Principle of explosion

In classical logic, intuitionistic logic and similar logical systems, the principle of explosion (Latin: ex falso [sequitur] quodlibet, 'from falsehood, anything [follows]'; or ex contradictione [sequitur] quodlibet, 'from contradiction, anything [follows]'), or the principle of Pseudo-Scotus, is the law according to which any statement can be proven from a contradiction.[1] That is, once a contradiction has been asserted, any proposition (including their negations) can be inferred from it; this is known as deductive explosion.[2][3]

The proof of this principle was first given by 12th-century French philosopher William of Soissons.[4] Due to the principle of explosion, the existence of a contradiction (inconsistency) in a formal axiomatic system is disastrous; since any statement can be proven, it trivializes the concepts of truth and falsity.[5] Around the turn of the 20th century, the discovery of contradictions such as Russell's paradox at the foundations of mathematics thus threatened the entire structure of mathematics. Mathematicians such as Gottlob Frege, Ernst Zermelo, Abraham Fraenkel, and Thoralf Skolem put much effort into revising set theory to eliminate these contradictions, resulting in the modern Zermelo–Fraenkel set theory.

As a demonstration of the principle, consider two contradictory statements—"All lemons are yellow" and "Not all lemons are yellow"—and suppose that both are true. If that is the case, anything can be proven, e.g., the assertion that "unicorns exist", by using the following argument:

In a different solution to these problems, a few mathematicians have devised alternate theories of logic called paraconsistent logics, which eliminate the principle of explosion.[5] These allow some contradictory statements to be proven without affecting other proofs.

In symbolic logic, the principle of explosion can be expressed schematically in the following way:

Reduction in proof strength of logics without ex falso are discussed in minimal logic.