What is a logic?

Logic is the construction and study of models for reasoning.

There are many logics, because we reason in different ways, depending on the context. Modality shows up because there are contexts in which out statements are qualified (I can be rich, but I can also be very rich, and there is a difference between the two statements, and a connection of modality between the two, not captured by classical logic); temporality shows up because we sometimes reason with time; linear logic shows up (among other reasons) because we need to reason about resources (If I have a dollar, I can buy one candy — but we all know that we do not get to use the statement "I have a dollar" more than once... and this is different from "There are infinitely many primes", which has not ceased to be true for a loooong time, and we do not expect it to); paraconsistent logics show up because we need to be able to reason in the presence of conflicting information, simply because we tend to have conflicting information; and so on.

Of course, one wonders why logicians do not get their act together and come up with one logic to rule them all... Well, they may be waiting for physicists to wrap up their grand unified theory first!

(N.B.: all of this is prefixed by a huge "In my opinion", of course)

I know nothing about this but I happened to come across it while reading about closure operators on wikipedia: Universal Logic.

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.