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