126: Principle of Explosion
July 30, 2017
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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, . Let be any proposition we want to prove. By conjunction elimination, we have , and then by disjunction introduction we have
However, since we also have , it follows that 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.