# Rule of Explosion

## Proof Rule

The **rule of explosion** is a valid argument in certain types of logic dealing with contradiction $\bot$.

This includes classical propositional logic and predicate logic, and in particular natural deduction, but for example not Johansson's minimal logic.

### Proof Rule

- If a contradiction can be concluded, it is possible to infer any statement $\phi$.

### 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

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

### 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**.