  The {\em tetravalent modal logic} ($\cal TML$) is one of the two logics
defined by Font and Rius (\cite{FR2}) (the other is the {\em normal tetravalent
modal logic} ${\cal TML}^N$) in connection with Monteiro's tetravalent modal
algebras. These logics are expansions of the well--known {\em Belnap--Dunn's
four--valued logic} that combine a many-valued character (tetravalence) with a
modal character. In fact, $\cal TML$ is the logic that preserve degrees of
truth with respect to tetravalent modal algebras. As Font and Rius observed,
the connection between the logic $\cal TML$ and the algebras is not so good as
in ${\cal TML}^N$, but, as a compensation, it has a better proof-theoretic
behavior, since it has a strongly adequate Gentzen calculus (see \cite{FR2}).
In this work, we prove that the sequent calculus given by Font and Rius does
not enjoy the cut--elimination property. Then, using a general method proposed
by Avron, Ben-Naim and Konikowska (\cite{Avron02}), we provide a sequent
calculus for $\cal TML$ with the cut--elimination property. Finally, inspired
by the latter, we present a {\em natural deduction} system, sound and complete
with respect to the tetravalent modal logic.
