My first thought was to look for the technical version of Priest's article, which turns out to be his book "In Contradiction", which turns out not to be in my university library. The Amazon preview tells me that he discusses Gödel's theorems but not the computational models that so many comments here talk about, and he gives a formalisation of some form of paraconsistent logic. However, the preview isn't enough to answer the basic question to ask about any non-standard logic: is it intertranslatable with classical logic, such that every truth of either is mapped to a truth of the other? If it is, then there is no philosophically interesting distinction between them, any more than there is between English and French, or C++ and Perl, or standard analysis and non-standard analysis.