What is the ruling on one who vowed to seek religious knowledge, but is unsure if he uttered the vow formula, and there is no strong presumption? And what is the fundamental principle in this case?

A vow (Nathr) is when a Muslim obligates himself to an act for the sake of Allah the Almighty. It is made binding through speech or writing, provided it is accompanied by the intention of vowing. Fulfillment of the vow becomes obligatory if it is an act of obedience to Allah, on condition that the one making the vow is an adult and legally competent at the time of making the vow; otherwise, it is not binding upon him.