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.