Decision theory and compatibilism

Here's a decision theoretic picture of how to make the decision between A and B. First, gain as much knowledge K as is reasonably possible about the laws and present conditions in the universe. The more information, the better our decision is likely to be (cf. Good's Theorem). Then calculate the conditional expected utility of the future given A with K, and do the same for B. Then do the action where the conditional expected utility is higher.

Let U(A,K) and U(B,K) be the two conditional expected utilities. (Note: I mean this to be neutral between causal and epistemic decision theories, but if I have to commit to one, it'll be causal.) We want to make our decision on U(A,K) and U(B,K) for the most inclusive K we can.

Now imagine that we could ask an angel for any piece of information I about the present and the laws (e.g., by asking "How many hairs do I have on my head?"), and then form a new set of information K₂ including I on which to calculate U(A,K₂) and U(B,K₂). Then we should ask for as much information as we can. But now here is a problem: if determinism holds, then once we get enough information, K_n will entail which of A and B happens. Let's say it entails A. Then U(B,K_n) is undefined. This informs one that one will do A, but makes decision-making impossible.

So how much cost-free information should we get from the angel? If we ask for so much that it entails what we're going to do, we won't be able to decide. If our choice is indeterministic, we have a simple principled answer: Ask for everything about the laws and the present. But if our choice is determined, we must stop short of full information. But where?

Perhaps we ask for full information about the laws and about everything outside our minds. But the contents of our minds are often highly relevant to our decisions. For instance, if we leave out in our decision-making the content of our minds, we won't have information on what we like and what we dislike. And in some decisions, such as when deciding whether to see a psychologist, information about our character is crucial.

Here's another interesting question. Our angel knows all about the present and the laws. It seems that he's got all the information we want to have about how we should act. So we just ask: Given all you know, does A or does B maximize utility? And he can't answer this question. For given all that he knows, only one of the two conditional utility values makes sense.

Of course, a similar problem comes up in asking an omniscient being in a case where our choices are indeterministic. We might think that we can make a better decision if that being tells about the future. ("What will AAPL close at tomorrow?") But there is a bright line that can be drawn. We cannot use in our decision any information that depends on things that depend on our decision, since then we have a vicious loop in the order of explanation. So an omniscient being metaphysically cannot give us information that essentially depends on our decisions. (In particular, if we're deciding whether to buy AAPL stock, he can't tell us what it will close at tomorrow, unless he has a commitment to make it close at that no matter what we do, since without such a commitment, what it will close at tomorrow depends—in a complex and stochastic and perhaps chaotic way—on whether we buy the stock today.)

Let me end with this curious question:

• If you have a character that determines you never to ask for help, isn't that a reason to get professional help?

I think this is an interesting question both for compatibilists and for incompatibilists.

The vagueness argument against restricted compositionality

Lewis and Sider have argued that if restricted compositionality is true—some but not all pluralities of two or more objects compose a whole—then there will be cases where it's vague how many objects there are. For instance, imagine two universes, A and B, each with the same finite set of n particles with the same intrinsic properties. But in A, the particles are neatly arranged into galaxies, trees, tables, buildings, etc. And in B there is just a blooming buzzing confusion. If restricted compositionality holds, then, assuming there are no immaterial objects, universe B has exactly n or at most n+1 objects—it's just too messy to have any cases of composition, except perhaps for the universe as a whole (that's why it might be n+1 rather than n). But A is much like our universe, and so we would expect lots of cases of composition, and hence the number of objects will be a lot more than n+1, say n+m for some large m. However, we can now imagine a continuous sequence of universes ranging from A to B, differing continuously in how the particles are arranged. As we move that continuous sequence, the number of objects will have to change from no more than n+m to n+1. But it is incredible that the object count should sharply change due to a very tiny shift in particle positions. Instead, the object count will at times be vague. But how many objects there are is a matter of which sentences using universal quantification, conjunction, negation and identity are true. But quantification, conjunction, negation and identity are not vague. So we have vagueness where we cannot have vagueness.

There may be some technical problems with the argument as I formulated it, given the assumption of no immaterial objects. Maybe we can't do without immaterial entities like God or numbers. One could reformulate the argument to restrict the counting to material entities, but "material" might actually be a vague term. Perhaps the best thing to do is to assume that these universes have no immaterial contingent entities, and then just count contingent entities. Contingency shouldn't be a vague matter, after all. The Aristotelian may balk at this. For it may well be that a necessary condition for a bunch of material entities to compose a whole that they have a form, and forms are immaterial but contingent. Maybe, though, "form" is not vague, and so we can just count the contingent non-forms.

But talking of forms suggests a more serious difficulty. If there are Aristotelian forms, then how many material objects there are may well not supervene on how material objects are spatiotemporally arranged and what intrinsic properties they have. For objects to come to compose a whole, there must come into existence a form. There is nothing absurd about there being sharp laws of nature specifying under which precise conditions a form comes into existence. There is no need for the laws of nature to be continuous (and the possibility of fundamental discreteness is empirically still open). Or perhaps God decides on a case-by-case basis whether to create a form. Then there is no vagueness as to how many material objects there are: the number of material objects equals the number of forms of material objects that in fact inform some matter (the souls of the resurrected are forms of material objects but temporarily fail to inform any matter). Of course in transitional cases we won't have much confidence whether some objects compose a whole, but that's just because we are unable to see forms except through their normal effects.

More on global nuclear war

The most surprising and important event of the second half of the 20th century was the nonoccurrence of a global nuclear war. In an earlier post, I suggested that the probability of such a war was fairly high given naturalism and fairly low given theism, and hence the nonoccurrence of such a war is evidence for theism. I want to add one more thing to that line of thought: the nonoccurrence of a global nuclear war was probably the most prayed-for event of the second half of the 20th century. All the prayers for world peace were first and foremost prayers for peace between East and West, prayers that there be no global nuclear war. And there is peace between East and West, and there was no global nuclear war.

Two legs, four legs and the Incarnation

In this picture from 30 Rock's "Brooklyn Without Limits" episode, Kenneth has only two legs and yet Kenneth has four legs. This sounds like a contradiction, but of course it is not--we can see that it's not.

How shall we resolve the contradiction? Perhaps: Kenneth has two human legs and four table legs. But that suggests that Kenneth has six legs, and that doesn't seem right to say. Maybe we can say that his human legs are also table legs, so he has only four legs: two of them doing double-duty for table legs and human legs, and two of them doing double-duty for table legs and human arms. Maybe. But even the statement "Kenneth has four legs" seems wrong or at least misleading without qualification.

Much better to qualify with a qua: Kenneth qua human has only two legs. Kenneth qua table has four legs.

This should remind us of one of the standard solutions to apparently contradictory talk of Christ incarnate. Christ is eternal. Christ is conceived in time. Christ has boundless knowledge. Christ's knowledge is bounded. And so on. The solution is to say things like: Christ qua God is eternal and has boundless knowledge. Christ qua human is conceived in time and has bounded knowledge--and has two legs.

The naturalness of qua talk in the case of Kenneth should make us less suspicious of the incarnational case.

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A cheap shot against Lewis on free will?

David Lewis thinks that even if determinism holds, we have the ability to act in ways other than what is entailed by the pre-human state of the universe, P, and the laws, L. Were we to have so acted, a "small miracle" would have happened. The actual world's law of nature would no longer have held. Perhaps in that world there wouldn't be enough laws for determinism or there would have been a new law with exception clauses. Here is a cheap shot:

• On this theory, billions and billions of people daily had the ability to to act such that were they to act so, the laws of nature would have been insufficient for determinism or would have had exception clauses. Why did none of them ever exercise that ability?

Is this just a cheap shot? Maybe. The best answer I see to it is:

• Look: this is all hypothetical. We don't actually live in a deterministic world. There will be deterministic worlds with neat exceptionless laws but there will be many more nearby-to-them worlds with many such exceptions. purposes, the world without exceptions. But for the purposes of considering the compatibility of free will with determinism, we posited this unlikely world.

This is a decent answer. But it does suggest that that we would never have reason to think we live in a softly deterministic world with very uniform laws. For near any such world there will be lots of worlds with less uniform laws, worlds that people had the power to actualize simply by acting differently.

Are we free to change the past?

Let P be a complete description of the ancient past and let L be the laws. Lewis agrees that our freedom includes the ability to act in a way that falsifies the conjunction P&L but denies that it includes the ability to act in such a way that were we to act so, P would be false.

But here is a plausible thesis. Fundamental particles are essentially tied to laws of nature. There would be no electrons or photons if the laws of electrodynamics were different. This is clear on the Aristotelian picture on which laws are grounded in the powers of objects, but is also plausible without that picture.

Given this plausible thesis, P entails L. And hence P&L is logically equivalent to P. Thus if we can act in a way that falsifies the conjunction P&L, we can act in a way that falsifies P. Lewis denies the thesis, but it is still plausible.

Compatibilism, trying and trying hard

Some compatibilists—e.g., Vihvelin and Fara—think that something that merely blocks the possibility of your trying to do A but doesn't block your disposition to do A when trying to do A does not take away your present power to do A. Two examples of such blocks are (a) Frankfurt cases where you'd be counterfactually prevented from doing A and (b) being determined not to do A.

But there is an interesting family of cases where you can only do something when you try hard enough. For instance, you can run distance D in time T when you try really hard, and you can only try that hard when you know a bear is chasing you. In a case like that, even though you are disposed to do A when you try hard enough, anything that blocks you from the possibility of trying hard enough also blocks you from being able to do A. Thus the absence of a bear, or even just ignorance of the presence of the bear, blocks you from being able to running D in T.

So where trying hard to do A is needed for you to do A, anything that blocks your possibility of trying hard blocks your ability to do A.

Now, anything that blocks you from the possibility of trying also blocks you from trying hard. So in cases where trying hard to do A is needed to do A, determinism and Frankfurt cases block you from being able to do A.

So in cases where success requires trying hard, blocks to trying remove the ability to succeed. But why should this only be true where success requires trying hard? So in cases where success requires trying, blocks to trying remove the ability to succeed, too.

Brains, souls and consciousness

1. (Premise) I am the only entity that has all the conscious states I presently have.
2. (Premise) I am breathing.
3. (Premise) My soul isn't breathing.
4. (Premise) My brain isn't breathing.
5. I am neither my soul nor my brain. (2-4)
6. Neither my soul nor my brain has all the conscious states I presently have. (1,5)
7. (Premise) If my soul or my brain is conscious, it has all the conscious states I presently have.
8. So, neither my soul nor my brain is conscious.

If my soul or my brain grounds my consciousness, it does not ground my consciousness by being conscious. It grounds my consciousness by having non-conscious states that ground my consciousness. These non-conscious states will then be more fundamental than my conscious states.

In particular, substance dualists should agree with naturalists that conscious states are non-fundamental. Only non-substance dualists, like hylomorphic dualists and property dualists, have a hope of saying that conscious states are fundamental. And of course a similar argument can be run for other mental states beside the conscious ones.

In practice, some substance dualists will say that I am my soul. If so, then I don't breathe (at most I cause breathing), I don't weigh anything, and so on.

Popper functions and infinite sequences of heads

Williamson gave a lovely argument that infinitesimals can't capture the probability of an infinite sequence of heads in fair independent trials: Let H_n be the event that we have heads in each of the trials n,n+1,n+2,.... Then, P(H₁)=(1/2)P(H₂). But P(H₁)=P(H₂) since they're both just the probability of getting an infinite sequence of heads. Thus, P(H₁)=(1/2)P(H₁) and so P(H₁) is zero, not infinitesimal.

It turns out that a somewhat similar result holds for Popper functions as well. For technical reasons, I need a bidirectionally infinite sequence of coin tosses, one for each integer (positive, zero or negative). Our probability space Ω of infinite sequences will then be the set of all functions from the integers Z to {H,T}. Let G be the group of transformations of Ω generated by translations defined by reflections on Z. In other words, G is generated by the transformations R_a, where a is an integer or a half-integer, and (R_as)(n)=s(2a−n) for any sequence s in Ω.

Let F be any G-invariant field in Ω that contains all the H_n. A very plausible symmetry condition on the Popper function on Ω representing the double sequence of heads then is:

1. For any g in G and any A and B in F, P(A|B)=P(gA|gB).

In other words, if we flip the sequences around, we don't change the probabilities. E.g., the probability of getting heads on tosses 2,3,4,5,... conditionally on B is the same as the probability of getting heads on tosses 1,0,-1,-2,... conditionally on R_1.5B. This symmetry condition is related to Williamson's symmetry assumption that P(H₂)=P(H₁).

A second obvious condition is that the probability of getting heads on tosses 1,2,3,... given that one has heads on 2,3,... is equal to the probability of getting heads on toss 1, i.e., is 1/2:

2. P(H₁|H₂)=1/2.

Theorem: There is no Popper function on F satisfying (1) and (2).

Utilitarianism and trivializing the value of life

Consider these scenarios:

• Jim killed ten people to save ten people and a squirrel.
• Sam killed ten people to save ten people and receive a yummy and healthy cookie that would have otherwise gone to waste.
• Frederica killed ten people to save ten people and to have some sadistic fun.

If utilitarianism is true, then in an appropriate setting where all other things are equal and no option produces greater utility, the actions of Jim, Sam and Frederica are not only permissible but are duties in their circumstances. But clearly these actions are all wrong.

I find these counterexamples against utilitarianism particularly compelling. But I also think they tell us something in deontological theories. I think a deontological theory, in order not to paralyze us, will have to include some version of the Principle of Double Effect. But consider these cases (I am not sure I can come up with a good parallel to the Frederica case):

• John saved ten people and a squirrel by a method that had the death of ten other people as a side-effect.
• Sally killed ten people and received a yummy and healthy cookie that w would have otherwise gone to waste by a method that had the death of ten other people as a side-effect.

These seem wrong. Not quite as wrong as Jim's, Sam's and Frederica's actions, but still wrong. These actions trivialize the non-fungible loss of human life. The Principle of Double Effect typically has a proportionality constraint: the bad effects must not be out of proportion to the good. It is widely accepted among Double Effect theorists that this constraint should not be read in a utilitarian way, and the above cases show this. Ten people dying is out of proportion to saving ten people and a squirrel. (What about a hundred to save a hundred and one? Tough question!)

Confidence and security

Two days ago, I posted a post on this topic based on an inequality that I had claimed to have proved but that was just plain false. Oops! The surprising numerical results in that post were also wrong.

But let's re-do it. Suppose that to count as being confident that p, you need P(p)≥α. But if you're confident that p, you still need not be confident that you will permanenly remain confident, even assuming that you are sure you will remain rational. For if you are confident but barely so, there may well be a good chance that future data will push you below the confidence level.

Say that you are secure from future rational defeat when you are rationally confident that future data won't rationally push you below the confidence level. Assume you know for sure that:

1. Your probabilities are coherent.
2. They will remain coherent.
3. You won't forget data.
4. If N is the number of pieces of evidence you will receive in your (earthly) life, then E[N]<∞.
5. When you receive a piece of evidence in your life, you also know for sure that you're receiving a piece of evidence in your life.

Assumptions (4) and (5) are merely technical. Basically, they say that your expectation for your length of (earthly) life is finite and that you're conscious of being in your earthly life when you are.

Under these assumptions, if the confidence level is α, i.e., if P(p)≥α is what it is to be confident in p, then it is enough for security in p that P(p)≥1−(1−α)². And this inequality is sharp. For any P(p) <1−(1−α)², one can imagine an experiment that has probability 1−α of pushing your credence below α.

Security from future rational defeat (in this life) normally requires quite a bit more than mere confidence. If your confidence level is 0.95, then security level might need to be as high as 1−(0.05)²=0.9975, but no higher.

Of course, you might know for sure that no more information will come in. Then the security level will equal the confidence level. But normally you don't know that. So let's let the absolute security level be that level of credence which suffices for security regardless of what you know about the kinds of future data you might receive. That level just is 1−(1−α)² for a confidence level α. The graph shows the absolute security level for each given confidence level.

Suppose confidence α is the credence you need for knowledge. Then absolute security ensures that no matter what expectations you have about future information, you have enough credence to know that your knowledge won't be defeated.

Absolute security is something akin to moral certainty.

More generally, the probability that given a current credence of β your credence will ever dip below α is less than (1−β)/(1−α). The above inequality for the security level is derived from this inequality, and this inequality follows from Doob's optional stopping theorem and the fact that your conditional expectations for p will be a martingale when we condition on an increasing sequence of σ-fields (the increasingness just encodes the fact that you don't forget) and then we can stop that martingale once we reach the end of information or hit either α or 1.

Of course, maybe once again I made a mistake!

Arranged marriage and love

This is an extract from my One Body book. I defend arranged marriage as a morally acceptable option, though of course only in the case where both members of the couple freely consent (without that, there is no marriage). But yet it seems that one ought to have a romantic love for the person one is going to marry. I respond:

[L]ove is always a duty, and the love needs to be appropriate to the relationship. Thus, it is one’s duty to love the person whom one is to marry, and it is a duty to love the person in the way appropriate to the person whom one is to marry. Of course, if one does not know anything about this person, the love cannot be very specifically developed. But it can involve the three aspects of all love: one has a disposition to benefit this person (should one find out what the person needs), one appreciates the other at least as a person, a creature of God, a fellow human being and someone with whom one can engage in sexual activity, and one intends such a union with this person. (The sexual aspects of this union may be the easiest to intend for a young and sexually curious person!) All the while, one can remain open to the mystery, the surprise of the other person. And in this way, the arranged marriage is not so different from an unarranged “love match”. In a love match, too, one must remain open to the enfolding mystery of the other person, traditionally including a lack of sexual knowledge of the other person. In any case, marriage and sex themselves can change people in unpredictable ways, and some of the knowledge of the person prior to marriage is likely irrelevant. Every love must involve a willingness to adjust its form to changes in the beloved and in the relationship, and must remain open to new things.

It is not so much wrong to marry someone that one does not love, as it is wrong not to love the person one marries. Love is required of us always, under all circumstances. It is wrong not to love the person one with whom one shakes hands, the person one sentences to two years in jail, the person one gives a free meal to, or the person one marries. Of course a different form of love is required in each case. However, what primarily distinguishes the different forms of love is the type of real union toward which the love is directed and the aspects under which the beloved is appreciated. If one marries, one ought to have a directedness toward sexual and personal union with the other person, and an appreciation of the other person insofar as this person can be united with. But for this one needs only to know the other person as a fellow human being of the opposite sex with whom one can unite sexually.

Ability and probability

You find a ticking bomb which will go off in five seconds. There is a pad on it, and if you enter the right five digit number in the time remaining you will defuse the bomb. You frantically enter "12345", "54321" and "91101", and none of these work. The bomb goes off. As it happens, had you entered 88479, a number that didn't occur to you, the bomb would have been defused. You surely could have entered 88479. Are you responsible for failing to defuse the bomb?

Of course not. But why not? You were in some sense able to.

In cases like this, a natural suggestion is one made by Gerald Harrison: you were unable to do it "because ... doing so is contingent upon something highly improbable happening, namely ... entering the right combination."

But I don't think this has much to do with improbability as such. Suppose I'm a habitual criminal and I come across a sure opportunity to steal a million dollars with almost no chance of being caught, and sure enough I avail myself of the opportunity. The probability that I refrain from this was nonzero, but it was very small—perhaps no bigger than the probability of entering the right combination in the above case. Yet despite the low probability, I am responsible.

Or vary the bomb case. There is time to enter only one combination. And you know it's either 12345 or 54321. You try 12345, and fail. You are not responsible for the failure to defuse, even though your defusing the bomb was not "contingent upon something highly improbable happening".

So why aren't you responsible in the two bomb cases? It sounds right to me to say that both in the original case and the bomb case you did your best, while in the criminal case, I failed to do my best. And neither the probabilities of success (low in the first bomb case and in the theft case, but moderate in the second bomb case) nor those of trying to do one best (high in the bomb cases but low in the theft case) seem to settle any of the cases either way.

This suggests the following principle:

• If you always tried to do the best you could as hard as you could, you are not culpable for the bad outcomes of any of your actions.

But if determinism rules out alternate possibility—as I think, pace Lewis and Hume and others—and if determinism is true, then we have always tried to do the best we could as hard as we could. For there never was anything else we could do, nor any other way of trying we could engage in.

Another argument against an infinite past?

I wonder if this very neat argument can't be used to provide another Grim Reaper style argument against an infinite past? The argument nicely fits with the intuition that Grim Reaper induces in me, namely that no event can have an infinite number of events in its causal history.