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"?

I think considerably more than two things have to go well for your interpretation to succeed in describing this post... I don't necessarily disagree with what you wrote, in that I don't see clear enough statements that I disagree with, and some things seem correct, but I don't understand it well.

Also, calculator is correct 99% of the time, so you've probably labeled things in a confusing way that could lead to incorrect solution, although the actual resulting numbers seem fine for whatever reason.

The reason I used a logical statement instead of a coin, was to compare logical and observational knowledge, since logical knowledge, in its usual understanding, applies mostly to logical statements, and doesn't care what you reason about using it. This can allow extending the thought experiment, for example, in this way.

The reason I used a logical statement instead of a coin, was to compare logical and observational knowledge, since logical knowledge, in its usual understanding, applies mostly to logical statements, and doesn't care what you reason about using it. This can allow extending the thought experiment, for example, in this way.

I'm not seeing why that extended thought experiment couldn't have used a coin and two scanners of different reliability.

0Tyrrell_McAllister15y

Indeed. That is because you needed more than two things to go right for your post to succeed in communicating your point ;). My confusion is over this sentence from your post: [...] My difficulty is that everything that I would call knowledge is like what you get when you look at a calculator display. Suppose that the test had asked you whether "2+2" reduced to an even number. Then you would perform certain mental operations on this expression, and you would answer in accordance with how those operations concluded. (For example, you might picture two sets of two dots, one set next to the other, and see whether you can pair off elements in one set with elements in the other. Or you might visualize a proof in Peano arithmetic in your mind, and check whether each line follows from the previous line in accordance with the rules of inference.) At any rate, whatever you do, it amounts to relying on the imperfect wetware calculator that is your brain. If a counterfactual version of you got a different answer with his brain, you would still want his test sheet to match his answer. So, what is the residue left over, after we set aside observational knowledge? What is this "logical knowledge"? Calling it "the usual kind" is not sufficing to pick out what you mean for me. My guess was that your "logical knowledge" includes (in your terminology) the "moral arguments" that "the agent can prove" in the "theory it uses". The analogous role in Wie Dai's "brute-force" UDT is served by the agent's computation of an expected utility EU(f) for an input-output map f. Is this a correct interpretation of what you meant by "logical knowledge"? (I know that I may need more than two things to go right to have interpreted you correctly. That is why I am giving you my interpretation of what you said. If I got it right, great. But my main motivation arises in the case where I am wrong. My hope is that you will then restate your claim, this time calibrating for the way that I am evidently

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