We shall be concerned with the modal logic BK — which is based on the Belnap–Dunn four-valued matrix, and can be viewed as being obtained from the least normal modal logic K by adding `strong negation'. Though all four values `truth', `falsity', `neither' and `both' are employed in its Kripke semantics, only the first two are expressible as terms. We show that expanding the original language of BK to include constants for `neither' or/and `both' leads to quite unexpected results. To be more precise, adding one of these constants has the effect of eliminating the respective value at the level of BK-extensions. In particular, if one adds both of these, then the corresponding lattice of extensions turns out to be isomorphic to that of ordinary normal modal logics.