What is the ruling on a father breaking his oath, and is expiation obligatory for him or for the son who caused him to break it, and do the rulings differ based on the intention of the one who swore the oath?

It is not permissible to insist with the father on what angers him. The oath of an angry person is considered binding as long as he comprehends what he is saying, as the majority of jurists are of the opinion that an angry person is accountable and held responsible for what emanates from him.

The oath is based on the intention of the one who swears it, so he should be asked about his intention; for intention specifies the general لفظ (utterance) and restricts the absolute. The swearer's retraction from his oath is permissible, even preferable, if there is a benefit in it, and he must offer expiation for breaking an oath.

The reason that prompts the oath is considered like the intention in specifying and restricting the oath. So, if the reason ceases to exist, the breach of the oath does not occur.