\noindent The research we will report on takes its move on an alternative presentation of classical and positive
modal logic where the coalgebraic modalities $\nabla$ and $\Delta$ are taken as
primitive. 
Analogously to the duality between $\Box$ and
$\Diamond$, $\Delta$ can be defined as the dual of $\nabla$, but also, 
interestingly, in the following, negation-free way:

\begin{equation}
\label{eq:delta for powerset} \Delta \alpha =\begin{cases}
\nabla\varnothing \vee \bigvee\{\nabla\{a\}\ |\ a\in \alpha\}\vee
\nabla\{\bigwedge\alpha,
\top\} & \mbox{ if } \alpha\neq \varnothing\\
\nabla\{\top\} & \mbox{ if } \alpha = \varnothing\\
\end{cases}
\end{equation}

In earlier
work, %~\cite{bilk:AMY08}, 
we introduced a one-sided Gentzen system
%(see Definition~\ref{def:G1nabla}),
for an expansion of the
Boolean propositional language with  the nabla operator only;  this
system was shown to be sound, complete w.r.t.\ the class of all
Kripke models, and cut-free. Moreover, we defined a sound and
complete two-sided Gentzen system
%(see Definition \ref{def:G2nabla})
for the positive fragment of the same language, the cut rule of
which was shown not to be eliminable. The main feature of both
Gentzen systems was their being generalizable to the coalgebraic
setting where $\F$ is an arbitrary weak pullback-preserving
$\mathsf{Set}$-endofunctor.

Here will report on an improved  two-sided sequent calculus for the $\nb\Delta$-based coalgebraic logic
over a standard and weak-pullback preserving functor $\F$: this sequent calculus is sound, complete, invertible and {\em cut free}. Moreover, sound, complete, invertible and cut-free variants of this system  include a simplified two-sided version for the powerset functor and  both a one-sided and a two-sided versions for the $\Delta$-free fragment of the language.