The authors of Beziau and Franceschetto work with logics that have the property of not satisfying any of the formulations of the principle of non contradiction, Béziau and Franceschetto also analyze, among the three-valued logics, which of these logics satisfy this property. They prove that there exist only four of such logics, but only two of them are worthwhile to study. The language of these logics does not consider implication as a connective. However, the enrichment of a language with an (...) implication connective leads us to more interesting systems, therefore we look for one implication for these logics and we study further properties that the logics obtain when this connective is added to these systems. (shrink)