Tuesday, September 16, 2008

Could it contingently be the case that the laws of nature hold necessarily?

Here is a nice cautionary tale about being careful with scope and modality. A friend asked me whether one could argue from the possibility of the laws of nature being necessary to laws of nature being actually necessary. I answered in the affirmative, thinking about S5, and imagining the following argument:

1. Possibly, the laws of nature hold necessarily. (Premise)
2. If possibly necessarily p holds, then necessarily p holds. (Premise: S5)
3. Therefore, the laws of nature hold necessarily.

But the argument is badly fallacious. One way to see the fallacy is to note that in (2) the proposition p is referred to rigidly, while in (3) "the laws of nature" are a definite description. Compare the fallacious argument:

1. Possibly, what I will write on the board holds necessarily. (Premise)
2. If possibly necessarily p holds, then necessarily p holds. (Premise: S5)
3. Therefore, what I will write on the board holds necessarily.
Claim (4) is true: it is possible that I will write on the board something that holds necessarily (e.g., "5+7=12"). But one cannot conclude to (6), since after all I might also write something that holds merely contingently or not at all.

Another way to see the fallacy is to see it as a scope ambiguity in (4). For (6) to follow, (4) must be read as saying that possibly p holds necessarily, where p is what I will write on the board. But that claim is unjustified. All I am justified in saying is that possibly I will write something that holds necessarily.

And in fact we can see that it is prima facie possible that it a merely contingent fact that the laws of nature hold necessarily. Suppose that the laws of nature come in two classes. Some laws of nature are metaphysically necessary and some are not. For instance, prima facie, it might be a metaphysically necessary law that electrons have electric charge, but a metaphysically contingent law that opposite charges attract. Then it might turn out that there is a world w where there are no metaphysically contingent laws. It would then be true at w that all the laws of nature hold necessarily. But this would be only contingently the case, because there are worlds that have some of the contingent laws as well.

Interestingly, we can make the argument from (1) and (2) to (3) work if we add the following premise:

1. Necessarily: (For all p, either it is a law that p is a law, or it is a law that p is not a law).
For then, by (1), suppose that w1 is a world where the laws hold necessarily. Suppose now that w2 is any world and p is any law in w2. Then, p either is or is not a law in w1. If p is a law in w1, then it is a law in w1 that p is a law. But the laws in w1 hold necessarily, so it follows that p holds necessarily. And if p is not a law in w1, then it is a law in w1 that p is not a law. Since the laws in w1 hold necessarily, p is not a law in w2, which is absurd. Hence, every law of w2 holds necessarily.

Note: The above argument assumed that "p is a law" entails p, and that the claim "law p holds necessarily" entails that p holds necessarily (I did not additionally assume that "law p holds necessarily" entails that necessarily p is a law, though that does follow from (7)).

[I fixed some typos, and more importantly edited (2) and (4) in response to Comment #1 below. The original version of the post had "p" instead of "necessarily p" in the consequents of (2) and (4), which made the arguments not work. This emphasizes again the first sentence of the post.]