Statistically grounded logic operators in fuzzy sets

In this study, we are concerned with a category of logic operators for fuzzy sets that is inherently associated with underlying statistical properties of membership grades they operate on. These constructs are referred to as statistically grounded logic operators, namely statistically grounded ORs (SORs, for short) and statistically grounded ANDs (SANDs, for short) operators. In essence, they arise as a solution to the optimization problem of the following form: View the MathML sourcearg minm∈[0,1]∑k=1Nw(xk)|xk-m|, Turn MathJax on where {xk}, k = 1, 2, …, N are the corresponding membership grades to be combined. The weight function w: [0, 1] → [0, 1] captures the nature of the logic and and or aggregation, respectively. We show that the weight function could be induced by t-norms and t-conorms. The weight functions could be also implied by some statistical characteristics of data (membership grades). The choice of the t-norm (t-conorm) depends upon the predefined form of the logic operators to be developed. We demonstrate that SANDs and SORs offer an efficient operational framework for constructing fuzzy rough sets; lead to the increased sensitivity of computing possibility and necessity measures, bring a new insight into fuzzy relational equations and deliver an interpretation vehicle for fuzzy clustering (that is provided in the form of dependency analysis).