Principle of explosion

The principle that, in classical logic, a contradiction entails every proposition. It is also called ex falso quodlibet.

Modus ponens: A rule of inference that derives a consequent from a conditional and its antecedent. Together with disjunction introduction, it derives any chosen proposition from a contradiction.
Paraconsistent logic: A family of logics in which contradictions do not entail every proposition. It rejects explosion while allowing some inconsistent theories to remain usable.
Classical logic: A family of logics characterized by principles such as excluded middle and double-negation elimination. Explosion is valid in the standard classical systems that frame the principle.
Inconsistent mathematics: Mathematical reasoning conducted with theories that contain contradictions without being discarded wholesale. Avoiding explosion lets mathematical work proceed from selected inconsistent assumptions.
Disjunction introduction: A rule of inference that allows a proposition to be inferred from that proposition alone or with a disjunction. Adding any proposition to a contradiction by disjunction enables explosion.
Relevant logic: A family of logics requiring premises to be relevant to the conclusions they support. Its relevance constraints block classical derivations that use unrelated premises to explode.
Logical consequence: A relation in which a conclusion follows from premises under specified rules or semantics. Explosion states a consequence relation: contradictory premises entail every proposition.
Database inconsistency: A condition in which stored data violate constraints or contain mutually incompatible claims. Classical query answering can become trivial if inconsistent data are treated as premises.
Contradiction: A conjunction of a proposition and its negation, or a state in which both are asserted. Explosion begins when a theory contains both a claim and its negation.
Minimal logic: A subsystem of intuitionistic logic that does not generally infer arbitrary conclusions from contradiction. It withholds the inference from falsehood to any proposition.

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