Is it permissible for someone who did not make the intention to fast from the night to complete their fast if they wake up and have not consumed anything that breaks the fast, taking into account the reliance on the Prophet's (PBUH) saying, "I intended to fast," and that "deeds are by intentions"? And what about someone who is fasting on a Thursday and inadvertently breaks their fast, then remembers that it is Thursday—should they complete their fast or break it?

It is permissible to form the intention for a supererogatory fast after Fajr for someone who has not consumed any fast-breaking things, because the majority of scholars do not stipulate forming the intention overnight for it, unlike an obligatory fast for which forming the intention overnight is a condition for its validity. The hadith "Deeds are by intentions" cannot be used to argue against the validity of a supererogatory fast in this case, because the intention is present even if it was not formed overnight. However, for someone who intended to fast after Fajr but had already consumed a fast-breaking thing, their fast is not valid, and they are not required to abstain.