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.

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