Law of noncontradiction

The principle that a proposition and its negation cannot both be true in the same respect and at the same time.

Aristotle: An ancient Greek philosopher whose works established influential foundations for logic, metaphysics, and natural philosophy. He gave the classic defense of noncontradiction in Metaphysics Book Γ.
Propositional logic: A formal system that represents propositions and combines them using logical connectives. It expresses the principle with a proposition and its negation.
Metaphysics (Aristotle): Aristotle's philosophical work examining being, substance, causation, and first principles. Book Γ contains his influential formulation and defense of noncontradiction.
Intuitionistic logic: A logic that interprets proof constructively and does not generally accept the law of excluded middle. It rejects excluded middle in general while retaining noncontradiction.
Law of identity: The logical principle that each thing is identical with itself. Together with identity, noncontradiction is often presented as a basic law of thought.
Negation: A logical operation that forms the denial of a proposition. The law concerns the impossibility of a proposition and its negation both being true.
Organon: The traditional collection of Aristotle's writings on logic and argument. Its logical framework helped transmit the principle through later philosophy.
Relevant logic: A family of logics requiring a meaningful connection between premises and conclusions. Some relevant systems resist classical inference from contradiction without accepting every classical principle.
Law of excluded middle: The principle that, for any proposition, either it or its negation is true. It is commonly grouped with noncontradiction, though it makes a distinct claim.
Contradiction: A pair of claims that cannot both be true under the same interpretation and conditions. A proposition and its negation form the central contradictory pair.

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Reductio ad absurdumRelated: A direct contradiction supplies the clearest endpoint for a reductio argument.