Type-k Neutrosophic Sets

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.