An improved paradox about thoughts and worlds

Yesterday, I offered a paradox about possible thoughts and pluralities of worlds. The paradox depends on a kind of recombination principle (premise (2) in the post) about the existence of thoughts, and I realized that the formulation in that post could be objected to if one has a certain combination of views including essentiality of origins and the impossibility of thinking a proposition that involves non-qualitative features (say, names or natural kinds) in a world where these features do not obtain.

So I want to try again, and use two tricks to avoid the above problem. Furthermore, after writing up an initial draft (now deleted), I realized I don’t need pluralities at all, so it’s just a paradox about thoughts and worlds.

The first trick is to restrict ourselves to (purely) qualitative thoughts. Technically, I will do this by supposing a relation Q such that:

1. The relation Q is an equivalence (i.e., reflexive, symmetric, and transitive) on worlds.

We can take this equivalence relation to be qualitative sameness or, if we don’t want to make the qualitative thought move after all, we can take Q to be identity. I don’t know if there are other useful choices.

We then say that a Q-thought is a (possible) thought θ such that for any world there aren’t two worlds w and w′ with Q(w,w′) such that θ is true at one but not the other. If Q is qualitative sameness, then this captures (up to intensional considerations) that θ is qualitative. Furthermore, we say that a Q-plurality is a plurality of worlds ww such that there aren’t two Q-equivalent worlds one of which is in ww and the other isn’t.

The second trick is a way of distinguishing a “special” thought—up to logical equivalence—relative to a world. This is a relation S(w,θ) satisfying these assumptions:

2. If S(w,θ) and S(w,θ′) for Q-thoughts θ and θ′, then the Q-thoughts are logically equivalent.

3. For any Q-thought θ and world w, there is a thought θ′ logically equivalent to θ and a world w such that S(w,θ′).

4. For any Q-thought θ and any Q-related worlds w and w′, if S(w,θ), there is a thought θ′ logically equivalent to θ such that S(w′,θ′).

Assumption (2) says that when a special thought exists at a world, it’s unique up to logical equivalence. Assumption (3) says that every thought is special at some world, up to logical equivalence. In the case where Q is identity, assumption (4) is trivial. In the case where Q is qualitative sameness, assumption (4) says that a thought’s being special is basically (i.e., up to logical equivalence) a qualitative feature.

We get different arguments depending on what specialness is. A candidate for a specialness relation needs to be qualitative. The simplest candidate would be that S(w,θ) iff at w the one and only thought that occurs is θ. But this would be problematic with respect (3), because one might worry that many thoughts are such that they can only occur in worlds where some other thoughts occur.

Here are three better candidates, the first of which I used in my previous post, with the thinkers in all of them implicitly restricted to non-divine thinkers:

1. S(w,θ) iff at w there is a time t at which θ occurs, and no thoughts occur later than t, and any other thought that occurs at t is entailed by θ

2. S(w,θ) iff at w the thought θ is the favorite thought of the greatest number of thinkers up to logical equivalence (i.e., there is a cardinality κ such that for each of κ thinkers θ is the favorite thought up to logical equivalence, and there is no other thought like that)

3. S(w,θ) iff at w the thought θ is the one and only thought that anyone thinks with credence exactly π/4.

On each of these three candidates for the specialness relation S, premises (2)–(4) are quite plausible. And it is likely that if some problem for (2)–(4) is found with a candidate specialness relation, the relation can be tweaked to avoid the relation.

Let L be a first-order language with quantifiers over worlds (Latin letters) and thoughts (Greek letters), and the above predicates Q and S, as well as a T(θ,w) predicate that says that the thought θ is true at w. We now add the following schematic assumption for any formula ϕ = ϕ(w) of L with at most the one free variable w, where we write ϕ(w′) for the formula obtained by replacing free occurrences of w in ϕ with w′:

6. Q-Thought Existence: If ∀w∀w′[Q(w,w′)→(ϕ(w)↔︎ϕ(w′))], there is a thought θ such that ∀w(T(θ,w)↔︎ϕ(w)).

Our argument will only need this for one particular ϕ (dependent on the choice of Q and S), and as a result there is a very simple way to argue for it: just think the thought that a world w such that ϕ(w) is actual. Then the thought will be actual and hence possible. (Entertaining a thought seems to be a way of thinking a thought, no?)

Fact: Premises (1)–(6) are contradictory.

Eeek!!

I am not sure what to deny. I suppose the best candidates for denial are (3) and (6), but both seem pretty plausible for at least some of the above choices of S. Or, maybe, we just need to deny the whole framework of thoughts as entities to be quantified over. Or, maybe, this is just a version of the Liar?

Proof of Fact

Let ϕ(w) say that there is a Q-thought θ such that S(w,θ) and but θ is not true at w.

Note that if this is so, and Q(w,w′), then S(w′,θ′) for some θ′ equivalent to θ by (4). Since θ is a Q-thought it is also not true at w′, and hence θ′ is not true at w′, so we have ϕ(w′).

By Q-Thought Existence (6), there is a Q-thought that is true at all and only the worlds w such that ϕ(w) and by (3) there is a Q-thought ρ logically equivalent to it and a world c such that S(c,ρ). Then ρ is also true at all and only the worlds w such that ϕ(w).

Is ρ true at c?

If yes, then ϕ(c). Hence there is a Q-thought θ such that S(c,θ) but θ is not true at w. Since S(c,ρ), we must have θ and ρ equivalent by (2), so ρ is is not true at c, a contradiction.

If not, then we do not have ϕ(c). Since we have S(c,ρ), in order for ϕ(c) to fail we must have ρ true at c, a contradiction.

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.

Proper classes as merely possible sets

This probably won’t work out, but I’ve been thinking about the Cantor and Russell Paradoxes and proper classes and had this curious idea: Maybe proper classes are non-existent possible sets. Thus, there is actually no collection of all the sets in our world, but there is another possible world which contains a set S whose members are all the sets of our world. When we talk about proper classes, then, we are talking about merely possible sets.

Here is the story about the Russell Paradox. There can be a set R whose members are all the actual world’s non-self-membered sets. (In fact, since by the Axiom of Foundation, none of the actual world’s sets are self-membered, R is a set whose members are all the actual world’s sets.) But R is not itself one of the actual world’s sets, but a set in another possible world.

The story about Cantor’s Paradox that this yields is that there can be a cardinality greater than all the cardinalities in our world, but there actually isn’t. And in world w₂ where such a cardinality exists, it isn’t the largest cardinality, because its powerset is even larger. But there is a third world which has a cardinality larger than any in w₂.

It’s part of the story that there cannot be any collections with non-existent elements. Thus, one cannot form paradoxical cross-world collections, like the collection of all possible sets. The only collections there are on this story are sets. But we can talk of collections that would exist counterfactually.

The challenge to working out the details of this view is to explain why it is that some sets actually exist and others are merely possible. One thought is something like this: The sets that actually exist at w are those that form a minimal model of set theory that contains all the sets that can be specified using the concrete resources in the world. E.g., if the world contains an infinite sequence of coin tosses, it contains the set of the natural numbers corresponding to tosses with heads.