Um. Yes it does. If you prove that you are not (not Satoshi), then you prove that you're Satoshi. There is this widespread myth that "you can't prove a negative" when, in fact, you can.

You might have been meaning to point out that the characters of evidence needed in either case are somewhat different: the problem with proving some negatives via an argument from silence is that an absence of evidence is expected. You see, it's an absence of expected evidence that's evidence of absence! If, given something were true, we wouldn't expect to have evidence of it, then such "arguments from silence" (we lack evidence, therefore it's not true) do not work.

So when I want to prove that someone is not a giraffe, I can reasonably expect that if they were, they would have a long neck, walk on four long legs, probably stand taller than a given human -- if I see no evidence that they have any of these, we have an absence of expected evidence. This functions as evidence of the absence of their giraffe-ness.

But when I want to prove that someone is not immortal, say, there is no expected evidence of immortality for me to hang my hopes on. I mean, I might hope that they'd been born long enough in the past that they'd have documentation of their existence lasting out hundreds of years, but I'd have to introduce more assumptions to expect this. Or I might hope that they'd been subjected to lethal force sometime during their lifespan and lived through it, but in the posh, cushy developed world there's not necessarily any reason to expect that the typical 30-year-old has suffered a grievous injury, so I'd have to introduce added assumptions about their rough past before I could expect this. Still, proof would still be possible in such a circumstance -- just ask them to kindly survive stepping in front of a train or so -- and the negative could be proved. It's just that it can't be proved by an argument from silence.

The usual way to make this semi-rigorous is to use the definition of conditional probability, where the event "A given B" is written "A | B" and its probability is defined as Pr(A | B) = Pr(both A and B) / Pr(B). Then we have some claim C with some new evidence N and some old evidence O; we can prove that three numbers that need to be known:

P = Pr(C | O)
the prior probability of C before adding evidence N;
Q = Pr(N | O and C)
the likelihood that C would generate this evidence N, given the old evidence,
R = Pr(N | O and not-C)
the likelihood that not-C would generate this evidence, given the same.

The expression for the probability that we want to know is then:

Q * P
Pr(C | N and O) = --------------------
Q * P + R * (1 - P)

An argument from silence is just an argument where the new evidence N to be incorporated is the lack of some observation, for example the lack of observed dying after being shot by a gun, or the lack of observing a long neck on a person claimed to be a giraffe.

We can then see that arguments from silence are still supportive (Pr(...) > P) whenever Q > R, but may not raise P to the level where we'd really bet on it. In particular if P is very unlikely (e.g. immortality would have a P of one in 10 billion or worse, given what we've seen about the world), then with Q=1 (e.g. not dying when shot, given immortal), you're still looking for R to be commensurately tiny (e.g. the shot must have survival odds 10 billion to 1 against) to raise the argument from silence to a decisive position. Any weaker sort of silence does not confirm it enough to raise it to the level of plausibility, although of course you can rinse and repeat with N now incorporated into O (so we can talk about someone who has been shot, poisoned, and hanged many times on many different occasions and each of them might have a 1/100 survival rate; still if we have her surviving 5 such things which are individually bad evidence, we may get a non-negligible probability that all together they are good evidence and either she's conspiring with her killers or else maybe she is, in fact, immortal.)