Counterfactual Calculation and Observational Knowledge

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20 Counterfactual Calculation and Observational Knowledge

by Vladimir_Nesov

31st Jan 2011

2 min read

188

20

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.)

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20 Counterfactual Calculation and Observational Knowledge

by Vladimir_Nesov

31st Jan 2011

2 min read

188

20

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"?

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Perplexed15y30

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:

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

Vladimir_Nesov15y10

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 it the case you originally presented?

So far, shokwave clearly gets it. Compare to any other sophisticated question asked in a language you aren't familiar with. Here, you need to be sufficiently comfortable with counterfactuals, ... (read more)

3Perplexed15y

Two more questions. As in the original scenario, but instead of an unreliable calculator, you have a reliable (so far) theorem prover. Type in a proposition to be proved and hit the "ProveIt" button, and immediately the display shows "Working". Then, an unpredictable amount of time later, the display may change to show either "Proven" or "Disproven". So, the base case here is that you type "Q is even" into the device and hit "ProveIt". You plan to only allow 5 minutes for the device to find a proof, and then to just guess, but fortunately the display changes to "Proven" in 4 minutes. But then just as you finish writing "Even" on your test paper, Omega appears. 1. This time, Omega asks you to consider the counterfactual world in which the device still shows "Working" after 5 minutes. Should counter-factual Omega still write "Even" on the test? 2. In a different Omega-suggested counterfactual world, a black swan flies in the window after 4 1/2 minutes and the display shows "Disproven". You know that this means that either a). Arithmetic is inconsistent. b). The theorem prover device is unreliable. or c). Omega is messing with you. Does thinking about this situation cause you to change your answer to the previous question? My opinion: Evidence, counter-evidence, and lack of evidence have no effect on the truth of necessary statements. They only impact the subjective probability of those statements. And subjective probabilities cannot flow backward in time (surviving the erasure of the evidence that produced those subjective probabilities). Even Omega cannot mediate this kind of paradoxical information flow.

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