Multi-Dimensional Semantics for Every Modal Language

After seeing so many multi-dimensional modal logics, the reader might wonder if not every modal logic is (or can be seen as) a multi-dimensional one. The answer is in some sense positive. We will show that for every modal similarity type S, the basic derivation system K S is strongly sound and complete with respect to a multidimensional class of frames. The dimension and the interpretation of the modalities depends only on the type S.