Pythagorean  fuzzy deductive system of BCL-algebra

Background The deductive system of BCL-algebra has not been thoroughly explored, and its fuzzification remains undefined. This study aims to the association of this gap by introducing a fuzzy extension of the deductive system of BCL-algebra using Pythagorean fuzzy sets. Methods We define a Pythagorean fuzzy set as a pair of membership and non-membership functions, where the sum of their squares lies between 0 and 1. These functions are used to extend the classical deductive system of BCL-algebra, allowing for reasoning based on fuzzy degrees. The system incorporates fuzzy operations such as union, intersection, and complement, while maintaining the structure of BCL-algebra. We also introduce key components such as the accuracy function, score function, degree of indeterminacy, and square deviation to measure the certainty, truth, uncertainty, and deviation of fuzzy sets. Results We prove that the intersection of two Pythagorean fuzzy deductive systems remains a valid Pythagorean fuzzy deductive system within BCL-algebra. However, we show that the union of such systems does not necessarily form a valid fuzzy deductive system. The study also provides detailed proofs using induction, logical derivations, and algebraic techniques. Conclusion The results disclose that while the intersection of Pythagorean fuzzy deductive systems preserves the system's structure, the union does not, offering new insights into the behavior and limitations of fuzzy systems in BCL-algebra.