Counterfactual Calculation and Observational Knowledge

Vladimir_Nesov

Consider the following thought experiment ("Counterfactual Calculation"):

You are taking a test, which includes a question: "Is Q an even number?", where Q is a complicated formula that resolves to some natural number. There is no a priori reason for you to expect that Q is more likely even or odd, and the formula is too complicated to compute the number (or its parity) on your own. Fortunately, you have an old calculator, which you can use to type in the formula and observe the parity of the result on display. This calculator is not very reliable, and is only correct 99% of the time, furthermore its errors are stochastic (or even involve quantum randomness), so for any given problem statement, it's probably correct but has a chance of making an error. You type in the formula and observe the result (it's "even"). You're now 99% sure that the answer is "even", so naturally you write that down on the test sheet.

Then, unsurprisingly, Omega (a trustworthy all-powerful device) appears and presents you with the following decision. Consider the counterfactual where the calculator displayed "odd" instead of "even", after you've just typed in the (same) formula Q, on the same occasion (i.e. all possible worlds that fit this description). The counterfactual diverges only in the calculator showing a different result (and what follows). You are to determine what is to be written (by Omega, at your command) as the final answer to the same question on the test sheet in that counterfactual (the actions of your counterfactual self who takes the test in the counterfactual are ignored).

Should you write "even" on the counterfactual test sheet, given that you're 99% sure that the answer is "even"?

This thought experiment contrasts "logical knowledge" (the usual kind) and "observational knowledge" (what you get when you look at a calculator display). The kind of knowledge you obtain by observing things is not like the kind of knowledge you obtain by thinking yourself. What is the difference (if there actually is a difference)? Why does observational knowledge work in your own possible worlds, but not in counterfactuals? How much of logical knowledge is like observational knowledge, and what are the conditions of its applicability? Can things that we consider "logical knowledge" fail to apply to some counterfactuals?

(Updateless analysis would say "observational knowledge is not knowledge" or that it's knowledge only in the sense that you should bet a certain way. This doesn't analyze the intuition of knowing the result after looking at a calculator display. There is a very salient sense in which the result becomes known, and the purpose of this thought experiment is to explore some of counterintuitive properties of such knowledge.)

Consider the following thought experiment ("Counterfactual Calculation"):

You are taking a test, which includes a question: "Is Q an even number?", where Q is a complicated formula that resolves to some natural number. There is no a priori reason for you to expect that Q is more likely even or odd, and the formula is too complicated to compute the number (or its parity) on your own. Fortunately, you have an old calculator, which you can use to type in the formula and observe the parity of the result on display. This calculator is not very reliable, and is only correct 99% of the time, furthermore its errors are stochastic (or even involve quantum randomness), so for any given problem statement, it's probably correct but has a chance of making an error. You type in the formula and observe the result (it's "even"). You're now 99% sure that the answer is "even", so naturally you write that down on the test sheet.

Then, unsurprisingly, Omega (a trustworthy all-powerful device) appears and presents you with the following decision. Consider the counterfactual where the calculator displayed "odd" instead of "even", after you've just typed in the (same) formula Q, on the same occasion (i.e. all possible worlds that fit this description). The counterfactual diverges only in the calculator showing a different result (and what follows). You are to determine what is to be written (by Omega, at your command) as the final answer to the same question on the test sheet in that counterfactual (the actions of your counterfactual self who takes the test in the counterfactual are ignored).

Should you write "even" on the counterfactual test sheet, given that you're 99% sure that the answer is "even"?

Let me try again, then, hopefully more clearly.

Suppose that I am asked to precommit to a strategy before I know the result of the calculation (such assumption removes the potential disagreement with CDT in Counterfactual Mugging). Also, I expect that Omega appears with certainty, no matter what result the calculator gives.

So, I know that I will be given the calculator result, which is 99% correct, and asked by Omega to imagine a counterfactual world where the result was the opposite, and that I am free to determine what should Omega write in that counterfactual world.

The only chance why I should care is when I think that Omega could rewrite my result in the actual world. But I was not sure what algorithm Omega would follow. From the description of the problem it seemed that Omega simply asks the question and "modifies the counterfactual world", which I interpret as "changing Omega's beliefs about the counterfactual world." But anybody can do that, there is no need for Omega's exceptional qualities here, and I am certainly not going to change my beliefs after being asked this question by a janitor in place of Omega.

So Omega must be following some distinct algorithm. He may scan my mind and always rewrite the result depending on how would I respond in the counterfactual world. Hence I have asked whether it rewrites the answers of the actual people, rather than only changing its fantasies about the counterfactual. Probably that interpretation was the natural one when Omega was included, but it didn't occur to me after reading the original post. I continue within this interpretation.

I have four pure strategies: Precommit to tell Omega to write down (in the counterfactual world)

the actual calculator output.
the counterfactual (i.e. opposite) output.
always even.
always odd.

The first one always leads Omega to rewrite my answer to the opposite, which leaves me with 99% chance of losing. The second one wins in 99% of cases. The remaining two are 50% successful. So, answer to your question is "odd".

Is this interpretation correct, or still I am misunderstanding something?

Also, I expect that Omega appears with certainty, no matter what result the calculator gives.

This could work if you give up control over your own test sheet to the counterfactual you mediated by Omega (and have your own decision control the counterfactual test sheet using counterfactual Omega). That's an elegant variant of the problem, with an additional symmetry. (In my thought experiment, the you that observed "odd" doesn't participate in the thought experiment at all, and the test sheet on "even" side is controlled by the you that... (read more)

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Counterfactual Calculation and Observational Knowledge

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