  Moisil logic, having as algebraic counterpart \L ukasiewicz-Moisil algebras,
provide an alternative way to reason about vague information based on the
following principle: a many-valued event is characterized by a family of
Boolean events. However, using the original definition of \L ukasiewicz-Moisil
algebra, the principle does not apply for subalgebras. In this paper we
identify an alternative and equivalent definition for the $n$-valued \L
ukasiewicz-Moisil algebras, in which the determination principle is also saved
for arbitrary subalgebras, which are characterized by a Boolean algebra and a
family of Boolean ideals. As a consequence, we prove a duality result for the
$n$-valued \L ukasiewicz-Moisil algebras, starting from the dual space of their
Boolean center. This leads us to a duality for MV$_n$-algebras, since are
equivalent to a subclass of $n$-valued \L ukasiewicz-Moisil algebras.
