Modeling holistic fuzzy implication using co-copulas

Abstract

Two related aggregation operators called copulas and co-copulas are introduced and various properties are described. The relationship, of these operators to t-norms and t-conorms is noted. Generalizations of these, respectively, called conjunctors and disjunctors, are introduced. We suggest the use of disjunctor operators for modeling the multi-valued implication operator in fuzzy logic. We point out that the selection of operators used in fuzzy logic, in addition to having appropriate pointwise properties, should be holistic, this requires consideration of the nature of the resulting fuzzy set as a whole. Focusing on the protoform of fuzzy modus ponens and looking at the information contained in the inferred fuzzy set we show that the use of co-copulas has some desirable properties. Taking advantage of the fact that the weighted sum of co-copulas is a co-copula we consider the problem of constructing customized implication operators.

On a Role for Copula's in Jeffrey's Rule with An Application to Decision Making

“…We recall that a two‐dimensional copula is a function C : [0, 1] 2 → [0, 1] satisfying the following conditions: (1)For all x , y ∈ [0, 1], C ( x , 0) = C (0, y ) = 0. (2)For all x , y ∈ [0, 1], C ( x , 1) = x and C (1, y ) = y . (3)For all x 1 ≤ x 2 and y 1 ≤ y 2, we have …”

Conditional information using copulas with an application to decision making

“…A beautiful result that can help us, at times, in this task is contained in the Sklar theorem [1]. This theorem provides a direct way of obtaining the joint cumulative distribution function from the two marginal cumulative distribution functions by a simple binary aggregation of these marginals using a copula [2][3][4][5][6][7], which is a kind of "anding"…”

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