The last thought

Entertain this thought:

1. There is a unique last thought that anyone ever thinks and it’s not true.

Or, more briefly:

2. The last thought is not true.

There are, of course, possible worlds where there is no last thought, because (temporal) thoughts go on forever, and worlds where there is a tie for last thought, and worlds where the last thought is “I screwed up” and is true. But, plausibly, there are also worlds where there is a unique last thought and it’s not true—say, a world where the last thought is “I see how to defuse the bomb now.”

In other words, (1) seems to be a perfectly fine, albeit depressing, contingent thought.

Is there a world where (1) is the last thought? You might think so. After all, it surely could be the case that someone entertains (1) and then a bomb goes off and annihilates everyone. But supposing that (1) is the last thought in w, then (1) either is or is not true in w. If it is true in w, then it is not, and if it is not true, then it is true. Now that’s a paradoxical last thought!

Over the last week, I’ve been thinking of a paradox about thoughts and worlds, inspired by an argument of Rasmussen and Bailey. I eventually came to realize that the paradox (apparently unlike their argument) seems to be just a version of the Liar Paradox, essentially the one that I gave above.

But we shouldn’t stop thinking just because we have hit upon a Liar. (You don’t want your last thought to be that you hit upon a Liar!) Let’s see what more we can say. First, the version of the Liar in (1) is the Contingent Liar: we only get paradoxicality in worlds where the last thought is (1) or something logically equivalent to (1).

Now, consider that (1) has unproblematic truth value in our world. For in our world, there is no last thought, given eternal life. And even if there were no eternal life, and there was a last thought, likely it would be something that is straightforwardly true or false, without any paradox. Now an unproblematic thought that has truth value has a proposition as a content. Let that proposition be p. Then we can see that neither p nor anything logically equivalent to it can be the content of the last thought in any world.

This is very strange. If you followed my directions, as you read this blog post, you began your reading by entertaining a thought with content p. It surely could have happened that at that exact time, t, no one else thought anything else. But since a thought with content p cannot be the last thought, it seems that some mysterious force would be compelling people to think something after t. Granted, Judaism, Christianity and Islam, there is such a mysterious force, namely God: God has promised eternal life to human beings, and this eternal life is a life that includes thinking. But we could imagine someone thinking a thought with content p at a time when no one else is thinking in a world where God has made no such promises.

So what explains the constraint that neither p nor anything logically equivalent to it can be the content of the last thought in a possible world? After all, we want to maintain some kind of a reasonable rearrangement or mosaic principle and it’s hard to think of one that would let one require that a world where a thought with content p happens at a time t when no one else is thinking, then a thought must occur later. Yet classical logic requires us to say this.

I think what we have to say is this. Take a world w₁ without any relevant divine promises or the like, where after a number of other thoughts, Alice finally thinks a thought with content p at a time when no one else is engaging in any mental activity, and then she permanently dies at t before anyone else can get to thinking anything else. Then at w₁ there will be other thoughts after Alice’s death. Now take a world w₂ that is intrinsically just like w₁ up to and including t, and then there is no thought. I think it’s hard to avoid saying that worlds like w₁ and w₂ are possible. This requires us to say that at w₂, Alice does not think a thought with content p before death, even though w₂ is intrinsically just like w₁ up to and including the time of her death.

What follows is that whether the content of Alice’s thought is p depends on what happens after her (permanent) death. In other words, we have a particularly controversial version of semantic externalism on which facts about the content of mental activity depend on the future, even in cases like p where the proposition does not depend on the identities of any objects or natural kinds other than perhaps ones (is thought a natural kind?) that have already been instantiated. Semantic externalism extends far!

The lastness in (1) and (2) functions to pick out a unique thought in some worlds without regard for its content. There are other ways of doing so:

• the most commonly thought thought

• the least favorite thought of anybody

• the one and only thought that someone accepts with credence π/4.

Each of these leads to a similar argument for a very far-reaching semantic externalism.

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.

Non-formal provability

A simplified version of Goedel’s first incompleteness theorem (it’s really just a special case of Tarski’s indefinability of truth) goes like this:

• Given a sound semidecidable system of proof that is sufficiently rich for arithmetic, there is a true sentence g that is not provable.

Here:

• sound: if s is provable, s is true

• semidecidable: there is an algorithm that given any provable sentence verifies in a finite number of steps that it is provable.

The idea is that we start with a precisely defined ‘formal’ notion of proof that yields semidecidability of provably, and show that this concept of proof is incomplete—there are truths that can’t be proved.

But I am thinking there is another way of thinking about this stuff. Suppose that instead of working with a precisely defined concept of proof, we have something more like a non-formal or intuitive notion of proof, which itself is governed by some plausible axioms—if you can prove this, you can prove that, etc. That’s kind of how intuitionists think, but we don’t need to be intuitionists to find this approach attractive.

Note that I am not explicitly distinguishing axioms.

The idea is going to be this. The predicate P is not formally defined, but it still satisfies some formal constraints or axioms. These can be formulated in a formal language (Brouwer wouldn’t like this) that has a way of talking about strings of symbols and their concatenation and allows one to define a quotation function that given a string of symbols returns a string of symbols that refers to the first string.

One way to do this is to have a symbol ′α′ for any symbol α in the original language which refers to α, and a concatenation operator +, so one can then quote αβγ as ′α′ + ′β′ + ′γ′. I assume the language is rich enough to define a quotation function Q such that Q(x) is the quotation of a string x.

To formulate my axioms, I will employ some sloppy quotation mark shorthand, partly to compensate for the difficulty of dealing with corner quotes on the web. Thus, ′αβγ′ is shorthand for ′α′ + ′β′ + ′γ′, and as needed I will allow substitution inside the quotation marks. If there are nested quotation marks, the inner substitutions are resolved first.

1. For all sentences ϕ and ψ, if P(′ϕ↔︎∼ψ′) and P(′ϕ′), then P(′∼ϕ′).

2. For all sentences ϕ and ψ, if P(′ϕ↔︎∼ψ′) and P(′ψ→ϕ′), then P(′ϕ′).

3. For all sentences ϕ, we have P(′P(′ϕ′)→ϕ′).

4. If ϕ has a formal intuitionistic proof from sufficiently rich axioms of concatenation theory, then P(′ϕ′).

Here, (1) and (2) embody a little bit of facts about proof, both of which facts are intuitionistically and classically acceptable. Assumption (3) is the philosophically heaviest one, but it follows from its being axiom that if ϕ is provable, then ϕ, together with the fact that all axioms count as provable. That a formal intuitionistic proof is sufficient for provability is uncontroversial.

Using similar methods to those used to prove Goedel’s first incompleteness theorem, I think we should now be able to construct a sentence g and the prove, in a formal intuitionistic proof in a sufficiently rich concatenation theory, that:

5. g ↔︎  ∼ P(′g′).

But these facts imply a contradiction. Since 5 can be proved in our formal way, we have:

6. P(′g↔︎∼P(′g′)′). By 4.

7. P(′P(′g′)→g′). By 3.

8. P(′g′). By 6, 7 and 2.

9. P(′∼g′). By 6, 8 and 1.

Hence the system P is inconsistent in the sense that it makes both g and  ∼ g are provable.

This seems to me to be quite a paradox. I gave four very plausible assumptions about a provability property, and got the unacceptable conclusion that the provability property allows contradictions to be proved.

I expect the problem lies with 3: it lets one ‘cross levels’.

The lesson, I think, is that just as truth is itself something where we have to be very careful with the meta- and object-language distinction, the same is true of proof if we have something other than a formal notion.

Chanceability

Say that a function P : F → [0,1] where F is a σ-algebra of subsets of Ω is chanceable provided that it is metaphysically possible to have a concrete (physical or not) stochastic process with a state space of the same cardinality as Ω and such that P coincides with the chances of that process under some isomorphism between Ω and the state space.

Here are some hypotheses ones might consider:

1. If P is chanceable, P is a finitely additive probability.

2. If P is chanceable, P is a countably additive probability.

3. If P is a finitely additive probability, P is chanceable.

4. If P is a countably additive probability, P is chanceable.

5. A product of chanceable countably additive probabilities is chanceable.

It would be nice if (2) and (4) were both true; or if (1) and (3) were.

I am inclined to think (5) is true, since if the P_i are chanceable, they could be implemented as chances of stochastic processes of causally isolated universes in a multiverse, and the result would have chances isomorphic to the product of the P_i.

I think (3) is true in the special case where Ω is finite.

I am skeptical of (4) (and hence of (3)). My skepticism comes from the following line of thought. Let Ω = ℵ₁. Let F be the σ-algebra of countable and co-countable subsets (A is co-countable provided that Ω − A is countable). Define P(A) = 1 for the co-countable subsets and P(A) = 0 for the countable ones. This is a countably additive probability. Now let < be the ordinal ordering on ℵ₁. Then if P is chanceable, it can be used to yield paradoxes very similar to those of a countably infinite fair lottery.

For instance, consider a two-person game (this will require the product of P with itself to be chanceable, not just P; but I think (5) is true) where each player independently gets an ordinal according to a chancy isomorph of P, and the one who gets the larger ordinal wins a dollar. Then each player will think the probability that the other player has the bigger ordinal is 1, and will pay an arbitrarily high fee to swap ordinals with them!

Independence conglomerability

Conglomerability says that if you have an event E and a partition {R_i : i ∈ I} of the probability space, then if P(E∣R_i) ≥ λ for all i, we likewise have P(E) ≥ λ. Absence of conglomerability leads to a variety of paradoxes, but in various infinitary contexts, it is necessary to abandon conglomerability.

I want to consider a variant on conglomerability, which I will call independence conglomerability. Suppose we have a collection of events {E_i : i ∈ I}, and suppose that J is a randomly chosen member of I, with J independent of all the E_i taken together. Independence conglomerability requires that if P(E_i) ≥ λ for all i, then P(E_J) ≥ λ, where ω ∈ E_J if and only if ω ∈ E_J(ω) for ω in our underlying probability space Ω.

Independence conglomerability follows from conglomerability if we suppose that P(E_J∣J=i) = P(E_i) for all i.

However, note that independence conglomerability differs from conglomerability in two ways. First, it can make sense to talk of independence conglomerability even in cases where one cannot meaningfully conditionalize on J = i (e.g., because P(J=i) = 0 and we don’t have a way of conditionalizing on zero probability events). Second, and this seems like it could be significant, independence conglomerability seems a little more intuitive. We have a bunch of events, each of which has probability at least λ. We independently randomly choose one of these events. We should expect the probability that our randomly chosen event happens to be at least λ.

Imagine that independence conglomerability fails. Then you can have the following scenario. For each i ∈ I there is a game available for you to play, where you win provided that E_i happens. You get to choose which game to play. Suppose that for each game, the probability of victory is at most λ. But, paradoxically, there is a random way to choose which game to play, independent of the events underlying all the games, where your probability of victory is strictly bigger than λ. (Here I reversed the inequalities defining independence conglomerability, by replacing events with their complements as needed.) Thus you can do better by randomly choosing which game to play than by choosing a specific game to play.

Example: I am going to uniformly randomly choose a positive integer (using a countably infinite fair lottery, assuming for the sake of argument such is possible). For each positive integer n, you have a game available to you: the game is one you win if n is no less than the number I am going to pick. You despair: there is no way for you to have any chance to win, because whatever positive integer n you choose, I am infinitely more likely to get a number bigger than n than a number less than or equal to n, so the chance of you winning is zero or infinitesimal regardless which game you pick. But then you have a brilliant idea. If instead of you choosing a specific number, you independently uniformly choose a positive integer n, the probability of you winning will be at least 1/2 by symmetry. Thus a situation with two independent countably infinite fair lotteries and a symmetry constraint that probabilities don’t change when you swap the lotteries with each other violates independence conglomerability.

Is this violation somehow more problematic than the much discussed violations of plain conglomerability that happen with countably infinite fair lotteries? I don’t know, but maybe it is. There is something particularly odd about the idea that you can noticeably increase your chance of winning by randomly choosing which game to play.

Truthteller's relative

The truthteller paradox is focused on the sentence:

1. This sentence is true.

There is no contradiction in taking (1) to be true, but neither is there a contradiction in taking (1) to be false. So where is the paradox? Well, one way to see the paradox is to note that there is no more reason to take (1) to be true than to be false or vice versa. Maybe there is a violation of the Principle of Sufficient Reason.

For technical reasons, I will take “This sentence” in sentences like (1) to be an abbreviation for a complex definite syntactic description that has the property that the only sentence that can satisfy the description is (1) is itself. (We can get such a syntactic description using the diagonal lemma, or just a bit of cleverness.)

But the fact that we don’t have a good reason to assign a specific truth value to (1) isn’t all there is to the paradox.

For consider this relative of the truthteller:

2. This sentence is true or 2+2=4.

There is no difficulty in assigning a truth value to (2) if it has one: it’s got to be true because 2+2=4. But nonetheless, (2) is not meaningful. When we try to unpack its meaning, that meaning keeps on fleeing. What does (2) say? Not just that 2+2=4. There is that first disjunct in it after all. That first disjunct depends for its truth value on (2) itself, in a viciously circular way.

But after all shouldn’t we just say that (2) is true? I don’t think so. Here is one reason to be suspicious of the truth of (2). If (2) is true, so is:

3. This sentence is true or there are stars.

But it seems that if (3) is meaningful, then it should should have a truth value in every possible world. But that would include the possible world where there are no stars. However, in that world, the sentence (3) functions like the truthteller sentence (1), to which we cannot assign a truth value. Thus (3) does not
have a sensible truth value assignment in worlds where there are no stars. But it is not the sort of sentence whose meaningfulness should vary between possible worlds. (It is important for this argument that the description that “This sentence” is an abbreviation for is syntactic, so that its referent should not vary between worlds.)

It might be tempting to take (2) to be basically an infinite disjunction of instances of “2+2=4”. But that’s not right. For by that token (3) would be basically an infinite disjunction of “there are stars”. But then (3) would be false in worlds where there are no stars, and that’s not clear.

If I am right, the fact that (1) wouldn’t have a preferred truth value is a symptom rather than the disease itself. For (2) would have a preferred truth value, but we have seen that it is not meaningful. This pushes me to think that the problem with (1) is the same as with (2) and (3): the attempt to bootstrap meaning in an infinite regress.

I don’t know how to make all this precise. I am just stating intuitions.

Saving infinitely many lives

Suppose there is an infinitely long line with equally-spaced positions numbered sequentially with the integers. At each position there is a person drowning. All the persons are on par in all relevant respects and equally related to you. Consider first a choice between two actions:

1. Save people at 0, 2, 4, 6, 8, ... (red circles).

2. Save people at 1, 2, 3, 5, 7, ... (blue circles).

It seems pretty intuitive that (1) and (2) are morally on par. The non-negative evens and odds are alike!

But now add a third option:

3. Save people at 2, 4, 6, 8, ... (yellow circles).

The relation between (2) and (3) is exactly the same as the relation between (1) and (2)—after all, there doesn’t seem to be anything special about the point labeled with the zero. So, if (1) and (2) are on par, so are (2) and (3).

But by transitivity of being on par, (1) and (3) are on par. But they’re not! It is better to perform action (1), since that saves all the people that action (3) saves, plus the person at the zero point.

So maybe (1) is after better than (2), and (2) is better than (3)? But this leads to the following strange thing. We know how much better (1) is than (2): it is better by one person. If (1) is better than (2) and (2) is better than (3), then since the relationships between (1) and (2) and between (2) and (3) are the same, it follows that (1) must be better than (2) by half a person and (2) must be better than (3) by that same amount.

But when you are choosing which people to save, and they’re all on par, and the saving is always certain, how can you get two options that are “half a person” apart?

Very strange.

In fact, it seems we can get options that are apart by even smaller intervals. Consider:

4. Save people at 0, 10, 20, 30, 40, ....

5. Save people at 1, 11, 21, 31, 41, ....

and so on up to:

14. Save people at 10, 20, 30, 40, ....

Each of options (4)–(14) is related the same way to the next. Option (4) is better than option (14) by exactly one person. So it seems that each of options (4)–(13) is better by a tenth of a person than the next!

I think there is one at all reasonable way out, and it is to say that in both the (1)–(3) series and the (4)–(14) series, each option is incomparable with the succeeding one, but we have comparability between the start and end of each series.

Maybe, but is the incomparability claim really correct? It still feels like (1) and (2) should be exactly on par. If you had a choice between (1) and (2), and one of the two actions involved a slight benefit to another person—say, a small probability of saving the life of the person at  − 17—then we should go for the action with that slight benefit. And this makes it implausible that the two are incomparable.

My own present preferred solution is that the various things here seem implausible to us because human morality is not meant for cases with infinitely many beneficiaries. I think this is another piece of evidence for the species-relativity of morality: our morality is grounded in human nature.

A variant of Thomson's Lamp

In the classic Thomson’s Lamp paradox, the lamp has a switch such that each time you press it, it toggles between on and off. The lamp starts turned off, say, before 10:00, and then the switch is pressed at 10:00, 10:30, 10:45, 10:52.5, 10:56.25, and so on ad infinitum. And the puzzle is: Is it on or off at 11? It’s a puzzle, but not obviously a paradox.

But here’s an interesting variant. Instead of a switch that toggles on or off each time you press, you have a standard slider switch, with an off position and an on position. Before 10:00, the lamp is off. At 10:00, 10:45, 10:56.25, and so on, the switch is pushed forcefully all the way to the on side. At 10:30, 10:52.5, and so on, the switch is pushed forcefully all the way to the off side.

The difference between the slider and toggle versions is this. Intuitively, in the toggle version, each switch press is relevant to the outcome—intuitively, it reverses what the outcome would be. In the slider variant, however, each slider movement becomes irrelevant as soon as the next time happens. At 10:45, the switch is pushed to the on side, and at 10:52.5, it is pushed to the off side. But if you skipped the 10:45 push, it doesn’t matter—the 10:52.5 push ensures that the switch is off, regardless of what happened at 10:45 or earlier.

Thus, on the slider version, each of the switch slides is causally irrelevant to the outcome at 11. But now we have a plausible principle:

1. If between t₀ and t₁ a sequence of actions each of which is causally irrelevant to the state at t₁ takes place, and nothing else relevant to the state takes place, the state does not change between t₀ and t₁.

Letting t₀ be 9:59 and t₁ be 11:00, it follows from (1) that the lamp is off at 11:00 since it’s off at 10:00, since in between the lamp is subjected to a sequence of caually irrelevant actions.

Letting t₀ be 10:01 and t₁ still be 11:00, it follows from (1) that the lamp is on at 11:00, since it’s on at 10:01 and is subjected to a sequence of causally irrelevant actions.

So it’s on and off at 11:00. Now that’s a paradox!

More on the interpersonal Satan's Apple

Let me take another look at the interpersonal moral Satan’s Apple, but start with a finite case.

Consider a situation where a finite number N of people independently make a choice between A and B and some disastrous outcome happens if the number of people choosing B hits a threshold M. Suppose further that if you fix whether the disaster happens, then it is better you to choose A than B, but the disastrous outcome outweighs all the benefits from all the possible choices of B.

For instance, maybe B is feeding an apple to a hungry child, and A is refraining from doing so, but there is an evil dictator who likes children to be miserable, and once enough children are not hungry, he will throw all the children in jail.

Intuitively, you should do some sort of expected utility calculation based on your best estimate of the probability p that among the N − 1 people other than you, M − 1 will choose B. For if fewer or more than M − 1 of them choose B, your choice will make no difference, and you should choose B. If F is the difference between the utilities of B and A, e.g., the utility of feeding the apple to the hungry child (assumed to be fairly positive), and D is the utility of the disaster (very negative), then you need to see if pD + F is positive or negative or zero. Modulo some concerns about attitudes to risk, if pD + F is positive, you should choose B (feed the child) and if its negative, you shouldn’t.

If you have a uniform distribution over the possible number of people other than you choosing B, the probability that this number is M − 1 will be 1/N (since the number of people other than you choosing B is one of 0, 1, ..., N − 1). Now, we assumed that the benefits of B are such that they don’t outweigh the disaster even if everyone chooses B, so D + NF < 0. Therefore (1/N)D + F < 0, and so in the uniform distribution case you shouldn’t choose B.

But you might not have a uniform distribution. You might, for instance, have a reasonable estimate that a proportion p of other people will choose B while the threshold is M ≈ qN for some fixed ratio q between 0 and 1. If q is not close to p, then facts about the binomial distribution show that the probability that M − 1 other people choose B goes approximately exponentially to zero as N increases. Assuming that the badness of the disaster is linear or at most polynomial in the number of agents, if the number of agents is large enough, choosing B will be a good thing. Of course, you might have the unlucky situation that q (the ratio of threshold to number of people) and p (the probability of an agent choosing B) are approximately equal, in which case even for large N, the risk that you’re near the threshold will be too high to allow you to choose B.

But now back to infinity. In the interpersonal moral Satan’s Apple, we have infinitely many agents choosing between A and B. But now instead of the threshold being a finite number, the threshold is an infinite cardinality (one can also make a version where it’s a co-cardinality). And this threshold has the property that other people’s choices can never be such that your choice will put things above the threshold—either the threshold has already been met without your choice, or your choice can’t make it hit the threshold. In the finite case, it depended on the numbers involved whether you should choose A or B. But the exact same reasoning as in the finite case, but now without any statistical inputs being needed, shows that you should choose B. For it literally cannot make any difference to whether a disaster happens, no matter what other people choose.

In my previous post, I suggested that the interpersonal moral Satan’s Apple was a reason to embrace causal finitism: to deny that an outcome (say, the disaster) can causally depend on infinitely many inputs (the agents’ choices). But the finite cases make me less confident. In the case where N is large, and our best estimate of the probability of another agent choosing B is a value p not close to the threshold ratio q, it still seems counterintuitive that you should morally choose B, and so should everyone else, even though that yields the disaster.

But I think in the finite case one can remove the counterintuitiveness. For there are mixed strategies that if adopted by everyone are better than everyone choosing A or everyone choosing B. The mixed strategy will involve choosing some number 0 < p_best < q (where q is the threshold ratio at which the disaster happens) and everyone choosing B with probability p_best and A with probability 1 − p_best, where p_best is carefully optimized allow as many people to feed hungry children without a significant risk of disaster. The exact value of p_best will depend on the exact utilities involved, but will be close to q if the number of agents is large, as long as the disaster doesn’t scale exponentially. Now our statistical reasoning shows that when your best estimate of the probability of other people choosing B is not close to the threshold ratio q, you should just straight out choose B. And the worry I had is that everyone doing that results in the disaster. But it does not seem problematic that in a case where your data shows that people’s behavior is not close to optimal, i.e., their behavior propensities do not match p_best, you need to act in a way that doesn’t universalize very nicely. This is no more paradoxical than the fact that when there are criminals, we need to have a police force, even though ideally we wouldn’t have one.

But in the infinite case, no matter what strategy other people adopt, whether pure or mixed, choosing B is better.

The interpersonal Satan's Apple

Consider a moral interpersonal version of Satan’s Apple: infinitely many people independently choose whether to give a yummy apple to a (different) hungry child, and if infinitely many choose to do so, some calamity happens to everyone, a calamity outweighing the hunger the child suffers. You’re one of the potential apple-givers and you’re not hungry yourself. The disaster strikes if and only if infinitely many people other than you give an apple. Your giving an apple makes no difference whatsoever. So it seems like you should give the apple to the child. After all, you relieve one child’s hunger, and that’s good whether or not the calamity happens.

Now, we deontologists are used to situations where a disaster happens because one did the right thing. That’s because consequences are not the only thing that counts morally, we say. But in the moral interpersonal Satan’s Apple, there seems to be no deontology in play. It seems weird to imagine that disaster could strike because everyone did what was consequentialistically right.

One way out is causal finitism: Satan’s Apple is impossible, because the disaster would have infinitely many causes.

Nano St Petersburg

In the St Petersburg game, you toss a fair coin until you get heads. If it took you n tosses to get to heads, you get a utility of 2ⁿ. The expected payoff is infinite (since (1/2) ⋅ 2 + (1/4) ⋅ 4 + (1/8) ⋅ 8 + ... = ∞), and paradoxes abound (e.g., this.

One standard way out is to deny the possibility of unboundedly large utilities.

Interestingly, though, it is possible to imagine St Petersburg style games without really large utilities.

One way is with tiny utilities. If it took you n tosses to get to heads, you get a utility of 2ⁿα, where α > 0 is a fixed infinitesimal. The expected payoff won’t be infinite, but the mathematical structure is the same, and so the paradoxes should all adapt.

Another way is with tiny probabilities. Let G(n) be this game: a real number is uniformly randomly chosen between zero and one, and if the number is one of 1, 1/2, 1/3, ..., 1/n, then you get a dollar. Intuitively, the utility of getting to play G(n) is proportional to n. Now our St Petersburg style game is this: you toss a coin until you get heads, and if you got heads on toss n, you get to play G(2ⁿ).

The paradox of the Jolly Givers

Consider the Grim Reaper (GR) paradox. Fred’s alive at midnight. Only a GR can kill him. Each GR has an alarm with a wakeup time. When the alarm goes off, the GR looks to see if Fred’s alive, and if he is, the GR kills him. Otherwise, the GR does nothing. Suppose the alarm times of the GR’s are 12:30 am, 12:15 am, 12:07.5 am, …. Then Fred’s got to be dead, but no GR could have killed him. If, say, the 12:15 GR killed him, that means Fred was alive at 12:07.5, which means the 12:07.5 GR would have killed him.

A Hawthorne answer to the GR paradox is that the GRs together killed Fred, though no one of them did.

Here’s a simple variant that shows this can’t be true. You hang up a stocking at midnight. There is an infinite sequence of Jolly Givers, each with a different name, and each of which has exactly one orange. There are no other oranges in the world, nor anything that would make an orange. When a JG’s alarm goes off, it checks if there is anything in the stocking. If there is, it does nothing. If there is nothing in the stocking, it puts its orange in the stocking. The alarm times are the same as in the previous story.

The analogous Hawthorne answer would have to be that the JGs together put an orange in the stocking. But then one of the JGs would need to be missing his orange. But no one of the JGs is missing his orange, since no one of them took it out of his pocket. So, the orange would have had to come out of nowhere.

And, to paraphrase a very clever recent comment, if it came out of nowhere, why would it be an orange, rather than, say, a pear?

I think the JG paradox also suggests an interesting link between the principle that nothing comes from nothing and the rejection of supertasks.

Leaving space

Suppose that we are in an infinite Euclidean space, and that a rocket accelerates in such a way that in the first 30 minutes its speed doubles, in the next 15 minutes it doubles again, in the next 7.5 minutes it doubles, and so on. Then in each of the first 30 minutes, and the next 15 minutes, and the next 7.5 minutes, and so on, it travels roughly the same distance, and over the next hour it will have traveled an infinite distance. So where will it be? (This is a less compelling version of a paradox Josh Rasmussen once sent me. But it’s this version that interests me in this post.)

The causal finitist solution is that the story is impossible, for the final state of the rocket depends on infinitely many accelerations, and nothing can causally depend on infinitely many things.

But there is another curious solution that I’ve never heard applied to questions like this: after an hour, the rocket will be nowhere. It will exist, but it won’t be spatially related to anything outside of itself.

Would there be a spatial relationship between the parts of the rocket? That depends on whether the internal relationships between the parts of the rocket are dependent on global space, or can be maintained in a kind of “internal space”. One possibility is that all of the rocket’s particles would lose their spatiality and exist aspatially. Another is that they would maintain spatial relationships with each other, without any spatial relationships to things outside of the rocket.

While I embrace the causal finitist solution, it seems to me that the aspatial solution is pretty good. A lot of people have the intuition that material objects cannot continue to exist without being in space. I don’t see why not. One might, of course, think that spatiality is definitive of materiality. But why couldn’t a material object then continue to exist after having lost its materiality?

Thomson's core memory paradox

This is a minor twist on the previous post.

Magnetic core memory (long obsolete!) stored bits in the magnetization of tiny little rings. It was easy to write data to core memory: there were coils around the ring that let you magnetize it in one of two directions, and one direction corresponded to 0 and the other to 1. But reading was harder. To read a memory bit, you wrote a bit to a location and sensed an electromagnetic fluctation. If there was a fluctuation, then it follows that the bit you wrote changed the data in that location, and hence the data in that location was different from the bit you wrote to it; if there was no fluctuation, the bit you wrote was the same as the bit that was already there.

The problem is that half the time reading the data destroys the original bit of data. In those cases—or one might just do it all the time—you need to write back the original bit after reading.

Now, imagine an idealized core not subject to the usual physics limitations of how long it takes to read and write it. My particular system reads data by writing a 1 to the core, checking for a fluctuation to determine what the original datum was, and writing back that original datum.

Let’s also suppose that the initial read process has a 30 second delay between the initial write of the 1 to the core and the writing back of the original bit. But the reading system gets better at what it’s doing (maybe the reading and writing is done by a superpigeon that gets faster and faster as it practices), and so each time it runs, it’s four times as fast.

Very well. Now suppose that before 10:00:00, the core has a 0 encoded in it. And read processes are triggered at 10:00:00, 10:00:45, 10:00:56.25, and so on. Thus, the nth read process is triggered 60/4ⁿ seconds before 10:01:00. This process involves the writing of a 1 to the core at the beginning of the process and a writing back of the original value—which will always be a 0—at the end.

Intuitively:

1. As long as the memory is idealized to avoid wear and tear, any possible number—finite or infinite—of read processes leaves the memory unaffected.

By (1), we conclude:

2. After 10:01:00, the core encodes a 0.

But here’s how this looks from the point of view of the core. Prior to 10:00:00, a 0 is encoded in the core. Then at 10:00:00, a 1 is written to it. Then at 10:00:30, a 0 is written back. Then at 10:00:45, a 1 is written to it. Then at 10:00:52.5, a 0 is written back. And so on. In other words, from the point of view of the core, we have a Thomson’s Lamp.

This is already a problem. For we have an argument as to what the outcome of a Thomson’s Lamp process is, and we shouldn’t have one, since either outcome should be as likely.

But let’s make the problem worse. There is a second piece of core memory. This piece of core has a reading system that involves writing a 0 to the core, checking for a fluctuation, and then writing back the original value. Once again, the reading system gets better with practice. And the second piece of core memory is initialized with a 1. So, it starts with 1, then 0 is written, then 1 is written back, and so on. Again, by premise (1):

3. After the end of the reading processes, we have a 1 in the core.

But now we can synchronize the reading processes for the second core in such a way that the first reading occurs at 9:59:30, and space out and time the readings in such a way that prior to 9:59:30, a 1 is encoded in the core. At 9:59:30, a 0 is written to the core. At 10:00:00, a 1 is written back to the core, thereby completing the first read process. At 10:00:30, a 0 is written to the core. At 10:00:45, a 1 is written back, thereby completing a second read process. And so on.

Notice that from around 10:00:01 until, but not including, 10:01:00, the two cores are always in the same state, and the same things are done to it: zeroes and ones are written to the cores at exactly the same time. But when, then, do the two cores end up in different final states? Does the first core somehow know that when, say, at 10:00:30, the zero is written into it, that zero is a restoration of the value that should be there, so that at the end of the whole process the core is supposed to have a zero in it?

Mystery and religion

Given what we have learned from science and philosophy, fundamental aspects of the world are mysterious and verge on contradiction: photons are waves and particles; light from the headlamp on a fast train goes at the same speed relative to the train and relative to the ground; objects persist while changing; we should not murder but we should redirect trolleys; etc. Basically, when we think deeper, things start looking strange, and that’s not a sign of us going right. There are two explanations of this, both of which are likely a part of the truth: reality is strange and our minds are weak.

It seems not unreasonable to expect that if there were a definitive revelation of God, that revelation would also be mysterious and verge on contradiction. Of the three great monotheistic religions, Christianity with the mystery of the Trinity is the one that fits best with this expectation. At the same time, I doubt that this provides much of an argument for Christianity. For while it is not unreasonable to expect that God’s revelation would be paradoxical, it is a priori a serious possibility that God’s revelation might be so limited that what was revealed would not be paradoxical. And it would also be a priori a serious possibility that while creation is paradoxical, God is not, though this last option is a posteriori unlikely given what we learn from the mystical experience traditions found in all the three monotheistic religions.

So, I am not convinced that there is a strong argument for Christianity and against the other two great monotheistic religions on the grounds that Christianity is more mysterious. But at least there is no argument against Christianity on the basis of its embodying mysteries.

Repentance and Satan's Apple

Suppose Alice is an misanthropic immortal who lives in a universe of happy people. Suppose, too, that Alice is an immortal. Then one day Alice does a really bad thing. She is unreasonably annoyed at all other people and instantly freezes everything besides herself.

What ought Alice to do? Well, she ought to unfreeze everything.

But when? If she delays unfreezing the universe by a week, she gets to enjoy a week without the annoyance of other people. And nobody will be any the worse for it. So, why not? But if a week, why not a month, or a millennium?

There seems to be nothing wrong with procrastinating when the action is just as well done later. So, why can’t Alice just continue procrastinating for eternity?

Maybe the thing to say is this. Alice ought to repent now. It is wrong to live unrepentantly, so one should repent as soon as possible. And repentance requires an intention to repair the damage that one has done insofar as one can.

But it is true that when the damage can be equally well repaired later, the repentant person does not need to do it immediately. We can even tweak the case so that the repair is better done later. Perhaps Alice will be slightly less grumpy each day, and so if she unfreezes people later, they will be better off as they will have a slightly less grumpy Alice to live with (this makes the case more like Satan’s Apple). And it’s clear that when the damage repair is better done later, it may be left for later.

I think what we need to say is this: The intention needs to have a reasonable level of specificity. When one is able to specify how and when one will do the repair, one needs to intend that. One cannot simply have the intention to do one of infinitely many things (unfreeze tomorrow or unfreeze the day after or …). Intentions, either in general or in the special case of the intentions of restitution that repentance calls for, must come with a plan of action. And so Alice needs to set herself a plan, rather than just vaguely leaving things for the future.

But can’t she just procratinate, even so? When I have an intention to do something, and a better idea comes along, there is nothing wrong with switching to the better idea. So, take the case where the repair is better done later. It seems that Alice can permissibly form the intention to unfreeze tomorrow, and tomorrow change her mind, and so on. But that would allow Alice to get away with never unfreezing, and yet without violating any further moral obligations (besides the ones she violated by the initial freezing).

It seems to me that to get out of this, one needs some way for making intentions be morally binding. Perhaps repentant Alice needs to promise herself or vow to God to unfreeze people on a particular day.

It seems that from our outlandish freezing scenario we can get some interesting conclusions:

• intentions of restitution need a significant amount of specificity; and

• there are ways of moral self-binding, such as self-promises or vows to god.

Bizarre consequences in bizarre circumstances

In strange physical circumstances, we would not be surprised by strange and unexpected behavior of a system governed by physical laws.

Under conditions a device was not designed for, we would not be surprised by odd behavior from the device.

Nor should we be surprised by bizarre behavior by an organism far outside its evolutionary niche.

Therefore, it seems that we should not be surprised by how an entity governed by moral or doxastic laws would behave in out-of-this-world moral or evidential circumstances.

In particular perhaps we should be very cautious—in ways that I have rarely been—about the lessons to be drawn from the ethics or epistemology in bizarre counterfactual stories. Instead, perhaps, we should think about how it could be that ethics or epistemology is tied to our niche, our proper environment, and we should be suspicious of Kantian-style ethics or epistemology grounded in niche- and kind-transcending principles, perhaps preferring a more Aristotelian approach with norms for behavior in our natural environment being grounded in our own nature.

[Thesis:] April Fool's Philosophy Post Generator

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The thesis that [thesis] has not received much of a defense[literature type]. But here is an argument for it:

1. This argument is valid.

2. Therefore, [thesis].

Let's see why this argument is not only valid but sound.

First, let’s see that it’s valid. Suppose for a reductio that it is invalid. But whether an argument is valid or not cannot be a contingent matter. Thus if, the argument is invalid, it is necessarily invalid. But if it is necessarily invalid, then necessarily its first premise is false (since the premise says that the argument is valid). But any argument which has a necessarily false premise is automatically valid. (An argument is valid if and only if it is impossible for the premises to be true and the conclusion false. This is trivially satisfied if it is impossible for a premise to be true.) But that would contradict the assumption that it’s invalid. So, the argument must be valid.

But if the argument (1)–(2) is valid, it’s also automatically sound. For a valid argument is sound provided its premises are true. But the only premise of the argument is (1), the statement that the argument is valid. If the argument is valid, then that premise is true, and so the argument is sound.

But the conclusion of a sound argument is true. Therefore, [thesis].

Yet another infinite hat-guessing story

Suppose first a countably infinite line of blindfolded people standing on tiles numbered 0,1,2,…, with the ones on a tile whose number is divisible by 10 having a red hat, and the others having blue hats. Suppose you’re in the line, with no idea where, but apprised of the above. It seems you should reasonably think: “Probably my hat is blue.”

But then the blindfolded people are shuffled, without any changes of hats, so that now it is the tiles with numbers divisible by 10 that have the blue hatters and the others have the red hatters. Such mere shuffling shouldn’t change what you think. So after being informed of the shuffle, it seems you should still think: “Probably my hat is blue.” It is already puzzling, though, why the first arrangement defined the probabilities and not the second. (What does temporal order have to do with these probabilities?)

Now suppose you gather the nine people after you (in the tile order—even though you are blindfolded, I suppose you can tell which direction the tile number numbers increase) along with yourself into a group of ten. In any group of ten successive people on the line, there is exactly one blue hat and nine red hats. Yet each of the ten of you thinks: “Probably my hat is blue.” And by a reasonable closure, you each also think: “Probably the other nine all have red hats.” You talk about it. You argue about it. “No, I am probably the one with the blue hat!” “No, my hat is probably the blue one.” “No, you’re probably both wrong: It’s probably mine.” I submit there is no rational room for any resolution to the disagreement, and indeed no budging of probabilities, no matter how much you pool your data, no matter how completely you recognize your epistemic peerhood, no matter how you apply exactly the same reasonable principles of reasoning. For nothing you learn from the other people is evidentially relevant. This is paradoxical.