On Rasmussen and Bailey's "How to build a thought"

Rasmussen and Bailey prove that under certain assumptions it follows that there are possible thoughts that are not grounded in anything physical.

I want to offer a version of the argument that is slightly improved in a few ways.

Start with the idea that an abstract object x is a “base” for types of thoughts. The bases might be physical properties, types of physical facts, etc. I assume that in all possible worlds exactly the same bases abstractly exist, but of course what bases obtain in a possible world can vary between worlds.

Consider these claims:

1. Independence: For any plurality xx of bases, there is a possible world where it is thought that exactly one of the xx exists and there is no distinct plurality yy of bases such it is thought that exactly one of the yy obtains.

2. Comprehension: For any formula ϕ(x) with one free variable x that is satisfied by at least one base, there is a plurality yy of all the bases that satisfy ϕ(x).

3. Plurality: There are at least two bases.

4. Basing: In every possible world, if there is a plurality xx of bases and it is thought that exactly one of the xx obtains, then there is a base z such that necessarily if z obtains, it is thought that exactly one of the xx obtains.

By the awkward locution “it is thought that p”, I mean that something or some plurality of things thinks that p, or there is a thinkerless thought that p. The reason for all these options is that I want to be friendly to early-Unger style materialists who think that there no thinkers. :-)

Theorem: If Independence, Comprehension, Plurality and S5 are true, Basing is false.

Here is how this slightly improves on Rasmussen and Bailey:

• RB’s proofs use the Axiom of Choice twice. I avoid this. (They could avoid it, too, I expect.)

• I don’t need a separate category of thoughts to run the argument, just an “it is thought that exactly one of the xx exists” predicate. In particular, I don’t need types of thoughts, but only abstract bases.

• RB use the concept of a thought that at least one of the xx exists. This makes their Independence axiom a little bit less plausible, because one might think that, say, someone who thinks that at least one of the male dogs exists automatically also thinks that at least one of the dogs exists. One might also reasonably deny this, but it is nice to skirt the issue.

• I replace grounding with mere entailment in Basing.

• I think RB either forgot to assume Plurality or are working with a notion of plurality where empty collections are possible.

Some notes:

• RB don’t explicitly assume Comprehension, but I don’t see how to prove their Cantorian Lemma 2 without it.

• Independence doesn’t fit with the necessary existence of an omniscient being. But we can make the argument fit with theism by replacing “it is thought” with “it is non-divinely thought”.

• I think the materialist could just hold that there are pluralities xx of bases such that no one could think about them.

Proofs

Write G(z,xx) to mean that z is a base, the xx are a plurality of bases, and necessarily if z obtains it is thought that exactly one of the xx obtains.

The Theorem follows from te following lemmas.

Lemma 1: Given Independence, Basing and S5, for every plurality of bases xx there is a z such that G(z,xx) and for every other plurality of bases yy it is not the case that G(z,yy).

Proof: Let w be a possible world like in Independence. By Basing, at w there is a base z such that G(z,xx). Suppose now that at w we have G(z,yy) with yy other than xx. Then at w, it is thought that exactly one of yy exists. But that contradicts the choice of w. Since the bases are the same in all worlds, by S5 we can move from the world w to the actual world to get the conclusion of the Lemma.

Lemma 2: Assume Comprehension and Plurality. Then there is no predicate H such that for every plurality of bases xx there is a z such that H(z,xx) and for every other plurality of bases yy it is not the case that H(z,yy).

Proof: Suppose we have such an H. Say that z is an admissible base provided that there is a unique plurality of bases xx such that H(z,xx). I claim that there is an admissible base z such that z is not among any xx such that H(z,xx). For suppose not. Then for all admissible bases z, z is among all xx such that H(z,xx). Let a and b be distinct bases. Let ff, gg and hh be the pluralities consisting of a, of b, and of both a and b respectively. Then the above assumptions show that we must have H(a,ff), H(b,gg) and either H(a,hh) or H(b,hh), and either of these options violates our assumptions on H. By Comprehension, then, let yy be the plurality of all admissible bases z such that z is not among any xx such that H(z,xx). Let z be an admissible base such that H(z,yy). Is z among the yy? If it is, then it’s not. If it is not, then it is. Contradiction!