Animal experimentation

We have an intuitive line as to where the suffering in animal’s life is so great compared to the goods in it that when we are able to do so, we euthanize animals when the suffering causes the value of the animal’s life to fall below that line. On the other hand, the life of an animal that falls above this line is a life that is a benefit to the animal.

It seems to me that this intuitive line could be a helpful discernment criterion for animal experimentation. Animals that are used for experiments are often bred for that purpose. Thus, they wouldn’t exist absent the practice of experimentation. It seems, hence, that experiments where the stresses on the animals make the animal’s life fall below this intuitive line are very easy to justify: the animals are benefitted by the practice, even if the experiments do impose some suffering on the animal. It seems plausible that such experiments could thus be justified by the intrinsic value of the knowledge gained for its own sake, or even by pedagogical benefits for students.

On the other hand, if the stresses in the animal’s life are such as to make their life fall below the line, then stronger justification is needed: the prospective benefits of the research need to be rather more significant.

I don’t know how good we are at discerning where that line goes. People with pets and farm animals do make hard decisions about this, though, so we seem to have some epistemic access to the line.

(I do think that it is permissible to be much more utilitarian about animal life than about human life. I certainly would not generalize what I say above to the case of humans.)

Three-dimensionality

It seems surprising that space is three-dimensional. Why so few dimensions?

An anthropic answer seems implausible. Anthropic considerations might explain why we don’t have one or two dimensions—perhaps it’s hard to have life in one or two dimensions, Planiverse notwithstanding—but thye don’t explain why don’t have thirty or a billion dimensions.

A simplicity answer has some hope. Maybe it’s hard to have life in one and two dimensions, and three dimensions is the lowest dimensionality in which life is easy. But normally when we do engage in simplicity arguments, mere counting of things of the same sort doesn’t matter much. If you have a theory on which in 2050 there will be 9.0 billion people, your theory doesn’t count as simpler in the relevant sense than a theory on which there will be 9.6 billion then. So why should counting of dimensions matter?

There is something especially mathematically lovely about three dimensions. Three-dimensional rotations are neatly representable by quaternions (just as two-dimensional ones are by complex numbers). There is a cross-product in three dimension (admittedly as well as in seven!). Maybe the three-dimensionality of the world suggests that it was made by a mathematician or for mathematicians? (But a certain kind of mathematician might prefer an infinite-dimensional space?)

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.

Comparing axiologies

Are there ways in which it would be better if axiology were different? Here’s a suggestion that comes to mind:

1. It would be better if cowardice, sloth, dishonesty, ignorance, suffering and all the other things that are actually intrinsic evils were instead great intrinsic goods.

For surely it would be better for there to be more goods!

On the other hand, one might have this optimistic thought:

2. The actually true axiology is better than any actually false axiology.

(Theists are particularly likely to think this, since they will likely think that the true axiology is grounded in the nature of a perfect being.)

We have an evident tension between (1) and (2).

What’s going on?

One move is to say that it makes no sense to discuss the value of impossible scenarios. I am inclined to think that this isn’t quite correct. One might think it would be really good if the first eight thousand binary digits of π encoded the true moral code in English using ASCII coding, even though this is impossible (I assume). Likewise, it is impossible for a human to know all of mathematics, but it would be good to do so.

The solution I would go for is that axiology needs to be kept fixed in value comparisons. Imagine that I am living a blessed life of constant painless joy, and dissatisfied with that I find myself wishing for the scenario where joyless pain is even better than painless joy and I live a life of joyless pain. If one need not keep axiology fixed in value comparisons, that wish makes perfect sense, but I think it doesn’t—unlike the wish about π or the knowledge of mathematics.

A way to be calmer

For years I would find myself periodically annoyed by shoelaces. Several times a day, I would have to engage in finicky fine-motor activity to tie my shoes. This made me a little angry, because I suspected that the reason why few adult shoes have alternate closures has to do with fashion rather than with any technological benefits of shoelaces (note, after all, that shoelaces come undone, as well as get caught in bike gears, so it's not all a matter of laziness), and I've always resented social pressures of fashion imposing burdens on us. 

I've thought about this for a long time, and then recently finally decided to do something about it. I pulled out some cord locks (in the photo are some heavy duty cord locks that I salvaged from something years ago), pulled my shoelaces through them, and after a day or two of experimental use, I cut the shoelaces down, and knotted them above the cord locks. No more regular annoyance and anger at society's fashion choices! 

To fasten, I just grab the cord lock with one hand, and pull the permanent knot with the other. To unfasten, I just grab the cord lock and pull it to the knot. At any time, I can easily adjust tension in either direction without untying. It doesn't come loose. It doesn't get stuck in bike gears. It's not quite as instantaneous as I had imaged, but it is pretty fast.

It has some minor down sides. Eventually a cord lock will break down--though I don't know if this will be sooner than the shoe. At the length of lace I settled for (a little shorter than in this photo), the shoes don't loosen quite as far for removal as I might ideally prefer. And one would probably need to cut the laces to launder the shoes, but I don't launder my shoes. 

The void between the atoms

Philoponus says:

When Democritus said that the atoms are in contact with each other, he did not mean contact, strictly speaking, which occurs when the surfaces of the things in contact fit on [epharmazousōn] one another, but the condition in which the atoms are near one another and not far apart is what he called contact. For no matter what, they are separated by void. (67A7)

This odd view would lead to three difficulties. First, the loveliness of the Democritean system is that everything is explained by atoms pushing each other around, without any mysterious action at a distance, without any weird forces like the love and strife posited by other Greek thinkers. But if two atoms are moving toward each other, and they must stop short of touching each other, it seems that we have some kind of a repulsion at a “near” distance. Second, the atomists thought everything happened of necessity. But why should two atoms heading for each other stop at distance x apart rather than distance x/2 or x/3, say? This seems arbitrary. And, third, what reason would Democritus have to say such a strange thing?

One solution is to simply say Philoponus was wrong about Democritus (cf. this interesting paper). One might, for instance, speculate that Democritus said something about how there will always be interstices of void when atoms meet, much like the triangle-like interstices when you tile the plane with circles in a hexagonal pattern, because their surfaces do not perfectly match like jigsaw pieces would, and Philoponus confused this with the claim that there is void between the atoms.

But I want to try something else. There is a famous problem—discussed by Sextus Empiricus, the Dalai Lama (!) and a number of people in between—about how impenetrable material objects can possibly touch. For if they touch, their surfaces are either separated by some distance or not.If their surfaces are separated, they don’t really touch. If their surfaces are not separated, then the surfaces are in the same place, and the objects have penetrated each other (albeit only infinitesimally) and hence they are not really impenetrable.

Suppose now that we think that Democritus was aware of this problem, and posited the following solution. Atoms occupy open regions of space, ones that do not include any of their boundaries or surfaces. For instance, atoms of fire, which are spherical, occupy the set of points in space whose distance to the center is strictly less than a radius r: the boundary, where the distance to the center is exactly r, is unoccupied. If two spherical atoms, each of radius r, come in contact, the distance between their centers is 2r, but the point exactly midway between their centers is not occupied by either atom. There is a single point’s worth of void there.

This immediately solves two of the three problems I gave for the void-between-atoms view. If I’m right, Democritus has very good reason to posit the view: it is needed to avoid the problem of interpenetration of surfaces. Furthermore, the arbitrariness problem disappears. Atoms heading for each other stop precisely when their boundaries would interpenetrate if they had boundaries in them. They stop at distance zero. There is no smaller distance they could stop at. The two spherical atoms stop moving toward each other when there is exactly one point of void between them: any more and they could keep on moving; any less is impossible.

We still have the problem of mysterious action at a distance requiring some force beyond mere contact. But Democritus might think—I don’t know if he would be right—that action at zero distance is less mysterious than action at positive distance, and on the suggestion I am offering the distance between objects that are touching is zero. There is a point’s (or a surface of points, if say we have two cubical atoms meeting with parallel faces) worth of distance, and that’s zero. Impenetrability at least explains why the atoms can’t go any further towards each other, even if it does not explain why they deflect each other’s motion as they do (which anyway, as we learn from Hume’s discussion of billiard balls, isn’t easy). So the remaining problem is reduced.

It wouldn’t surprise me at all if this was in the literature already.

One-thinker colocationism

Colocationists about human beings think that in my chair are two colocated entities: a human person and a human animal. Both of them are made of the same stuff, both of them exhibit the same physical movements, etc.

The standard argument against colocationism is the two thinkers argument. Higher animals, like chimpanzees and dogs, think. The brain of a human animal is more sophisticated than that of a chimpanzee or a dog, and hence human animals also have what it takes to think. Thus, they think. But human persons obviously think. So there are two thinkers in my chair, which is innately absurd, plus leads to some other difficulties.

If I were a colocationist, I think I would deny that any animals think. Instead, the same kind of duplication that happens in the human case happens for all the higher animals. In my chair there is a human animal and a person, and only the person thinks. In the doghouse, there is a dog and a “derson”. In the savanna, one may have a chimpanzee and a “chimperson”. The derson and the chimperson are not persons (the chimperson comes closer than the derson does), but all three think, while their colocated animals do not. We might even suppose that the person, the derson and chimperson are all members of some further kind, thinker.

Suppose one’s reason for accepting colocationism about humans is intuitions about the psychological components of personal identity: if one’s psychological states were transfered into a different head, one would go with the psychological states, while the animal would stay behind, so one isn’t an animal. Then I think one should say a similar thing about other higher animals. If we think that that an interpersonal relationship should follow the psychological states rather than the body of the person, we should think similarly about a relationship with one’s pet: if one’s pet’s psychological states are transfered into a different body, our concerns should follow. If Rover is having a vivid dream of chasing a ball, and we transfer Rover’s psychological states into the body of another dog, Rover would continue the dream in that other body. I don’t believe this in the human case, and I don’t believe it in the dog case, but if I believed this in the human case, I’d believe it in the dog case.

What are the reasons for the standard colocationist’s holding that the human animal thinks? One may say that because both the animal and the person have the same brain activity, that’s a reason to say that either both or neither thinks. But the brain also has the same brain activity, and so if this is one’s reason for saying that the animal thinks, we now have three thinkers. And, if there are unrestricted fusions, the mereological sum of the person with their clothes also has the same brain activity, thereby generating a fourth thinker. That’s absurd. Thus thought isn’t just a function of hosting brain activity, but hosting brain activity in a certain kind of context. And why can’t this context be partly characterized by modal characteristics, so that although both the animal and the person have the same brain activity, they provide a different modally characterized context for the brain activity, in such a way that only one of the two thinks?

This one-thinker colocationism can be either naturalistic or dualistic. On the dualistic version, we might suppose that the nonphysical mental properties belong to only one member of the pair of associated beings. On the naturalistic version, we might suppose that what it is to have a mental property is to have a physical property in a host with appropriate modal properties—the ones the person, the derson and the chimperson all have.

I think there is one big reason why a colocationist may be suspicious of this view. Ethologists sometimes explain animal behavior in terms of what the animal knows, is planning, and more generally is thinking. These explanations are all incorrect on the view in question. But the one-thinker co-locationist has two potential answers to this. The first is to weaken her view and allow animals to think, but not consciously. It is only the associated non-animal that has conscious states, that has qualia. But the conscious states need not enter into behavioral explanations. The second is to say that the scientists’ explanations while incorrect can be easily corrected by replacing mental properties with their neural correlates.

Reducing de re to de dicto modality

In my previous post, I gave an initial defense of a theory of qualitative haecceities in terms of qualitative origins: qualitative haecceities encapsulate complete qualitative descriptions of an entity’s initial state and causal history. I noted that among the advantages of the theory is that it can allow for a reduction of de re modality to de dicto modality, without “the mystery of non-qualitative haecceities”.

I want to expand on this, and why qualitative-origin haecceities are superior to non-qualitative haecceities here. A haecceitistic account of de re modality proceeds in something like the following vein. First, introduce the predicate H(Q,x) which says that Q is a haecceity of x. Then we reduce de re claims as follows:

• x is essentially F ↔︎ ∀Q(H(Q,x)→□(∀y(Qy→Fx)))

• x is accidentally F ↔︎ ∃Q(H(Q,x)∧◊(∃y(Qy∧Fx))).

Granted, this involves de re modality for second-order variables like Q. But this de re modality is less problematic because we can suppose the Barcan and converse Barcan formulas to hold as axioms for the second-order quantifiers, and we can treat the second-order entities as necessary beings. De re modality is particularly difficult for contingent beings, so if we can reduce to a modal logic where only necessary beings are subject to de re modal claims, we have made genuine progress.

We will also need some axioms. Here are two that come to mind:

• ∀x∀Q(H(Q,x)→Qx) (things have their haecceities)

• ∀x∃Q(H(Q,x)) (everything has a haecceity).

Now, here is why I think that qualitative-origin haecceities are superior to non-qualitative haecceities. Given qualitative-origin haecceities, we can give an account of what H(Q,x) means without using de re modality. It just means that Qy attributes to y all of the actual qualitative causal origins of x, including x’s initial qualitative state. On the other hand, if we go for non-qualitative haecceities, we seem to have two options. We could take H(Q,x) to be primitive, which always should be a last resort, or we could try to define in some way like:

• H(Q,x) ↔︎ (□(Ex→Qx) ∧ □∀y(Qy→y=x))

where Ex says that x exists (it might be a primitive in a non-free logic, or it might just be an abbreviation for ∃y(y=x)). But this definition uses de re modality with respect to x, so it is not satisfactory in this context, and I can’t think of any way to do it without de re modality with respect to potentially contingent individuals like x.

Qualitative haecceities

A haecceity H of x is a property of an entity such that necessarily x exists if and only if x instantiates H.

Haecceities are normally thought of as non-qualitative properties. But one could also have qualitative haecceities. Of course, if an entity has a qualitative haecceity then it cannot be duplicated, so one can only suppose that everything has a qualitative haecceity provided one is willing to agree with Leibniz’s Identity of Indiscernibles.

I am personally drawn to the idea that everything does have a qualitative haecceity, and specifically that the qualitative haecceity of x encapsulates x’s qualitative causal history: a complete qualitative description of x’s explanatorily initial state and of all of its causal antecedents. One might call such properties “qualitative origins”. The view that every entity has a qualitative origin is a haecceity is a particularly strong version of the essentiality of origins: everything in an entity’s causal history is essential to it, and the causal history is sufficient for the entity’s existence.

I suppose the main reason not to accept this view is that it implies that two distinct objects couldn’t have the same qualitative origin, but it seems possible that God could create two objects ex nihilo with the same qualitative initial state Q. I am not so sure, though. How would God do that? “Let there be two things satisfying Q?” But this is too indeterminate (I disagree with van Inwagen’s idea that God can issue indeterminate decrees). If there can be two, there can be three, so God would have to specify which two things satisfying Q to create. But that would require a way of securing numerical reference to specific individuals prior to their creation, and that in turn would require haecceities, in this case non-qualitative haecceities. So the objection to the view requires non-qualitative haecceities.

But what started us on this objection was the thought that God could say “Let there be two things satisfying Q.” But if God could say that, why couldn’t he say “Let there be two things satisfying H”, where H is a non-qualitative haecceity? I suppose one will say that this is nonsense, because it is nonsense to suppose two things share a non-qualitative haecceity. But isn’t there a double-standard here? If it is nonsense to suppose two things share a non-qualitative haecceity, why can’t it be nonsense to suppose two things share a qualitative haecceity? It seems that “what does the explaining” of why two things can’t share a non-qualitative haecceity is the obscurity of non-qualitative haecceities, and that’s not really an explanation.

So perhaps we can just say: Having a distinct qualitative origin is what it is to be a thing, and it is impossible for two things to share one. This does indeed restrict the space of possible worlds. No exactly similar iron spheres or anything like that. That’s admittedly a little counterintuitive. But on the other hand, we have a lovely explanation of intra- and inter-world identity of objects, as well as a reduction of de re modality to de dicto, all without the mystery of non-qualitative haecceities. Plus we have Leibniz’s zero/one picture of the world on which all of reality is described by zeroes and ones: we put a zero beside an uninstantiated qualitative haecceity and a one besides an initiated one, and then that tells us everything that exists. This is all very appealing to me.

Existence and causation

Start with these plausible claims:

1. If x causes y, the causal relation between x and y is not posterior to the existence of y.

2. A relation between two entities is never prior to the existence of either entity.

So, the causal relation between x and y is neither prior nor posterior to the existence of y.

But the causal relation is, obviously, intimately tied to the existence of y. What is this tie? The best answer I know is that the causal relation is the existence of y or an aspect of that existence: for y to exist is at least in part for y to have been caused by x.

Appropriateness of memory chains

A lot of discussion of memory theories of personal identity invokes science-fictional thought experiments, such as when memories are swapped between two brains.

One of the classic papers is Shoemaker’s “Persons and their Pasts”. There, Shoemaker accounts for personal identity across time, at least in the absence of branching, in terms of appropriate causal connections between apparent memories, not just any causal connections.

This matters. Imagine that Alice and Bob both get total memory wipes, so on the memory theory they cease to exist. But the person inhabiting the Alice body then reads Bob’s vividly written diary, which induces in her apparent memories of Bob’s life. I think most memory theorists will want to deny that after the reading of the diary, Bob comes back to life in Alice’s body. Not only would this be a highly counterintuitive consequence, but it would violate the plausible principle that whether someone is dead does not depend on future events, absent something like time travel. For suppose this sequence:

• Monday: Memory wipe

• Tuesday: Person inhabiting Alice’s body lives a confused life

• Wednesday: Person inhabiting Alice’s body reads Bob’s diary, comes to think she’s Bob, and gains all sorts of “correct” apparent memories of Bob’s life.

On Wednesday, the person inhabiting Alice’s body has memories of the person inhabiting Alice’s body on Tuesday, so by the memory theory they are the same person. But if on Wednesday, it is Bob who inhabits Alice’s body, then Bob also already existed on Tuesday by transitivity of identity. On the other hand, if Alice hadn’t read the diary on Wednesday, Bob would not have existed either on Wednesday or on Tuesday. So whether Bob is alive on Tuesday depends on future events, despite the absence of anything like time travel, which is absurd.

To get around diary cases, memory theorists really do need to have an appropriateness condition on the causal connections. Shoemaker’s own appropriateness condition appears inadequate: he thinks that what is needed is the kind of connection that makes a later apparent memory and an earlier apparent memory be both of the same experience. But Alice’s induced apparent memories are of the experiences that Bob so vividly described in his diary, which are the same experiences that Bob set down his memories of.

What the memory theorist should insist on are causal chains that are of the right kind for the transmission of memories, modulo any sameness-of-person condition. But now it is far from clear that the science-fictional scenarios in the literature satisfy this condition. Certainly, the scanning of memories in a brain and the imposition of the same patterns on a brain isn’t the normal way for memories to be causally transmitted over time. That it’s not the normal way does not mean that it’s not an appropriate way, but at least it’s far from clear that it is an appropriate way.

It would be interesting what one should say about a memory theory on which the appropriate causal chain condition is sufficiently strict that the only way to transfer memories from one head to another would be by physically moving the brain. (Could one move a chunk of the brain instead? Maybe, but only if it turns out that memories can be localized. And even so it’s not clear whether coming along with a mere chunk of the brain is the appropriate way to transmit memories; the appropriate way may require full cerebral context.) Such a version of the memory theory would not do justice to “memory swapping” intuitions about the memories from one brain being transferred to another. And I take it that such memory swapping intuitions are important to the case for the memory theory.

Here’s another implausible consequence of this kind of memory theory. Suppose aliens are capturing people, and recording their brain data using a method that destroys the memories. However, being somewhat nice, the aliens then use the recording to restore the memories, and then return the person to earth. On the memory theory, anybody coming back to earth is a new individual. That doesn’t seem quite right.

A challenge for the memory theorist, thus, is to have an account of the appropriate causal chain condition that is sufficiently lax to allow for the memory swap intuitions that often motivate the theory but is strict enough to rule out diary cases. This is hard.

Restitution

Suppose Bob paid professional killer Alice to kill him on a day of her choice in the next month. Next day, Bob changes his mind, but has no way of contacting Alice. A week later, Bob sees Alice in the distance aiming a rifle at him. Is it permissible for him to shoot Alice in self-defense?

I take it (somewhat controversially) that killing a juridically innocent person is murder even if the victim consents. Thus, Alice is attempting murder, and normally it is permissible to shoot someone who is trying to murder one. But it seems rather dastardly for Bob to shoot Alice in this case.

On the other hand, though, if Bob hired Alice to kill Carl, and then repented, shooting Alice when Alice is trying to murder Carl does seem the right thing for Bob to do if there is no other way to save Carl’s life.

What is the exact moral difference between the two cases? In both cases, Alice is trying to commit a murder, and in both cases Bob bears a responsibility for this.

I think the difference has something with duties of restitution. When one has done something wrong, and then repented, one needs to do one’s best to “undo” the wrong, repaying the victims in a reasonable manner. But there is a gradation of priority, and in particular even if one is oneself among the victims (Socrates thinks the wrongdoer is the chief victim, since in doing wrong one damages one’s virtue), restitution to others takes priority. In both cases, Bob has harmed Alice by tempting her to commit murder. In the case where Alice was hired to murder Bob, restitution to Alice takes precedence over restitution to Bob, and refraining from killing Alice in self-defense seems a precisely appropriate form of restitution. In the case where Alice was hired to murder Carl, however, restitution to Carl takes precedence, and Bob owes it to Carl to shoot Alice.

In fact, I suspect that in the case where Bob hired Alice to kill Carl, if the only way to save Carl’s life is for Bob to leap into the line of fire and die protecting Carl, other things being equal that would be Bob’s duty. Normally to sacrifice one’s life to save another is supererogatory, but not so when the danger to the other comes from one’s own murderous intent.

The morality of restitution is difficult and complex.

Independent invariant regular hyperreal probabilities: an existence result

A couple of years ago I showed how to construct hyperreal finitely additive probabilities on infinite sets that satisfy certain symmetry constraints and have the Bayesian regularity property that every possible outcome has non-zero probability. In this post, I want to show a result that allows one to construct such probabilities for an infinite sequence of independent random variables.

Suppose first we have a group G of symmetries acting on a space Ω. What I previously showed was that there is a hyperreal G-invariant finitely additive probability assignment on all the subsets of Ω that satisfies Bayesian regularity (i.e., P(A) > 0 for every non-empty A) if and only if the action of G on Ω is “locally finite”, i.e.:

• For any finitely generated subgroup H of G and any point x in G, the orbit Hx is finite.

Here is today’s main result (unless there is a mistake in the proof):

Theorem. For each i in an index set, suppose we have a group G_i acting on a space Ω_i. Let Ω = ∏_iΩ_i and G = ∏_iG_i, and consider G acting componentwise on Ω. Then the following are equivalent:

1. there is a hyperreal G-invariant finitely additive probability assignment on all the subsets of Ω that satisfies Bayesian regularity and the independence condition that if A₁, ..., A_n are subsets of Ω such that A_i depends only on coordinates from J_i ⊆ I with J₁, ..., J_n pairwise disjoint if and only if the action of G on Ω is locally finite

2. there is a hyperreal G-invariant finitely additive probability assignment on all the subsets of Ω that satisfies Bayesian regularity

3. the action of G on Ω is locally finite.

Here, an event A depends only on coordinates from a set J just in case there is a subset A′ of ∏_j ∈ JΩ_j such that A = {ω ∈ Ω : ω|_J ∈ A′} (I am thinking of the members of a product of sets as functions from the index set to the union of the Ω_i). For brevity, I will omit “finitely additive” from now on.

The equivalence of (b) and (c) is from my old result, and the implication from (a) to (b) is trivial, so the only thing to be shown is that (c) implies (a).

Example: If each group G_i is finite and of size at most N for a fixed N, then the local finiteness condition is met. (Each such group can be embedded into the symmetric group S_N, and any power of a finite group is locally finite, so a fortiori its action is locally finite.) In particular, if all of the groups G_i are the same and finite, the condition is met. An example like that is where we have an infinite sequence of coin tosses, and the symmetry on each coin toss is the reversal of the coin.

Philosophical note: The above gives us the kind of symmetry we want for each individual independent experiment. But intuitively, if the experiments are identically distributed, we will want invariance with respect to a shuffling of the experiments. We are unlikely to get that, because the shuffling is unlikely to satisfy the local finiteness condition. For instance, for a doubly infinite sequence of coin tosses, we would want invariance with respect to shifting the sequence, and that doesn’t satisfy local finiteness.

Now, on to a sketch of the proof from (c) to (a). The proof uses a sequence of three reductions using an ultraproduct construction to cases exhibiting more and more finiteness.

First, note that without loss of generality, the index set I can be taken to be finite. For if it’s infinite, for any finite partition K of I, and any J ∈ K, let G_J = ∏_i ∈ JG_i, let Ω_J = ∏_i ∈ JΩ_i, with the obvious action of G_J on Ω_J. Then G is isomorphic to ∏_J ∈ KG_J and Ω to ∏_J ∈ KΩ_J. Then if we have the result for finite index sets, we will get a regular hyperreal G-invariant probability on Ω that satisfies the independence condition in the special case where J₁, ..., J_n are such that J_i and J_j for distinct i and j are such that at least one of J_i ∩ J and J_j ∩ J is empty for every J ∈ K. We then take an ultraproduct of these probability measures with respect to K and an ultrafilter on the partially ordered set of finite partitions of I ordered by fineness, and then we get the independence condition in full generality.

Second, without loss of generality, the groups G_i can be taken as finitely generated. For suppose we can construct a regular probability that is invariant under H = ∏_iH_i where H_i is a finitely generated subgroup of G_i and satisfies the independence condition. Then we take an ultraproduct with respect to an ultrafilter on the partially ordered set of sequences of finitely generated groups (H_i)_i ∈ I where H_i is a subgroup of G_i and where the set is ordered by componentwise inclusion.

Third, also without loss of generality, the sets Ω_i can be taken to be finite, by replacing each Ω_i with an orbit of some finite collection of elements under the action of the finitely generated G_i, since such orbits will be finite by local finiteness, and once again taking an appropriate ultraproduct with respect to an ultrafilter on the partially ordered set of sequences of finite subsets of Ω_i closed under G_i ordered by componentwise inclusion. The Bayesian regularity condition will hold for the ultraproduct if it holds for each factor in the ultraproduct.

We have thus reduced everything to the case where I is finite and each Ω_i is finite. The existence of the hyperreal G-invariant finitely additive regular probability measure is now trivial: just let P(A) = |A|/|Ω| for every A ⊆ Ω. (In fact, the measure is countably additive and not merely finitely additive, real and not merely hyperreal, and invariant not just under the action of G but under all permutations.)