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 dilemma for best-systems accounts of laws

Here is a dilemma for best-systems accounts of laws.

Either:

1. law-based scientific explanations invoke the lawlike generalization itself as part of the explanation, or

2. they invoke the further fact that this generalization is a law.

Thus, if it is a law that all electrons are charged, and Bob is an electron, on (1) we explain Bob’s charge as follows:

3. All electrons are charged.

4. Bob is an electron.

5. So and that’s why Bob is charged.

But on (2), we replace (3) with:

6. It is a law that all electrons are charged.

Both options provide the Humean with problems.

If it is just the lawlike generalization that explains, then the explanation is fishy. The explanation of why Bob is charged in terms of all electrons being charged seems too close to explaining a proposition by a conjunction that includes it:

7. Bob is charged because Bob is charged and Alice is charged.

Indeed both (3)–(5) and (7) are objectionably cases of explaining the mysterious by the more mysterious: the conjunction is more mysterious than its conjunct and the universal generalization is more mysterious than its instances.

On the other hand, suppose that our explanation of why Bob is charged is that it’s a law that all electrons are charged. This sounds correct in general, but is not appealing on a best-systems view. For on a best-systems view, what the claim that it’s a law that all electrons are charged adds to the claim that all electrons are charged is that the generalization that all electrons are charged is sufficiently informative and brief to make it into the best system. But the fact that it is thus informative and brief does not help it explain anything.

Moreover, if the problem with (3)–(5) was that universal generalizations are too much like conjunctions, the problem will not be relieved by adding more conjuncts to the explanation, namely that the generalization is sufficiently informative and brief.

Merging Lewisian worlds

According to Lewis, any pair (or, more generally, plurality) of concrete (he doesn't even restrict it this way) of objects has a mereological sum. Now, suppose that x and y are concrete objects in worlds w₁ and w₂ respectively. Let z be the mereological sum of x and y. According to Lewis, worlds are maximal spatiotemporally connected sums of objects. Now, here are some plausible principles:

1. Spatiotemporal connection is transitive and symmetric.
2. If a is spatiotemporally connected to a part of b, then a is spatiotemporally connected to b.

Consider any concrete objects a and b in w₁ and w₂, respectively. Then a is connected with x, since all objects in a world are connected. And y is connected with b. Moreover, by 2, a is connected with z since x is a part of z. And by 2, b is connected with z. Thus, by 1, a is connected with b. Thus, all objects in w₁ and w₂ are mutually connected, and so by Lewis's account of worlds, there is only one world. Which is absurd.