I am trying to explain why double negation elimination $\neg \neg \phi \vdash \phi$ is invalid in intuitionistic logic, but introduction $\phi \vdash \neg \neg \phi$ is valid.
The latter is easy: if I can prove $\phi$, then I can prove that '$\phi$ is contradictory' is contradictory. I might explain this by saying that "if I can see that your coat is blue, then I can immediately construct a contradiction from (your coat is not blue)".
For elimination, one can see how having a proof that `$\phi$ is contradictory' is contradictory doesn't furnish you with a proof of $\phi$ itself, but in my experience people don't find examples like "'(your coat is not blue) is contradictory' does not give a proof that your coat is blue" very convincing.
What I would like is a similar simple natural language example that makes it clear that elimination is not straightforwardly valid.
