Using negative results to prove positive results in computability theory
In some sense, this is what the theory of parametricity is all about. 

Data abstraction is how we ensure that no client of a module can access the elements of a module except according to the interface exposed by the module. 
We rely on this to ensure that the internal invariants of data structures cannot be broken by the clients of a module -- eg, if you access a balanced tree only via the published interface, then it follows that the tree will always be balanced. 

So we use a negative property -- that no possible client can break the abstraction boundary -- to deduce a positive one -- that the data representation invariant always holds.