Inverted Fuzzy Implications in Approximate Reasoning

Abstract

In 1973 Lotfi Zadeh introduced the theory of fuzzy logic [17]. Fuzzy logic was an extension of Boolean logic so that it allowed using not only Boolean values to express reality. One kind of basic logical operations in fuzzy logic are so-called fuzzy implications. From over eight decades a number of different fuzzy implications have been described [3] - [16]. In the family of all fuzzy implications the partial order induced from [0,1] interval exists. Pairs of incomparable fuzzy implications can generate new fuzzy implications by using min(inf) and max(sup) operations. As a result the structure of lattice is created ([1], page 186). This leads to the following question: how to choose the correct functions among basic fuzzy implications and other generated as described above. In our paper, we propose a new method for choosing implications. Our method allows to compare two fuzzy implications. If the truth value of the antecedent and the truth value of the implication are given, by means of inverse fuzzy implications we can easily optimize the truth value of the implication consequent. In other words, we can choose the fuzzy implication, which has the greatest or the smallest truth value of the implication consequent or which has greater or smaller truth value than another implication. Primary results regarding this problem are included in the paper [14].