When you hear powerful evidence or arguments that should get you to revise your beliefs, not only do all sorts of cognitive biases fight the changes but so do the social factors of status and face saving. Perhaps I've long been a vocal proponent of X which implies Y, and you show me that Y isn't always true. It's very hard to just straight up admit "ok, I'm not a hardcore Xist anymore." There's a status loss in letting yourself be convinced.
For a long time I thought that I was stronger than this, that saving face only mattered as much as I let it matter. I wish I could freely admit when I've been convinced, but I often can't manage to. [1] Instead I'll finish a conversation defending my earlier beliefs and only later start acting on my new ones.
After a discussion where someone didn't admit to any change of mind, I'll often see them later having changed their behavior. So now if I'm trying to persuade someone I don't focus on securing verbal agreement. Instead I just try to be as convincing as possible, and notice if they come around later. [2]
(I also posted this on my blog)
[1] This is not a helpful trait: I'd like other people to let me know when I'm wrong or when they have evidence I'm not considering, but if they never get the satisfaction of knowing they've convinced me they may just feel like they've wasted their time, and not try in the future. So I'm working on it.
[2] Keeping people from feeling personally invested in one side or the other of an argument is probably also helpful: I understand discussions are much more likely to convince bystanders than participants.
I've observed the same behavior, and I've also found that if you give someone an "out" they're more likely to agree:
"I support X!" "Why?" "Well, A, B, C." "Oh! You mean you support Y? Because A+B+C implies Y, so obviously you support Y" "Right, of course. What did you think I meant?"
I've done this a fair few times, and it often produces some absurd about-faces in people. Being confrontational doesn't produce the same changes, so I don't think this is just a matter of them having actually meant Y. I'll often spend some time clarifying they really mean X, not Y, and then do this, and then clarify that yes, they really mean Y, not X now. It's baffling at times.
I haven't followed-up a ton on it, but the changes seem to generally "stick" - they don't just agree with Y to get me to leave them alone.
Basically, the trick is to lead the conversation such that you're not implying that they were ever wrong. It's a weird bit of double-think at first, and can be tricky to adjust to, but I find it's fairly useful quite frequently. If nothing else, by avoiding being confrontational, I seem to leave myself more open to hearing that I'm wrong and really meant Y all along :)