un problema y los resultados obtenidos. Este número (36) de "Neutrosophic Computing and Machine Learning" (NCML) presenta una colección focalizada de artículos que exploran la aplicación práctica de la teoría neutrosófica en varios dominios. Los trabajos abordan desafíos clave en áreas como el diagnóstico médico, el análisis de riesgos y el análisis de redes sociales, demostrando cómo los enfoques neutrosóficos pueden manejar eficazmente la incertidumbre y la indeterminación. Un tema central es el desarrollo de modelos neutrosóficos novedosos para resolver problemas del mundo real. El número presenta investigaciones sobre mapas cognitivos neutrosóficos para la evaluación multicriterio, un enfoque integrado neutrosófico-TOPSIS para la selección de personal y un modelo neutrosófico único para la detección del cáncer de mama. Además, los artículos profundizan en el uso de conjuntos neutrosóficos refinados para analizar el sentimiento en las redes sociales y su aplicación en la clasificación de rasgos de personalidad. Esta colección de investigaciones demuestra la utilidad en expansión de la computación neutrosófica en la creación de soluciones más robustas y matizadas para sistemas complejos.

"Neutrosophic Computing and Machine Learning" (NCML) es una revista académica dedicada a la publicación de estudios avanzados en neutrosofía, conjuntos neutrosóficos, lógica neutrosófica, probabilidad neutrosófica, estadística neutrosófica y enfoques neutrosóficos para el aprendizaje automático y sus aplicaciones en diversos campos. La revista da la bienvenida a trabajos profesionales en inglés o español que incluyan un breve resumen de un problema y los resultados obtenidos. Este número (38) de "Neutrosophic Computing and Machine Learning" (NCML) es una edición especial dedicada a explorar la "Neutrosofía aplicada a los dilemas bioéticos: una mirada transdisciplinaria desde la medicina, el derecho y la educación". Los artículos del número utilizan una variedad de métodos neutrosóficos para abordar una amplia gama de problemas complejos, principalmente en el contexto de la sociedad ecuatoriana. El número se centra en gran medida en los desafíos legales y sociales, presentando análisis neutrosóficos de temas críticos como los derechos de la familia y personales, la eficacia legal de la adopción homoparental, la revictimización de las víctimas de violencia sexual y la protección de los derechos de los niños durante los procedimientos de divorcio y cuando nacen en penitenciarías. También se abordan temas relacionados con el sistema de justicia, como la eficacia de la justicia restaurativa, la despenalización del aborto, el impacto de los medios en los "juicios paralelos" y la exclusión de las mujeres indígenas del sistema de justicia. En cuanto a los asuntos civiles y públicos, se analizan los derechos de los migrantes, la discriminación contra los Testigos de Jehová, la seguridad jurídica del comercio digital y las cuestiones procesales relacionadas con los casos de pensión alimenticia y la adjudicación de bienes abandonados. Además de los asuntos legales, la investigación se extiende a otros campos, demostrando la versatilidad de los enfoques neutrosóficos. Los artículos evalúan la reforma del sistema penitenciario, analizan las lagunas legales en la gestión de áreas de conservación y evalúan el derecho a la educación en Ecuador. La investigación en salud y bienestar aplica métodos neutrosóficos para evaluar la eficacia de los adhesivos dentales y el impacto de la leucemia viral felina. En los campos de negocios y economía, los trabajos exploran estrategias comerciales para cafeterías especializadas, analizan la viabilidad financiera para pequeñas empresas y proponen un modelo de innovación gastronómica para la cocina vegetariana. El número también incluye un análisis de la aplicación de la inteligencia artificial en la lucha contra el terrorismo y un estudio sobre el uso de la lógica neutrosófica para evaluar los marcos legales para el ciberacoso.

"Neutrosophic Computing and Machine Learning" (NCML) es una revista académica dedicada a la publicación de estudios avanzados en neutrosofía, conjuntos neutrosóficos, lógica neutrosófica, probabilidad neutrosófica, estadística neutrosófica y enfoques neutrosóficos para el aprendizaje automático y sus aplicaciones en diversos campos. La revista da la bienvenida a trabajos profesionales en inglés o español que incluyan un breve resumen de un problema y los resultados obtenidos. Este número (39) de "Neutrosophic Computing and Machine Learning" (NCML) presenta una amplia gama de investigaciones que demuestran la vasta aplicación de la teoría neutrosófica en múltiples campos, con un enfoque significativo en la resolución de problemas complejos del mundo real que implican incertidumbre e indeterminación. La colección comienza con una discusión fundamental sobre el concepto de indeterminación en las teorías neutrosóficas. A partir de esta base teórica, los trabajos aplican métodos neutrosóficos a una amplia variedad de dominios. En medicina y salud, la investigación incluye la evaluación de la función respiratoria en atletas profesionales, la comparación de procedimientos quirúrgicos para la obesidad mórbida y una revisión sistemática del impacto del sistema neuroendocrino en la función inmunológica. El número también profundiza en el análisis legal y social, con artículos sobre el poder discrecional en el derecho tributario, los aspectos legales de la familia y las pensiones alimenticias, y la responsabilidad del estado con respecto a las víctimas de feminicidio. Un tema importante en este volumen es el uso de modelos neutrosóficos para el análisis estratégico y la toma de decisiones, especialmente en negocios y políticas públicas. Los trabajos cubren la aplicación de modelos neutrosóficos PEST-FODA para pequeñas y medianas empresas (PyMEs) y para analizar la industria audiovisual cubana. Otras contribuciones exploran tecnología y educación, como el uso de métodos neutrosóficos para la gestión de procesos municipales, el desarrollo de sistemas de recomendación para estudiantes y el análisis de expresiones emocionales en la educación de estudiantes con Trastorno del Espectro Autista (TEA). En general, el número muestra la versatilidad de la computación neutrosófica al proporcionar soluciones sólidas para sistemas complejos tanto en un contexto teórico como aplicado.

Misinformation and biased information significantly impact public perception and political decisions, especially on critical issues such as climate change and environmental conservation. This study aims to understand how indeterminacy and contradiction influence public perception and policy formulation by applying neutrosophic theory to model the complexity and multi-dimensionality of ignorance. Using neutrosophic Likert scales, we capture a nuanced spectrum of opinions on the scientific certainty of human impact on climate change. The results are analyzed through a k-means clustering algorithm to identify patterns and segment participants into groups based on their levels of truth, indeterminacy, and falsehood. This approach reveals deeper insights into public perceptions and aids in evaluating their implications for effective communication and policy-making.

A special issue of the International Journal in Information Science and Engineering “Neutrosophic Sets and Systems” (vol. 69/2024) is dedicated to the Neutrosophic approaches in research, on the occasion of the international and multidisciplinary conference held at the Universidad César Vallejo in Lima, Peru, on July 8 and 9. This event marks a significant milestone, as it is the first time that the Andean region and Latin America host scholars and researchers dedicated to studying various theoretical and applicative issues in the expansive and diverse field of Neutrosophic approaches. Since its conception, Neutrosophic theory has proven to be an interdisciplinary and innovative field, notably growing with the introduction of several generalizations of Neutrosophic Sets, such as Plithogenic Sets, Hypersoft Sets, IndetermSoft Sets, SuperHyperSoft Sets, and MultiAlism. These advanced conceptualizations have further expanded the versatility and application range of Neutrosophic theory, allowing its adoption in an ever-increasing spectrum of disciplines. The conference, with its international and multidisciplinary character, has brought together experts and scholars from various fields, providing a unique platform for the discussion and exchange of ideas on the multiple applications of Neutrosophic approaches. This special issue also addresses how scientific production in Neutrosophy focuses on social issues specific to Latin American philosophy. In the regional context of Latin America, it is possible to state that Neutrosophic tools and knowledge are used for the identification, analysis, and resolution of social problems, offering unique approaches or distinctive contributions to the field of Neutrosophy, influenced by its cultural and philosophical context. Neutrosophic science in Latin America shows a clear pattern of how scientific production addresses social problems, standing out for its innovative approaches that reflect the cultural and philosophical particularities of the region. This approach has allowed Neutrosophy not only to advance in theoretical terms but also to provide practical and contextually relevant solutions to social challenges. This special issue compiles works presented at the conference, reflecting the richness and diversity of current research in this field. We hope that these articles not only contribute to the advancement of knowledge in Neutrosophic theory but also inspire new research and applications in multiple disciplines.

A special issue of the International Journal in Information Science and Engineering “Neutrosophic Sets and Systems” (vol. 71/2024) is dedicated to the Conference on NeutroGeometry, NeutroAlgebra, and Their Applications, organized by the Latin American Association of Neutrosophic Sciences. This event, which took place on August 12-14, 2024, in Havana, Cuba, was made possible by the valuable collaboration of the University of Havana, the University of Physical Culture and Sports Sciences "Manuel Fajardo," the José Antonio Echeverría University of Technology, University of Informatics Sciences and the Cuban Academy of Sciences among other institutions. In 2019 Smarandache generalized the classical Algebraic Structures to NeutroAlgebraic Structures (or NeutroAlgebras) {whose operations and axioms are partially true, partially indeterminate, and partially false} as extensions of Partial Algebra, and to AntiAlgebraic Structures (or AntiAlgebras) {whose operations and axioms are totally false} and on 2020 he continued to develop them. The NeutroAlgebras & AntiAlgebras are a new field of research, which is inspired from our real world. In classical algebraic structures, all operations are 100% well-defined, and all axioms are 100% true, but in real life, in many cases these restrictions are too harsh, since in our world we have things that only partially verify some operations or some laws. Similarly, a classical Geometry structure has all axioms totally (100%) true. A NeutroGeometry structure has some axioms that are only partially true, and no axiom is totally (100%) false. Whereas an AntiGeometry structure has at least one axiom that is totally (100%) false. And in general, in any field of knowledge one has: Structure, NeutroStructure, and AntiStructure which were inspired from our real world where the laws (axioms) do not equally apply to all people and in the same degree. This special issue aims to highlight the most recent advances and applications in the fields of NeutroGeometry and NeutroAlgebra, two areas that are at the forefront of contemporary mathematical and scientific thought. During the conference, the mathematical foundations and practical applications of these disciplines were explored, as well as their relevance in the MultiAlism system and other interdisciplinary areas. The content of this special issue has been carefully selected to reflect the diversity and depth of the topics discussed at the conference. This event and the subsequent publication of these works underline the growing importance of neutrosophic theories in the current scientific landscape. We are confident that the ideas and discoveries shared in these pages will be of great value to researchers, academics, and professionals interested in these innovative areas of knowledge.

This is the fourth volume of the Encyclopedia of Neutrosophic Researchers, edited from materials offered by the authors who responded to the editor’s invitation. The authors are listed alphabetically. The introduction contains a short history of neutrosophics, together with links to the main papers and books. Neutrosophic set, neutrosophic logic, neutrosophic probability, neutrosophic statistics, neutrosophic measure, neutrosophic precalculus, neutrosophic calculus and so on are gaining significant attention in solving many real life problems that involve uncertainty, impreciseness, vagueness, incompleteness, inconsistent, and indeterminacy. In the past years the fields of neutrosophics have been extended and applied in various fields, such as: artificial intelligence, data mining, soft computing, decision making in incomplete / indeterminate / inconsistent information systems, image processing, computational modelling, robotics, medical diagnosis, biomedical engineering, investment problems, economic forecasting, social science, humanistic and practical achievements.

Este es el cuarto volumen de la Enciclopedia de Investigadores Neutróficos, editados a partir de materiales ofrecidos por los autores que respondieron a la invitación del editor. Los autores se enumeran alfabéticamente. La introducción contiene una breve historia de la neutrosófica, y en especial de su impacto en Latinoamérica junto con enlaces a los principales artículos y libros. Conjunto neutrosófico, lógica neutrosófica, probabilidad neutrosófica, estadística neutrosófica, medida neutrosófica, precálculo neutrosófico, cálculo neutrosófico, psicología neutrosófica, sociología neutrosófica etc., están ganando una atención significativa en resolver muchos problemas de la vida real que implican incertidumbre, imprecisión, vaguedad, incompletitud, inconsistencia e indeterminación.