Validity, consistency and the meaning of the recovery operators

In this paper, we argue that the connective  present in the Logics of Formal inconsistency (LFIs) are better understood as expressing the notion of classicality. We present the modal logic LFI1S0:5, whose modalities  and  captue he formal concepts of logical validity and logical consistency of the logic LFI1, the strongest LFI. We also prove that these modalities can also work as recovery operators. Keywords: paraconsistency; consistency; modal logics; recovery operators; logics of formal inconsistency.