Balanced fuzzy negation: Theory, generators and motivations

Balanced fuzzy sets (BFS), introduced in 2006, offer a new perspective on describing reality.In fuzzy logic, the negation of partial membership becomes another form of partial membership, which leads to the indistinguishability of a set from its complement.In BFS, where the sign of a value determines the nature of membership -positive or negative -this ambiguity is removed, and negation regains its semantic role as an opposite.This is consistent with the natural interpretation of human reasoning, in which the strengths of positive and negative premises are treated as equally important.It allows for the modelling of positive, negative, and neutral information.Moreover, they also account for the fact that the shape of the membership and non-membership functions can change.The innovation of the presented work lies in its new approach to negation, which takes into account the need to generate the negations adapted to the observed processes representing human perception of reality.This allows understanding that the negation of a value symbolizing partial excess is a partial deficiency, meaning that the negation of values representing the information that something is partially positive becomes the fact that something is partially negative.In other words, the negation in BFS reflects the transition from positive information to its opposite, in a manner observed in natural processes.This negation determines how fuzzy membership in a given set translates into the uncertainty of membership in its complement.Balanced fuzzy negation was presented, demonstrating the construction of BNF from classical negations and generating functions.Its various properties and the differences between supplement and complement in BFS were discussed.Fundamental relationships with operators in classical fuzzy sets were presented.Properties and examples (including Yager, power, and trigonometric operators) were provided.The results clarify the concepts of negation and complement in BFS and suggest directions for further research on balanced fuzzy operations in [-1,1].This enables more informed decision-making, risk analysis, sentiment assessment, and other engineering tasks that require considering positive and negative ratings, as well as neutral or no information.