Anybody here familiar with paraconsistent logic?

I’d love to help you out and it does sound interesting, but I didn’t even understand the definition of paraconsistent logic that I found.

Graham Preist has written a number of primers to dialetheism and the paraconsistent logics (note the plural, they are a number of different systems). I would say he’s the “go to” guy on this topic.

As @hiphiphopflipflapflop says, Graham Priest is typically considered the leading expert on these issues. It is my opinion, however, that he sometimes ties paraconsistent logic and dialetheism too closely together. I have no problem with paraconsistent logic as a tool for certain purposes, but I do not find dialetheism to be at all plausible.

The only motivation for the latter, after all, is that it supposedly gives us a solution to the liar paradox. It cannot give a solution to Curry’s paradox, however, whereas some of the alternative solutions that have been proposed to the liar paradox can also resolve Curry’s paradox while still preserving classical logic. Thus I see no reason to accept the unpalatable notion that there are true contradictions (a statement that itself is a contradiction, though Priest would insist it is a true one).

What is different about paraconsistent logic, then? Paraconsistent logic does not include the principle of explosion—which is, the rule that says anything follows from a contradiction—so it is not immediately trivial to investigate sets of claims that contain a contradiction. It sort of allows us to “ignore” the contradiction and figure out what does and does not follow non-trivially from a set of claims. I quite like how the following sentence from the Wikipedia article on paraconsistent logic puts it:

The primary motivation for paraconsistent logic is the conviction that it ought to be possible to reason with inconsistent information in a controlled and discriminating way.

I should point out, however, that there is a debate about whether paraconsistent logic is the correct logic (meaning we should dispense with the principle of explosion altogether) or whether it is simply a logic to be used for special purposes (meaning there might be nothing wrong with the principle of explosion, even if it is sometimes unhelpful or even obstructive). I have already given myself away as holding the latter, though without committing myself to the view that something else is the correct logic or that there even is such a thing as the correct logic. I do believe that anything follows from a contradiction, but I also hold that we can reason about internally inconsistent world views in a sensible way.