> if you want to claim "forall x in X, P(x) is true" then you need to exhibit a particular element of x for which P holds
I don’t mean to be pedantic (although it’s in keeping with constructivism) but in the case you describe, you don’t have to provide a particular x but rather you have to provide a function mapping all x in X to P(x). It may very well be that X is uninhabited but this is still a valid constructive proof (anything follows from nothing, after all).
If instead of “for all” you’d said “there exists”, then yes constructivism requires that you deliver the goods you’ve promised.
I don’t mean to be pedantic (although it’s in keeping with constructivism) but in the case you describe, you don’t have to provide a particular x but rather you have to provide a function mapping all x in X to P(x). It may very well be that X is uninhabited but this is still a valid constructive proof (anything follows from nothing, after all).
If instead of “for all” you’d said “there exists”, then yes constructivism requires that you deliver the goods you’ve promised.