Rule of Explosion

Proof Rule
The rule of bottom-elimination is a valid deduction sequent in propositional logic: If a contradiction can be concluded, it is possible to infer any statement.

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

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 logic.

Variants
The following can be used as variants of this theorem:

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 follows whatever you like.

Also see

 * False Statement implies Every Statement

Technical Note
When invoking the Rule of Bottom-Elimination in a tableau proof, use the BottomElimination template:



or:

where:
 * is the number of the line on the tableau proof where Rule of Bottom-Elimination is to be invoked
 * is the pool of assumptions (comma-separated list)
 * is the statement of logic that is to be displayed in the Formula column, without the  delimiters
 * is the line of the tableau proof upon which this line directly depends, the one with $\bot$ on it
 * is the (optional) comment that is to be displayed in the Notes column.