Florentin Smarandache

This book addresses a fundamental epistemological limitation of contemporary artificial intelligence systems: their inability to represent and communicate uncertainty. Current AI architectures, particularly large language models, generate outputs based on probabilistic pattern matching without an intrinsic mechanism to distinguish between verified knowledge, partial knowledge, and fabricated information. This results in what the authors define as “confident machines,” capable of producing fluent yet potentially unreliable responses without signaling epistemic boundaries. To address this structural deficiency, the book introduces a practical and conceptual framework grounded in neutrosophic logic, a triadic system developed by Florentin Smarandache, in which every proposition is characterized by degrees of truth (T), indeterminacy (I), and falsity (F). The authors extend this formalism beyond mathematics, integrating it with philosophical traditions from Latin America and other non-Western epistemologies that embrace ambiguity, complementarity, and productive doubt. The work develops an operational methodology—referred to as “The Third Answer”—which enables users to critically evaluate AI-generated outputs through structured questioning, classification of uncertainty, and decision protocols. By translating abstract logical principles into accessible tools (e.g., decision frameworks, uncertainty mapping, and prompt strategies), the book provides a practical guide for professionals interacting with AI systems across domains. Ultimately, the book argues that improving AI reliability is not merely a matter of increasing data or model size, but requires a paradigm shift toward uncertainty-aware reasoning. It positions neutrosophic logic as a bridge between ancient philosophical insights and modern computational systems, offering a pathway toward more transparent, trustworthy, and epistemically responsible artificial intelligence.

This book explores the development and importance of these mathematical approaches in managing uncertainty and imprecision in decision-making, and focuses on the practical applications of these theories in decision-making, demonstrating how they improve uncertainty modeling and facilitate more effective decision-making across various fields. By leveraging these frameworks, decision-makers can better navigate complexity, leading to more accurate and dependable outcomes. Ultimately, it highlights the progression from fuzzy to intuitionistic fuzzy to neutrosophic set theories as a significant advancement in capturing and analyzing uncertainty within decision-making contexts.

The domain of neutrosophic set theory has been undergoing significant and rapid expansion. New approaches, theories, strategies, and optimization techniques are developed at a swift rate. An important development in neutrosophic theory is its integration with existing set theories, such as soft set theory, rough set theory and more. Neutrosophic set has been extended to neutrosophic hyperset which is vast and extends across multiple disciplines and problem domains, from decision support and Artificial Intelligence (AI) to data analysis and complex systems modelling. A neutrosophic hypersoft set extends the flexibility of neutrosophic logic and soft set theory by considering multidimensional or hypersoft sets to address real-world complexities where information is incomplete, uncertain, or inconsistent. Neutrosophic theories have emerged as essential tools in a diverse range of disciplines, including, MCDM, data mining, biomedical research, social studies, and beyond. The fourth volume of “New Trends in Neutrosophic Theories and Applications” focuses on theories, strategies, optimizing techniques for MCDM within neutrosophic frameworks. Some topics deal with introducing of Pythagorean hypersoft sets with possibility degree, quadripartitioned neutrosophic Lie-ideal of Lie-algebra, quadripartitioned neutrosophic quasi coincident topological space, neutrosophic supra-open set in neutrosophic supra topological space and neutrosophic soft matrices. Some topics deal with medical diagnosis, organ transplantation success using neutrosophic superhyperstructure and artificial intelligence. Some topics deal with revenue management, social situation. Some topics deal with MCDM in single valued neutrosophic set environment, rough set environment, and interval trapezoidal neutrosophic environment.

“Neutrosophics Knowledge” has been created for publications on advanced studies in neutrosophy, neutrosophic set, neutrosophic logic, neutrosophic probability, neutrosophic statistics that started in 1995 and their applications in any field, such as the neutrosophic structures developed in algebra, geometry, topology, etc. The submitted papers should be professional, in good English and Arabic, containing a brief review of a problem and obtained results. Neutrosophy is a new branch of philosophy that studies the origin, nature, and scope of neutralities, as well as their interactions with different ideational spectra.

In this paper, we have developed an Excel package to be utilized for calculating neutrosophic data and analyze them. The use of object oriented programming techniques and concepts as they may apply to the design and development a new framework to implement neutrosophic data operations, the c# programming language, NET Framework and Microsoft Visual Studio are used to implement the neutrosophic classes. We have used Excel as it is a powerful tool that is widely accepted and used for statistical analysis. Figure 1 shows Class Diagram of the implemented package. Figure 2 presents a working example of the package interface calculating the complement. Our implemented Neutrosophic package can calculate Intersection, Union, and Complement of the neutrosophic set. Figure 3 presents our neutrosophic package capability to draw figures of presented neutrosophic set. Figure 4 presents charting of Union operation calculation, and figure 5 Intersection Operation. neutrosophic set are characterized by its efficiency as it takes into consideration the three data items: True, Intermediate, and False.

In this paper we introduce the notion of ideals on neutrosophic set which is considered as a generalization of fuzzy and fuzzy intuitionistic ideals studies in [9,11] , the important topological neutrosophic ideals has been given in [4]. The concept of neutrosophic local function is also introduced for a neutrosophic topological space. These concepts are discussed with a view to fiind new neutrosophic topology from the original one in [8]. The basic structure, especially a basis for such generated neutrosophic topologies and several relations between different topological neutrosophic ideals and neutrosophic topologies are also studied here. Possible application to GIS topology rules are touched upon.