Suppose that there is a predictor – let us call him Peter – who is an astoundingly reliable predictor of what beliefs human beings will form. He is not infallible, but has in fact never made an incorrect prediction about what belief a human being will form. On tens of thousands of occasions, he has predicted what belief a human being will form, and on every single one of these occasions, his prediction is correct.
Now suppose that you know about Peter’s track record. Suppose that you also know that Peter has filled a big jar with a number of coins, and has made a prediction about what belief (if any) you will form about whether there is an odd or even number of coins in the jar, and then acted on his prediction as follows:
- If he predicted that you would believe that there is an odd number, he made sure that there was an odd number of coins in the jar;
- If he predicted that you would form the belief that there is an even number, he made sure that there was an even number of coins in the jar;
- If he predicted that you would not form any belief about the number of coins in the jar, then he filled the jar randomly, not caring whether there was an odd or an even number of coins in it.
In this case, I claim, whatever belief you form about the number of coins in the jar, it will be rational for you to form that belief. If you form the belief that there is an odd number, that belief is very probably true, and so it is rational for you to form this belief; and if you form the belief that there is an even number, that belief is also very probably true, and so it is rational for you to form that belief too.
If my claim is true, then it seems to show that the correct theory of rational belief is more like Evidential Decision Theory (EDT) than like Causal Decision Theory: The relevant probability that should guide one in rationally forming one’s beliefs with respect to a given proposition p is not the unconditional probability of p given the evidence, but the conditional probability that p has on the assumption that one believes it.
This also explains the following striking fact: Even though it might be highly probable on my evidence that It’s raining but I don’t believe it’s raining, this is not a belief that it is rational for me to form, because the conditional probability of this proposition on the assumption that I believe it is extremely low.