  Tetravalent modal logic (T ML) was introduced by Font and Rius in 2000; and
it is an expansion of the Belnap-Dunn four{valued logic FOUR, a logical system
that is well{known for the many applications it has been found in several
fields. Besides, T ML is the logic that preserve degrees of truth with respect
to Monteiro's tetravalent modal algebras. Among other things, Font and Rius
showed that T ML has a strongly adequate sequent system, but unfortunately this
system does not enjoy the cut-elimination property. However, in [16], it was
presented a sequent system for T ML with the cut{elimination property. Besides,
in this same work, it was also presented a sound and complete natural deduction
system for this logic. In the present work, we continue with the study of T ML
under a proof-theoretic perspective. In first place, we show that the natural
deduction system given in [16] admits a normalization theorem. In second place,
taking advantage of the contrapositive implication for the tetravalent modal
algebras introduced in [17], we define a decidable tableau system adequate to
check validity in the logic T ML. Finally, we provide a sound and complete
tableau system for T ML in the original language.These two tableau systems
constitute new (proof-theoretic) decision procedures for checking validity in
the variety of tetravalent modal algebras.
