Principle of explosion

Ex falso quodlibet, also known as ex contradictione (sequitur) quodlibet or the principle of explosion is the rule of classical logic that states that anything follows from a contradiction. In more formal terms, from any proposition of the form P ∧ ¬P, any arbitrary A can be derived. "Explosion" refers to the fact that the acceptance of a single contradiction into one's system makes the number of overall theorems "explode".

Besides the general prima facie implausibility of contradictions, this is the primary logical argument for not allowing P ∧ ¬P to be true in a formal system: systems in which any arbitrary formula is a theorem are trivial. Thus explosion justifies the law of noncontradiction.

Explosion is based on several of the fundamental formal properties of disjunction, the logical operator corresponding to the English "or". Consider the following proof:

Subscribers to paraconsistent logics reject the above reasoning, usually citing the invalidity of either disjunction introduction or disjunctive syllogism. Dialetheism, one particular paraconsistent logic, rejects the argument in order to accept certain instances of P ∧ ¬P.