Florentin Smarandache

Maikel Yelandi Leyva-Vázquez, Florentin Smarandache

Large Language Models (LLMs) are predominantly governed by probabilistic frameworks in which the sum of outcome probabilities is constrained to unity. This architectural limitation, often imposed by Softmax layers, leads to a collapse of uncertainty that makes it difficult to differentiate between epistemic uncertainty, paradox, and vagueness. We present an empirical investigation of the application of Neutrosophic Logic, a framework that treats Truth (T), Indeterminacy (I), and Falsity (F) as three independent dimensions, to model epistemic states in LLMs. We conducted experiments on a family of four OpenAI GPT models across five linguistic phenomena: logical paradoxes, epistemic ignorance, vagueness, ethical contradictions, and future contingencies, under three prompting strategies: neutrosophic, probabilistic, and entropy-derived. Our findings reveal that the neutrosophic approach, by allowing T+I+F > 1, a state we term hyper-truth, provides a richer representation of a model's internal state. In 35% of evaluations, hyper-truth emerged spontaneously, predominantly under ethical contradiction and logical paradox. We demonstrate that this approach preserves truth values in fuzzy contexts and offers a robust method for identifying and quantifying internal model conflict. We conclude that the integration of neutrosophic evaluation layers is a critical step toward more transparent, reliable, and ethically aware AI systems.

Takaaki Fujita, Florentin Smarandache

Rough set theory models uncertainty by approximating target concepts through lower and upper sets induced by indiscernibility, or more generally, by granulation relations in data tables. This perspective captures vagueness caused by limited observational resolution and supports set-theoretic reasoning about what can be determined with certainty and what remains only possible. This book is written as a map of models. Rather than developing a single algorithmic pipeline in depth, it provides a systematic survey of the main rough set paradigms and their extension routes. More specifically, representative variants are organized according to (i) the underlying granulation mechanism, such as equivalence-based, tolerance-based, covering-based, neighborhood-based, and probabilistic approximations, and (ii) the uncertainty semantics attached to data and relations, such as crisp, fuzzy, intuitionistic fuzzy, neutrosophic, and plithogenic settings. The book also explains how each choice changes the form of approximations and the interpretation of boundary regions. Throughout the book, small illustrative examples are used to clarify modeling intent and typical use cases in classification and decision support. Finally, an important clarification of scope should be noted. Since the main purpose of this book is to provide a map of models, the Abstract and Introduction should not lead readers to expect that feature reduction and rule induction are primary objectives. Although these topics are central in the rough set literature, they are treated here mainly as motivating applications and as entry points to the broader research landscape. The principal aim of the book is to survey and position rough set models and their extensions in a systematic and coherent manner.

Takaaki Fujita, Florentin Smarandache

Decision-making in real applications is often affected by vagueness, incomplete information, heterogeneous data, and conflicting expert opinions. This survey reviews uncertainty-aware multi-criteria decision-making (MCDM) and organizes the field into a concise, task-oriented taxonomy. We summarize problem-level settings (discrete, group/consensus, dynamic, multi-stage, multi-level, multiagent, and multi-scenario), weight elicitation (subjective and objective schemes under fuzzy/linguistic inputs), and inter-criteria structure and causality modelling. For solution procedures, we contrast compensatory scoring methods, distance-to-reference and compromise approaches, and non-compensatory outranking frameworks for ranking or sorting. We also outline rule/evidence-based and sequential decision models that produce interpretable rules or policies. The survey highlights typical inputs, core computational steps, and primary outputs, and provides guidance on choosing methods according to robustness, interpretability, and data availability. It concludes with open directions on explainable uncertainty integration, stability, and scalability in large-scale and dynamic decision environments.

Takaaki Fujita, Florentin Smarandache

Real-world phenomena often exhibit vagueness, partial truth, and incomplete information. To model such uncertainty in a mathematically rigorous way, many generalized set-theoretic frameworks have been introduced, including Fuzzy Sets [1], Intuitionistic Fuzzy Sets [2], Neutrosophic Sets [3,4], Vague Sets [5], Hesitant Fuzzy Sets [6], Picture Fuzzy Sets [7], Quadripartitioned Neutrosophic Sets [8], Penta-Partitioned Neutrosophic Sets [9], Plithogenic Sets [10], HyperFuzzy Sets [11], and HyperNeutrosophic Sets [12]. Within these frameworks, a wide range of notions has been proposed and studied, particularly in the settings of fuzzy, intuitionistic fuzzy, neutrosophic, and plithogenic set theories. This extensive literature underscores both the significance of these theories and the breadth of their application areas. As a result, many ideas, constructions, and structural patterns recur across these four major families of uncertainty-oriented models. In this book, we provide a comprehensive, large-scale survey of Fuzzy, Intuitionistic Fuzzy, Neutrosophic, and Plithogenic Sets. Our goal is to give readers a systematic overview of existing developments and, through a unified exposition, to stimulate new insights, further conceptual extensions, and additional applications across a wide range of disciplines.

Takaaki Fujita, Florentin Smarandache

Soft set theory provides a direct framework for parameterized decision modeling by assigning to each attribute (parameter) a subset of a given universe, thereby representing uncertainty in a structured way [1, 2]. Over the past decades, the theory has expanded into numerous variants-including hypersoft sets, superhypersoft sets, TreeSoft sets, bipolar soft sets, and dynamic soft sets-and has been connected to diverse areas such as topology and matroid theory. In this book, we present a survey-style overview of soft sets and their major extensions, highlighting core definitions, representative constructions, and key directions of current development.

Takaaki Fujita, Florentin Smarandache

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.

Takaaki Fujita, Florentin Smarandache

Many real-world phenomena are naturally modeled by graphs and networks. However, classical graph models are often limited to pairwise interactions and may not adequately capture the richer structures that arise in practice. Higher-order graph formalisms extend this framework by incorporating multiway, hierarchical, temporal, multilayer, recursive, and tensor-based interactions, thereby providing more expressive representations of complex systems. This book presents a comprehensive overview of mathematical notions that can be used to model higher-order networks. It surveys foundational concepts, extensional frameworks, and newly introduced formalisms, with an emphasis on their structural principles, relationships, and modeling roles. The aim is to provide a unified perspective that helps readers compare diverse higher-order network models and identify appropriate tools for theoretical study and practical applications. This book is Edition 2.0. It mainly includes the addition of several concepts, as well as corrections and improvements of typographical errors and explanations.

Florentin Smarandache

O the third version of this response-paper to Imamura criticism, we recall that NonStandard Neutrosophic Logic was never used by neutrosophic community in no application, that the quarter of century old neutrosophic operators (1995) criticized by Imamura were never utilized since they were improved shortly after but he omits to tell their development, and that in real world applications we need to convert/approximate the NonStandard Analysis hyperreals, monads and binads to tiny intervals with the desired accuracy, otherwise they would be inapplicable. We point out several errors and false statements by Imamura with respect to the inf/sup of nonstandard subsets, also Imamura 'rigorous definition of neutrosophic logic' is wrong and the same for his definition of nonstandard unit interval, and we prove that there is not a total order on the set of hyperreals (because of the newly introduced Neutrosophic Hyperreals that are indeterminate), whence the transfer principle is questionable.

Florentin Smarandache

In this book we introduce the plithogenic set (as generalization of crisp, fuzzy, intuitionistic fuzzy, and neutrosophic sets), plithogenic logic (as generalization of classical, fuzzy, intuitionistic fuzzy, and neutrosophic logics), plithogenic probability (as generalization of classical, imprecise, and neutrosophic probabilities), and plithogenic statistics (as generalization of classical, and neutrosophic statistics). Plithogenic Set is a set whose elements are characterized by one or more attributes, and each attribute may have many values. An attribute value v has a corresponding (fuzzy, intuitionistic fuzzy, or neutrosophic) degree of appurtenance d(x,v) of the element x, to the set P, with respect to some given criteria. In order to obtain a better accuracy for the plithogenic aggregation operators in the plithogenic set, logic, probability and for a more exact inclusion (partial order), a (fuzzy, intuitionistic fuzzy, or neutrosophic) contradiction (dissimilarity) degree is defined between each attribute value and the dominant (most important) attribute value. The plithogenic intersection and union are linear combinations of the fuzzy operators tnorm and tconorm, while the plithogenic complement, inclusion, equality are influenced by the attribute values contradiction (dissimilarity) degrees. Formal definitions of plithogenic set, logic, probability, statistics are presented into the book, followed by plithogenic aggregation operators, various theorems related to them, and afterwards examples and applications of these new concepts in our everyday life.

Florentin Smarandache, Surapati Pramanik

Neutrosophic theory and applications have been expanding in all directions at an astonishing rate especially after the introduction the journal entitled Neutrosophic Sets and Systems. New theories, techniques, algorithms have been rapidly developed. One of the most striking trends in the neutrosophic theory is the hybridization of neutrosophic set with other potential sets such as rough set, bipolar set, soft set, hesitant fuzzy set, etc. The different hybrid structure such as rough neutrosophic set, single valued neutrosophic rough set, bipolar neutrosophic set, single valued neutrosophic hesitant fuzzy set, etc. are proposed in the literature in a short period of time. Neutrosophic set has been a very important tool in all various areas of data mining, decision making, e-learning, engineering, medicine, social science, and some more. The book New Trends in Neutrosophic Theories and Applications focuses on theories, methods, algorithms for decision making and also applications involving neutrosophic information. Some topics deal with data mining, decision making, e-learning, graph theory, medical diagnosis, probability theory, topology, and some more.

W. B. Vasantha Kandasamy, Ilanthenral K, Florentin Smarandache

In this book authors for the first time introduce the notion of strong neutrosophic graphs. They are very different from the usual graphs and neutrosophic graphs. Using these new structures special subgraph topological spaces are defined. Further special lattice graph of subgraphs of these graphs are defined and described. Several interesting properties using subgraphs of a strong neutrosophic graph are obtained. Several open conjectures are proposed. These new class of strong neutrosophic graphs will certainly find applications in Neutrosophic Cognitive Maps (NCM), Neutrosophic Relational Maps (NRM) and Neutrosophic Relational Equations (NRE) with appropriate modifications.

Florentin Smarandache

Neutrosophic Over-/Under-/Off-Set and -Logic were defined by the author in 1995 and published for the first time in 2007. We extended the neutrosophic set respectively to Neutrosophic Overset {when some neutrosophic component is over 1}, Neutrosophic Underset {when some neutrosophic component is below 0}, and to Neutrosophic Offset {when some neutrosophic components are off the interval [0, 1], i.e. some neutrosophic component over 1 and other neutrosophic component below 0}. This is no surprise with respect to the classical fuzzy set/logic, intuitionistic fuzzy set/logic, or classical/imprecise probability, where the values are not allowed outside the interval [0, 1], since our real-world has numerous examples and applications of over-/under-/off-neutrosophic components. For example, person working overtime deserves a membership degree over 1, while a person producing more damage than benefit to a company deserves a membership below 0. Then, similarly, the Neutrosophic Logic/Measure/Probability/Statistics etc. were extended to respectively Neutrosophic Over-/Under-/Off-Logic, -Measure, -Probability, -Statistics etc. [Smarandache, 2007].

Florentin Smarandache

Symbolic (or Literal) Neutrosophic Theory is referring to the use of abstract symbols (i.e. the letters T, I, F, or their refined indexed letters Tj, Ik, Fl) in neutrosophics. We extend the dialectical triad thesis-antithesis-synthesis to the neutrosophic tetrad thesis-antithesis-neutrothesis-neutrosynthesis. The we introduce the neutrosophic system that is a quasi or (t,i,f) classical system, in the sense that the neutrosophic system deals with quasi-terms (concepts, attributes, etc.). Then the notions of Neutrosophic Axiom, Neutrosophic Deducibility, Degree of Contradiction (Dissimilarity) of Two Neutrosophic Axioms, etc. Afterwards a new type of structures, called (t, i, f) Neutrosophic Structures, and we show particular cases of such structures in geometry and in algebra. Also, a short history of the neutrosophic set, neutrosophic numerical components and neutrosophic literal components, neutrosophic numbers, etc. We construct examples of splitting the literal indeterminacy (I) into literal subindeterminacies (I1, I2, and so on, Ir), and to define a multiplication law of these literal subindeterminacies in order to be able to build refined I neutrosophic algebraic structures. We define three neutrosophic actions and their properties. We then introduce the prevalence order on T,I,F with respect to a given neutrosophic operator. And the refinement of neutrosophic entities A, neutA, and antiA. Then we extend the classical logical operators to neutrosophic literal (symbolic) logical operators and to refined literal (symbolic) logical operators, and we define the refinement neutrosophic literal (symbolic) space. We introduce the neutrosophic quadruple numbers (a+bT+cI+dF) and the refined neutrosophic quadruple numbers. Then we define an absorbance law, based on a prevalence order, in order to multiply the neutrosophic quadruple numbers.

Florentin Smarandache

The author has pledged in various papers, conference or seminar presentations, and scientific grant applications (between 2004-2015) for the unification of fusion theories, combinations of fusion rules, image fusion procedures, filter algorithms, and target tracking methods for more accurate applications to our real world problems - since neither fusion theory nor fusion rule fully satisfy all needed applications. For each particular application, one selects the most appropriate fusion space and fusion model, then the fusion rules, and the algorithms of implementation. He has worked in the Unification of the Fusion Theories (UFT), which looks like a cooking recipe, better one could say like a logical chart for a computer programmer, but one does not see another method to comprise/unify all things. The unification scenario presented herein, which is now in an incipient form, should periodically be updated incorporating new discoveries from the fusion and engineering research.

Florentin Smarandache, Mumtaz Ali, Muhammad Shabir

Study of soft sets was first proposed by Molodtsov in 1999 to deal with uncertainty in a non-parametric manner. The researchers did not pay attention to soft set theory at that time but now the soft set theory has been developed in many areas of mathematics. Algebraic structures using soft set theory are very rapidly developed. In this book we developed soft neutrosophic algebraic structures by using soft sets and neutrosophic algebraic structures. In this book we study soft neutrosophic groups, soft neutrosophic semigroups, soft neutrosophic loops, soft neutrosophic LA-semigroups, and their generalizations respectively.

Florentin Smarandache

In this paper we present a short history of logics: from particular cases of 2-symbol or numerical valued logic to the general case of n-symbol or numerical valued logic. We show generalizations of 2-valued Boolean logic to fuzzy logic, also from the Kleene and Lukasiewicz 3-symbol valued logics or Belnap 4-symbol valued logic to the most general n-symbol or numerical valued refined neutrosophic logic. Two classes of neutrosophic norm (n-norm) and neutrosophic conorm (n-conorm) are defined. Examples of applications of neutrosophic logic to physics are listed in the last section. Similar generalizations can be done for n-Valued Refined Neutrosophic Set, and respectively n- Valued Refined Neutrosopjhic Probability.

Florentin Smarandache

Neutrosophic Statistics means statistical analysis of population or sample that has indeterminate (imprecise, ambiguous, vague, incomplete, unknown) data. For example, the population or sample size might not be exactly determinate because of some individuals that partially belong to the population or sample, and partially they do not belong, or individuals whose appurtenance is completely unknown. Also, there are population or sample individuals whose data could be indeterminate. In this book, we develop the 1995 notion of neutrosophic statistics. We present various practical examples. It is possible to define the neutrosophic statistics in many ways, because there are various types of indeterminacies, depending on the problem to solve.

Florentin Smarandache

In this paper, we introduce for the first time the notions of neutrosophic measure and neutrosophic integral, and we develop the 1995 notion of neutrosophic probability. We present many practical examples. It is possible to define the neutrosophic measure and consequently the neutrosophic integral and neutrosophic probability in many ways, because there are various types of indeterminacies, depending on the problem we need to solve. Neutrosophics study the indeterminacy. Indeterminacy is different from randomness. It can be caused by physical space materials and type of construction, by items involved in the space, etc.