Illustrating the burden of proof - pay me!

[Belacqua]

I think 'I don't believe you've met your burden of proof' is a perfectly cromulent reason for me not believing some claim is justified. Not in the philosophical sense, but in the general sense of what sorts of things I should believe. If you convince me that you have met your burden of proof, I'll believe the claim. Conversely, if you convince me that something I do believe isn't rationally justifed (or in the case of something necessarrily not completely rational, like politics, not consistent with my own values), I'll stop believing it. Like I stopped believing in big L Libertarianism. When it comes down to what I do or don't believe, there's an unavoidable personal element. Is my skepticism set at the ideal level to 'let the most true stuff in and keep the most rubbish stuff out'? I'm not sure, I'm just making my best guess in each situation.

Here I think you're using haven't "met the burden of proof" to mean that a person's arguments aren't sufficient to persuade you. 

That seems perfectly sensible to me. After we've listened carefully to the other side, we all judge whether they've sufficiently made their case. Then we can decide whether it's worthwhile continuing the conversation.

Some people, though, use a Burden of Proof Rule (perhaps handed down by Moses) to say that the side making the claim has to do all the work, and the side rejecting the claim doesn't have to give any reasons for their rejection. 

Again, I hold that thinking adults on both sides in a good faith conversation both have a case to make. Though I understand that the "thinking adults" thing may be a little too optimistic.