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When I've encountered arguments about the brain in a vat problem, they went something like

We could be brains in a vat, with a simulated reality, how could we know we are not?

But isn't this ignoring the fact that to make an argument, one has to present a case for the argument, instead of presenting a thesis, and then asking the other person how can we know it is not true?

For example, if a person would say

Joe Biden could be actually an alien from outer space\ a Russian spy. How can we know it is not true?

This second argument has the same structure, but it is clear that the burden of proof is not to disprove that Joe Biden is an alien \ a russian spy. The burden of proof is on the person making this claim.

The brain in a vat argument puts the person who is against this theory on the defensive, which is the very definition of shifting the burden of proof.

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    I believe this applies to radical skepticism in general. Radical skepticism effectively shifts the burden of proof by asking "How can we be sure we know anything", instead of making an actual argument that we can know nothing. (Had it made an actual argument, then its self-contradiction would be much more evident).
    – Sam
    Aug 21 at 16:05
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    I think the point is simply to show that there isn't anything self-contradictory about the 'brain in a vat' scenario. Not to show that it is a good thing to believe in it. The point is simply to show that we do not have direct certain knowledge of an outside world. The people presenting the brain in a vat scenario don't actually believe in it. They're simply showing that it's possible. Aug 21 at 16:16
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    "Radical skepticism effectively shifts the burden of proof by asking 'How can we be sure we know anything'"... so are you saying we can be certain we're not a brain in a vat? Aug 21 at 16:18
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    Not sure the issue here is 'shifting the burden of proof'. For example we could say the regular person is making a positive claim, "there's an outside world" while the radical skeptic is making no positive claims. Now does the regular person have the burden of proving the existence of the outside world? Aug 21 at 16:38
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    No. Radical skepticism does not shift the burden of proof because radical skepticism has no burden of proof. It presents no theses. That would be far worse than shifting the burden, it would be incoherent. What becomes of the thesis that nothing is justified when applied to itself? Radical skeptics are more subtle. They only challenge any thesis to the contrary and urge suspension of judgment on everything. I know, it seems unfair, but it is what it is. One can charge them with insincerity as, in practice, they must judge to act and live, but logically they are unassailable.
    – Conifold
    Aug 21 at 20:26
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The purpose of the brain-in-a-vat scenario is not to convince anyone that we might really be a brain in a vat; the purpose is to illustrate the fundamental disconnection between perception and reality. Before the brain-in-a-vat scenario, Descartes made the same point by discussing dreams.

The classical view of perception is that when you, for example, look at a tree, you are seeing the tree; you are experiencing the tree; the tree has become present to your mind. But as the brain-in-a-vat or dreams show, you can have exactly the same visual experience of the tree even when there is no tree there. What this means is that there is some disconnect between what is really out there (which we might call the noumenon) and what is perceived (what we might call the phenomenon). Arguably then, you are not experiencing the tree; the tree is not present to your mind; rather, you are experiencing a mental image; it is the image that is present to your mind, not the tree itself.

Apparently, there is a sort of chain of causality like this:

tree --> light --> eye --> ... --> [X factor] --> experience of a tree,

and another chain of causality like this:

[something that is not a tree] --> [X factor] --> experience of a tree.

The experience of a tree is something that is somehow constructed by the [X factor], not by the tree itself. One could assume that there are two entirely different mechanisms, one that transfers the image of a tree into mind and another that constructs the image of a tree, but there doesn't seem to be a good reason to assume this.

So, if the experience of the tree is something that is manufactured in the brain (or the mind) by [X factor], then we have a real problem: what reason is there to think that the tree is anything at all like the way we experience it? In other words, the image of the tree that I have in my mind is a useful way to interact with the world, but what reason do I have to think that the noumenal tree is anything at all like the phenomenal tree? If I am a brain in a vat, the noumenal tree might just be a data structure in a computer program. If I am dreaming, the tree may be a bit of undigested beef, a blot of mustard, a crumb of cheese, a fragment of an underdone potato (in the words of Ebeneezer Scrooge). Physics even backs this up. According to physics, the tree is really just a swarm of subatomic particles that our brain presents to us as a complete unified tree.

So, the answer is, no, the brain-in-a-vat scenario does not shift the burden of proof, because the point is not to prove that we are (or might be) brains in a vat; it is just a thought experiment intended to show that we have no reason to think that the noumenal world is really what it appears to be. An idealist would say that there is no noumenal world at all; there are only the experiences. A Kantian would say that although the noumenal world may exist, we can't know anything about it through perception; we can only know about the phenomenal world.

I should note that there are a few writers who are trying to prove that the universe is a simulation, but these are mostly physicists and computer scientists, not philosophers, and they claim to have evidence beyond the brain-in-a-vat scenario, but as far as I know, almost no one takes them seriously.

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The most straightforward way to refute an argument is to provide a counter example, and what you're calling "the brain in a vat argument" isn't an argument at all, so here's what I suspect is happening (I'm going out on a limb here, so if I'm wrong please just tell me instead of downvoting and I'll delete this answer):

Most likely, you're making an actual argument and your interlocutor is trying to bring in BIV as a counter example to "refute" you. Since BIV isn't an argument (it's just the straightforward possibility that you could be a brain in a vat), there's nothing to push back against, so it feels like a cheat—you can't counter-example a counter example.

Perhaps your frustration isn't due to an unacknowledged shift in the burden of proof, but to the entire discussion being shifted out of the realm of productivity and driven into a ditch. This would be fortunate. After all, there are very few pressing matters that hang upon our determining whether or not we are all brains in vats. If it's a practical matter, there are probably ways to argue for it in spite of BIV scenarios. Elon Musk, for example, famously subscribes to something like Nick Bostrom's simulation argument while managing to be an exceptionally productive human being.

Alternatively, if you're having an ivory-tower argument about the nature of reality then you really do need to either do what no philosopher in history has yet been able to do, i.e., come up with an argument that casts radical skepticism back into the fiery chasm from whence it came, or do what every secular philosopher in history has been doing: painstakingly eke out some form of philosophical subsistence in its shadow.

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