Anybody here familiar with paraconsistent logic?

I just read a description intended for the layman, and would like to know more. It is an interesting concept.

Most of us are not particularly bothered by paradoxical statements like the well known liar paradox, “This statement is false.” However, those who work on mathematical foundations get very upset by such statements. In classical mathematics, if any system contains both a statement and its contradiction, then anything at all in the system can be proved. The system is labeled inconsistent and is of no value. In order to avoid paradoxical statements, there is an elaborate process for formally building systems.

Paralogic gets around this by permitting some contradictory statements. What I am trying to find out, without going too deep into the subject, is what the rules are for permitting contradictory statements.