Rule of Explosion

Proof Rule

The Rule of Explosion is a valid deduction sequent in propositional logic.

Proof Rule

If a contradiction can be concluded, it is possible to infer any statement.

Sequent Form

$\bot \vdash \phi$

Explanation

The Rule of Explosion can be expressed in natural language as:

If you can prove a contradiction, you can prove anything.

Compare this with the colloquial expression:

"If England win the World Cup this year, then I'm a kangaroo."

The assumption is that the concept of England winning the world cup is an inherent contradiction (it being taken worldwide as a self-evident truth that England will never win the World Cup again). Therefore, if England does win the World Cup this year, then this would imply a falsehood as the author of this page is certainly human.

This rule is denied validity in the system of Johansson's minimal logic.

Variants

The following can be used as variants of this theorem:

Variant 1

$\vdash p \implies \paren {\neg p \implies q}$

Variant 2

$\vdash \left({p \land \neg p}\right) \implies q$

Variant 3

$p, \neg p \vdash q$

Also known as

The Rule of Explosion is also known as the rule of bottom-elimination.

Those who fancy Latin may like ex falso (sequitur) quodlibet, which literally means from a falsehood (follows) whatever you like.