Well—look, you can’t possibly fix your argument by reformulating BIP*, because your paper gives a correct mathematical proof that its version of BIP*, plus the “quadripartite disjunction”, are enough to imply the simulation argument! :-)
(For people who haven’t read the paper: the quadripartite disjunction is H1 = almost no human civilizations survive to posthumanOR H2 = almost no posthuman civilizations run ancestor simulationsOR H3 = almost all humans live in a simulationOR SIM = I live in a simulation. To get the right intuition, note that this is logically equivalent to “If I live in the real world, i.e. if not SIM, then H1 or H2 or H3”.)
More formally, the argument in your paper shows that BIP* plus Cr(quadripartite disjunction) ~ 1 implies Cr(SIM) >~ 1 - Cr(H1 or H2).
I think that (a) there’s a confusion about what the symbol Cr(.) means, and (b) what you’re really trying to do is to deny Bostrom’s original BIP.
Your credence symbol must be meant to already condition on all your information; recall that the question your paper is examining is whether we should accept that Cr(SIM) ~ 1, which is only an interesting question if this is supposed to take into account our current information. A conditional credence, like Cr(SIM | A), must thus mean: If I knew all I know now, plus the one additional fact that A is true, what would my credence be that I live in a simulation?
[ETA: I.e., the stuff we’re conditioning on is not supposed to represent our state of knowledge, it is hypothetical propositions we’re taking into account in addition to our knowledge! The reason I’m interested in the BIP* from the paper is not that I consider (A or B) a good representation of our state of knowledge (in which case Cr(SIM | A or B) would simply be equal to Cr(SIM)); rather, the reason is that the argument in your paper shows that together with the quadripartite disjunction it is sufficient to give the simulation argument.]
So Bostrom’s BIP, which reads Cr(SIM | f_sim = x) = x, means that given all your current information, if you knew the one additional fact that the fraction of humans that live in a simulation is x, then your credence in yourself living in a simulation would be x. If you want to argue that the simulation argument fails even if our current evidence supports the quadripartite disjunction, because the fact that we observe what we observe gives us additional information that we need to take into account, then you need to argue that BIP is false. I can see ways in which you could try to do this: For example, you could question whether the particle accelerators of simulated humans would reliably work in accordance with quantum mechanics, and if one doesn’t believe this, then we have additional information suggesting we’re in the outside world. More generally, you’d have to identify something that we observe that none (or very very close to none) of the simulated humans would. A very obvious variant of the DNA analogy illustrates this: If the gene gives you black hair, and you have red hair, then being told that 60% of all humans have the gene shouldn’t make you assign a 60% probability to having the gene.
The obvious way to take this into account in the formal argument would be to redefine f_sim to refer to, instead of all humans, only to those humans that live in simulations in which physics etc. looks pretty much like in the outside world; i.e., f_sim says how many of those humans actually live in a simulation. Then, the version of BIP referring to this new f_sim should be uncontroversial, and the above counterarguments would become an attack on the quadripartite disjunction (which is sensible, because they’re arguments about the world, and the quadripartite disjunction is where all the empirically motivated input to the argument is supposed to go).
H2 = almost no posthuman civilizations run ancestor simulations
isn’t nearly specific enough to serve the purposes of the simulation argument. We need to say something about what kinds of evidence would appear to the simulations.
Well—look, you can’t possibly fix your argument by reformulating BIP*, because your paper gives a correct mathematical proof that its version of BIP*, plus the “quadripartite disjunction”, are enough to imply the simulation argument! :-)
(For people who haven’t read the paper: the quadripartite disjunction is H1 = almost no human civilizations survive to posthuman OR H2 = almost no posthuman civilizations run ancestor simulations OR H3 = almost all humans live in a simulation OR SIM = I live in a simulation. To get the right intuition, note that this is logically equivalent to “If I live in the real world, i.e. if not SIM, then H1 or H2 or H3”.)
More formally, the argument in your paper shows that BIP* plus Cr(quadripartite disjunction) ~ 1 implies Cr(SIM) >~ 1 - Cr(H1 or H2).
I think that (a) there’s a confusion about what the symbol Cr(.) means, and (b) what you’re really trying to do is to deny Bostrom’s original BIP.
Your credence symbol must be meant to already condition on all your information; recall that the question your paper is examining is whether we should accept that Cr(SIM) ~ 1, which is only an interesting question if this is supposed to take into account our current information. A conditional credence, like Cr(SIM | A), must thus mean: If I knew all I know now, plus the one additional fact that A is true, what would my credence be that I live in a simulation?
[ETA: I.e., the stuff we’re conditioning on is not supposed to represent our state of knowledge, it is hypothetical propositions we’re taking into account in addition to our knowledge! The reason I’m interested in the BIP* from the paper is not that I consider (A or B) a good representation of our state of knowledge (in which case Cr(SIM | A or B) would simply be equal to Cr(SIM)); rather, the reason is that the argument in your paper shows that together with the quadripartite disjunction it is sufficient to give the simulation argument.]
So Bostrom’s BIP, which reads Cr(SIM | f_sim = x) = x, means that given all your current information, if you knew the one additional fact that the fraction of humans that live in a simulation is x, then your credence in yourself living in a simulation would be x. If you want to argue that the simulation argument fails even if our current evidence supports the quadripartite disjunction, because the fact that we observe what we observe gives us additional information that we need to take into account, then you need to argue that BIP is false. I can see ways in which you could try to do this: For example, you could question whether the particle accelerators of simulated humans would reliably work in accordance with quantum mechanics, and if one doesn’t believe this, then we have additional information suggesting we’re in the outside world. More generally, you’d have to identify something that we observe that none (or very very close to none) of the simulated humans would. A very obvious variant of the DNA analogy illustrates this: If the gene gives you black hair, and you have red hair, then being told that 60% of all humans have the gene shouldn’t make you assign a 60% probability to having the gene.
The obvious way to take this into account in the formal argument would be to redefine f_sim to refer to, instead of all humans, only to those humans that live in simulations in which physics etc. looks pretty much like in the outside world; i.e., f_sim says how many of those humans actually live in a simulation. Then, the version of BIP referring to this new f_sim should be uncontroversial, and the above counterarguments would become an attack on the quadripartite disjunction (which is sensible, because they’re arguments about the world, and the quadripartite disjunction is where all the empirically motivated input to the argument is supposed to go).
isn’t nearly specific enough to serve the purposes of the simulation argument. We need to say something about what kinds of evidence would appear to the simulations.