# Today I Learned

Some of the things I've learned every day since Oct 10, 2016

## 126: Principle of Explosion

In logic systems such as classical logic, the principle of explosion is the principle that from a contradiction any proposition can be inferred to be true.

To show this, suppose we’ve arrived at a contradictory statement, $P \wedge \neg P$. Let $Q$ be any proposition we want to prove. By conjunction elimination, we have $P$, and then by disjunction introduction we have

$P \vee Q$.

However, since we also have $\neg P$, it follows that $Q$ is true.

This principle suggests an interesting alternative attitude towards contradiction in systems that exhibit this behavior: contradiction is not undesirable because of any ‘intuitive’ nonsensicalness, but simply because a set of assumptions leading to contradiction gives us no information.