Axiomatic fuzzy set theory and its applications

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“Fuzziness”—represented through information granules and fuzzy sets—is a key characteristic of human cognition and our understanding of reality. Fuzzy phenomena are prevalent in nature and society. The concept of fuzzy sets was introduced by L. A. Zadeh in 1965 to formalize human concepts and facilitate the representation of natural language and reasoning with words. Fuzzy sets and fuzzy logic enable modeling of imprecise reasoning, which is crucial for human decision-making in uncertain environments. The rise of fuzzy set applications stems from an “empirical-semantic” approach, emphasizing practical interpretations over the mathematical structures of fuzzy sets. For example, in control theory, fuzzy sets have demonstrated significant practical relevance. While fuzzy sets can be viewed as abstract concepts with formal foundations, their operational meaning varies across different contexts. This variability highlights the importance of understanding how membership functions operate in various applications, underlining the need for a nuanced approach to fuzzy sets in practice.