Two variants of noncontingency operator

By slightly adapting two equivalent semantics of noncontingency operator, we obtain two variants, $\boxdot$ and $\boxplus$, with non-equivalent semantics. We show that on the class of models satisfying any of five basic properties (i.e. seriality, reflexivity, transitivity, symmetry, Euclidicity), the logic $\mathcal{L}(\boxdot)$, which has $\boxdot$ as the sole modal primitive, is less expressive than the logic $\mathcal{L}(\boxplus)$, which has $\boxplus$ as the sole modal primitive. We investigate the frame definability of both languages. We then axiomatize $\mathcal{L}(\boxplus)$ and $\mathcal{L}(\boxdot)$ over various classes of bimodal frames. Among other results, a notion of morphisms, called `$\boxdot$-morphisms', are provided to show the completeness of axiomatizations of $\mathcal{L}(\boxdot)$ over serial frames and also over symmetric frames.