Rule of Explosion

Context
The rule of bottom-elimination is one of the axioms of natural deduction.

The rule
If we can conclude a contradiction, we may infer any statement:
 * $\bot \vdash p$

It can be written:
 * $\displaystyle{\bot \over p} \bot_e$


 * Abbreviation: $\bot \mathcal E$
 * Deduced from: The pooled assumptions of $\bot$.
 * Depends on: The line containing $\bot$.

Explanation
What this says is: 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 Dutchman."

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 does not hail from Nederland.

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

Also known as
This is also known as the rule of explosion.

Those who fancy Latin may like ex falso sequitur quodlibet, which literally means: "from falsity anything follows".