I imagine the following point is well known, but it might be news enough to some people to be worth posting here. The following principle is inconsistent.
Trust Experts. If you know there is someone who is (a) perfectly rational and (b) strictly better informed than you, i.e. they know everything you know and know more as well, whose credence in p is x, then your credence in p should be x.
The reason this is inconsistent is that there can be multiple experts. Here’s one way to generate a problem for Trust Experts.
There are two coins, 1 and 2, to be flipped. The coin flipping procedure is known to be fair, so it is known that for each coin, the chance of it coming up heads is 1/2. And the coins are independent. Let H1 be that the first coin lands heads, and H2 be that the second coin lands heads.
The coins are now flipped, but you can’t see how they are flipped. There are sixteen people, called witnesses, in an adjacent room, plus an experimenter, who knows how the coin lands. Using a randomising device, the experimenter assigns each of the sixteen different propositions that are truth-functions of H1 and H2 to a different witness, and tells them what their assigned proposition is, and what its truth value is. The witnesses know that the experimenter is assigning propositions at random, and that the experimenter always tells the truth about the truth value of propositions. So the witnesses simply conditionalise on the truth of the information that they receive.
Consider first the witnesses who are told the truth values of (H1 v T2) and (H1 v ~T2). One of these will be told that their assigned proposition is true. Whoever that is, call them W1, will have credence 2/3 in H1.
Consider next the witnesses who are told the truth values of (~H1 v T2) and (~H1 v ~T2). Again, one of these will be told that their assigned proposition is true. Whoever that is, call them W2, will have credence 1/3 in H1.
So if you were following Trust Experts, you’d have to have credence 2/3 in H1, because of the existence of W1, and have credence 1/3 in H1, because of the existence of W2. That’s inconsistent, so Trust Experts is inconsistent.
Posted by Brian Weatherson in Uncategorized