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principle of explosion

n. (label en logic) (l en The principle or axiom of classical logic stating that if a contradiction or a false proposition) is proven to be true, then it proves that everything is true. In symbols: bot to P

Principle of explosion

The principle of explosion ( Latin: ex falso (sequitur) quodlibet (EFQ), "from falsehood, anything (follows)", or ex contradictione (sequitur) quodlibet (ECQ), "from contradiction, anything (follows)"), or the principle of Pseudo-Scotus, is the law of classical logic, intuitionistic logic and similar logical systems, according to which any statement can be proven from a contradiction. That is, once a contradiction has been asserted, any proposition (including their negations) can be inferred from it.

As a demonstration of the principle, consider two contradictory statements – “All lemons are yellow” and "Not all lemons are yellow", and suppose (for the sake of argument) that both are simultaneously true. If that is the case, anything can be proven, e.g. "Santa Claus exists", by using the following argument:

  1. We know that "All lemons are yellow" as it is defined to be true.
  2. Therefore, the statement that (“All lemons are yellow" OR "Santa Claus exists”) must also be true, since the first part is true.
  3. However, if "Not all lemons are yellow" (and this is also defined to be true), Santa Claus must exist – otherwise statement 2 would be false. It has thus been "proven" that Santa Claus exists. The same could be applied to any assertion, including the statement "Santa Claus does not exist".

The principle is not a universal rule; rather it exists as a consequence of a choice of which logic to use. It does not appear in some paraconsistent logics which allow localised 'gluts' of contradictory statements to be proved without affecting other proofs. In artificial intelligence and models of human reasoning it is common for such logics to be used. Truth maintenance systems are AI models which try to capture this process.