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.

I think there is supposed to be a correspondence between logics and kinds of category, e.g.,

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.