It seems intuitive that finding a counter-example might be easier than proving a generality, since you're starting from a concrete goal ("build a foo that has properties X, Y & Z") that you can branch out from, identify sub-problems, etc.

Proving a generality seems much more difficult since you don't know what you are trying to build, although I suppose in some cases you can prove it by proving that it's impossible to construct a counter-example.