In this talk we present some of the latest theoretical results obtained for the f-inclusion index. These results motivate its use as a new way of representing the inclusion between two fuzzy sets and as a logical inference operator. In this summary we recall two: the Sinha-Dougherty axioms are satisfied (conveniently adapted to the theoretical framework of the f-inclusion index) and, moreover, it corresponds to an optimal choice of a fuzzy residuated implication to carry out Modus Ponens inference.