Falsification-aware Twist Structures in Description Logic

We introduce falsification-aware twist semantics for the standard description logic
$\mathcal{ALC}$
with an extended language endowed with concept subtraction and implication operators. The proposed falsification-aware twist semantics is constructed based on the idea of N-lattices with twist structures, which can appropriately represent positive and negative information simultaneously. We prove the equivalence between the proposed falsification-aware twist semantics and the standard semantics for
$\mathcal{ALC}$
. We also observe that the proposed semantics is a generalization of the previously proposed twist semantics for quasi-classical, four-valued, and paraconsistent description logics.