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

There is widespread acknowledgement that the law of non-contradiction is an important logical principle. However, there is less-than-universal agreement on exactly what the law amounts to. This unclarity 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 law of non-contradiction is violated in these logics. However, in many paraconsistent logics, statement (it is not the case that and not) is still provable. In this sense, the law of non-contradiction is upheld. This paper attempts to clarify the different readings of the law of non-contradiction, in particular taking cues from the tradition of relevant logics. A further guiding principle will be the natural duality between the law of non-contradiction and rejection on the one hand and the law of the excluded middle and acceptance on the