Modal Logics with Intersection Modality

We give a simple proof of a result recently obtained in [12] on the completeness of modal logics with modality that corresponds to the intersection of accessibility relations in a Kripke model. Completeness is proved for logics in modal languages of two types: one has modalities \({{\square }_{1}}, \ldots ,{{\square }_{n}}\) for relations \({{R}_{1}}, \ldots ,{{R}_{n}}\) that satisfy a unimodal logic L and modality \({{\square }_{{n + 1}}}\) for the intersection \({{R}_{{n + 1}}} = {{R}_{1}} \cap \ldots \cap {{R}_{n}}\); the other language has modalities \({{\square }_{i}}(i \in \Sigma )\) for relations R_i that satisfy the logic L, and, for every nonempty subset of indices \(I \subseteq \Sigma \), the modality \({{\square }_{I}}\) for the intersection \(\bigcap\nolimits_{i \in I} {{R}_{i}}\). While in [12] the completeness is proved only for logics over \({\mathbf{K,KD,KT,K4,S4}}\), and S5, we give a “uniform” construction that enables us to obtain completeness for logics with intersection over 15 “traditional” modal logics KΛ for \(\Lambda \subseteq \{ {\mathbf{D,T,B,4,5}}\} \). The proof method is based on unraveling a frame and then taking the Horn closure of the resulting frame.