Thoughts and pluralities of worlds: A paradox

These premises are plausible if the quantifiers over possible thoughts are restricted to possible non-divine thoughts and the quantifiers over people are restricted to non-divine thinkers:

1. For any plurality of worlds ww, there is a possible thought that is true in all and only the worlds in ww.

2. For any possible thought θ, there is a possible world w at which there is a time t such that

   1. someone thinks a thought equivalent to θ at t,
   2. any other thought that anyone thinks at t is entailed by θ, and
   3. nobody thinks anything after t.

In favor of (1): Take the thought that one of the worlds in ww is actual. That thought is true in all and only the worlds in ww.

In favor of (2): It’s initially plausible that there is a possible world w at which someone thinks θ and nothing else. But there are reasons to be worried about this intuition. First, we might worry that sometimes to think a thought requires that one have earlier thought some other thoughts that build up to it. Thus we don’t require that there is no other thinking than θ in w, but only that at a certain specified t—the last time at which anyone thinks anything—there is a limitation on what one thinks. Second, one might worry that by thinking a thought one also thinks its most obvious entailments. Third, Wittgensteinians may deny that there can be a world with only one thinker. Finally, we might as well allow that instead of someone thinking θ in this world, they think something equivalent. The intuitions that led us to think there is a world where the only thought is θ, once we account for these worries, lead us to (2).

Next we need some technical assumptions:

3. Plurality of Worlds Comprehension: If ϕ(w) is a formula true for at least one world w, then there is a plurality of all the worlds w such that ϕ(w).

4. There are at least two worlds.

5. If two times are such that neither is later than the other, then they are the same.

(It’s a bit tricky how to understand (5) in a relativistic context. We might suppose that times are maximal spacelike hypersurfaces, and a time counts as later than another provided that a part of that time is in the absolute future of a part of the other time. I don’t know how plausible the argument will then be. Or we might restrict our attention to worlds with linear time or with a reference frame that is in some way preferred.)

Fact: (1)–(5) are contradictory.

So what should we do? I myself am inclined to deny (3), though denying (1) is also somewhat attractive.

Proof of Fact

Write T(w,uu) for a plurality of worlds uu and a world w provided that for some possible thought θ true in all and only the worlds of uu at w there is a time t such that (a)–(c) are true.

Claim: If T(w,uu) and T(w,vv) then uu = vv.

Proof: For suppose not. Let θ₁ be true at precisely the worlds of uu and θ₂ at precisely the worlds of vv. Let t_i be such that at t conditions (a)–(c) are satisfied at w for θ = θ_i. Then, using (5), we get t₁ = t₂, since by (c) there are no thoughts after t_i and by (a) there is a thought at t_i for i = 1, 2. It follows by (b) that θ₁ entails θ₂ and conversely, so uu = vv.

It now follows from (1) and (2) that T defines a surjection from some of the worlds to pluralities of worlds, and this violates a version of Cantor’s Theorem using (3). More precisely, let C(w) say that there is a plurality uu of worlds such that T(w,uu) and w is not among the uu.

Suppose first there is no world w such that C(w). Then for every world w, if T(w,uu) then the world w is among the uu. But consider two worlds a and b by (4). Let uu, vv and zz be pluralities consisting of a, b and both a and b respectively. We must then have T(a,uu), T(b,vv) and either T(a,zz) or T(b,zz)—and in either case the Claim will be violated.

So there is a world w such that C(w). Let the uu be all the worlds w such that C(w) (this uses (3)). By the surjectivity observation, there is a world c such that T(c,uu). If c is among the uu, then we cannot have C(c) since then there would be a plurality vv of worlds such that T(c,vv) with c not among the vv, from which we would conclude that c is not among the uu by the Claim, a contradiction. But if c is not among the uu, then we have C(c), and so c is among the uu, a contradiction.