According to Rabbi Akiva in the Mishna (19b), if someone takes a shevua, saying that he will not eat, if he eats even a tiny amount he has broken his oath and he will be obligated to bring a sacrifice. The Sages disagree with Rabbi Akiva, arguing that the minimum amount that a person must eat in order to be obligated is a ke-zayit – food the size of an olive – and we never find that someone is liable for eating less than that amount. Rabbi Akiva responds to that argument saying that we never find that someone is obligated to bring a sacrifice simply because of something that he said, yet we find that to be true in the case of a shevua. Therefore we can conclude that the sacrifice is not brought because of what he ate so much as because he has broken his oath, and the amount that he ate is irrelevant.
The dialogue between Rabbi Akiva and the Sages continues in the Gemara on today’s daf. Several more cases are brought where we find that someone will bring a sacrifice simply because of statements that he made. For example:
Megaddef – someone who blasphemes, who curses God accidentally. In that case Rabbi Akiva rules that the person who did this must bring a sacrifice. In response Rabbi Akiva argues that the megaddef brings the sacrifice because of the sin that he did, not because of the statement that he made.
Nazir – a Nazirite brings a sacrifice at the close of his nezirut. Here Rabbi Akiva explains that the sacrifice does not come because of his statement; its purpose is to allow him to drink wine.
Hekdesh – someone who consecrates an animal to the Temple will have to bring a sacrifice if he makes use of it for mundane purposes. Rabbi Akiva explains that in that case once he has consecrated the animal the same rule applies to all, and the sacrifice that he brings is not unique to him.
Konamot – when someone makes a neder he must “attach” it to some object that represents the source of the prohibition. Thus a typical neder would be “This will be forbidden to me like a korban.” The Mishna in Massekhet Nedarim teaches that the word konam was considered an appropriate substitute for korban. Taking such a vow would create a situation where for the person who made the neder the object would be considered consecrated, while for the rest of the world it remains ordinary. To this challenge Rabbi Akiva responds that he does not view the case as being literally a korban, and the person who made something konam would not be required to bring a sacrifice.