I think there is supposed to be a correspondence between logics and kinds of category, e.g.,
(higher order?) classical logic elementary topos with some extra properties? (higher order?) intuitionistic logic elementary topos linear logic symmetric monoidal category with a dualizing object modal logic ?
I'm not sure exactly how much one can say about the entries on the right, but as a start, they are all 2-categories. So maybe a logic can be viewed as a (certain kind of) 2-category.
I would be grateful if an expert on the subject could expand this into a real answer! There is something similar on the nlab page for internal logic, but it does not seem to be geared specifically for the question as phrased here.