MODAL LOGICS WITH THE INTERSECTION MODALITY

We give a simple proof of a recently obtained in [12] result on the completeness of modal logics with the modality that corresponds to the intersection of accessibility relations in a Kripke model. In epistemic logic, this is the so-called distributed knowledge operator. We prove completeness for the logics in the modal languages of two types: one has the modalities □1,...,□n for the relations R1,...,Rn that satisfy a unimodal logic L, and the modality □n+1 for the intersection Rn+1=R1 ∩...∩ Rn; the other language has the modalities □i (i ∈ Σ) for the relations Ri that satisfy the logic L, and, for every nonempty subset of indices I ⊆ Σ, the modality □I for the intersection ∩i∈I Ri. While in [12] the completeness is proved only for the logics over K, KD, KT, K4, S4, and S5, here we give a "uniform" construction that enables us to obtain completeness for the logics with intersection over the 15 so-called "traditional" modal logics KΛ, for Λ ⊆ {D, T, B, 4, 5}. The proof method is based on unravelling of a frame and then taking the Horn closure of the resulting frame.