Tuesday, October 8, 2019

Humean accounts of modality

Humean accounts of modality, like Sider’s, work as follows. We first take some privileged truths, including all the mathematical ones, and an appropriate collection of others (e.g., ones about natural kind membership or the fundamental truths of metaphysics). And then we stipulate that to be necessary is to follow from the collection of privileged truths, and the possible that whose negation isn’t necessary.

Here is a problem. We need to be able to say things like this:

  1. Necessarily it’s possible that 2+2=4.

For that to be the case, then:

  1. It’s possible that 2+2=4

has to follow from the privileged truths. But on the theory under consideration, (2) means:

  1. That 2 + 2 ≠ 4 does not follow from the privileged truths.

So, (3) has to follow from the privileged truths. Now, how could it do that? Suppose first that the privileged truths include only the mathematical ones. Then (3) has to be a mathematical truth: for only mathematical truths follow logically from mathematical truths. But this means that “the privileged truths”, i.e., “the mathematical truths”, has to have a mathematical description. For instance, there has to be a set or proper class of mathematical truths. But that “the mathematical truths” has a mathematical description is a direct violation of Tarski’s Indefinability of Truth theorem, which is a variant of Goedel’s First Incompleteness Theorem.

So we need more truths than the mathematical ones to be among the privileged ones, enough that (3) should follow from them. But it unlikely that any of the privileged truths proposed by the proponents of Humean accounts of modality will do the job with respect to (3). Even the weaker claim:

  1. That 2 + 2 ≠ 4 does not follow from the mathematical truths

seems hard to get from the normally proposed privileged truths. (It’s not mathematical, it’s not natural kind membership, it’s not a fundamental truth of metaphysics, etc.)

Consider this. The notion of “follows from” in this context is a formal mathematical notion. (Otherwise, it’s an undefined modal term, rendering the account viciously circular.) So facts about what does or does not follow from some truths seem to be precisely mathematical truths. One natural way to make sense of (4) is to say that there is a privileged truth that says that some set T is the set of mathematical truths, and then suppose there is a mathematical truth that 2 + 2 ≠ 4 does not follow from T. But a set of mathematical truths violates Indefinability of Truth.

Perhaps, though, we can just add to the privileged truths some truths about what does and does not follow from the privileged truths. In particular, the privileged truths will contain, or it will easily follow from them, the truth that they are mutually consistent. But now the privileged truths become self-referential in a way that leads to contradiction. For instance:

  1. No x such that F(x) follows from the privileged truths.

will make sense for any F, and we can choose a predicate F such that it is provable that (5) is the only thing that satisfies F (cf. Goedel’s diagonal lemma). Now, if (5) follows from the privileged truths, then it also follows from the privileged truths that (5) doesn’t follow from the privileged truths, and hence that the privileged truths are inconsistent. Thus, from the fact that the privileged truths are consistent, which itself is a privileged truth or a consequence thereof, one can prove (5) doesn’t follow from the privileged truths, and hence that (5) is true, which is absurd.

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