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You have a hypothesis h = "it's raining somewhere in England" and evidence e = "it's raining in London". You have an empirical theory that e implies h, which m

You have a hypothesis h = "it's raining somewhere in England" and evidence e = "it's raining in London". You have an empirical theory that e implies h, which may or may not be true depending on whether London is in England in your universe, but you don't know which universe you live in. There are also universes where London is not in England but your theory is still true for some complicated meteorological reasons that are unknown. For all h and e, you can always write (using C bitwise operations to denote logic) h = (h | ~e) & (h | e). The second factor (h | e) is the part which logically follows from the evidence, that is, it is true in all universes in which e is true. The first factor is the part that is not logically implied by the evidence, that is, there are universes where it is raining in London and yet h is false because London is not in England. Now the real question: somebody tells you that it is raining in London, so your credence in e goes up. What happens to the probability of (h | ~e)? It should go down, because as e becomes "more true", ~e becomes "more false", and thus (h | ~e) becomes more false in the sense that there are fewer worlds where (h | ~e) is true. But (h | ~e) is the same as "e implies h", which is your empirical theory. So your belief in the theory should go down as you gather more evidence. Another way to say it is that, as the evidence becomes stronger, the part logically implied by e becomes more likely, and whatever remains (h | ~e) becomes a smaller set of possibilities, so it is less likely. Note that your belief in (h | ~e) goes down, but your belief in h goes up. I think Deutsch's criticism is that people confuse the two, and they think that evidence increases the credence in the theory instead of the hypothesis.

justinpombrio 4y ago
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?
im3w1l 4y ago
There is a subtlety going on here. You should really parameterize by time. So we have e.g. h(t) - it's raining somewhere in England at time t. Now our empirical theory is forall t e(t) -> h(t). We are told that e(T) for some particular T. This makes us believe that h(T) | ~e(T), but should barely change anything about our belief in the theory. Edit: I noticed you realized the part of higher order in a side thread, but a specific important point to add here. You mentioned that "logically necessary part is sort of trivial (a mere theorem) whereas (h|~e) has actual physical content". Our theory makes predictions about the future - so it's impossible to observe e(t) for all t. We can't just rely on observing all instances of the actual physical thing. We have to make use of logical necessity.

justinpombrio 4y ago

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?

im3w1l 4y ago

There is a subtlety going on here. You should really parameterize by time. So we have e.g. h(t) - it's raining somewhere in England at time t. Now our empirical theory is forall t e(t) -> h(t). We are told that e(T) for some particular T. This makes us believe that h(T) | ~e(T), but should barely change anything about our belief in the theory. Edit: I noticed you realized the part of higher order in a side thread, but a specific important point to add here. You mentioned that "logically necessary part is sort of trivial (a mere theorem) whereas (h|~e) has actual physical content". Our theory makes predictions about the future - so it's impossible to observe e(t) for all t. We can't just rely on observing all instances of the actual physical thing. We have to make use of logical necessity.