Nisf al-Nahar

The modern midday of twelve noon and its midnight counterpart have no role within the Islamic Shariah. Instead, when the classical fiqh manuals speak of a "midday" and a "midnight", they are speaking of specifically calculated times. The Islamic midday - termed nisf al-nahar - is the exact point of time between the start of Fajr and the start of Maghrib and will alter every single day due to the altering nature of both Fajr and Maghrib. The Islamic midnight, similarly, is the exact point in time betweeen Maghrib and the start of Fajr and this too has some implications within the Shariah.

See here for further details on Isha and midnight.

Nisf al-nahar has some implications in terms of the intention of the fast. The intention for the following three types of fasts must be made prior to nisf al-nahar, although it is mustahab (commendable) to intend to fast the night before.

• The fasts of Ramadan
• Nafl or Sunnah fasts
• Fasts for a vow of fast made for a specific day

There are four fasts that require ones intention to be specific and to be made the night prior:

• Qadha of Ramadan
• Qadha of Nafl after it was broken before completion
• Kaffarah fasts
• Fasts of vows in general, non-specififed ways

There are some misunderstandings about the nature of an intention. For many, the "niyyah" to pray Salah or fast remains merely a verbal expression in the sense of "nawaytu an usalliya ..." (I intend to pray.. ) or, "wa bisawmin ghadin nawaytu ..." such that many will repeat these expressions before drinking the last sip of water. Those that oversleep, or forget to drink with this expression, often are in doubts as to whether their fast is proper. This is incorrect.

It simply means that one is making a conscious decision to perform an action whether that is to perform wudhu, or salah, or fast. Consider the following narration:

Our mother Aisha reports: "The Messenger of Allah, peace and blessings be upon him, said to me one day, 'O Aisha, do you have anything to eat?' I said, 'O Messenger of Allah, I have nothing.' The Prophet then said, 'Then I am fasting'”.
Ṣaḥīḥ Muslim, 1154.

This is the intention to fast, prior to which, the Messenger, peace and blessings be upon him, had not decided to fast. In a nutshell, "intention" means your decision to do an act. When you're standing on the prayer mat to prayer Dhuhr salah, for example, that there is your intention as the reason you are standing on the mat is to pray and not to decide whether to pray or not. To utter out loud that you are now going to pray Dhuhr, and that you are facing Qibla, which was the precise reason you are here, is considered an innovation by many scholars.

Consider a person that intended to fast all thirty days of Ramadan: they fall asleep at night and wake up at 2pm. Is their fast valid-? If they had gone to sleep knowing that it is Ramadan and that they will be fasting, that there is their intention. However, if for some reason or other they were not sure if they were going to fast, and then overslept until 2pm, then in this instance, their decision to fast upon waking will be too late. This is what it means to "intend" to fast.

On the other hand, if someone were to wake up early in the morning and not commit any action that would normally violate a fast such as eating and drinking, then they have until the time of nisf al-nahar to decide if they wish to turn their current fast (as they haven't had breakfast yet) into a Shar'i fast. This option will expire after this time. Similarly, if a person had intended to not fast a day in Ramadan, which would be extremely sinful, but a minute before nisf al-nahar decided to change their mind and keep the fast, their fast will be valid so long as they did not commit any actions that otherwise would violate a fast.

Note, however, that missing obligatory prayers or other sinful acts does not render the fast as invalid but will deprive it of blessings and rewards.
And Allah knows best.