A Study On The Contrapositive Symmetrisation Of Fuzzy Implications

As we know, contrapositive symmetry is the tautology in classical logic. However,not all fuzzy implications satisfy contrapositive symmetry with respect to a given strong negation in fuzzy logic. Contrapositive symmetrisation is to transform a fuzzy implication without contrapositive symmetry into a fuzzy implication with contrapositive symmetry. In this paper, we carry out a detailed comparative study on the contrapositive symmetrisation methods .Firstly, we recall some definitions of fuzzy logic connectives including fuzzy negations, t ?norms, t ?conorms and fuzzy implications. Meanwhile, we give a brief introduction to the definitions and properties of S ?implications, R ?implications and the two new classes of fuzzy implications ( f ?implication and g ?implication)suggested by Yager in 2004.Secondly, we systematically discuss the properties of the three contrapositive symmetrisation methods existing in the literature and the one proposed in this paper. Some usual properties of fuzzy implications are involved, such as neutrality property, exchange principle, ordering property, strong fuzzy negation principle, consequent boundary, identity principle, continuity and so on. We present some conditions under which the four methods satisfy these properties and give the counterexamples if the involved properties are not satisfied.Then, we investigate the N -compatibility of these contrapositive symmetrisation methods and get some equivalent conditions under which the natural negation of the contrapositivised implication is equal to the strong negation involved.Finally, observing that every S -implication has contrapositive symmetry, we study whether a contrapositivised implication can be written as an S ?implication constructed from some binary operator. We obtain some sufficient conditions for the binary operator to become a t ?conorm.