Suppose at t1 there are countably infinitely many people with red hats and countably infinitely many people with black hats. You’re one of them and you can’t see anybody hat (including your own). What probability you should attach to the proposition that your hat is red depends on the causal history rather than on what the world is like at t1.
For consider two causal histories, each of which results in the same time slice at t1. In the first history, we start with infinitely many hatless people at t0, and for each one we flip a fair coin to see if they get a red or a black hat. Then we arrange the people in a (bidirectionally infinite) line of alternating hat colors. In the second history, we start the same way, but now our coin is unfair, so any given person has only a 1/4 chance of getting a red hat and a 3/4 chance of getting a black hat. But again after the fact the people are arranged in an infinite line of alternating hat colors.
In the two scenarios, the outputs at t1 are relevantly alike—an infinite line of people of alternating hat colors—but what probability you should assign to the proposition that your hat is red depends on which causal history actually took place. So probabilities don’t just depend on how things are now, but also on how things were. At least when we’re dealing with infinities.
7 comments:
I agree, but I would put it more generally. In setting your credences, you should consider all the information you have. This could include information about the past. Of course, it may turn out that much of your information is irrelevant.
In a finite case of M red hats among N people, we might like to say that that each person should have credence M/N in my hat is red. But to justify this without invoking indifference, the people would have to know that the way the hats were assigned was suitably symmetric. This would be a fact about the past.
That's a really good point, though indifference will appeal to some.
There is, however, a difference between the finite and infinite cases. In the finite case, once you know or assume (indifference?) that the assignment was symmetric between you and the others, if you know the distribution at time t1, then any further information is irrelevant. If 50% of the hats at t1 are red, your credence that your hat is red should be 1/2, whether or not the hats were assigned with fair or unfair coins. The facts about the distribution at t1 trump the facts about the earlier stuff, assuming symmetry. But in the infinite case there is no such trumping.
That’s true. It’s interesting to think about why it works.
What symmetric information about the outcomes could there be in the infinite case? The obvious candidates are that there was a particular finite number of red hats, an unspecified finite number of red hats (similarly for black), or an infinite number of both red and black.
In the last case, any particular person will note that ‘both infinite’ has probability 1 whatever her outcome, so straightforward Bayes says no trumping.
The other cases have probability zero so Bayes (and standard probability theory) won’t work. In the ‘unspecified finite’ case, any particular person will note that ‘unspecified finite’ is true or false whatever her outcome, so causal independence suggests no trumping. (Though this may seem unintuitive.)
For ‘particular finite’ any particular person will note that its truth does depend on her outcome. Both intuition and an argument I have given elsewhere suggest that told ‘particular finite red’, the people should give credence 1 to black. ‘Particular finite’ trumps the original objective chances.
Now here is the question that interests me (it may be silly). Is there any other infinite case of trumping?
Can there really be a causal history (a history composed of a causal chain of events), which makes it to "infinity", such that these strange effects occur? Isn't this thought experiment about the hats just another evidence that this can never happen (which has the non-trivial result that the past is finite and had a beginning)?
I am thinking here of an infinite number of simultaneous causes--so the story only requires two times, t0 and t1.
You could imagine the coins being flipped simultaneously (say on a command from an angel). But the arranging would surely be an infinite causal process. In my variations, the counting would be an infinite causal process.
That's true.
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