  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.
