Counterfactual Calculation and Observational Knowledge

by Vladimir_Nesov 1 min read31st Jan 2011188 comments

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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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