I'm trying to understand what happens when quantifiers in the prediate are swapped.
Let's assume the following:
Two sets A and a D and a predicate H(a, d), where a ∈ A and d ∈ D.
There exists d ∈ D for all a ∈ A such that H(a, d).
This one I can figure out. It means that there exists a single d for all a such that H(a, d).
For all a ∈ A there exists d ∈ D such that H(a, d).
This one I can't figure out. Does it mean that for every a there exists a unique d, i.e a1 - d1, a2 - d2, a3 - d3 etc. or does it mean that d's can be shared by some a's, i.e. a1 - d1, a2 - d2, a3 - d1, a4 - d2 etc.
Any help is appreciated.