This is one of the most common logical misconceptions, even among those whose grasp of logic is solid, and it's depressing how often ground is conceded just because of a tiny slip in the parsing of a fallacy. It's pretty simple to grasp, once you see what's happening, but it isn't obvious, which is why it causes problems. It also leads to other slips such as, having accepted an unearned burden, we assert that 'you can't prove a negative' (you can) or 'you can't prove a non-existence postulate' (you can), all trying to avoid fulfilling a burden to demonstrate the truth of an incoherent statement. Let's unpack it.
It stems from a bit of confusion between a declarative and an affirmative. Specifically, it's treating ALL declaratives as affirmative claims. Negating (or denying) claims are also declarative, but they're the precise negation of an affirmative, and this has interesting logical consequences.
In particular, not all declaratives carry a burden of proof. The assertion that god exists and the assertion that god does not exist are both declarative assertions, but only one of them is an affirmative declarative. The reason is obvious, if you think about it, because the negating declarative is incoherent absent the affirming declarative. If I were to assert that furzlewurgle doesn't exist, I'd be talking gibberish. This is exactly as true of God as it is for furzlewurgle, and for exactly the same reasons*. It's undefined and no referent has been demonstrated.
Similarly, if I were to assert that the Oort cloud is composed entirely of cinnamon-scented teddy-bears, and you were to say 'you're talking bollocks', the burden would still be mine. Indeed, the negating declarative is the same whether I say 'you're talking bollocks' or 'The Oort cloud is NOT made of cinnamon-scented teddy-bears', because they're logically and semantically identical (in context) and either statement is absurd absent the affirmative. Accepting the burden of proof on the negating declarative is, therefore, logically equivalent to treating saying 'you're talking bollocks' to nobody in particular as a coherent affirmative claim. It clearly isn't.
The full Latin moniker for the fallacy of shifting the burden of proof is Onus probandi incumbit ei qui dicit, non ei qui negat which translates directly into English as 'the burden of proof is incumbent on him who affirms, not him who denies'. Only the claim of existence is an affirmative claim.
There is a case in which the burden of proof seems to shift, but it's really a change in affirmative claim, and that's when the affirmative claim has already been proven or put on a sound evidential footing, at which point denying the affirmative and well-supported claim becomes the affirmative claim that the evidence/proof is wrong or flawed.
This is a subtlety oft-overlooked, and I see many instances of the burden being accepted by the negating claim just because they think that having argued that the affirmative claim is bullshit means they've shouldered a burden which is not nor was it ever theirs.
*There is actually a deep logical problem with the phrase 'x exists' (for any given x) that leads to some absurdities. The reasons for this have to do with how arguments are structured, and require some grasp of predicate logic. The problems can loosely be summarised in the statement 'existence is not a valid predicate'. The statement 'x exists' is logically equivalent in predicate logic to 'x exists exists', which leads to some really thorny issues. We'll be exploring this in the future of the 'In On The Secret' series exploring jargon and notation, when we get to predicate logic. We've already started laying the foundations in 'Bad Form, What?'
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