Against Imprecise Probabilities #1
Some people think we should have sets of probability functions to represent our credence in a proposition, rather than a single such function. They are wrong. Here is one reason.
You have two fair coins. H1 says that the first coin will be heads when flipped, H2 says that the second coin will be heads when flipped. Since the coins are fair, P(H1)=1/2, and P(H2)=1/2. You have no idea whether or not the coins are correlated. Since you are ignorant about the correlation, advocates of the imprecise approach say you should have functions P* and P^ such that P*(H1&H2)=1/2 (perfect correlation) and P^(H1&H2)=0 (perfect anti-correlation) in your set of probabilities, as well as everything in between.
Suppose H1 is true, so that the first coin came up heads. Then P(H2|H1)=P(H2&H1)/P(H1). In other words, on the imprecise approach which updates by conditionalization on every member of your set of probabilities, Pnew(H2)=[0,1]. But look, suppose H1 is false. Then, again Pnew(H2)=[0,1]. So whatever you learn about H1, your credence in H2 is [0,1].
So, now, we see a conflict between reflection and the principal principle. The principal principle says that if you know the objective chance about some proposition, your credence should equal that objective chance. So P(H2) (before the first coin gets tossed) should equal 1/2. But reflection says that if you know what your future credence will be, you should set your current credence to that value. Generalized to the imprecise case, since whatever happens regarding the first coin your credence in H2 is [0,1], your credence in H2 before the first coin is tossed should be [0,1]. So we have a conflict. Your credence cannot be both 1/2 and [0,1]. You have to give up one or the other. So the imprecise approach to credence requires you to reject either the principal principle or reflection. Both are very, very plausible epistemic requirements. So, by requiring you to reject one of them, the imprecise approach is implausible. (There are, of course, answers based on complementarity to this problem. But they are unsuccessful).
You have two fair coins. H1 says that the first coin will be heads when flipped, H2 says that the second coin will be heads when flipped. Since the coins are fair, P(H1)=1/2, and P(H2)=1/2. You have no idea whether or not the coins are correlated. Since you are ignorant about the correlation, advocates of the imprecise approach say you should have functions P* and P^ such that P*(H1&H2)=1/2 (perfect correlation) and P^(H1&H2)=0 (perfect anti-correlation) in your set of probabilities, as well as everything in between.
Suppose H1 is true, so that the first coin came up heads. Then P(H2|H1)=P(H2&H1)/P(H1). In other words, on the imprecise approach which updates by conditionalization on every member of your set of probabilities, Pnew(H2)=[0,1]. But look, suppose H1 is false. Then, again Pnew(H2)=[0,1]. So whatever you learn about H1, your credence in H2 is [0,1].
So, now, we see a conflict between reflection and the principal principle. The principal principle says that if you know the objective chance about some proposition, your credence should equal that objective chance. So P(H2) (before the first coin gets tossed) should equal 1/2. But reflection says that if you know what your future credence will be, you should set your current credence to that value. Generalized to the imprecise case, since whatever happens regarding the first coin your credence in H2 is [0,1], your credence in H2 before the first coin is tossed should be [0,1]. So we have a conflict. Your credence cannot be both 1/2 and [0,1]. You have to give up one or the other. So the imprecise approach to credence requires you to reject either the principal principle or reflection. Both are very, very plausible epistemic requirements. So, by requiring you to reject one of them, the imprecise approach is implausible. (There are, of course, answers based on complementarity to this problem. But they are unsuccessful).