What if I went by the Sea?
This is a follow-up to the previous post on Shangri La. As before, the story is that a fair coin decides which path you take to Shangri La: on heads, you travel by the Mountains, on tails, by the Sea. If you arrive at Shangri La via the Sea, the guardians will replace your Sea memories with Mountain memories.
In the other post, I said that if you actually traveled by the Mountains, you should remain confident that you traveled by the Mountains, even though you would have ended up with the same evidence had you traveled by the Sea.
What if you actually travel by the Sea? In this case, we have to be more precise about what it means that your Sea memories are replaced by Mountain memories.
On one reading, what gets replaced is merely your evidence about the journey: you get to have vivid images of traveling by the Mountains, you'll have sensations as of strongly believing that you went by the Mountains, etc. In this version of the story, I think you should remain confident that you traveled by the Sea, despite all the Mountain evidence. The reason is the same as before: given your knowledge that you will have Mountain evidence either way, these experiences are irrelevant to the question which path you took. You should discard them and stick to whatever you believed before.
In another version of the story, the guardians alter not only your evidence but also your credence. That is, just before you enter Shangri La on the Sea path, they implant in you the belief that you traveled by the Mountains. (It must be just before you enter, otherwise Arntzenius' Mountain traveler could easily figure out that she went by the Mountains by noticing that her credence in heads is 1/2, whereas it would have been near 1 if she went by the Sea.) In this case, if you follow the conservative principle to only revise your beliefs in the light of relevant evidence, you will end up believing that you came by the Mountains.
This is the type of case where a Cartesian agent who proportions their belief to their present evidence alone might outperform a Conservative: imagine the coin is biased 9:1 towards tails, and the experiment is carried out many times. Conservative agents will presumably accept bets that they came by the Mountains, and lose 90% of the time.
How bad is that? Not too bad, I think. It is not surprising that you end up losing bets if you irrationally acquire beliefs that contradict all your evidence (like the Mountain beliefs on the Sea path). Nevertheless, the Cartesian does seem to have the better recovery strategy here.
On the other hand, perhaps this is also a case where credence and betting behaviour should come apart. In particular, if the Conservative agent is an evidential decision theorist, she could reason that she ought not to bet much on Mountains even though she is certain that she came this way, because she knows that whatever she bets now she will also bet in the other cases in which she traveled by the Sea.
Anyway, here is another fun variation (due to David Chalmers). This time, when you arrive at Shangri La on either path, a second coin is tossed. If it lands heads, nothing happens; if it lands tails, your memories (credence and evidence) get replaced by memories of traveling the other path.
Suppose in fact you travel by the Mountains and the second coin lands heads, so nothing happens to you. As a Conservative, you should stick to your Mountain beliefs, take your Mountain evidence as veridical and conclude that the second coin must have landed heads. For consider the moment just before you arrived. At this point, you were certain that you went by the Mountains, but uncertain about your mental state in the near future: either you will continue to believe and remember that you went by the Mountains, or you will suddenly begin to falsely believe and (quasi-)remember that you went by the Sea. Which of these will happen, you don't know. Once you enter Shangri La, you can exclude the second possibility, as you notice that you don't have Sea memories. Thus you can conclude that the second coin has landed heads.
Again we can sharpen the point by making the coins biased: if both coins have a 9:1 bias towards tails, there is an initial 0.81 chance that you go by the Sea and end up with Mountain memories, and a 0.01 chance that you go by the Mountains and end up with Mountain memories. If your Mountain memories were all you had to go by (say, because you slept during the journey), you should be fairly certain that you went by the Sea. But they are not. You also learn, while traveling by the Mountains, that the first coin definitely landed heads. Given this further information, the only way you could end up with Mountain memories is for the second coin also to have landed heads.
What if the second coin is tails-only, that is, what if your memories are certain to be replaced on either path? In this version (also due to Dave), it surely seems irrational to stick to your implanted memories.
Suppose you went by the Sea. Notice that the following three propositions are inconsistent: i) the setup is as described, ii) you went by the Mountains, iii) you end up with Mountain memories. For the setup implies that if you went by the Mountains, you will end up with Sea memories. I assume the guardians do not implant inconsistent beliefs in you, hence they can't make you believe (i)--(iii). (If they do, the first thing you should do is make your beliefs consistent, presumably by lowering the credence in all three propositions.) Now when you arrive in Shangri La, you gain very strong evidence for (iii). Conditional on (iii), (i) and (ii) are incompatible. So your credence in at least one of them must now be fairly low. Which one that is depends on what the guardians made you believe -- whether they made your credence in (i)&(iii) higher or lower than your credence in (ii)&(iii). On the assumption that you remain confident of the setup, your credence in (ii) will go close to zero. So even as a Conservative, you should indeed revise your implanted beliefs about the Journey.
A somewhat analogous, but simpler case: forget about the two coin tosses and suppose you will definitely travel by the Sea, and once you arrive, your memories (still understood including credence) will be replaced by Mountain memories. What should you believe?
This time, the implanted belief that you went by the Mountains directly contradicts your knowledge of the setup, which entails that you go by the Sea. Hence the only way the guardians could raise your credence in Mountains (without introducing inconsistency) is by lowering your credence in the setup. If they raise your credence in Mountains to near 1 and thereby lower your credence in the setup to near 0, you should stick to your Mountain beliefs: it would be irrational to change your mind if you're certain you're not being manipulated. On the other hand, if they keep your credence in the setup high and only give you strong evidence for Mountains (leaving your Mountain credence near 0), you should discard the evidence and continue to believe that you went by the Sea.