Normal practice is not that formal (except for logicians ;), but would definitely be second order logic. It's common to quantify over "all functions such that..." or over set of sets for example. Second order logic is more expressive.
Thanks. I dont understand why for all elements of a set is first order and for all functions is second order - since functions (lets say R->R) are elements of the powerset of the cartesian product of RxR. So functions in math are usually part of first order logic, no?
As said by @dwohnitmok it depends on the domain you consider. If you have a proposition quantifying over objects and functions operating on said objects, it's second order. But if you have a proposition considering only said functions as objects, it is first order (with a different domain: not the base objects, but functions over those objects).
And again: you can ignore such subtleties for most of mathematics.
