Is there only one logic in mathematics, or are there multiple variants of logic? Is there only one logic in mathematics, or are there multiple variants of logic? I heard that some people believed in logical pluralism. I am wondering if mathematics provide the tools necessary for alternative logic to be explored and used in mathematical abstractions. What are some or all of these variants, and is there a root logic to all these variants or do wayward logics exist that are completely unrelated to the more conventional logic we know?
 A: Welcome to MSE!
The answer to this question depends on how philosophical you want to be. I'll start with a formalist idea.
If by "logic" you mean "a formal method of proving things", then yes. Lots of people study lots of systems in which theorems can be proven. And it's very interesting (at least to me) what the differences in these systems are. What theorems can you prove in one but not another? What do proofs look like in the different systems? Which (if any) of these systems could be adapted to work as a "bottom floor" on which the rest of mathematics is built? After all, when we prove things as mathematicians, what system are we working in!?
Some common things that people think about include intuitionistic logic, which restricts the notion of "theorem" to only encompass those things which have a witness to their truth. Another popular one in the CS community is linear logic, which restricts the notion of "theorem" to only those things which (roughly) "use each assumption exactly once".
Of course, we can prove things about these proof systems. Some people call this "metamathematics", and it brings up an obvious question (which I hinted at before): In which system are we proving facts about these proof systems?
It is at this point that the question becomes philosophical in nature, and I'm not sure MSE is the best place for it.

It's useful to know that mathematics can be formalized inside some proof system, because it lets us use theorems from logic in order to prove things in other branches of math. However, I personally, don't really care what that system is, and I don't think it matters. Maybe one day I'll regret saying that, but in the moment it's how I feel.
Of course, many people disagree with me. We know that math can all be done internal to the theory of sets (ZFC, or others). So in order to formally encode all mathematics (and thus, potentially argue that all mathematicians are really working inside some logical system) we need exactly enough logic to talk about ZFC.... In fact we can go deeper. We need enough logic to describe a logic which can describe ZFC.
I'm not sure how deep the rabbit hole goes, but there are papers who are trying to build the weakest system possible (which minimizes the possibility of incompleteness happening) which can still, at some point, do set theory (and thus mathematics). Whether we're "really" working in one of these systems or not is unanswerable, of course. So it doesn't particularly interest me personally.

So, to summarize: Yes. There are lots of different logics which people study, and mathematics absolutely furnishes us with the tools to study and compare what is provable in the different systems. Asking for "some or all" of the variants in a MSE answer is a bit of a tall order. Lots and lots of ink has been spilled on this topic. The question of if a "root logic" exists is a very subtle one, and personally I think it's best left to the mathematical philosophers.
As one last aside, if you're interested in different logical systems, and want to see these ideas applied to mathematics that nonlogicians care about, you might be interested in reverse mathematics it's a relatively popular subject right now, and it's full of lots of interesting questions (and some answers!) in this vein.
Edit: I should clarify that the "logical systems" that are studied in reverse mathematics are different from the proof systems I was describing earlier. Rather than studying different ways of proving things, we fix a way of proving things (a classical one) and instead ask what theorems we can prove when we vary our axioms. These are "alternative logic"s in the sense that most of them are not strong enough to prove everything an analyst (say) is interested in. I wasn't sure if this was relevant, but since it's something I'm interested in I thought I would mention it anyways.

I hope this helps ^_^
