But is he rewarded for a full day, or only from the time he formed the intention?

The scholars have two views on this:

The first view is that he is rewarded from the beginning of the day, because the legally prescribed fast must begin at the start of the day.

The second view is that he is rewarded only from the time of his intention. (1) Thus, if he forms the intention at noon, his reward is that of half a day.

This second view is the stronger one, based on the Prophet’s ﷺ statement: “Actions are only by intentions, and every person shall have only what he intended.” (2) This person formed no intention except during the day, so his reward is counted from the time of his intention.

According to the stronger view, if the virtue of a fast is connected to a particular day—such as fasting on Monday, Thursday, the white days, or three days of every month—and one forms the intention during the day, he does not attain the reward of fasting on that day.

For example, if he fasts on a Monday but forms the intention during the day, he is not rewarded as one who fasted Monday from the beginning of the day, because it cannot truly be said of him that he fasted on Monday.

Likewise, if he begins the day not fasting and is then told, “Today is the thirteenth of the month, the first of the white days,” and he says, “Then I am fasting,” he is not rewarded with the reward of fasting the white days, because he did not fast an entire day. This is an issue