Here's your chance to prove it. You've got several people saying that you need to re-evaluate the way you're treating/interacting with your girlfriend, and your conviction that she's irrational and you're rational. Why not try taking that viewpoint on, and doing a 180 degree shift so you can see things from her POV?
Here's the story of the runners then:
Hi reddit, my boyfriend and I have a problem. I've asked him not to wash his runners in the kitchen sink as that's where we wash our dishes and occasionally items of clothing. There's mud and possibly fecal matter on them, and I just don't feel comfortable with that. He could wash them outside the door or use wipes for the purpose, but apparently these are not rational options (because I suggested them, and he can only imagine washing his runners in the sink). He spent half an hour escalating the conflict by telling me the many ways in which I was wrong for not wanting runners to be washed in the sink when there were more hygienic options, and promising me a substantial cleaning routine that I'm just not sure he'll deliver each time to a standard where I feel like I'm not washing dishes in his foot mud.
Unfortunately, he insisted that since he'd provided what he believed was a rational response, that I should change my mind or at least admit that my response was irrational and stupid. I don't think he did actually provide a response that made sense for the situation, but he appears to be unable to see my POV. I'm beginning to feel that his desire to be right in every situation is irrational.
It doesn't help that whenever we argue, he'll repeat the same narrow points over and over, while ignoring what I'm saying, until I feel like I need to agree with him to get him to stop, because he just won't stop talking until he thinks he's convinced me. How can I get him to understand that just because we disagree, doesn't mean I'm wrong?
Spoiler: Neither of you are right. Neither of those situations is 100% correct, accurate or how things went down, but that means that yours isn't either. You are in all likelihood wrong sometimes, certainly failing at recognising the legitimacy of someone's feelings, and incapable of dropping of a fight because of a need to prove you're right. She's not very good at expressing her POV, has a tendency to blow up, and may also be wrong sometimes.
Solution? Learn to listen and stop feeling the need to insist that you're right when the end result doesn't matter. Take the time to re-evaluate and look at it from her POV and see if maybe you're wrong. For the love of all that is good, stop telling her she's irrational or a dope.