A problem for probabilistic best systems accounts of laws

Suppose that we live in a Humean universe and the universe contains an extremely large collection of coins scattered on a flat surface. Statistical analysis of all the copper coins fits extremely well with the hypothesis that each coin was independently randomly placed with the chance of heads being 1/16 and that of tails being 15/16.

Additionally, there is a gold coin where you haven’t observed which side it’s on.

And there are no other coins.

On a Lewisian best systems account of laws of nature, if the number of coins is sufficeintly large, it will be a law of nature that all coins are independently randomly placed with the chance of heads being 1/16 and that of tails being 15/16. This is true regardless of whether the gold coin is heads or tails. If you know the information I just gave, and have done the requisite statistical analysis of the copper coins, you can be fully confident that this is indeed a law of nature.

If you are fully confident that it is a law of nature that the chance of tails is 15/16, then your credence for tails for the unobserved gold coin should also be 15/16 (I guess this is a case of the Principal Principle).

But that’s wrong. The fact that the coin is of a different material from the observed coins should affect your credence in its being tails. Inductive inferences are weakened by differences between the unobserved and the observed cases.

One might object that perhaps the Lewisian will say that instead of a law saying that the chance of tails on a coin is 15/16, there would be a law that the chance of tails on a copper coin is 15/16. But that’s mistaken. The latter law is not significantly more informative than the former (given that all but one coin is copper), but is significantly less brief. And laws are generated by balancing informativeness with brevity.

A quick argument against subjective Bayesianism

1. You should assign a prior probability less than 1/2 to the hypothesis that over the lifetime of the universe there were exactly 100 tosses of a fair coin and they were all heads.

2. The hypothesis in (1) is contingent.

3. If there is a contingent hypothesis to which you should assign a prior probability less than 1/2, then subjective Bayesianism is false.

4. So, subjective Bayesianism is false.

Molinism and the Principal Principle

Molinism says that there are non-trivial conditionals of free will and that God providentially decides what to create on the basis of his knowledge of them. I shall assume, for simplicity of examples, that what Molinism says about free will it says about other non-determined events. The Principal Principle says that when you know for sure that the objective chance of some future event is r, the epistemic probability you should assign to that event is r.

Suppose a dictator has set up a machine what will flip an indeterministic and fair coin. On heads, the coin will trigger a bomb that will kill almost everyone on earth. On tails, it will do nothing. Since the coin is fair, the objective chance of heads is 1/2. But suppose you are sure that Molinism is true. Then you should think: "Likely, God would only allow this situation to happen if he knew that the coin flipped in these circumstances would land tails. So, probably, the coin will land tails." Maybe you aren't too convinced by this argument--maybe God would allow the coin to land tails and then miraculously stop the bomb or maybe God is fed up with 99% of humankind. But clearly taking into account your beliefs about God and Molinism will introduce some bias in favor of tails.

This seems to be a violation of the Principal Principle: the objective chance of heads is 1/2 but the rational epistemic probability of heads is at least a little less than 1/2.

Not so fast! The "objective chances" need to be understood carefully in cases where foreknowledge of the future is involved. An assumption behind the Principal Principle is that our evidence only includes information about the past and present. If we know that a true prophet prophesied tails, then our credence in tails should be high even if the coin is fair. Given Molinism, the fact that God allowed the coin toss to take place is information about the future, since it indicates that the coin is less likely to land heads given the disastrous consequences.

So, Molinism is compatible with the Principal Principle, but it renders the Principal Principle inapplicable in cases where it matters to God how a random process will go. But everything that matters matters to God. So the Principal Principle is inapplicable in cases where the outcomes of the random process matter, if Molinism is true. This renders the Principal Principle not very useful. Yet it seems that we need the Principal Principle to reason about the future, and we need it precisely in the cases that matter. So we have a bit of a problem for Molinists.