In classical logic, if logic program S is inconsistent, then S |= p for all p. That is, any proposition p may be derived from S, since "any conclusion follows a false premise." While logically sound, this is not particularly useful. What's more, it is somewhat contrary to ordinary, everyday analytical reasoning: people (normally) do not give up and allow for any assertion whenever they are confronted with inconsistent or conflicting information. Rather, they find some means to resolve the conflicts in order to arrive at some definite conclusion.
As an example, consider the jury process. Jurors in a trial are often presented with evidence that both support and refute a proposition ("the defendant is guilty") but usually still manage to come to a verdict ("guilty" / "not guilty"). Needless to say, there is a variety of methods and skills employed by the jurors to accomplish this task, such as determining the credibility of the witnesses, considering the solidity of forensic evidence, etc.
What is of interest here is one possible method with which jurors resolve conflicting evidence: by isolating subsets of evidence which are consistent and which tend to support some intermediate propositions. For instance, the jurors may decide that evidence A, B, and C may point to a sufficient 'window of opportunity' and P, Q, and R may support the theory that the defendant committed the crime by X means. Now, it's possible that A contradicts P so that the body of evidence as a whole is inconsistent. Nevertheless, such interim decisions can be used as a foundation for arriving at the final verdict.
The logical concepts introduced in [Fisher96] represent an attempt to help formalize this process of first identifying subsets of consistent information in order to (possibly) make further derivations from them. The new logical concepts provide a framework for dealing with logic programs which may be inconsistent as a whole but which may nevertheless contain interesting sub-programs that are in themselves consistent.