I'm interested in Priest's solution using paraconsistent logic, but before I embark on that, I wanted to know if there was a generally agreed upon solution in more "classical" schools of thought. Looking up things online, I only see just different arbitrary ways of looking at it. If this is so, I'm inclined to think that maybe paraconsistent logic is required for a better way of systemizing thought. I'm familiar with the property/relation aspect of this problem the most, so this context is preferred.
The short answer is: no, there's no generally accepted solution of this 'problem' (if it is one). Some simply reject Bradley's argument(s) since they reject some of its (their) assumptions, for instance, that particulars are bundels of qualities or that qualities are tropes (e.g. Russell). Those that accept Bradley's assumptions have responded in various ways:
- Relata are related by non-relational links (Bergmann, the earlier Armstrong)
- The complex entities themselves such as facts or states of affairs unify their constituents (the later Armstrong)
- Relata are related via some sort of mutual inter-dependence (Frege, Simons)
There's some discussion of these options in the SEP-entry on the topic:
Bradley did not advocate an infinite regress but gave us a way to avoid one. He denied the fundamental reality of the distinctions that lead us into this problem. As sequitor indicates above, there is no other known solution.
I have yet to see a successful argument to justify the use in philosophy of a paraconsistent logic and feel Priest is muddled on this issue. His logic is not a solution for anything but is, rather, the claim that the Universe is incomprehensible. If you can find a way to make paraconsistent logic useful in philosophy you might become mildly famous since so far Priest has not succeeded.
His problem is that he interprets metaphysics, and in particular Buddhist metaphysics, as requiring a non-ordinary form of logic where this is not the case. His mistake is exactly the mistake I refer to in a recent question here Do Most Philosophers Ignore the Rules of Dialectical Logic?
If you delve into the issues raised by this question you'll see that Priest makes a basic mistake, and one that is more or less ubiquitous is Western thought. If we study Nagarjuna's metaphysics without making this mistake we should see that it solves all problems of existential regression for a fundamental theory with no need to modify or change ordinary logic. His solution is the same as Bradley's. The idea that Reality requires a paraconsistent logic for its description is ad hoc when there are solutions that do not require it.
It's a big topic and this is just a sketch of an answer. A longer answer is here https://philpapers.org/archive/JONANA-6.pdf.