And that did it. For the rest of the day, I wreaked physical havoc, and emotionally alienated everyone I interacted with. I even seriously contemplated suicide. I wasn't angry at my friend in particular for having said that. For the first time, I was angry at an idea: that belief systems about certain things should not be internally consistent, should not follow logical rules.

This emotional reaction seems abnormal. Seriously, somebody says something confusing and you contemplate suicide?
What are you, a Straw Vulcan computer that can be disabled with a Logic Bomb ?

Unless you are making this up, I suggest you consider seeking professional help.

It was extremely difficult to construct an argument against, because all of my arguments had logically consistent bases, and were thus invalid in its face.

Actually, it's rather easy: just tell them that ex falso quodlibet.