Counterfactual Calculation and Observational Knowledge

The correct decision for the counterfactual you in that world is to decide Odd World.

I believe the above is correct updateless analysis of the thought experiment. (Which is a natural step to take in considering it, but not the point of the post, see its last paragraph.)

It occurs to me that with Omega telling you about the counterfactual world, you are receiving a second observation. For this understanding, you would specify Even World with 99% confidence in the factual world and either Even or Odd World depending on how the coin landed for the counterfactual world.

Vladimir says that "Omega doesn't touch any calculator". If the counterfactual is entered at the point where the computation starts and Omega tells you that it results in Odd (ETA2: rereading Vladimir's comment, this is not the case), then it is a second observation contributed by Omega running the calculator and should affect both worlds. If on the other hand the counterfactual is just about the display, then the counterfactual Omega will likely write down Odd (ETA3: not my current answer). So I agree with your analysis. I see it this way: real Omegas cannot write on counterfactual paper.

ETA: -- the "counterfactual" built as "being in another quantum branch of exactly the same universe" strikes me as being of the sort where Omega does run the calculator again, so it should affect both worlds as another observation.

ETA2: I've changed my mind about there being an independent observation.

Ignoring Bostrom's book on how to deal with observer selection effects (did Omega go looking for a Wrong Calculator world and report it? Did Omega go looking for an Odd World to report to you? Did Omega pick at random from all possible worlds? Did Omega roll a four-sided die to determine which world to report?)

Actually, isn't this the very heart of the matter? In my other comment here I assumed Omega would always ask what the correct answer is if the calculator shows The Other Result; if that's not the case everything changes.

This would be correct if Q could be different, but Q is the same both in the counterfactual and the actual word. There is no possibility for the actual world being Even World and the counterfactual Odd World.

The possibilities are:

1. Actual: Even World, Right Calculator (99% of Even Words); Counterfactual: Even World, Wrong Calculator (1% of Even Worlds).

2. Actual: Odd World, Wrong Calculator (1% of Odd Words); Counterfactual: Odd World, Right Calculator (99% of Odd Words).

The prior probability of either is 50%. If we assume That Omega randomly picks one you out of 100% of possible words(either 100% of all Even Worlds or 100% of all Odd Words) to decide for all possible words where the calculator result is different (but the correct answer is the same), then there is a 99% chance all worlds are Even and your choice affects 1% of all worlds and a 1% chance all words are Odd and your choice affects 99% of all worlds. The result of the calculator in the counterfactual world doesn't provide any evidence on whether all words are Even or all worlds are Odd since in either case there would be such a world to talk about.

If we assume that Omega randomly visits one world and randomly mentions the calculator result of one other possible world and it just happened to be the case that in that other world the result was different; or if Omega randomly picks a world, then randomly picks a world with the opposite calculator result and tosses a coin as to which world to visit and which to mention then the calculator result in the counterfactual word is equally relevant and hearing Omega talk about it just as good as running the calculator twice. In this case you are equally likely to be in a Odd world and might just as well toss a coin as to which result you fill in yourself.

You are not asked to update your belief about the answer being "even" upon observing Omega (in any sense of "knowledge" of those discussed in the post). You knew that the other possibility existed all along, you don't need Omega to see that. You are asked to decide what to do in the counterfactual.

Consider uncertainty about when Omega visits you part of the problem statement, but clearly if a tricky condition such as "it only visits you when your decision will make it worse for you" was assumed, it would be stated.

I don't think it's any different. You could have a Q in the box, and include a person that types it in a calculator as part of the scanning device. Does your variant evoke different intuitions about observational knowledge? It looks similar in all relevant respects to me.

It is no different, as far as I can tell. You can't go from the coin landed the same as it did in your world to only considering worlds where the coin is heads - which is the premise you need if you want to conclude that this is the 1% case of scanner going wrong.

In my opinion the original post (barring the later comment by the author) does not imply that Q is the same in the real world and in the counterfactual world. Am I wrong here?

Then, if Omega is a trustworthy all-powerful device, it would not construct a counterfactual that is straight-out impossible just to play a trick on me. Therefore I conclude that the counterfactual amounts to running an identical scanner another time and getting a different result. But now I no longer think that it is an independent copy of the scanner -- actually it is completely dependent (it is determined to return a different answer), so I no longer think that the conclusion about the coins is fifty-fifty, but that we shouldn't update.

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Calculator is stochastic, Omega doesn't touch any calculator, it assists you with determining what gets written on the counterfactual test sheet (not by counterfactual you, but by Omega personally, counterfactual you is ignored). The worlds diverge at a point where the calculator happens to display different answers to the same question.

Indeed. Consider a variant of the thought experiment where in the "actual" world you used a very reliable process, that's only wrong 1 time in a trillion, while in the counterfactual you're offered to control, you know only of an old calculator that is wrong 1 time in 10, and indicated a different answer from what you worked out. Updateless analysis says that you still have to go with old calculator's result.

Knowledge seems to apply only to the event that produced it, even "logical" knowledge. Even if you prove something, you can't be absolutely sure, so in the counterfactual you trust an old calculator instead of your proof. This would actually be a good variant of this thought experiment ("Counterfactual Proof"), interesting in its own right, by showing that "logical knowledge" has the same limitations, and perhaps further highlighting the nature of these limitations.

Is it correct to to interpret this as "if-counterfactual the calculator had showed odd, Omega would have shown up and (somehow knowing what choice you would have made in the "even" world) altered the test answer as you specify"?

Yes.

Yes, clearly in some counterfactuals such knowledge works.

The reason we cannot use the knowledge from this particular counterfactual is that we have no knowledge about how the counterfactual was selected.

What do you additionally need to know about the counterfactuals? Where is the ambiguity (among what two examples of possible interpretations that change the analysis)? What do you mean by "selected"?

I agree with this answer. I believe in the question as given the answer is probably "even", but if Omega clarifies that in a counterfactual world randomly selected from the pool of all counterfactual worlds the calculator displayed "odd", then you should have a 50% probability each way.

The reason observational evidence works in your world but not in other not randomly selected possible worlds is that if Omega selected the world in any way other than at random, then we're talking about a world that may have been specifically selected for being improbable.

Q is most likely even,

Derived from the likelihood of the calculator being in error

so in the counterfactual the calculator is most likely in error,

You can't conclude this - think about what evidence you have that the calculator is in error!

It seems like one answer to "Why isn't anything you can think of evidence?" might be that "anything you can think of" becomes incomputable very quickly.

Let's say you were to ask a computer to consider "Anything you can think of" with respect to this problem. Imagine each unique hard drive configuration is a thought, And it can process 1 thought per second per hertz. Let's make it a 5ghz computer.

It can think of anything on a 32 bit drive in a bit less then 1 second since 2^32 is 4,294,967,296, which is less then 5 billion.

The problem is, in uncompressed Ascii where you would need 8bits for a character, you can't even fit the thought "32bit" onto a 32 bit harddrive, since it's 5 bytes/40 bits long.

If we double the harddrive to 64 bits to give ourselves more room for longer thoughts, our 5ghz computer goes from being able to calculate all possible thoughts in less then a second to being able to calculate it in around a human lifetime, because of the exponential growth involved. (At least, assuming I've made no math errors.)

We actually have computers do this when we try to have them crack passwords with brute force. A computer trying to brute force a password is essentially trying "Anything it can think of" to open the password protected data.

Cheating.

Huh?

Omega writes the final answer on the counterfactual test sheet, it doesn't rewrite the question. The question is the same, Q, everywhere, as is the process of typing it in the calculators. It writes what you say it to write, correctness doesn't matter. Clarified in the post.

OK, my final understanding is that the question is whether to build the two world models with a shared Q node or with separate Q nodes. We have separate calculator nodes so by analogy I see no strong reason for there to be a shared Q node, but also no strong reason for separate Q nodes since the counterfactual calculator is severed from the Q node. My inclination is that sharing nodes (as opposed to structure+parameters) between counterfactual worlds is the wrong thing to do, but sharing nodes is a limiting case of sharing structure+parameters... so the "logical" nodes should be shared and I've been the most wrong (by entertaining all other solutions). (But then the "logical" here is defined exactly as what is shared between all legitimate counterfactuals, so it is weaker than the "classically logical"; not all formulas are logical in this sense, but the ones that a mere calculator can compute probably are.)

This is how I interpreted it as well. I'm assuming something else is going on with the "updateless" part, but I don't know what it is.

This is based on my understanding of counterfactual as considering what you would do in some hypothetical alternate branch 'what-if'.

Vladmir explicitly ruled out caring what your counterfactual self would do:

(the actions of your counterfactual self who takes the test in the counterfactual are ignored).

(Apologies for a long string of mutually-contradictory replies I made to this and then deleted. Apparently I'm not in the best shape now, and the parent comment pattern-matches to elements of the correct solution, while still not making sense on further examination. One point that's clearly wrong is that risk-attitude matters for which solution is correct, whatever the other elements of this analysis mean.)

Correct, assuming you're only talking about the possible worlds included in the counterfactual (I didn't see this assumption first, so wrote some likely incorrect comments which are now removed).

See the disclaimer in the last paragraph. The topic of the post is not how to solve the thought experiment, that must be obvious with UDT. It's about the nature of our apparently somewhat broken intuition of observational knowledge.

Still wrong (99%/1% figures are incorrect), although maybe starting from a correct intuition.

Why has nobody posted a careful UDT analysis yet, just to see what actually goes on in the problem? I expected better, hence didn't include such analysis myself. The topic of the post is not how to solve the thought experiment, that must be obvious with UDT. It's about the nature of our apparently somewhat broken intuition of observational knowledge. Although at least one should clearly see the UDT analysis first, in order to discuss that.

Edit: (Although the 99% correct/1% wrong you give are the wrong figures, I wonder if I should retract this comment...)

Yes, "odd" is the correct answer, and you seem to have arrived at it by updateless analysis of the decision problem (without making logical assumptions about which answer is correct, only considering possible observations) which I disclaimed about in the last paragraph.

The question that the post poses, using this thought experiment, is not which answer is correct (we already have necessary tools to reliably tell), but what is the nature of observational knowledge, which apparently fails in this thought experiment but is a crucial element of most other reasoning, and in what sense logical knowledge is different.

(Note that this analysis doesn't face any under-specification problems that too many of the other commenters complained about without clearly explaining what examples of relevant ambiguity remain.)

Well, from the lack of a reply and the four downvotes, I take it that the question is sincere and that at least four people believe it is meaningful. So, I have two questions:

1. How many of the people who have responded so far seem to have understood the question?

2. Suppose (counterfactually) that counterfactual Omega asked counterfactual you what factual Omega should write in the factual test (ignoring what factual you actually does, of course). Should the answer (the instruction to Omega to write either "even" or "odd") be the opposite in this counterfactual case than in the case you originally presented?

I don't understand the problem, but it seems that you think that the result on the calculator affects some kind of objective probability that Q is even - a probability that is the same in both factual and counterfactual worlds. It doesn't, of course. All probability is subjective. Evidence observed in one world has no influence on counterfactual worlds where the evidence did not appear.

But since I suspect you already know this, it seems likely that I simply don't have a clue what your question was and why you decided to ask it in that way.