It rests on the proposed truth, itself. It doesn't matter who proposes it.
And if you propose something to be the truth, you better believe I'm gonna ask you to support it.
You have no such entitlement. The burden exists whether you do or not, whether it is challenged or not. It rests with the claim made.
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And if someone makes a claim, you better believe I'm gonna ask that person to support it.
Asking for proof is all well and fine, as long as you acknowledge the reason you're asking for it. And in the case of "oh, I was just checking if your reasoning agrees with mine," that's just a cop-out.
No. You can't start complaining about having to bear the burden of proof just because you don't like the reasons why whoever asked you to provide the evidence was asking you for that evidence.
If you make a claim, you must support it. It doesn't matter who asks you to support it.
And if you can't produce a well-reasoned and rational argument to support a claim that you make, then it deserves to be called out, even by people who have the same beliefs you do.
For example, I'm an atheist. But if someone is an atheist because they hate God and feel that God let them down, then I'm gonna call them out on that, because that's not a valid reason to be an atheist.
The burden has, as its determinant, truth. It is the proposed truth that requires proof, not you (or me).
But the claim itself can't provide evidence to support itself, just like it can't proclaim itself. It needs a person to do that. And the person who is required to provide the proof for the claim is the person who makes the claim.
That is to say that the truth and the requisite burden exist objectively, regardless of you or I.
So what? It doesn't matter what the truth is here, we aren't talking about truth, we are talking about claims and who has the responsibility of producing evidence to support those claims.
And I keep telling you that the person who has the responsibility to support a claim that has been made is the person who made the claim in the first place.
You seem to be trying to evade this responsibility by claiming that the only people who can demand evidence for a claim be provided are those who hold opinions counter to the claim that was made. This is not true. It is the responsibility of EVERYONE involved in a discussion to do everything they can to maintain intellectual honesty in that discussion, no matter what side of the argument they are on.
If I see a person in a debate about whether God exists or not, and that person just keeps repeating that God doesn't exist without giving any evidence or reasoning to support his claim, I'm gonna say to him, "Hey, you';re gonna need to make a better argument than that." And I'm on the same side as him!