Just to note that Zach Weber has set up a blog for the Melbourne based, ARC-funded project: Paraconsistent Foundations of Mathematics. Zach Weber is a post doc on the project, working together with Chief Investigators Graham Priest and Greg Restall. In addition, Francesco Berto, from the University of Aberdeen, NIP, is a research associate.
You can read about the aim and background on the blog. If you want the real thing, go look at Zach's PhD thesis Paradox and Foundation here.