(A ∧ ¬A)→ B

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". Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. ... Broadly speaking, a contradiction is when two or more statements, ideas, or actions are seen as incompatible. ... Proposition is a term used in logic to describe the content of assertions, content which may be taken as being true or false, and which are a non-linguistic abstraction from the linguistic sentence that constitutes an assertion. ...

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. Prima facie is a Latin expression meaning at first sight, used in common law regions to denote a case that is strong enough to justify further discovery and possibly a full trial. ... In mathematics and in the sciences, a formula is a concise way of expressing information symbolically (as in a mathematical or chemical formula), or a general relationship between quantities. ... A theorem is a proposition that has been or is to be proved on the basis of explicit assumptions. ... In logic, the law of noncontradiction judges as false any proposition P asserting that both proposition Q and its denial, proposition not-Q, are true at the same time and in the same respect. In the words of Aristotle, One cannot say of something that it is and that it...

Explosion is based on several of the fundamental formal properties of disjunction, the logical operator corresponding to the English "or". Consider the following proof: In logical calculus, logical operators or logical connectors serve to connect statements into more complicated compound statements. ...

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. In logic, conjunction elimination is the inference that, if the conjunction A and B is true, then A is true, and B is true. ... Disjunction introduction is the logic principle that, if A is true, then its true that either A or B is true. ... A disjunctive syllogism is one valid, simple argument form: A or B If not A Then B In logical operator notation: ¬ where represents the logical assertion. ... A paraconsistent logic is a logic which attempts to deal with contradictions. ... Dialetheism is a paraconsistent logic typified by its tolerance of at least some contradictions. ...