A simple approach towards recapturing consistent theories in paraconsistent settings

I believe that, for reasons elaborated elsewhere (Beall, 2009; Priest, 2006a, 2006b), the logic LP (Asenjo, 1966; Asenjo & Tamburino, 1975; Priest, 1979) is roughly right as far as logic goes.(1) But logic cannot go everywhere; we need to provide nonlogical axioms to specify our (axiomatic) theories. This is uncontroversial, but it has also been the source of discomfort for LP-based theorists, particularly with respect to true mathematical theories which we take to be consistent. My example, throughout, is arithmetic; but the more general case is also considered.