A Method of Generating Fuzzy Implications from n Increasing Functions and n + 1 Negations

Abstract

In this paper, we introduce a new construction method of a fuzzy implication from n increasing functions g i : [ 0 , 1 ] → [ 0 , ∞ ) ,   ( g ( 0 ) = 0 ) ( i = 1 , 2 , … , n ,   n   ∈ ℕ ) and n + 1 fuzzy negations N i ( i = 1 , 2 , … , n + 1 ,   n   ∈ ℕ ). Imagine that there are plenty of combinations between n increasing functions g i and n + 1 fuzzy negations N i in order to produce new fuzzy implications. This method allows us to use at least two fuzzy negations N i and one increasing function g in order to generate a new fuzzy implication. Choosing the appropriate negations, we can prove that some basic properties such as the exchange principle (EP), the ordering property (OP), and the law of contraposition with respect to N are satisfied. The worth of generating new implications is valuable in the sciences such as artificial intelligence and robotics. In this paper, we have found a novel method of generating families of implications. Therefore, we would like to believe that we have added to the literature one more source from which we could choose the most appropriate implication concerning a specific application. It should be emphasized that this production is based on a generalization of an important form of Yager’s implications.