New types of contrapositivisation of fuzzy implications with respect to fuzzy negations

Among the most studied properties of fuzzy implication functions, the so-called contrapositive symmetry or law of contraposition with respect to a fuzzy negation is of special interest because it is a key property in approximate reasoning, deductive systems and formal methods of proof. Even when contraposition is not satisfied, there exist some methods for modifying an implication function with the aim that the new implication satisfies this property, these methods are known as contrapositivisations. In this paper some new methods of contrapositivisation with respect to any fuzzy negation are presented and the properties preserved by these new methods are studied. Along this study we will see that these new methods not only preserve the usual properties preserved by the already known methods, but they also have some additional properties.