  Involutive Stone algebras (or {\bf S}--algebras) were introduced by R.
Cignoli and M. Sagastume in connection to the theory of $n$-valued \L
ukasiewicz--Moisil algebras.
  In this work we focus on the logic that preserves degrees of truth associated
to involutive Stone algebras, named {\bf \em Six}. This follows a very general
pattern that can be considered for any class of truth structure endowed with an
ordering relation, and which intends to exploit many--valuedness focusing on
the notion of inference that results from preserving lower bounds of truth
values, and hence not only preserving the value $1$.
  Among other things, we prove that {\bf \em Six} is a many--valued logic (with
six truth values) that can be determined by a finite number of matrices (four
matrices). Besides, we show that {\bf \em Six} is a paraconsistent logic.
Moreover, we prove that it is a genuine LFI (Logic of Formal Inconsistency)
with a consistency operator that can be defined in terms of the original set of
connectives. Finally, we study the proof theory of {\bf \em Six} providing a
Gentzen calculus for it, which is sound and complete with respect to the logic.
