Moral fetishism and intentional aiming

For a long time I’ve had an odd fascination with cases where you have to intentionally aim at something that is neither your end nor a means to your end. The first case to come to my mind was something like this: Let’s say you want to send a nerve signal from your brain to your forearm (e.g., maybe you are hooked up to a device that detects these nerve signals and dispenses chocolate covered almonds). What do you? You wiggle your fingers! Wiggling your fingers, however, is not your end. But neither is it a means to the sending of the nerve signals. On the contrary, the nerve signals are the cause of the finger motions. But you can’t directly aim at the nerve signals, so you have to aim at finger wiggling instead.

A more ordinary kind of case I’ve found is follow-through in racquet sports, where you continue racquet motion after hitting the ball (or shuttle), because your brain will make the swing weaker at the time of contact unless you’re trying to continue the movement after the swing. But there is no point to the movement after the time of contact—the ball isn’t somehow magically steered by the racquet once it no longer touches it. So the movement of the racquet after impact is neither means nor end.

Outside weird laboratory setups and some sports cases, it is hard to think of cases of this odd phenomenon where one takes aim at an action that one neither instrumentally or finally cares about. But I’ve just realized a very interesting application. There is a philosophical literature about what people call “moral fetishism”. Those who push this line of thought think that there is something wrong with aiming your actions specifically at rightness of action, instead of at the thick reasons (your friend’s need, your promise, etc.) that make the action right.

Now, I think there are cases where you need to aim at rightness. The cases that come to mind are ones where you need to rely on a moral expert to figure out what is the right thing to do and why. One family of cases is where you are a small child and are relying on parental authority. Another is when you are a medical professional, are dealing with a morally complex case, and are relying on the advice of an ethics committee. And probably the most common case is when you are a religious believer and you are relying on what you take to be divine revelation about what is right (a different case is where you are a believer and are relying on supposed revelation about what is commanded by God). One may take these cases to be a refutation of the objections to moral fetishism, since in these cases one may be driven to pursue rightness by genuine conscientiousness rather than by any fetishism.

However, over the last couple of days I’ve realized that there may be a way of acting in these cases in a way that gives the objectors to moral fetishism what they want—and that I actually rather like this way of acting in the cases. When an action is right, there are reasons why it is right. In straightforward cases, we can easily say what these are: it helps a friend in need, it fulfills a promise, etc. But the cases in the previous paragraph are ones where the agent cannot give these reasons. Nonetheless, these reasons exist, and the adviser is thought to have them.

We can now imagine that the agent aims at rightness not because the agent values rightness in and of itself, but because the agent values the thick but unknown reasons for which the action is right. This could be rather like the finger-wiggling and racquet-sport cases. For it could be that just as the agent doesn’t care about the finger-wiggling and follow-through as an end, and neither is a means to what the agent cares about, similarly the agent doesn’t care about the rightness, and the rightness is not a means to what the agent cares about, by aiming at rightness the agent gets what they do care about, which is acting in accordance with thick (but unknown) reasons. Furthermore, like in the finger-wiggling case, the thing one really cares about—the nerve signal or the satisfaction of thick reasons—is explanatorily prior to the thing one aims at. (The follow-through case is a bit more complicated; probably what aims at is a whole swing, of which the good hit is a part, so the good hit is a part of the whole swing and in that way prior to it.)

It may help to think about a specific moral theory. Suppose utilitarianism is correct, and one has a moral oracle that tells one what the right action is, and one acts on the deliverances of this oracle. One need not care about the rightness of these actions—but they are right if and only if they maximize utility, and it is the utility maximization one cares about.

Thus in the case where one relies on testimony to do what is right, but one cares about rightness because one cares about the values that rightness yields, one is no more and no less a rightness fetishist than the typical racquet-sport coach is a follow-through fetishist. But in any case, what is going on is not problematic.

All that said, I think caring about rightness as such to some degree is also appropriate. That’s because one should care about oneself, and acting rightly is good for one.

A flip side to omnirationality

Suppose I do an action that I know benefits Alice and harms Bob. The action may be abstractly perfectly justified, but if I didn’t take into account the harm to Bob, if I didn’t treat the harm to Bob as a reason against the action in my deliberation, then Bob would have a reason to complain about what my deliberation if he somehow found out. If I was going to perform the action, I should have performed it despite the harm to Bob, rather than just ignoring the harm to Bob. I owed it to Bob not to ignore him, even if I was in the end going to go with the benefit to Alice.

But suppose that I am perfectly virtuous, and the action is one that I owed Alice in a way that constituted a morally conclusive reason for the action. (The most plausible case will be where the action is a refraining from something absolutely wrong.) Once I see that I have morally conclusive reason for the action, it seems that taking other reasons into account is a way of toying with violating the conclusive reason, and that kind of toying is not compatible with perfect virtue.

Still, the initial intuition has some pull. Even if I have an absolute duty to do what I did for Alice, I should be doing it despite the harm to Bob, rather than just ignoring the harm to Bob. I don’t exactly know what it means not to just ignore the harm to Bob. Maybe in part it means being the sort of person who would have been open to avoiding the action if the reasons for it weren’t morally conclusive?

If I stick to the initial intuition, then we get a principle of perfect deliberation: In perfect deliberation, the deliberator does not ignore any reasons—or, perhaps, any unexcluded reasons—against the action one eventually chooses.

If this is right, then it suggests a kind of a flip side to divine omnirationality. Divine omnirationality says that when God does something, he does it for all the unexcluded reasons that favor it.

Paternalistically enhancing autonomy

Sometimes you are about to tell someone something, and they say: “I don’t want to hear about it.” Yet in some cases, the thing one wanted to tell them is actually rationally relevant to a decision they need to make, and without the information their decision will be less truly theirs.

Imagine, for instance, you have a friend who needs an organ transplant and is planning to travel to China to get the organ transplant. You start to tell them that you read that China engages (or at least recently engaged) in forced organ harvesting among executed prisoners, but they try to shut you up. Yet you keep on speaking. In doing so, you are being paternalistic, but your paternalism enables them to make a more truly informed, and hence autonomous, decision.

It sounds strange to think of paternalism as supporting autonomy, but if we think of autonomy in a Kantian way as tied to genuine rationality, rather than in a shallow desire-fulfillment way, then we will realize that a person can (e.g., through deliberate ignorance) act against their own autonomy, and there may be room for a healthy paternalism in restoring them to autonomy against their own desires. This kind of thing should be rare (except in the case of literal parents!), but it is also the kind of thing friends need to do for friends at times.

Defining deceit

A plausible definition of deceit is an action aiming to get someone to believe something one takes to be false.

But I wonder if that’s right. Here are two possible counterexamples.

1. Socratic conversation: One of my students believes some proposition p that I take to be false. Through Socratic questioning, I attempt to get the student to draw the natural conclusion q from p. Even if I take q to be false, it doesn’t seem I am deceiving my student.

2. Mitigation of error: Suppose that Alice believes Bob to be culpable for some enormity. You know that Bob never committed the enormity, but you also know it’s hopeless to try to convince Alice of this. But you think you have some hope showing Alice that instead of her evidence supporting the claim that Bob is culpable for the enormity, it only supports the claim that Bob has inculpably committed enormity. You show this to Alice, in the hope that she will come to believe Bob to have innocently committed the enormity, even though that is also false.

In both cases, one is working along with the evidence available to one’s interlocutor. It seems that deception requires one to get someone to believe something true by means of hiding or masking the truth. And here there is no such thing going on. There is nothing underhanded. In both cases, for instance, it would be quite possible for the other party to know what one is really thinking about the case. I need not hide from the student that I disagree with q and you need not hide from Alice that you don’t think Bob committed the enormity at all.

We can add an underhandedness condition to the account of deceit, but I don’t exactly know what underhandedness is.

It is well-known that defining lying is tricky. It looks like defining deceit is also tricky.

Highlighted outcome structures

In the previous two posts, I have been arguing that seeing action as pursuing ends does not capture all of the richness of the directed structure of action.

Here is my current alternative to the end-based approach. Start with the idea of a “highlighted outcome structure”, which is a partial ordering ≤ on the set O of possible outcomes together with a distinguished subset S of O with the property that if x ∈ S and y ∈ O and x ≤ y, then y ∈ S. The idea is that x ≤ y means one pursues y at least as much as x, and that one’s action is successful provided that one gets an outcome in S.

To a first approximation, directed action is action aligned along a highlighted outcome structure. But that doesn’t quite capture all the phenomena. For one might aim along a non-existent outcome structure. For instance, I may mistakenly think there is such a borogrove, and seek to know about borogroves, the more the better, but in fact the word “borogrove” is nonsense, and there is no set of outcomes corresponding to different degrees of knowledge about borogroves.

So, at least to a second approximation, direction action is action aligned along a conception of a highlighted outcome structure. Ideally, there actually is a highlighted outcome structure that fits the conception.

Note that this allows for the following interesting phenomenon: one can defer to another person with regard to a highlighted outcome structure. Thus, a Christian might pursue the structure that God has in mind for one, and do so as such.

More on directed activity without ends

In my previous post I focused on how the phenomenon of games with score undercuts the idea that activity is for an end, for some state of affairs that one aims to achieve. For no matter how good one’s score, one was aiming beyond that.

I want to consider an objection to this. Perhaps when one plays Tetris, one has an infinite number of ends:

• Get at least one point.

• Get at least two points.

• Get at least three points.

• ….

And similarly if one is running a mile, one has an infinite number of ends, namely for each positive duration t, one aims to run the miles in at most t.

My initial worry about this suggestion was that it has the implausible consequence that no matter how well one does, one has failed to achieve infinitely many ends. Thus success is always muted by failure. In the Tetris case, in fact, there will always be infinitely many failures and finitely many successes. This seemed wrong to me. But then I realized it fits with phenomenology to some degree. In these kinds of cases, when one comes to the end of the game, there may always be a slight feeling of failure amidst success—even when one breaks a world record, there is the regret that one didn’t go further, faster, better, etc. Granted, the slightness of that feeling doesn’t match the fact that in the Tetris case one has always failed at infinitely many ends and succeeded only at finitely many. But ends can be prioritized, and it could be that the infinitely many ends have diminishing value attached to them (compare the phenomenon of the “stretch goal”), so that even though one has failed at infinitely many, the finitely many one has succeeded at might outweigh them (perhaps the weights decrease exponentially).

So the game cases can, after all, be analyzed in the language of ends. But there are other cases that I think can’t. Consider the drive to learn about something. First, of course, note that our end is not omniscience—for if that were our end, then we would give up as soon as we realized it was unachievable. Now, some of the drive for learning involves known unknowns: there are propositions p where I know what p is and I aim to find out if p is true. This can be analyzed by analogy with the the infinitely-many-ends account of games with score: for each such p, I have an end to find out whether p. But often there are unknown unknowns: before I learn about the subject, I don’t even know what the concepts and questions are, so I don’t know what propositions I want to learn about. I just want to learn about the subject.

We can try to solve this by positing a score. Maybe we let my score be the number of propositions I know about the subject. And then I aim to have a score of at least one, and a score of at least two, and a score of at least three, etc. That’s trivial pursuit, not real learning, though. Perhaps, then, we have a score where we weight the propositions by their collective importance, and again I have an infinite number of ends. But in the case of the really unknown unknowns, I don’t even know how to quantify their importance, and I have no concept of the scale the score would be measured on. Unlike in the case of games, I just may not even know what the possible scores are.

So in the case of learning about a subject area, we cannot even say that we are positing an infinite number of ends. Rather, we can say that our activity has a directedness—to learn more, weighted by importance—but not an end.

Of Tetris, ends, and the beatific vision

Suppose I am playing Tetris seriously. What am I aiming at?

It’s not victory: one cannot win Tetris.

A good score, yes. But I wouldn’t stop playing after reaching a good score. So a merely good score isn’t all I am aiming at. An excellent score? But, again, even if I achieved an excellent score, I wouldn’t stop, so it’s not all I am aiming for. A world-record score? But I wouldn’t stop as soon as my score exceeded the record. An infinite score? But then I am aiming at an impossibility.

A phrase we might use is: “I am trying to get the best score I can.” But while that is how we speak, it doesn’t actually describe my aim. For consider what “the best score I can get” means. Does the “can” take into account my current skill level or not? If it does take into account my skill level, then I could count as having achieved my end despite getting a really miserable score, as long as that maxed out my skills. And that doesn’t seem right. But if it does not take into account my current skill level, but rather is the most that it could ever be possible for me, then it seems I am aiming at something unrealistic—for my current skill level falls short of what I “can” do.

What is true of Tetris is true of many other games where one’s aims align with a score. In some of these games there is such a thing as victory in addition to score. Thus, while one can time one’s runs and thus have just a score, typical running races include victory and a time, and sometimes both enter into the runner’s aims. This is not true of all games: some, like chess, only have victory (positions can be scored, but the scores are only indicative of their instrumentality for victory).

It’s worth noting that a score can be either absolute, such as time in running, and relative, such as one’s place among the finishers. In the case of place among finishers, one may be aiming for victory—first place—but one need not be. One might, for instance, make a strategic decision that one has no realistic hope for first place, and that aiming at first place will result in a poorer placement than simply aiming to “place as well as one can” (bearing in mind that this phrase is misleading, as already mentioned).

Insofar as aims align with a score, we can say that we have directed activity, but there seems to be no end, so the activity is not end-directed. We might want to say that the score is the “end”, but that would be misleading, since an end is a state you are aiming at. But typically you are not just aiming at the state of having a score—in Tetris, you get a score no matter what you do, though it might be zero. In timed fixed-distance sports, you need to finish the distance to have a time, and for some endurance races that in itself is a serious challenge, though for “reasonable” distances finishing is not much of an accomplishment.

I think what we should say is that in these activities, we have a direction, specified by increasing score, but not an end. The concept of a direction is more general than that of an end. Wherever there is an end, there is a direction defined by considering a score which is 1 if one achieves the end and 0 if one fails to do so.

So far all my examples were games. But I think the distinction between direction and end applies in much more important cases, and helps make sense of many phenomena. Consider our pursuits of goods such as health and knowledge. Past a certain age, perfect health is unachievable, and hence is not what one is aiming at. But more health is always desirable. And at any age, omniscience is out of our grasp, but more knowledge is worth having. Thus the pursuits of health and knowledge are examples of directed but not always end-directed activities. (Though, often, there are specific ends as well: the amelioration of a specific infirmity or learning the answer to a specific question.)

(Interesting question for future investigation: What happens to the maxim that the one who wills the end wills the means in the case of directed but not end-directed activity? I think it’s more complicated, because one can aim in a direction but not aim there at all costs.)

I think the above puts is in a position to make progress on a very thorny problem in Thomistic theology. The beatific vision of God is supremely good for us. But at the same time, it is a supernatural good, one that exceeds our nature. Our nature does not aim at this end, since for it to aim at this end, it would need to have the end written into itself, but its very possibility is a revealed mystery. Our desire for the beatific vision is itself a gift of God’s grace. But if our nature does not aim at the beatific vision, then it seems that the beatific vision does not fulfill us. For our nature’s aims specify what is good for us.

However, we can say this. Our nature directs us in the direction of greater knowledge and greater love of the knowable and the lovable. It does not limit that directedness to natural knowledge and love, but at the same time it does not direct us to supernatural knowledge and love as such. As far as we naturally know, it might be that natural knowledge and love is all that’s possible, and if so, we need go no further. But in fact God’s grace makes the beatific vision possible. The beatific vision is in the direction of greater knowledge and love from all our natural knowledge and love, and so it fulfills us—even though our nature has no concept of it.

Imagine that unbeknownst to me, a certain sequence of Tetris moves, which one would only be able to perform with the help of Alexey Pajitnov, yields an infinite score. Then if I played Tetris with Pajitnov’s help and I got that infinite score, I would be fulfilled in my score-directed Tetris-playing. However, it would also be correct that if I didn’t know about the possibility of the infinite score, it wasn’t an end I was pursuing. Nonetheless, it is fulfilling because it is objectively true that this score lies in the direction that I was pursuing.

Similarly, our nature, as it were, knows nothing of the beatific vision, but it directs us in a direction where in fact the beatific vision lies, should God’s grace make it possible for us.

This also gives a nice explanation of the following related puzzle about the beatific vision. When one reads what the theologians say about the beatific vision, it appears attractive to us. That attractiveness could be the result of God’s grace, but it is psychologically plausible that it would appear attractive even without grace. The idea of a loving union of understanding with an infinite good just is very attractive to humans. But how can it be naturally attractive to us when it exceeds our nature? The answer seems to me to be that we can naturally know that if the beatific vision is possible, it lies in the direction we are aimed at. But, absent divine revelation, we don’t know if it is possible. And, trivially, it’s only a potential fulfillment of our nature—i.e., a good for us to seek—if it is possible.

Does this mean that we should reject the language of “end” with respect to the beatific vision? Yes and no. It is not an end in the sense of something that our nature aims at as such. But it is an end in the sense that it is a supreme achievement in the direction at which our nature aims us. Thus it seems we can still talk about it as a supernatural end.

Divine desire ethical theories are false

On divine desire variants of divine command ethics, necessarily an action is right just in case it accords with God’s what God wants.

But it seems:

1. Necessarily, if God commands an action, the action is right.

2. Possibly, God commands an action but does not want one to do it.

Given (1) and (2), divine desire ethics is false.

I think everyone (and not just divine command theorists) should agree about (1): it is a part of the concept of God that he is authorative in such a way that whatever he commands is right.

What about (2)? Well, consider a felix culpa case where a great good would come from obedience to God and an even greater one would come from disobedience, and in the absence of a command one would have only a tiny good. Given such a situation, God could command the action. However, it seems that a perfectly good being’s desires are perfectly proportioned to the goods involved. Thus, in such a situation, God would desire that one disobey.

This is related to the important conceptual point about commands, requests and consentings that these actions can go against the characteristic desires that go with them. In the case of a human being, when there is a conflict between what a human wants and what the human commands, requests or consents to, typically it is right to go with what is said, but sometimes there is room for paternalistically going with the underlying desire (and sometimes we rightly go against both word and desire). But paternalism to God is never right.

Another argument for animals in heaven

1. All embodied humans are animals.

2. Some embodied humans will be in heaven.

3. So, there are animals in heaven.

But of course given the subject heading, you were likely interested whether there are non-human animals in heaven. That, too, can be argued for on the basis of the fact that we are animals.

4. The complete fulfillment of an animal requires it to be in an appropriate ecosystem.

5. Humans are animals.

6. An ecosystem appropriate to humans includes plants and non-human animals.

7. After the resurrection, the human beings in heaven will be completely fulfilled.

8. Thus, the human beings in heaven will be among plants and non-human animals.

Are there animals in heaven?

This argument is not a complete answer to the question, but is a start:

1. It is unlikely that all non-human earth animals will go extinct before the Second Coming.

2. It is unfitting that the Second Coming be a time where all non-human earth animals go extinct.

3. What is unfitting is unlikely.

4. So, it is likely that some non-human earth animals will survive the Second Coming.

Thomism and presentism

According to Thomism:

1. That I exist is explanatorily prior to all the other facts about me.

Obviously:

2. That yesterday I safely crossed a street is explanatorily prior to the fact that I presently exist.

3. If presentism is true, the fact that I presently exist is the same fact as that I exist.

4. There are no circles of explanatorily priority.

5. That yesterday I safely crossed a street is a fact about me.

It logically follows from these that:

6. Presentism is not true.

(For from 2 and 3, if presentism is true, that I safely crossed a street is prior to the fact that I exist. But by 1 and 5 that I exist is prior to the fact that I exist. If presentism is true, we thus have a priority circle, so by 4, we don’t have presentism.)

More steps in the open future and probability dialectics

I’ve often defended a probabilistic objection to open future views on which either future-tensed contingents are all false or are neither true nor false. If T(q) is the proposition that q is true, then:

1. P(T(q)) = P(q).

But on the open future views, the left-hand-side is zero, since it’s certain that q is not true. So the right-hand-side is zero. But then both q and its negation have zero probability, and we can’t make any predictions about the future.

An open futurist might push the following response. First, deny (1). Then insist that P(q) for a future contingent q is the objective tendency or chance towards q turning true. Thus, P(coin will be heads) is 1/2 for a fair indeterministic coin, since the there is an objective tendence of magnitude 1/2 for the coin to end up heads.

In this post I want to discuss my next step in the dialectics. I think there may be a problem with combining the objective tendency response with epistemic probabilities. Suppose that yesterday a fair coin was flipped. If the coin was heads, then tomorrow two fair indeterministic coins will be flipped, and if the coin is tails, then tomorrow one fair indeterministic coin will be flipped. Let H be the proposition that tomorrow at least one coin will be heads. If yesterday we had heads, then the objective tendency of H is 3/4. If yesterday we had tails, then the objective tendency of H is 1/2. But we need to be able to say:

2. P(H) = (1/2)(3/4) + (1/2)(1/2) = 5/8.

Now note that we are quite certain that 5/8 is not the objective tendency of H. The objective tendency of H is either 1/2 or 3/4.

So the open futurist needs a more sophisticated story. Here seems the right one. We say that P(q) is the average of the objective tendencies towards q weighted by the subjective probabilities of these tendencies. This is basically causal probability. The story requires that there be a present fact about all the objective tendencies.

On the technical side, this works. But here is a philosophical worry. If P(H) = 5/8 neither represents the objective tendency of H (which is either 1/2 or 3/4) nor one’s credence that H is true (which is zero on open-futurism), why is it that we should be making our decisions about the future in the light of P(H)?

Having multiple sufficient causes

It would be useful for discussions of causal exclusion arguments for physicalism to have a full taxonomy of the kinds of cases in which one effect E can have two sufficient causes C₁ and C₂.

Here is my tentative list of the cases:

1. Overdetermination: C₁ and C₂ overdetermine E

2. Chaining: C_i sufficiently causes C_j which sufficiently causes E (where i = 1 and j = 2 or i = 2 and j = 1)

3. Constitution: C_i sufficiently causes E by being partly constituted by C_j which sufficiently causes E (where i = 1 and j = 2 or i = 2 and j = 1)

4. Parthood: C_i sufficiently causes E by having the part C_j which sufficiently causes E (where i = 1 and j = 2 or i = 2 and j = 1).

If parthood is a special case of constitution, then (4) is a special case of (3). Moreover (2)–(4) are all species cases of:

5. Instrumentality: C_i sufficiently causes E by means of C_j sufficiently causing E (where i = 1 and j = 2 or i = 2 and j = 1).

Note that the above cases are not mutually exclusive. We can, for instance, imagine a case where we have both chaining and overdetermination. Let’s say I aim a powerful heat gun at a snowball. Just in front of the snowball is a stick of dynamite. The heat melts the snowball. But it also triggers an explosion which blows the snowball apart. Thus, we have overdetermination of the destruction of the snowball by two causes: heat and explosion. However, we also have chaining because the heat causes the explosion.

I wonder if we can come up with an argument that (1)–(4), or maybe (1) and (5), are the only options. That seems right to me.

Force-realism and simultaneous causation

If a charged particle is an electromagnetic field, the field exerts a Lorentz force F = qE + qv × B, where q and v are the charge and velocity of the particle, E is the electric field and B is the magnetic field. All of these quantities are taken at one location in spacetime. Thus, if realism about forces is correct, we have simultaneous causation: the electromagnetic field simultaneously causes the force.

Not everyone is a realist about forces, though. One might think that the electromagnetic field directly causes the subsequent change in velocity instead of causing a force which in turn causes the change in velocity.

Theism and the absolute present

Some people believe in an absolute present. An absolute present would define a privileged absolute reference frame. Suppose that there is an absolute present. Would we have any reason to think that the privileged absolute reference frame is anywhere close to our reference frame? If not, then for all we know, the things around us have an absolute geometry quite different from the one we think they have: that clock on the wall isn’t absolutely a circle, but an oval, say.

If the reason for accepting an absolute present is doing justice to common sense, then we not only need an absolute presnet, but an absolute present that defines a frame close to our frame. And that would be almost literally a version of anthropocentrism.

Of course, if we are in the image and likeness of God, the anthropocentrism may be defensible. And maybe only then.

If this is right, then the A-theory of time (which seems to require an absolute present) makes a lot more sense on theism. (Anecdotally, there is a correlation between being a theist and accepting the A-theory of time.) But on the other hand, the A-theory of time requires God’s beliefs to be changing.

Causing via a part

Assume this plausible principle:

1. If a part x of z causes w, then z causes w.

Add this controversial thesis:

2. For any x and y, there is a z that x and y are parts of.

Thesis (2) is a consequence of mereological universalism, for instance.

Finally, add this pretty plausible principle:

3. All the parts of a physical entity are physical.

Here is an interesting consequence of (1)–(3):

4. If there is any non-physical entity, any entity that has a cause has a cause that is not a physical entity.

For if w is an entity that has a cause x, and y is any non-physical entity, by (2) there is a z that x and y are both parts of. By (3), z is not physical. And by (1), z causes w.

In particular, given (1)–(3) and the obvious fact that some physical thing has a cause, we have an argument from causal closure (the thesis that no physical entity has a non-physical cause) to full-strength physicalism (the thesis that all entities are physical). Whatever we think of causal closure and physicalism, however, it does not seem that causal closure should entail full-strength physicalism.

Here is another curious line of thought. Strengthen (2) to another consequence of mereological universalism:

5. The cosmos exists, i.e., there is an entity c such that every entity is a part of c.

Then (1) and (5) yield the following holistic thesis:

6. Every item that has a cause is caused by the cosmos.

That sounds quite implausible.

We could take the above lines of thought to refute (1). But (1) sounds pretty plausible. A different move is to take the above lines of thought to refute (2) and (5), and thereby mereological universalism.

All in all, I suspect that (1) fits best with a view on which composition is quite limited.

Semantic determinacy and indeterminacy

There are arguments that our language is paradoxically indeterminate. For instance, Wittgenstein-Kripke arguments for underdetermination of rules by cases, Quine’s indeterminacy of translation arguments, or Putnam’s model-theoretic arguments.

There are also arguments that our language is paradoxically determinate. First order logic shows that there is a smallest number of grains of sand that’s still a heap.

In other words, there are cases where we want determinacy, and we find indeterminacy threatening, and cases where we want indeterminacy, and we find determinacy puzzling. I wonder if there is any relevant difference between these cases other than the fact that we have different intuitions about them.

If we are to go with our intuitions, we need to bite the bullet on, or refute, both sets of arguments, in their respective cases. But if we embrace determinacy everywhere or embrace indeterminacy everywhere, then it’s neater: we only need to bite the bullet on, or refute, one family of arguments.

I find embracing determinacy everywhere rather attractive.

Species relativity of priors

1. It would be irrational for us to assign a very high prior probability to the thesis that spiky teal fruit is a healthy food.

2. If a species evolved to naturally assign a very high prior probability to the thesis that spiky teal fruit is a healthy food, it would not be irrational for them to do this.

3. So, what prior probabilities are rational is species relative.

Reducing exact similarity

It is a commonplace that while Platonists need to posit a primitive instantiation relation for a tomato to stand in to the universal redness, trope theorists need an exact similarity relation for the tomato’s redness to stand in to another object’s redness, and hence there is no parsimony advantage to Platonism.

This may be mistaken. For the Platonist needs a degreed or comparative similarity relation, too. It seems to be a given that maroon is more similar to burgundy than blue is to pink, and blue is more similar to pink than green is to bored. But given a degreed or comparative similarity relation, there is hope for defining exact similarity in terms of it. For we can say that x and y are exactly similar provided that it is impossible for two distinct objects to be more similar than x and y are.

That said, comparative similarity is perhaps too weird and mysterious. There are clear cases, as above, but then there are cases which are hard to make sense of. Is maroon more or less similar to burgundy than middle C is to middle B? Is green more or less similar to bored than loud is to quiet?

Do particles have a self-concept?

Of course not.

But consider this. A negatively charged substance has the power to attract other substances to itslf. Its causal power thus seems to have a centeredness, a de se character. The substance’s power somehow distinguishes between the substance itself and other things.

Put this way, Leibniz's (proto?)panpsychism doesn’t seem that far a departure from a more sedate commitment to causal powers.

Morality and the gods

In the Meno, we get a solution to the puzzle of why it is that virtue does not seem, as an empirical matter of fact, to be teachable. The solution is that instead of involving knowledge, virtue involves true belief, and true belief is not teachable in the way knowledge is.

The distinction between knowledge and true belief seems to be that knowledge is true opinion made firm by explanatory account (aitias logismoi, 98a).

This may seem to the modern philosophical reader to confuse explanation and justification. It is justification, not explanation, that is needed for knowledge. One can know that sunflowers turn to the sun without anyone knowing why or how they do so. But what Plato seems to be after here is not merely justified true belief, but something like the scientia of the Aristotelians, an explanatorily structured understanding.

But not every area seems like the case of sunflowers. There would be something very odd in a tribe knowing Fermat’s Last Theorem to be true, but without anybody in the tribe, or anybody in contact with the tribe, having anything like an explanation or proof. Mathematical knowledge of non-axiomatic claims typically involves something explanation-like: a derivation from first principles. We can, of course, rely on an expert, but eventually we must come to something proof-like.

I think ethics is in a way similar. There is something very odd about having justified true belief—knowledge in the modern sense—of ethical truths but not knowing why they are true. Yet it seems humans are often in this position. They know the ethical truths but not why they are true. Yet they have correct, and maybe even justified, moral judgments about many things. What explains this?

Socrates’ answer in the Meno is that it is the gods. The gods instill true moral opinion in people (especially the poets).

This is not a bad answer.

Saving a Newtonian intuition

Here is a Newtonian intuition:

1. Space and time themselves are unaffected by the activities of spatiotemporal beings.

General Relativity seems to upend (1). If I move my hand, that changes the geometry of spacetime in the vicinity of my hand, since gravity is explained by the geometry of spacetime and my hand has gravity.

It’s occurred to me this morning that a branching spacetime framework can restore the Newtonian intuition of the invariance of space. Suppose we think of ourselves as inhabiting a branching spacetime, with the laws of nature being such as to require all the substances to travel together (cf. the traveling forms interpretation of quantum mechanics). Then we can take this branching spacetime to have a fixed geometry, but when I move my hand, I bring it about that we all (i.e., all spatiotemporal substances now existing) move up to a branch with one geometry rather than up to a branch with a different geometry.

On this picture, the branching spacetime we inhabit is largely empty, but one lonely red line is filled with substances. Instead of us shaping spacetime, we travel in it.

I don’t know if (1) is worth saving, though.

From a determinable-determinate model of location to a privileged spacetime foliation

Here’s a three-level determinable-determinate model of spacetime that seems somewhat attractive to me, particularly in a multiverse context. The levels are:

1. Spatiotemporality

2. Being in a specific spacetime manifold

3. Specific location in a specific spacetime manifold.

Here, levels 2 and 3 are each a determinate of the level above it.

Thus, Alice has the property of being at spatiotemporal location x, which is a determinate of the determinable of being in manifold M, and being in manifold M is a determinate of the determinable of spatiotemporality.

This story yields a simple account of the universemate relation: objects x and y are universemates provided that they have the same Level 2 location. And spatiotemporal structure—say, lightcone and proper distance—is somehow grounded in the internal structure of the Level 2 location determinable. (The “somehow” flags that there be dragons here.)

The theory has some problematic, but very interesting, consequences. First, massive nonlocality, both in space and in time, both backwards and forwards. What spacetime manifold the past dinosaurs of Earth and the present denizens of the Andromeda Galaxy inhabit is partly up to us now. If I raise my right hand, that affects the curvature of spacetime in my vicinity, and hence affects which manifold we all have always been inhabiting.

Second, it is not possible to have a multiverse with two universes that have the same spacetime structure, say, two classical Newtonian ones, or two Minkowskian ones.

To me, the most counterintuitive of the above consequences is the backwards temporal nonlocality: that by raising my hand, I affect the level 2 locational properties, and hence the level 3 ones as well, of the dinosaurs. The dinosaurs would literally have been elsewhere had I not raised my hand!

What’s worse, we get a loop in the partial causal explanation relation. The movement of my hand affects which manifold we all live in. But which manifold we all live in affects the movement of the objects in the manifold—including that of my hand.

The only way I can think of avoiding such backwards causation on something like the above model is to shift to some model that privileges a foliation into spacelike hypersurfaces, and then has something like this structure:

1. Spatiotemporality

2. Being in a specific branching spacetime

3. Being in a specific spacelike hypersurface inside one branch

4. Specific location within the specific spacelike hypersurface.

We also need some way to handle persistence over time. Perhaps we can suppose that the fundamentally located objects are slices or slice-like accidents.

I wonder if one can separate the above line of thought from the admittedly wacky determinate-determinable model and make it into a general metaphysical argument for a privileged foliation.

Achievement in a quantum world

Suppose Alice gives Bob a gift of five lottery tickets, and Bob buys himelf a sixth one. Bob then wins the lottery. Intuitively, if one of the tickets that Alice bought for Bob wins, then Bob’s win is Alice’s achievement, but if the winning ticket is not one of the ones that Alice bought for Bob, then Bob’s win is not Alice’s achievement.

But now suppose that there is no fact of the matter as to which ticket won, but only that Bob won. For instance, maybe the way the game works is that there is a giant roulette wheel. You hand in your tickets, and then an equal number of depressions on the wheel gets your name. If the ball ends in a depression with your name, you win. But they don’t write your name down on the depressions ticket-by-ticket. Instead, they count up how many tickets you hand them, and then write your name down on the same number of depressions.

In this case, it seems that Bob’s win isn’t Alice’s achievement, because there is no fact of the matter that it was one of Alice’s tickets that got Bob his win. Nor does this depend on the probabilities. Even if Alice gave Bob a thousand tickets, and Bob contributed only one it seems that Bob’s win isn’t Alice’s achievement.

Yet in a world run on quantum mechanics, it seems that our agential connection to the external world is like Alice’s to Bob’s win. All we can do is tweak the probabilities, perhaps overwhelmingly so, but there is no fact of the matter about the outcome being truly ours. So it seems that nothing is ever our achievement.

That is an unacceptable consequence, I think.

I think there are two possible ways out. One is to shift our interpretation of “achievement” and say that Bob’s win is Alice’s achievement in the original case even when it was the ticket that Bob bought for himself that won. Achievement is just sufficient increase of probability followed by the occurrence of the thus probabilified event.

The second is heavy duty metaphysics. Perhaps our causal activity marks the world in such a way that there is always a trace of what happened due to what. Events come marked with their actual causal history. Sometimes, but not always, that causal history specifies what was actually the cause. Perhaps I turn a quantum probability dial from 0.01 to 0.40, and you turn it from 0.40 to 0.79, and then the event happens, and the event comes metaphysically marked with its cause. Or perhaps when I turn the quantum probability dial and you turn it, I embue it with some of my teleology and when you turn it, you embue it with some of yours, and there is a fact of the matter as to whether a further on down effect comes from your teleology or mine.

I find the metaphysical answer hard to believe, but I find the probabilistic one conceptually problematic.

Continuity, scoring rules and domination

Pettigrew claimed, and Nielsen and I independently proved (my proof is here) that any strictly proper scoring rule on a finite space that is continuous on the probabilities has the domination property that any non-probability is strictly dominated in score by some probability.

An interesting question is how far one can weaken the continuity assumption. While I gave necessary and sufficient conditions, those conditions are rather complicated and hard to work with. So here is an interesting question: Is it sufficient for the domination property that the scoring rule be continuous at all the regular probabilities, those that assign non-zero values to every point, and finite?

I recently posted, and fairly quickly took down, a mistaken argument for a negative answer. I now have a proof of a positive answer. It took me way too long to get that positive answer, when in fact it was just a simple geometric argument (see Lemma 1).

Slightly more generally, what’s sufficient is that the scoring rule be continuous at all the regular probabilities as well as at every point where the score is infinite.

Two tweaks for Internet Arcade games on archive.org

The Internet Arcade has a great collection of browser-playable arcade games. But I ran into two problems: (1) the aspect ratio goes bad in full-screen mode (no black bars on the sides of 4:3 content), and (b) mouse support is turned off, which makes trackball games like Marble Madness less fun and spinner games like Blasteroids not great for those of us who have a USB spinner (I just made one myself using a magnetic rotary sensor). 

It took me a while to find a solution, but finally I managed to inject options into the underlying MAME emulator and thereby fix both problems:

First, go to a game, but don't click the start button.

Then put the following into the URL bar and press enter: 

javascript:MAMELoader.extraArgs=(z)=>({extra_mame_args:['-mouse','-keepaspect']});AJS.emulate()

(Note that Chrome will not let you past the "javascript:" prefix into the URL bar. You can copy and paste the rest, but you have to type "javascript:" manually.) The game will start, and the mouse will be enabled. You can add other MAME options if you like.

You can also turn this into a bookmarklet. Indeed, you can just drag this to your bookmark bar: Emularity Mouse .

There is still a minor aspect ratio problem. If you go to fullscreen and go back to windowed view, the aspect ratio will be bad in windowed mode.

Curmudgeonly griping

One of the standard gripes about modern manufacturing is how many items break down because the manufacturer saved a very small fraction of the price, sometimes only a few cents. I find myself frequently resoldering mice and headphones, presumably because the wires were too thin, but there at least there is a functionality benefit from thin wires.

The most recent is our GE dryer where the timer knob always felt flimsy, and finally the plastic holding the timer shaft cracked. The underside revealed thin-walled plastic holding the timer shaft, reinforced with some more chambers made of thin-walled plastic. Perhaps over-reacting, I ended up designing a very chunky 3D-printed one.

I suppose there is an environmental benefit from using less plastic, but it needs to be offset against the environmental cost of repair and replacement. Adding ten grams of plastic would have easily made the knob way, way stronger, and that's about 0.01 gallons of crude oil, which is probably an order of magnitude less than the crude oil someone might use to drive to the store for a replacement (or a repair person being called in; in our case, it wasn't obvious that the knob was the problem; I suspected the timer at first, and disassembled it, before finally realizing that the knob had an internal crack that made it impossible for it to exert the needed torque).

Strictly proper scoring rules in an infinite context

This is mostly just a note to self.

In a recent paper, I prove that there is no strictly proper score s on the countably additive probabilities on the product space 2^κ where we require s(p) to be measurable for every probability p and where κ is a cardinal such that 2^κ > κ^ω (e.g., the continuum). But here I am taking strictly proper scores to be real- or extended-real valued. What if we relax this requirement? The answer is negative.

Theorem: Assume Choice. If 2^κ > κ^ω and V is a topological space with the T₁ property and ≤ is a total pre-order on V. Let s be a function s from the extreme probabilities on Ω = 2^κ, with the product σ-algebra, concentrated on points to the set of measurable functions from Ω to V. Then there exist extreme probabilities r and q concentrated at distinct points α and β respectively such that s(r)(α) ≤ s(q)(α).

A space is T₁ iff singletons are closed.
Extreme probabilities only have 0 and 1 values. An extreme probability p is concentrated at α provided that p(A) = 1 if α ∈ A and p(A) = 0 otherwise. (We can identify extreme probabilities with the points at which they are concentrated, as long as the σ-algebra separates points.) Note that in the product σ-lagebra on 2^κ singletons are not measurable.

Corollary: In the setting of the Theorem, if E_p is a V-valued prevision from V-valued measurable functions to V such that (a) E_p(c⋅1_A) = cp(A) always, (b) if f and g are equal outside of a set with p-measure zero, then E_pf = E_pg, and (c) E_ps(q) is always defined for extreme singleton-concentrated p and q, then the strict propriety inequality E_rs(r) > E_rs(q) fails for some distinct extreme singleton-concentrated p and q.

Proof of Corollary: Suppose p is concentrated at α and E_pf is defined. Let Z = {ω : f(ω) = f(α)}. Then p(Z) = 1 and p(Z^c) = 0. Hence f is p-almost surely equal to f(α) ⋅ 1_Z. Thus, E_pf = f(α). Thus if r and q are as in the Theorem, E_rs(r) = s(r)(α) and E_rs(q) = s(q)(α), and our result follows from the Theorem.

Note: Towards the end of my paper, I suggested that the unavailability of proper scores in certain contexts is due to the limited size of the set of values—normally assumed to be real. But the above shows that sometimes it’s due to measurability constraints instead, as we still won’t have proper scores even if we let the values be some gigantic collection of surreals.

Proof of Theorem: Given an extreme probability p and z ∈ V, the inverse image of z under s(p), namely L_p, z = {ω : s(p) = z}, is measurable. A measurable set A depends only on a countable number of factors of 2^κ: i.e., there is a countable set C ⊆ κ such that if ω, ω′ ∈ κ agree on C, and one is in A, so is the other. Let C_p ⊂ κ be a countable set such that L_{p, s(p)(ω)} depends only on C_p, where ω is the point p is concentrated on (to get all the C_p we need the Axiom of Choice). The set of all the countable subsets of κ has cardinality at most κ^ω, and hence so does the set of all the C_p. Let Z_p = {q : C_q = C_p}. The union of the Z_p is all the extreme point-concentrated probabilities and hence has cardinality 2^κ. There are at most κ^ω different Z_p. Thus for some p, the set Z_p has cardinality bigger than κ^ω (here we used the assmption that 2^ω > κ^ω).

There are q and r in Z_p concentrated at α and β, respectively, such that α∣_Cp = β∣_Cp (i.e., α and β agree on C_p). For the function (⋅)∣_Cp on the concentration points of the probabilities in Z_p has an image of cardinality at most 2^ω ≤ κ^ω but Z_p has cardinality bigger than κ^ω. Since α∣_Cp = β∣_Cp, and L_{q, s(q)(α)} and L_{r, s(r)(β)} both depend only on the factors in C_p, it follows that s(q)(β) = s(q)(α) and s(r)(α) = s(r)(β). Now either s(q)(β) ≥ s(r)(α) or s(q)(β) ≤ s(r)(α) by totality. Swapping (q,α) with (r,β) if necessary, assume s(q)(α) = s(q)(β) ≤ s(r)(α).

Fundamentality and anthropocentrism

Say an object is grue if it is observed before the year 2100 and green, or it is blue but not observed before 2100. Then it is reasonable to do induction with “green” but not with “grue”: our observations of emerald color fit equally well with the hypotheses that emeralds are green and that they are grue, but it is the green hypothesis that is reasonable.

A plausible story about the relevant difference between “green” and “grue” is that “green” is significantly closer to being a “perfectly natural” or “fundamental” property than “grue” is. If we try to define “green” and “grue” in fundamental scientific vocabulary, the definition of “grue” will be about twice as long. Thus, “green” is projectible but “grue” is not, to use Goodman’s term.

But this story has an interesting problem. Say that an object is pogatively charged if it is observed before Planck time 2ⁿ and positively charged or it negatively charged but not observed before Planck time 2ⁿ. By the “Planck time”, I mean the proper time from the beginning of the universe measured in Planck times, and I stipulate that n is the smallest integer such that 2ⁿ is in our future. Now, while “pogatively charged” is further from the fundamental than “positively charged”, nonetheless “pogatively charged” seems much more fundamental than “green”. Just think how hard it is to define “green” in fundamental terms: objects are green provided that their emissive/refractive/reflective spectral profile peaks in a particular way in a particular part of the visible spectrum. Defining the “particular way” and “particular part” will be complex—it will make make reference to details tied to our visual sensitivities—and defining “emissive/refractive/reflective”, and handling the complex interplay of these, is tough.

One move would be to draw a strong anti-reductionist conclusion from this: “green” is not to be defined in terms of spectral profiles, but is about as close to fundamentality as “positively charged”.

Another move would be to say that projectibility is not about distance to fundamentality, but is legitimately anthropocentric. I think kind of anthropocentrism is only plausible on the hypothesis that the world is made for us humans.

Eliminating contingent relations

Here’s a Pythagorean way to eliminate contingent relations from a theory. Let’s say that we want to eliminate the relation of love from a theory of persons. We suppose instead that each person has two fundamental contingent numerical determinables: personal number and love number, both of which are positive integers, with each individual’s personal number being prime and an essential property of theirs. Then we say that x loves y iff x’s love number is divisible by y’s personal number. For instance suppose we have three people: Alice, Bob and Carl, and Alice loves herself and Carl, Bob loves himself and Alice, and Carl loves no one. This can be made true by the following setup:

	Alice	Bob	Carl
Personal number	2	3	5
Love number	10	6	1

While of course my illustration of this in terms of love is unserious, and only really works assuming love-finitism (each person can only love finitely many people), the point generalizes: we can replace non-mathematical relations with mathematizable determinables and mathematical relations. For instance, spatial relations can be analyzed by supposing that objects have a location determinable whose possible values are regions of a mathematical manifold.

This requires two kinds of non-contingent relations: mathematical relations between the values and the determinable–determinate relation. One may worry that these are just as puzzling as the contingent ones. I don't know. I've always found contingent relations really puzzling.

Rethinking priors

Suppose I learned that all my original priors were consistent and regular but produced by an evil demon bent upon misleading me.

The subjective Bayesian answer is that since consistent and regular original priors are not subject to rational evaluation, I do not need to engage in any radical uprooting of my thinking. All I need to do is update on this new and interesting fact about my origins. I would probably become more sceptical, but all within the confines of my original priors, which presumably include such things as the conditional probability that I have a body given that I seem to have a body but there is an evil demon bent upon misleading me.

This answer seems wrong. So much the worse for subjective Bayesianism. A radical uprooting would be needed. It would be time to sit back, put aside preconceptions, and engage in some fallibilist version of the Cartesian project of radical rethinking. That project might be doomed, but it would be my only hope.

Now, what if instead of the evil demon, I learned of a random process independent of truth as the ultimate origin of my priors. I think the same thing would be true. It would be a time to be brave and uproot it all.

I think something similar is true piece by piece, too. I have a strong moral intuition that consequentialism is false. But suppose that I learned that when I was a baby, a mad scientist captured me and flipped a coin with the plan that on heads a high prior in anti-consequentialism would be induced and on tails it would be a high prior in consequentialism instead. I would have to rethink consequentialism. I couldn’t just stick with the priors.

Socrates and thinking for yourself

There is a popular picture of Socrates as someone inviting us to think for ourselves. I was just re-reading the Euthyphro, and realizing that the popular picture is severely incomplete.

Recall the setting. Euthyphro is prosecuting a murder case against his father. The case is fraught with complexity and which a typical Greek would think should not be brought for multiple reasons, the main one being that the accused is the prosecutor’s father and we have very strong duties towards parents, and a secondary one being that the killing was unintentional and by neglect. Socrates then says:

most men would not know how they could do this and be right. It is not the part of anyone to do this, but of one who is far advanced in wisdom. (4b)

We learn in the rest of the dialogue that Euthyphro is pompous, full of himself, needs simple distinctions to be explained, and, to understate the point, is far from “advanced in wisdom”. And he thinks for himself, doing that which the ordinary Greek thinks to be a quite bad idea.

The message we get seems to be that you should abide by cultural norms, unless you are “far advanced in wisdom”. And when we add the critiques of cultural elites and ordinary competent craftsmen from the Apology, we see that almost no one is “advanced in wisdom”. The consequence is that we should not depart significantly from cultural norms.

This reading fits well with the general message we get about the poets: they don’t know how to live well, but they have some kind of a connection with the gods, so presumably we should live by their message. Perhaps there is an exception for those sufficiently wise to figure things out for themselves, but those are extremely rare, while those who think themselves wise are extremely common. There is a great risk in significantly departing from the cultural norms enshrined in the poets—for one is much more likely to be one of those who think themselves wise than one of those who are genuinely wise.

I am not endorsing this kind of complacency. For one, those of us who are religious have two rich sets of cultural norms to draw on, a secular set and a religious one, and in our present Western setting the two tend to have sufficient disagreement that complacency is not possible—one must make a choice in many cases. And then there is grace.

From strict anti-anti-Bayesianism to strict propriety

In my previous post, I showed that a continuous anti-anti-Bayesian accuracy scoring rule on probabilities defined on a sub-algebra of events satisfying the technical assumption that the full algebra contains an event logically independent of the sub-algebra is proper. However, I couldn’t figure out how to prove strict propriety given strict anti-anti-Bayesianism. I still don’t, but I can get closer.

First, a definition. A scoring rule on probabilities on the sub-algebra H is strictly anti-anti-Bayesian provided that one expects it to penalize non-trivial binary anti-Bayesian updates. I.e., if A is an event with prior probability p neither zero nor one, and Bayesian conditionalization on A (or, equivalently, on A^c) modifies the probability of some member of H, then the p-expected score of finding out whether A or A^c holds and conditionalizing on that is strictly better than if one adopted the procedure of conditionalizing on the complement of the actually obtaining event.

Suppose we have continuity, the technical assumption and anti-anti-Bayesianism. My previous post shows that the scoring rule is proper. I can now show that it is strictly proper if we strengthen anti-anti-Bayesianism to strict anti-anti-Bayesianism and add the technical assumption that the scoring rule satisfies the finiteness condition that E_ps(p) is finite for any probability p on H. Since we’re working with accuracy scoring rules and these take values in [−∞,M] for finite M, the only way to violate the finiteness condition is to have E_ps(p) =  − ∞, which would mean that s is very pessimistic about p: by p’s own lights, the expected score of p is infinitely bad. The finiteness condition thus rules out such maximal pessimism.

Here is a sketch of the proof. Suppose we do not have strict propriety. Then there will be two distinct probabilities p and q such that E_ps(p) ≤ E_ps(q). By propriety, the inequality must be an equality. By Proposition 9 of a recent paper of mine, it follows that s(p) = s(q) everywhere (this is where the finiteness condition is used). Now let r = (p+q)/2. Using the trick from the Appendix here, we can find a probability p′ on the full algebra and an event Z such that r is the restriction of p′ to H, p is the restriction of the Bayesian conditionalization of p′ on Z to H, and q is the restriction of the Bayesian conditionalization of q on Z^c to H. Then the scores of p and q will be the same, and hence the scores of Bayesian and anti-Bayesian conditionalization on finding out whether Z or Z^c is actual are guaranteed to be the same, and this violates strict anti-anti-Bayesianism.

One might hope that this will help those who are trying to construct accuracy arguments for probabilism—the doctrine that credences should be probabilities. The hitch in those arguments is establishing strict propriety. However, I doubt that what I have helps. First, I am working in a sub-algebra setting. Second, and more importantly, I am working in a context where scoring rules are defined only for probabilities, and so the strict propriety inequality I get is only for scores of pairs of probabilities, while the accuracy arguments require strict propriety for pairs of credences exactly one of which is not a probability.

From anti-anti-Bayesianism to propriety

Let’s work in the setting of my previous post, including technical assumption (3), and also assume Ω is finite and that our scoring rules are all continuous.

Say that an anti-Bayesian update is when you take a probability p, receive evidence A, and make your new credence be p(⋅|A^c), i.e., you conditionalize on the complement of the evidence. Anti-Bayesian update is really stupid, and you shouldn’t get rewarded for it, even if all you care about are events other than A and A^c.

Say that an H-scoring rule s is anti-anti-Bayesian providing that the expected score of a policy of anti-Bayesian update on an event A whose prior probability is neither zero nor one is never better than the expected score of a policy of Bayesian update.

I claim that given continuity, anti-anti-Bayesianism implies that the scoring rule is proper.

First, note that by continuity, if it’s proper at all the regular probabilities (ones that do not assign 0 or 1 to any non-empty set) on H, then it’s proper (I am assuming we handle infinities like in this paper, and use Lemma 1 there).

So all we need to do is show that it’s proper at all the regular probabilities on H. Let p be a regular probability, and contrary to propriety suppose that E_ps(p) < E_ps(q) for another probability q. For t ≥ 0, let p_t be such that tq + (1−t)p_t = p, i.e., let p_t = (p−tq)/(1−t). Since p is regular, for t sufficiently small, p_t will be a probability (all we need is that it be non-negative). Using the trick from the Appendix of the previous post with q in place of p₁ and p_t in place of p₂, we can set up a situation where the Bayesian update will have expected score:

• tE_qs(q) + (1−t)E_pts(p_t)

and the anti-Bayesian update will have the expected score:

• tE_qs(p_t) + (1−t)E_pts(q).

Given anti-anti-Bayesianism, we must have

• tE_qs(p_t) + (1−t)E_pts(q) ≤ tE_qs(q) + (1−t)E_pts(p_t).

Letting t → 0 and using continuity, we get:

• E_p0s(q) ≤ E_p0s(p₀).

But p₀ = p. So we have propriety.

Open-mindedness and propriety

Suppose we have a probability space Ω with algebra F of events, and a distinguished subalgebra H of events on Ω. My interest here is in accuracy H-scoring rules, which take a (finitely-additive) probability assignment p on H and assigns to it an H-measurable score function s(p) on Ω, with values in [−∞,M] for some finite M, subject to the constraint that s(p) is H-measurable. I will take the score of a probability assignment to represent the epistemic utility or accuracy of p.

For a probability p on F, I will take the score of p to be the score of the restriction of p to H. (Note that any finitely-additive probability on H extends to a finitely-additive probability on F by Hahn-Banach theorem, assuming Choice.)

The scoring rule s is proper provided that E_ps(q) ≤ E_ps(p) for all p and q, and strictly so if the inequality is strict whenever p ≠ q. Propriety says that one never expects a different probability from one’s own to have a better score (if one did, wouldn’t one have switched to it?).

Say that the scoring rule s is open-minded provided that for any probability p on F and any finite partition V of Ω into events in F with non-zero p-probability, the p-expected score of finding out where in V we are and conditionalizing on that is at least as big as the current p-expected score. If the scoring rule is open-minded, then a Bayesian conditionalizer is never precluded from accepting free information. Say that the scoring rule s is strictly open-minded provided that the p-expected score increases of finding out where in V we are and conditionalizing increases whenever there is at least one event E in V such that p(⋅|E) differs from p on H and p(E) > 0.

Given a scoring rule s, let the expected score function G_s on the probabilities on H be defined by G_s(p) = E_ps(p), with the same extension to probabilities on F as scores had.

It is well-known that:

1. The (strict) propriety of s entails the (strict) convexity of G_s.

It is easy to see that:

2. The (strict) convexity of G_s implies the (strict) open-mindedness of s.

Neither implication can be reversed. To see this, consider the single-proposition case, where Ω has two points, say 0 and 1, and H and F are the powerset of Ω, and we are interested in the proposition that one of these point, say 1, is the actual truth. The scoring rule s is then equivalent to a pair of functions T and F on [0,1] where T(x) = s(p_x)(1) and F(x) = s(p_x)(0) where p_x is the probability that assigns x to the point 1. Then G_s corresponds to the function xT(x) + (1−x)F(x), and each is convex if and only if the other is.

To see that the non-strict version of (1) cannot be reversed, suppose (T,F) is a non-trivial proper scoring rule with the limit of F(x)/x as x goes to 0 finite. Now form a new scoring rule by letting T * (x) = T(x) + (1−x)F(x)/x. Consider the scoring rule (T*,0). The corresponding function xT * (x) is going to be convex, but (T*,0) isn’t going to be proper unless T* is constant, which isn’t going to be true in general. The strict version is similar.

To see that (2) cannot be reversed, note that the only non-trivial partition is {{0}, {1}}. If our current probability for 1 is x, the expected score upon learning where we are is xT(1) + (1−x)F(0). Strict open-mindedness thus requires precisely that xT(x) + (1−x)F(x) < xT(1) + (1−x)F(0) whenever x is neither 0 nor 1. It is clear that this is not enough for convexity—we can have wild oscillations of T and F on (0,1) as long as T(1) and F(1) are large enough.

Nonetheless, (2) can be reversed (both in the strict and non-strict versions) on the following technical assumption:

3. There is an event Z in F such that Z ∩ A is a non-empty proper subset of A for every non-empty member of H.

This technical assumption basically says that there is a non-trivial event that is logically independent of everything in H. In real life, the technical assumption is always satisfied, because there will always be something independent of the algebra H of events we are evaluating probability assignments to (e.g., in many cases Z can be the event that the next coin toss by the investigator’s niece will be heads). I will prove that (2) can be reversed in the Appendix.

It is easy to see that adding (3) to our assumptions doesn’t help reverse (1).

Since open-mindedness is pretty plausible to people of a Bayesian persuasion, this means that convexity of G_s can be motivated independently of propriety. Perhaps instead of focusing on propriety of s as much as the literature has done, we should focus on the convexity of G_s?

Let’s think about this suggestion. One of the most important uses of scoring rules could be to evaluate the expected value of an experiment prior to doing the experiment, and hence decide which experiment we should do. If we think of an experiment as a finite partition V of the probability space with each cell having non-zero probability by one’s current lights p, then the expected value of the experiment is:

4. ∑_A ∈ Vp(A)E_pAs(p_A) = ∑_A ∈ Vp(A)G_s(p_A),

where p_A is the result of conditionalizing p on A. In other words, to evaluate the expected values of experiments, all we care about is G_s, not s itself, and so the convexity of G_s is a very natural condition: we are never oligated to refuse to know the results of free experiments.

However, at least in the case where Ω is finite, it is known that any (strictly) convex function (maybe subject to some growth conditions?) is equal to G_u for a some (strictly) proper scoring rule u. So we don’t really gain much generality by moving from propriety of s to convexity of G_s. Indeed, the above observations show that for finite Ω, a (strictly) open-minded way of evaluating the expected epistemic values of experiments in a setting rich enough to satisfy (3) is always generatable by a (strictly) proper scoring rule.

In other words, if we have a scoring rule that is open-minded but not proper, we can find a proper scoring rule that generates the same prospective evaluations of the value of experiments (assuming no special growth conditions are needed).

Appendix: We now prove the converse of (2) assuming (3).

Assume open-mindedness. Let p₁ and p₂ be two distinct probabilities on H and let t ∈ (0,1). We must show that if p = tp₁ + (1−t)p₂, then

5. G_s(p) ≤ tG_s(p₁) + (1−t)G_s(p₂)

with the inequality strict if the open-mindedness is strict. Let Z be as in (3). Define

6. p′(A∩Z) = tp₁(A)

7. p′(A∩Z^c) = (1−t)p₂(A)

8. p′(A) = p(A)

for any A ∈ H. Then p′ is a probability on the algebra generated by H and Z extending p. Extend it to a probability on F by Hahn-Banach. By open-mindedness:

9. G_s(p′) ≤ p′(Z)E_p′Zs(p′_Z) + p′(Z^c)E_p′Z^cs(p′_Z^c).

But p′(Z) = p(Ω∩Z) = t and p′(Z^c) = 1 − t. Moreover, p′_Z = p₁ on H and p′_Z^c = p₂ on H. Since H-scores don’t care what the probabilities are doing outside of H, we have s(p′_Z) = s(p₁) and s(p′_Z^c) = s(p₂) and G_s(p′) = G_s(p). Moreover our scores are H-measurable, so E_p′Zs(p₁) = E_p1s(p₁) and E_p′Z^cs(p₂) = E_p2s(p₂). Thus (9) becomes:

10. G_s(p) ≤ tG_s(p₁) + (1−t)G_s(p₂).

Hence we have convexity. And given strict open-mindedness, the inequality will be strict, and we get strict convexity.

Scoring rules and publication thresholds

One of the most problematic aspects of some science practice is a cut-off, say at 95%, for the evidence-based confidence needed for publication.

I just realized, with the help of a mention of p-based biases and improper scoring rules somewhere on the web, that what is going on here is precisely a problem of a reward structure that does not result in a proper scoring rule, where a proper scoring rule is one where your current probability assignment is guaranteed to have an optimal expected score according to that very probability assignment. Given an improper scoring rule, one has a perverse incentive to change one’s probabilities without evidence.

To a first approximation, the problem is really, really bad. Insofar as publication is the relevant reward, it is a reward independent of the truth of the matter! In other words, the scoring rule has a reward for gaining probability 0.95 (say) in the hypothesis, regardless of whether the hypothesis is true or false.

Fortunately, it’s not quite so bad. Publication is the short-term reward. But there are long-term rewards and punishments. If one publishes, and later it turns out that one was right, one may get significant social recognition as the discoverer of the truth of the hypothesis. And if one publishes, and later it turns out one is wrong, one gets some negative reputation.

However, notice this. Fame for having been right is basically independent of the exact probability of the hypothesis one established in the original paper. As long as the probability was sufficient for publication, one is rewarded for fame. Thus if it turns out that one was right, one’s long-term reward is fame if and only if one’s probability met the threshold for publication and one was right. And one’s penalty is some negative reputation if and only if one’s probability met the threshold for publication and yet one was wrong. But note that scientists are actually extremely forgiving of people putting forward evidenced hypotheses that turn out to be false. Unlike in history, where some people live on in infamy, scientists who turn out to be wrong do not suffer infamy. At worst, some condescension. And it barely varies with your level of confidence.

The long-term reward structure is approximately this:

• If your probability is insufficient for publication, nothing.

• If your probability meets the threshold for publication and you’re right, big positive.

• If your probability meets the threshold for publication and you’re wrong, at worst small negative.

This is not a proper scoring rule. It’s not even close. To make it into a proper scoring rule, the penalty for being wrong at the threshold would need to be way higher than the reward for being right. Specifically, if the threshold is p (say 0.95), then the ratio of reward to penalty needs to be (1−p) : p. If p = 0.95, the reward to penalty ratio would need to be 1:19. If p = 0.99, it would need to be a staggering 1:99, and if p = 0.9, it would need to be a still large 1:9. We are very, very far from that. And when we add the truth-independent reward for publication, things become even worse.

We can see that something is problematic if we think about cases like this. Suppose your current level of confidence is just slightly above the threshold, and a graduate student in your lab proposes to do one last experiment in her spare time, using equipment and supplies that would otherwise go to waste. Given the reward structure, it will likely make sense for you to refuse this free offer of additional information. If the experiment favors your hypothesis, you get nothing out of it—you could have published without it, and you’d still have the same longer term rewards available. But if the experiment disfavors your hypothesis, it will likely make your paper unpublishable (since you were at the threshold), but since it’s just one experiment, it is unlikely to put you into the position of yet being able to publish a paper against the hypothesis. At best it loses you the risk of the small negative reputation for having been wrong, and since that’s a small penalty, and an unlikely one (since most likely your hypothesis is true by your data), so that’s not worth it. In other words, the the structure rewards you for ignoring free information.

How can we fix this? We simply cannot realistically fix it if we have a high probability threshold for publication. The only way to fix it while keeping a high probability threshold would be by having a ridiculously high penalty for being wrong. But we should’t do stuff like sentencing scientists to jail for being wrong (which has happened). Increasing the probability threshold for publication would only require the penalty for being wrong to be increased. Decreasing probability thresholds for publication helps a little. But as long as there is a larger reputational benefit from getting things right than the reputational harm from getting things wrong, we are going to have perverse incentives from a probability threshold for publication bigger than 1/2, no matter where that threshold lies. (This follows from Fact 2 in my recent post, together with the observation that Schervish’s characterization of scoring rules shows implies that any reward function corresponds to a unique up to additive constant penalty function.)

What’s the solution? Maybe it’s this: reward people for publishing lots of data, rather than for the data showing anything interestingly, and do so sufficiently that it’s always worth publishing more data?