We develop C4 a logic for reasoning with information containing non-logical conflicts, where the information is encoded in the form of normal logic programs and the conflicts are represented using a construct called "contestation." We prove that the C4 logic is inferentially conflict-free in the sense that the set of entailments of a normal logic program augmented with a set of contestations are guaranteed to be free of the conflicts specified by the set of contestations. We provide a sound and complete procedure for answering ground queries to a ground and finite normal logic program augmented with a set of ground contestations. We show that C4 provides a new semantics for normal logic programs that subsumes both the stable model semantics and the well-founded semantics for normal logic programs. We use C4 to provide a new account of integrity constraint satisfaction for databases that may be inconsistent with their integrity constraints. We extend C4 to C5, a five valued logic, which is used to provide a new semantics for extended logic programs. We show that C5 can be used to provide an inferentially conflict-free logic for reasoning with information containing both logical and non-logical conflicts.