What is the ruling on someone who swore to do something on a specific day, then forgot the details of what he swore to do, and should he act on the غالب الظن (preponderance of probability) in this case?

If you doubt whether your oath included a partial diet on the mentioned day or not, the basic principle is that the oath did not cover it. You are not at fault for making that day open, and no expiation is required of you, because the original principle is freedom from liability, and certainty is not removed by doubt.