TheLawOfNoncontradiction

Variously formulated, the gist of the Law of Noncontradiction is that a proposition and its denial cannot both be true at the same time and "in the same respect." If we're being careful, we won't formulate it this way, in terms of truth (it might then might be confused with ThePrincipleOfBivalence). Instead, we'll say the law is: for any proposition P, it is not both the case that P and not-P. In Aristotle's formulation: "One cannot say of something that it is and that it is not in the same respect and at the same time."

The Law of Noncontradiction is almost universally affirmed by philosophers. See also TheLawOfExcludedMiddle and ThePrincipleOfBivalence.

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However, see [1] for a paper (in PDF format) on "paraconsistent" logics and non-contradiction:

Abstract: There is widespread agreement that the law of non-contradiction is an important logical principle. There is less 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 (in the sense that not everything need follow from a contradiction, and that there are "worlds" in which contradictions are true) but in which the statement [not (A and not-A)] (it is not the case that A and not-A) is still provable. This paper attempts to clarify the connection between different readings of the law of non-contradiction, the duality between the law of non-contradiction and the law of the excluded middle, and connections with logical consequence in general.

...and [2] for more discussion of this "law".