Four statements are built from the same two-place predicate P(x, y). Order them from the weakest claim, which is the easiest to satisfy, to the strongest.

1. there exists x and there exists y with P(x, y)
2. for all x there exists y with P(x, y)
3. there exists y such that for all x, P(x, y)
4. for all x and for all y, P(x, y)

One pair suffices for the first; the second wants a witness for each x, and it may be a different y each time; the third demands a single y that works for every x, which is far harder; the fourth demands every pair. 'Every person has a mother' is of the second kind and true, while 'there is one mother of everyone' is of the third kind and false, swapping the quantifier order silently changes the meaning.