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I'm having trouble following this hypothetical world where we don't know if London is in England, so let me try translating to a historical theory: h = "this p

I'm having trouble following this hypothetical world where we don't know if London is in England, so let me try translating to a historical theory: h = "this person has cancer" e = "this person is a smoker" For simplicity, let's ignore the fact that not everyone who smokes gets cancer, and that those who do get cancer might not get it on their first cigarette. Just: smoking -> cancer, is the theory. h = (h | ~e) & (h | e) -- this is in fact, always true Now we learn that the person is a smoker. So e is 1, and (h | e) goes up. And (h | ~e) = (e -> h) goes down. Because the person's a smoker so---uh oh---our theory is getting tested and it could be proved false right now. This... doesn't seem that surprising? If we observe lots of people smoking, but haven't checked anyone for cancer, it does make the strict theory (smoking -> cancer) less likely because we're making the very strong prediction that this large set of people must all have cancer. I wouldn't expect to get evidence confirming the implication (smoking -> cancer) until we started checking whether smokers have cancer. And once you start checking that, then (h | ~e) = (e -> h) will rise (assuming your theory is true), right?

MatteoFrigo 4y ago
Yeah, now that I think more about it, I think I was confused myself. Specifically, I think I got the math right but the example wrong (and I confused you too). Let me try again. I think that Deutsch is saying that h is the proposition "smoker implies cancer", and e is a specific instance of a person where the hypothesis holds (either a nonsmoker or a smoker with cancer). He is talking about e being instances of h, so h must be a higher order proposition about instances. But now h can be decomposed as we said into a logically necessary part (h|e) and a part (h|~e) that may be true or false depending upon which universe you live in. By the argument above, finding more instances of the theory should decrease our belief in the (h|~e) part. Since h|~e is the same as e->h, gathering more e should decrease our faith that the evidence validates the hypothesis. Presumably Deutsch is saying that the logically necessary part is sort of trivial (a mere theorem) whereas (h|~e) has actual physical content, so why do we believe that the evidence increases our confidence in the physical portion of the hypothesis? By the way, I got all this math from the unapproachable paper, which is not that unapproachable if one looks at the math alone. Like you, I am trying to figure out how this math applies to the real world.
- justinpombrio 4y ago
  So if we observe e, then: - h|e becomes 1 - h|~e = e->h decreases - h increases I notice I'm confused, though. Is h|e about this specific instance of e, or the general e? How about e->h? It feels like we're moving the goal-posts: at first h|e = 1 because we're talking about a specific e (we found a smoker with cancer), and e->h decreases because we're talking about that specific e, but then later we start to draw conclusions about the general e->h such as describing it as whether "the evidence validates the hypothesis". And I don't know how to formally relate the specific e->h to the general e->h.
- justinpombrio 4y ago
  > Specifically, I think I got the math right but the example wrong (and I confused you too). Don't worry, I was much more confused before you came along :-). I also followed the math in the paper but didn't know what devastating contradiction for Bayesianism it was oh-so-darkly hinting at. I keep trying to describe what happens if e and h are both general, and bouncing off. Let's try a different assumption: e is the statement "the first person tested is consistent with smoking->cancer". And in fact, we just tested them, and they're a smoker and have cancer. So e is unambiguously true. And h is the hypothesis "smoking -> cancer". Then: - e is 1 - h has increased - h|e has increased, in fact it is 1 - h|~e has decreased h|e is tautologically true, once we have observed e. And h|~e is e->h, the amount to which this observation implies "smoking->cancer". Which seems odd, though I don't have an immediate intuition about how an observation is supposed to effect, not the hypothesis that it supports, but the implication between itself and that hypothesis. But maybe that's the darkly-hinted-at problem?

MatteoFrigo 4y ago

Yeah, now that I think more about it, I think I was confused myself. Specifically, I think I got the math right but the example wrong (and I confused you too). Let me try again. I think that Deutsch is saying that h is the proposition "smoker implies cancer", and e is a specific instance of a person where the hypothesis holds (either a nonsmoker or a smoker with cancer). He is talking about e being instances of h, so h must be a higher order proposition about instances. But now h can be decomposed as we said into a logically necessary part (h|e) and a part (h|~e) that may be true or false depending upon which universe you live in. By the argument above, finding more instances of the theory should decrease our belief in the (h|~e) part. Since h|~e is the same as e->h, gathering more e should decrease our faith that the evidence validates the hypothesis. Presumably Deutsch is saying that the logically necessary part is sort of trivial (a mere theorem) whereas (h|~e) has actual physical content, so why do we believe that the evidence increases our confidence in the physical portion of the hypothesis? By the way, I got all this math from the unapproachable paper, which is not that unapproachable if one looks at the math alone. Like you, I am trying to figure out how this math applies to the real world.

justinpombrio 4y ago
So if we observe e, then: - h|e becomes 1 - h|~e = e->h decreases - h increases I notice I'm confused, though. Is h|e about this specific instance of e, or the general e? How about e->h? It feels like we're moving the goal-posts: at first h|e = 1 because we're talking about a specific e (we found a smoker with cancer), and e->h decreases because we're talking about that specific e, but then later we start to draw conclusions about the general e->h such as describing it as whether "the evidence validates the hypothesis". And I don't know how to formally relate the specific e->h to the general e->h.
justinpombrio 4y ago
> Specifically, I think I got the math right but the example wrong (and I confused you too). Don't worry, I was much more confused before you came along :-). I also followed the math in the paper but didn't know what devastating contradiction for Bayesianism it was oh-so-darkly hinting at. I keep trying to describe what happens if e and h are both general, and bouncing off. Let's try a different assumption: e is the statement "the first person tested is consistent with smoking->cancer". And in fact, we just tested them, and they're a smoker and have cancer. So e is unambiguously true. And h is the hypothesis "smoking -> cancer". Then: - e is 1 - h has increased - h|e has increased, in fact it is 1 - h|~e has decreased h|e is tautologically true, once we have observed e. And h|~e is e->h, the amount to which this observation implies "smoking->cancer". Which seems odd, though I don't have an immediate intuition about how an observation is supposed to effect, not the hypothesis that it supports, but the implication between itself and that hypothesis. But maybe that's the darkly-hinted-at problem?