In On the Question of Absolute Undecidability, Peter Koellner investigates whether it is possible to prove or disprove $V = L$ using (EDIT: both first and second-order) reflection principles, ie. statements of the form*
$$V \vDash \varphi(A) \to \exists \alpha \ V_\alpha \vDash \varphi^\alpha(A^\alpha)$$
and shows that it cannot be done.
Starting from $ZF$, can these principles be used to prove or disprove the Axiom of choice or some of its weaker variants (eg. Dependent, Countable choice)?
* definition on page 13.