A concrete extension principle for fuzzy set theory

In Fuzzy Set Theory (FST), Zadeh's Extension Principle (ZEP) has been a precise method for extending mathematical functions from classical crisp sets to fuzzy sets. However, some of the key components of ZEP are difficult to interpret, especially for users without deep FST expertise. Additionally, applying ZEP to fundamental tasks, such as extending classical arithmetic to fuzzy settings, can produce results too imprecise for practical applications. This article proposes an alternative FST extension principle with several advantages: (i) Well-Defined Procedure: It offers a clear method for extending classical mathematical functions to their fuzzy counterparts. (ii) Concrete and Comprehensible Ingredients: The principle consists of elements understandable to FST users, particularly non-experts. This ensures users not only receive a solution but also understand the steps leading to it. (iii) Adaptive Fuzzy Functions: The proposed principle allows users to construct fuzzy functions that adapt to their problem domains. (iv) Versatility in Application: To demonstrate its versatility as a unified approach to solving non-trivial tasks, we apply the principle to (a) calculate the probabilities of fuzzy events and (b) illustrate how to derive user-interpretable fuzzy-valued statistical measures systematically. © 2025 The Author