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.

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

[Revised 11/21/2025 to fix a few issues.]

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. I also assume that for objects, like bases, that are invariant between worlds, their pluralities are also invariant 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 obtains and there is no distinct plurality yy of bases such that 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: Necessarily, if there is a plurality xx of bases and it is thought that exactly one of the xx obtains, then there obtains 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 a “it is thought that exactly one of the xx exists” predicate. In particular, I don’t need types of thoughts, just 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 the 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 obtains a base z such that G(z,xx). By S5 and the bases and pluralities thereof being the same at all worlds, we have G(z,xx) at the actual world, too. Suppose now that we actually 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. Thus, actually, we have G(z,xx) but not G(z,yy).

Lemma 2: Assume Comprehension and Plurality. Then there is no formula ϕ(z,xx) open only in z and xx such that for every plurality of bases xx there is a z such that ϕ(z,xx) while for every other plurality of bases yy it is not the case that ϕ(z,yy).

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

Divine Belief Simplicity

Divine Belief Simplicity is the thesis that all of God's acts of belief are the same act of belief, the same belief token. While my belief that 2+2=4 seems distinct from my belief that the sky is blue, God's believings are all one. This is a special case of divine simplicity.

Here is an argument for Divine Belief Simplicity. The primary alternative to Divine Belief Simplicity is:

• Divine Belief Diversity: God's act of believing p is distinct from God's act of believing q whenever p and q are different.

But Divine Belief Diversity is false. The argument may be based on an anonymous referee's objection to a paper by Josh Rasmussen—I can't remember very well now—or to some comments by Josh Rasmussen. Here are some assumptions we'll need:

1. For any plurality, the Fs, there is a distinct proposition that the Fs exist or don't exist.

For instance, there is the proposition that the world's dogs exist or don't exist, and the proposition that the French exist or don't exist, and so on. Next:

2. Separation: Given any plurality, the Fs, and a predicate, P, that is satisfied by at least one of the Fs, there is a plurality of all and only the Fs satisfying P.
3. Plurality of Believings: If Divine Belief Diversity holds, then there is a plurality of all divine acts of believing.

But this is enough to run a Russell paradox.

Say that a divine believing b is settish provided that there is a plurality, the Fs, such that b is a believing that the Fs exist or don't exist. For any settish divine belief b, there is the plurality of things that b affirms the existence or nonexistence of. Say that a divine believing b is nonselfmembered provided that b is settish and is not in the plurality of things that b affirms the existence or nonexistence of. By (1), Separation and Plurality of Believings, let p be the proposition that affirms existence-or-nonexistence of the nonselfmembered believings. Now p is true. So there is a divine believing b in p. This is settish. Moreover, this b either is among the nonselfmembered believings or not. If it is, then it's not. If it's not, then it is. So we have a contradiction.

Moreover, this argument does not need to take propositions ontologically seriously. It only needs divine believings to be taken ontologically seriously.

Denying Divine Belief Diversity, however, denies that there is such a thing as the plurality of things that b affirms the existence or nonexistence of.

More on evil thoughts

When I say, in class or blog, that some thoughts are immoral, there is much resistance. Sometimes, this resistance is formulated as: "Only actions, not thoughts, are immoral." I have argued earlier that voluntary thoughts are actions. In this post, I just want to give a quick argument that some mental activities are immoral.

Suppose Maurice sincerely and prejudicedly believes a racist claim R about members of group G, and he says: "It is my opinion that R." We think Maurice deserves criticism. Even those who think that only physical actions are subject to moral evaluation are likely to agree that something has been done wrong—after all, Maurice has done something physical, namely he has spoken.

But what has gone wrong, and in what way? First of all, it need not be the case that Maurice's speech-act was either morally or rationally criticizable. After all, what he said was true: he said that it was his opinion that R, and indeed it was his opinion that R. He sincerely spoke the truth. What is wrong with that?

We should not criticize Maurice for telling us what his opinion was. We should, instead, criticize him for having that opinion. Now, we have two possibilities here. We can criticize him for epistemic failure, or for both epistemic failure and immorality. If we criticize him for both n the holding of the opinion that R, then we have agreed that mental activities, such as coming to the opinion that R, are subject to immorality.

So, let us explore the option of criticizing Maurice merely epistemically. But if so, then we cannot criticize him any more than we would criticize other people who hold equally or more irrational beliefs, such as that the earth is flat, or that the works of Shakespeare are a 19th century fake, or that the Law of Excluded Middle is false (the last of these is a denial of a law of logic—how much worse can one get than that?). Yet those holding racist beliefs are criticized in a special way, with the criticism not just having to do with the evidential weakness of their views. This suggests that there is something other than mere epistemic failure that is at issue. There is something immoral about coming to racist beliefs in an epistemically deficient way.[note 1]

And in fact I can say a bit about that immorality. We have a special moral obligation—surely not an epistemic one—not to form false beliefs about matters that are socially important, and particularly in cases where having such false beliefs is likely to result in prejudiced unjust behavior. Thus, Maurice should have been particularly careful epistemically in forming the belief R. And this "should have" is a moral "should have".

[Edited: The one time "C" occurred in the original post, it should have been "R".]