Principle of explosion

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In classical logic, intuitionistic logic, and similar logical systems, the principle of explosion^[a][b] is the law according to which any statement can be proven from a contradiction.^[1][2][3] That is, from a contradiction, any proposition (including its negation) can be inferred; this is known as deductive explosion.^[4][5]

The proof of this principle was first given by 12th-century French philosopher William of Soissons.^[6] 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.^[7] 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:

1. We know that "Not all lemons are yellow", as it has been assumed to be true.
2. We know that "All lemons are yellow", as it has been assumed to be true.
3. Therefore, the two-part statement "All lemons are yellow or unicorns exist" must also be true, since the first part of the statement ("All lemons are yellow") has already been assumed, and the use of "or" means that if even one part of the statement is true, the statement as a whole must be true as well.
4. However, since we also know that "Not all lemons are yellow" (as this has been assumed), the first part is false, and hence the second part must be true to ensure the two-part statement to be true, i.e., unicorns exist (this inference is known as the Disjunctive syllogism).
5. The procedure may be repeated to prove that unicorns do not exist (hence proving an additional contradiction where unicorns do and do not exist), as well as any other well-formed formula. Thus, there is an explosion of true statements.

In a different solution to the problems posed by the principle of explosion, some mathematicians have devised alternative theories of logic called paraconsistent logics, which allow some contradictory statements to be proven without affecting the truth value of (all) other statements.^[7]

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