An improved version of the worlds, thoughts and pluralities paradox

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. For then a world where someone thinks that Bucephalus does not exist has to be a world that not only contains Bucephalus but also contains all the thoughts that are causally prior to Bucephalus, such as the thoughts of the horse-breeders who bred Bucephalus’ parents, and in certain cases involving a forwards-temporal infinite sequence of non-qualitative features there may thus be a counterexample to the premise.

So I want to try again, and introduce two tools to avoid the above problem. The first tool 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. I then assume:

2. For any Q-plurality ww there is a (possible) thought true precisely at the worlds of ww.

This thought is then necessarily a Q-thought.

The second tool 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:

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

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

5. 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 (3) says that when a special thought exists at a world, it’s unique up to logical equivalence. Assumption (4) says that every thought is special at some world, up to logical equivalence. In the case where Q is identity, assumption (5) is trivial. In the case where Q is qualitative sameness, assumption (5) 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 (4), 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 (3)–(5) are quite plausible. And it is likely that if some problem for (3)–(5) 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, thoughts and pluralities, and the predicates Q, S, ∈ (where a ∈ uu means that a is among the uu) and TrueAt (where TrueAt(θ, w) says that θ is true at w). Add the following schematic assumption for any formula ϕ = ϕ(w) of L that has its free variables among w, w₁, ..., w_n, where we write ϕ(w′) for the formula obtained by replacing w in ϕ with w′:

6. World Comprehension_Q: For all w₁, ..., w_n, if ∃wϕ(w) and ∀w∀w′(Q(w,w′) → (ϕ(w)↔︎ϕ(w′)), there is a plurality uu such that for ∀w(w∈uu↔︎ϕ(w)).

7. There are two worlds that are not Q-equivalent.

Fact: Premises (1)–(7) are contradictory.

What’s controversial? I think (2) and (6).

Proof of Fact

Write T(w,uu) for a Q-plurality uu and a world w provided that there is a thought θ true at all and only worlds of uu such that S(w,θ).

It follows from (3) that:

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

From (2), (4) and (6) it follows that:

9. For any Q-plurality uu, there is a world w such that T(w,uu).

Finally, from (5) it follows that:

10. If Q(w,w′), then T(w,uu) iff T(w′,uu).

In other words, T defines a surjection from some of the worlds to all the Q-pluralities of worlds. In the case where Q is identity, this violates a version of Cantor’s Theorem in exactly the same way as I proved in my previous post. Where Q is not identity, we need to be a bit more careful. For the sake of completeness, I give the whole proof.

Let C(w) say that there is a Q-plurality uu 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 Q-inequivalent worlds a and b by (7). Let uu, vv and zz be Q-pluralities consisting of all the worlds Q-equivalent to a, to b, or to either one of a and b, respectively. (These exist by (6).) Let a world w be such that T(w,uu) (by (9)). Then since we do not have C(w), we must have w among the uu, and hence w must be Q-equivalent to a, and so by (10) we have T(a,uu). By the same argument, T(b,vv). Finally, the same argument shows that either T(a,zz) or T(b,zz), and so we have violated (8).

So there is a world w such that C(w). Note that if Q(w,w′), then C(w′) by (10). Let the uu be all the worlds w such that C(w) by (6). This is a Q-plurality. By (9), 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 Q-plurality vv such that T(c,vv) with c not among the vv, from which we would conclude that c is not among the uu by (8), a contradiction. But if c is not among the uu, then we have C(c), and so c is among the uu, a contradiction.