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.
