Resolving contradictions: A plausible semantics for inconsistent systems

The purpose of a knowledge system S is to represent the world W faithfully. If S turns out to be inconsistent containing contradictory data, its present state can be viewed as a result of information pollution with some wrong data. However, we may reasonably assume that most of the system content still reflects the world truthfully, and therefore it would be a great loss to allow a small contradiction to depreciate or even destroy a large amount of correct knowledge. So, despite the pollution, S must contain a meaningful subset, and so it is reasonable to assume (as adopted by many researchers) that the semantics of a logic system is determined by that of its maximally consistent subsets, mc-subsets. The information contained in S allows deriving certain conclusions regarding the truth of a formula F in W. In this sense we say that S contains a certain amount of semantic information and provides an evidence of F. A close relationship is revealed between the evidence, the quantity of semantic information of the system, and the set of models of its mc-subsets. Based on these notions, we introduce the semantics of weighted mc-subsets as a way of reasoning in inconsistent systems. To show that this semantics indeed enables reconciling contradictions and deriving plausible beliefs about any statement including ambiguous ones, we apply it successfully to a series of justifying examples, such as chain proofs, rules with exceptions, and paradoxes.