Annotated Logic with Rough Double Stone Algebraic Interpretation

In this study, we present rough annotated logic
$R_{\tau}$
which adopts rough double stone algebra (RDSA) as semantics interpretation for an annotation. Furthermore, we propose natural deduction system for
$NR_{\tau}$
, and we show
$R_{\tau}$
is non-alethic since it is both paracomplete and paraconsistent. To incorporate an annotation of RDSA into annotated logic
$P_{\tau}$
, we apply logical features of rough logic based on RDSA and its semantic structure represented with a lattice
$\tau$
which is an annotation of
$NR_{\tau}$
. In addition, we present some features of
$R_{\tau}$
and show a sketch of proof for soundness and completeness proof for
$NR_{\tau}$
.