Laws of Non-Contradiction, Laws of the Excluded Middle, and Logics

There is widespread acknowledgement that the law of non-contradiction (LNC) is an important logical principle. However, there is less-than-universal agreement on exactly what the law amounts to. This lack of clarity is brought to light by the emergence of paraconsistent logics in which contradictions are tolerated: from the point of view of proofs, not everything need follow from a contradiction. From the point of view of models, there are 'worlds' in which contradictions are true. In this sense, the LNC is violated in these logics. However, in many paraconsistent logics, statement <(A & <A) (it is not the case that A and not-A) is still provable. In this sense, the LNC is upheld. This chapter attempts to clarify the different readings of the LNC, in particular taking cues from the tradition of relevant logics. A further guiding principle will be the natural duality between the LNC and rejection on the one hand and the law of the excluded middle and acceptance on the other.