Actually they are engaged in an emotional battle. Usually the subject matter is part of their self-definition, part of "who they are" so they can't really budge on their opinion because it would shake their foundation. Instead they will fight back with everything they have. Illogical arguments, anger etc. Another scenario I see is when "winning" is the important factor, i.e the ego is central. Then also budging is not really an option.
I'm getting better at realizing when the person I'm talking to is having a battle instead of a discussion and disengage.