A solution to various problems in decision theory

There are various games where one cannot assign finite expected utilities in the standard way. St Petersburg is an extreme case, but there are milder and in some ways more interesting cases. Besides these cases from the literature, there interesting cases of gambling on nonmeasurable events (which I've used to argue for incommensurability).

One might put these cases aside as idle curiosities—when was the last time someone offered you a bet on the St Petersburg game?—were it not for the fact (noted by Hajek and others) that they threaten contagion to ordinary decisions as soon as there is some non-zero probability that such a game will happen. For if the utility of game A is undefined, and B is some perfectly ordinary game, but one thinks there is some tiny non-zero probability p of game A actually occurring, then one's expected utility for playing B won't be E[B] but E[B]+pE[A], which will be undefined. (A similar contagion problem applies to Pascal's Wager.)

The problems I am interested in all take place on some honest to goodness probability space (P,Ω,F), with P being a perfectly standard countably additive probability that can be used to define a classical expectation E. However, the problem comes when one needs to make decisions that involve gambles G (a gamble is just a real-valued function on Ω) which don't have a classically defined expectation E(G), due to either convergence problems or nonmeasurability.

Here is a solution. Say that a function E* defined for all gambles G on Ω is "extended expectation" provided that:

1. E* has values in some totally ordered extension of the reals (say, the hyperreals)
2. E*(aG)=aE*(G) for any real number a and gamble G
3. E*(G+H)=E*(G)+E*(H) for any gambles G and H
4. E*(G)≥0 if the gamble G is nowhere negative
5. E*(G)=E(G) if G has a finite classical expectation.

Now we can do our decision theory. Say that H is at least as good as G provided that E*(G)≤E*(H) for all extended expectations E*. And say that H is strictly better than G provided that H is at least as good as G but G is not at least as good as H. (There is a variant definition here that one might want to consider: E*(G)<E*(H) for all extended expectations E*.)

Note that by condition (5), this is going to give us the same answers as the classical theory when the classical theory gives answers. But it's also going to give answers in cases where the classical theory gives none. For instance, if A is one of those games with an undefined or infinite utility, and B is eating a cookie while C is getting a mild electric shock, then the classical theory won't be able to define the expected value of A+B and A+C and won't be able to conclude that A+B is better than A+C, even though clearly B is better than C. But the above works, as long as some E* exists. For E*(A+C)=E*(A)+E*(C)=E*(A)+E(C)>E*(A)+E(B)=E*(B)+E*(B)=E*(B+C), since E(B) and E(C) are defined. So, A+C is better than B+C.

The point generalizes, and the contagion problem is solved.

Of course, for this approach to be non-trivial, there have to actually exist extended expectations. I have a rough sketch of a proof of the following claim:

• For any classical probability space (P,Ω,F) there is an extended expectation E*.

The proof is a slightly challenging (to me) ultrafilter argument, so I am not absolutely confident, but I am pretty sure I can make something like it work, if only at the cost of weakening (5). But don't use this result until I have a proof written out. :-)

How to run an infinite fair lottery

Let <* be any well-ordering of the real numbers. Run a countable infinity of independent random processes each of which picks out a random real number in exactly the same way and with the same continuous (or just atom-free) distribution. For instance, maybe in each of an infinite number of universes you toss a dart at a target in exactly the same way and measure the x-coordinate. Number the processes 1,2,3,....

Let X_n be the real number picked out by the nth process. Almost surely, all the numbers X₁,X₂,... are different: the probability of a repetition in a countable number of trials given a continuous distribution is zero. Thus, almost surely, there is an N such that X_N<*X_n for all n≠N. This N counts as the choice in our lottery. All the processes being on par, it seems that we now have an infinite fair lottery outcome N, where the lottery tickets are 1,2,3,....

This process isn't guaranteed to work, since sometimes we will get a repetition. But most of the time the process will succeed.

Now, mathematically speaking, N is not going to be measurable on our product probability space.

But don't think about this mathematically. Think about it physically: Given an infinite multiverse, this could actually happen!

Maybe the lesson to be learned is that an infinite number of independent random trials is impossible and hence an infinite multiverse is impossible?

Conditional probabilities and infinite fair lotteries

Suppose infinitely many independent fair coins are tossed. Let H be the event that exactly one of the coins is heads. Then conditionally on H, it looks like we have an infinite fair lottery: every coin is equally likely to be the heads one! So if one thinks (as one should) that infinite fair lotteries are incoherent, one also cannot conditionalize on null probability events like H.

Another argument against divine command theory

1. If Divine Command Theory is true, necessarily, A is obligatory if and only if God commands A.
2. Necessarily, if there are created persons, obedience to God is obligatory.
3. Possibly, God creates persons but does not command them to obey him.
4. So, if Divine Command Theory is true, possibly obedience to God is not obligatory. (1 and 3)
5. So Divine Command Theory is not true. (2 and 4)

Premise (2) seems quite intuitive. Premise (3) seems to follow from divine freedom and the fact that God is under no obligation to command creatures.

A form of explanation in ethics

Why are murder and incest wrong? Here are plausible things to say:

1. Murder is wrong because it cuts short a future life of valuable agential activity.
2. Incest is wrong because it leads to genetic defects in offspring.

Now an explanation needs to be true. But murder doesn't always cut short such a future life (e.g., think of the murder of a severely disabled person) and incest doesn't always lead to genetic defects (e.g., suppose the two parties are past the age of childbearing or are of the same sex). So the explanations in (1) and (2) must be understood as making Aristotelian categorical claims, like "Sheep have four legs", claims that are true in normal or paradigm cases. On the other hand, the claims on the left side, that murder and incest are wrong, are meant to be true in all cases. Thus, (1) and (2) can be made more explicit as:

3. Murder is always wrong because normally it cuts short a future life of valuable agential activity.
4. Incest is always wrong because normally it leads to genetic defects in offspring.

Now, it seems that there is something deeply fishy about (3) and (4). How can the fact that normally murder and incest result in certain grave harms explain the fact that they are always wrong? Yet, murder and incest are always wrong, and the harms cited seem to have something to do with their wrongness. One could say that the harms cited explain only the wrongness of normal cases of murder and incest, and other cases are wrong due to other harms. But I do not think this is desirable. Similar worries are likely to apply to the other harms. Maybe one can find a set of such harms such that every possible case of murder or incest is made wrong by something in the set, but that is not so likely. One could, I suppose, abandon the claims that murder and incest are always wrong, but that's a serious moral mistake.

But on a number of ethical theories the explanations in (3) and (4) are perfectly fine.

Rule utilitarianism: Here the point is clear: an act can be wrong precisely because most of the time it is seriously harmful.

Divine command: Because the acts result in such terrible harms in most cases, God wants us to stay far away from these acts and wisely forbids them to us in all cases. So the fact that normally great harms result explains God's universal prohibition, which in turn grounds the universal wrogness.

Natural law: The natures of things support their flourishing as individuals and as a kind. That an action type is normally harmful to the individual or the kind makes it likely that the action type is unnatural, and hence wrong. This explanation becomes more satisfactory, I think, on a theistic natural law theory. For then we can explain why it is that the natures of things support their flourishing. On a view on which God designs natures, one can say that God is unlikely to design a nature that fails to support the flourishing of an individual or kind. On a view on which God finds (in his mind) natures and then decides which natures should be exemplified in creatures, one can say that God is unlikely to choose to exemplify natures that do not support the flourishing of the individuals or kinds. On a non-theistic natural law theory, it may be a bit more puzzling why the natures of things support their flourishing. Maybe an evolutionary explanation can be given, though.

Notice an interesting difference. In an appropriate rule utilitarianism the harm facts might ground the wrongness facts. In the divine command and natural law cases, they don't ground the wrongness facts, but explain them in a less direct way.

In all of these cases, the same line of thought that leads to the explanations allows for the following argument:

5. Action type A normally produces great harms.
6. So, A is always wrong.

On an appropriate rule utilitarianism, this could be a deductively valid argument. But on the other theories, it is only a defeasible inference.

The God quantifier

Hypothesis: There is no fundamental quantifier that includes within its domain both God and something other than God. (Obviously, this is inspired by Jon Jacobs' work on apophaticism.)

The hypothesis is compatible with saying in ordinary English that both God and human beings exist, and that nothing (not even God) is a unicorn. But if we speak Ontologese, a language where all our quantifiers are fundamental, we will need to modify these locutions. Perhaps we will have a fundamental divine existential quantifier D and a fundamental creaturely quantifier ∃, and if in Ontologese we want to give the truth conditions for the ordinary English "Nothing is a unicorn", we may say something like:

• ~Dx(Unicorn(x)) & ~∃x(Unicorn(x)).

And if we want to give truth conditions for "Something is alive", we may say something like:

• Dx(Alive(x)) or ∃x(Alive(x)).

(Assuming that Alive(x) is a predicate of Ontologese.)

Of course, it could be that Ontologese doesn't just have a single quantifier for creatures. It might, for instance, have "metaphysically Aristotelian quantification": a quantifier ∃ over (created) substances and a subscripted quantifier ∃_x over the accidents of the substance x. In that case, "Nothing is a unicorn" will have truth conditions:

• ~Dx(Unicorn(x)) & ~∃x(Unicorn(x)) & ~∃x∃_xy(Unicorn(y)).

(It might seem excessive to say that no accident is a unicorn, but better be safe than sorry.) Likewise, "Something is alive" has the truth conditions:

• Dx(Alive(x)) or ∃x(Alive(x)) or ∃x∃_xy(Alive(x)).

Now, it may seem wacky to think of a quantifier D that quantifies only over God. But it shouldn't seem so wacky if we recall that Montague-inspired linguistic classifies names as quantifiers (they correspond to functors that lower the arity of a predicate, after all).

Now this leads to an interesting question. Speaking in the ontology room, where we insist that our language cut at the joints, should we say "God exists"? That's a choice. We could adapt the English "exists" when used in the ontology room to go with the fundamental quantifier D or the fundamental quantifier ∃.

We might want to, this being the ontology room after all, make the decision that we will adapt words to the most fundamental meanings we can. But in some sense surely the divine quantifier D is more fundamental than the creaturely quantifier ∃, so in the ontology room we could say: "Only God exists." It is said that Jesus said to St Catherine of Siena: "I am he who is, and you are she who is not." Maybe St Catherine's mystical theology room wasn't that different from the ontology room.

Or we might want to keep as many of the ordinary existence claims unchanged, and so say "Photons exists". Then we might want to say something like "God does not exist but divinely-exists."

But since the ontology room isn't the ordinary context, this is really a matter of decision. My own preference would be to say "Only God exists" in the maximally fundamental ontology room, but to spend a lot of time in less fundamental ontology rooms, ones in which one can say "God exists" and "Photons exist" but not "Holes exist" or "Tables exist."

Brainlink on sale

I got an email earlier this week from Surplus Shed about the Brainlink being on sale for $20, in the aftermath of its discontinuation. It looks like a really cool device. It can hook up via Bluetooth to a computer or an Android phone on one end, and to many things on the other end: it has two PWM motor controllers, some DAC I/O, some analogue I/O (low resolution but the firmware is user-upgreadeable), a proximity sensor, accelerometers, and IR transmitter. The last of these is supposed to make it capable of controlling Roombas, TVs, DVD players and toy robots (I plan to try it with some of our IR helicopters, though the range of the IR on the Brainlink is supposed to be short, and maybe with our Pleo if we can make its battery pack work), and you can control it with Java code (there is an SDK). It's all beautifully open and well-documented. Very sad it's discontinued, but the original price was way more than a Raspberry Pi, so it's not surprising it didn't fly. For $20 it's a steal. The official website for the product is here. My eldest daughter and I are really looking forward to it! (Of course we may end up disappointed.) Techie readers may want to check it out.

Merely justifying reasons

A lot of philosophers think that there are "merely justifying reasons", reasons that do not require action but can justify it. The defining feature of a merely justifying reason is that if one has a merely justifying reason to A, one can rationally refrain from Aing without needing any reason to do so. On the other hand, if one has a requiring reason to even a pro tanto one, to rationally refrain from Aing one needs a contrary reason.

I will argue against this based mainly on five plausible theses:

1. One only acts rationally when one acts for reasons.
2. When one has to do what one does not have rationally compelling reason to do, one is in bondage.
3. One does not come to act in bondage simply by not having reasons to act otherwise.
4. Rationally compelling reasons are not merely justifying reasons.
5. The status of a reason R as merely justifying does not depend on what other options are rationally available.

For my view of action, (1) is rock bottom. Claims (2) and (3) concern a concept of "bondage" that I don't have a very good characterization of. It is the opposite of the kind of freedom that Augustine and Leibniz talk about (Leibniz defines freedom as doing the best thing for the best reasons). Brainwashing produces bondage. There is bondage whenever a reason's action-causing force significantly exceeds its rational force. On the other hand, being compelled by one's virtue to do the right thing is not a case of bondage, even though a libertarian might worry that it's not a case of freedom (or only derivatively a case of freedom). Bondage is not necessarily opposed to responsibility. For our own freely chosen vicious activities can cause us to be in bondage. A compatibilist may think lack of bondage is necessary and sufficient for freedom. The libertarian is apt to think that it's necessary but not sufficient. Claim (4) seems very plausible. Now, maybe (5) can be disputed. One might think that whether a reason to A is merely justifying will depend on what reasons one has for other options. But that seems mistaken: the reason to A may become more or less opposed by the presence or absence of other options, but that shouldn't affect the status of the reason.

Now, imagine that I am the sort of being that can only act rationally (probably the notion I have in mind is something like minimal rationality). This surely does not make me be in bondage. Suppose that I rationally and freely choose to A for a reason R over some option B for which I have some other reason S. And consider a similar world W where I do not in fact have any reason to choose otherwise than to A. In that world, S doesn't support my choosing B. For instance, maybe in this world I choose to watch a movie for fun (and "for fun" seems to be a paradigm case of a merely justifying reason, if there are merely justifying reasons) over going to bed early to rest up more. But in W, going to bed early is known by me not to be restful. By (3), I don't come to be in bondage just by losing reasons, so in W my choice to A is still a choice not made in bondage. But in W, I have only one choice available supported by reasons, namely to A, and hence only one rational choice by (1). So if I can only act rationally, I have only one possibility available: to A. Since I am not in bondage, by (2) it follows that my reason R to A is rationally compelling. But a rationally compelling reason is not merely justifying, by (4). So, my reason R to A is not merely justifying in W. Hence, it is not merely justifying in the actual world. Thus, one does not rationally choose to A on the basis of a merely justifying reason.

From relationalism about times to infinitesimal lengths of time

Assume that simultaneity is a reflexive and symmetric relation between events. I will, however, not think of it as transitive. This lets me say that an event that goes from 2 pm to 3 pm is simultaneous with one that goes from 2:30 pm to 3:30 pm. (This is important if there is to be any hope of the thesis that all causation is simultaneous being true.)

Can one construct times out of the simultaneity relation between events? Well, a natural attempt is to say that any maximal set T of pairwise simultaneous events is a time (we can use the Axiom of Choice to show that every event is contained in such a maximal set), and an event E happens at a time T if and only if E is a member of T.

This account, however, has a curious consequence. Consider some event E_n that starts right after noon, and ends right at noon plus 1/n hours. Thus, E_n takes place on the time interval (12,12+1/n] (non-inclusive at 12, inclusive at 12+1/n). Let T be any maximal set of pairwise simultaneous events that contains the E_n. (By the Axiom of Choice, T exists.) By the above account of times, T is a time, and all the events E_n occur at T. But when is T? It's not noon: none of the events E_n occur at noon. But for any positive real number u, most of the events E_n occur before 12+u, so T is not 12+u.

In other words, T is a time between 12 and 12+u for every positive real u>0. It is, thus, a time that is infinitesimally after noon. Thus, curiously, the natural construction of times out of the simultaneity relation very naturally leads to times that are infinitesimally close together, as long as there are events like E_n.

This is quite interesting, because it suggests that a hyperreal timeline may not be such an outlandish hypothesis (Rosinger has also suggested this hypothesis in a number of preprints, e.g., this one). It is a hypothesis that one is led to quite naturally from a relationalist picture, a hypothesis that given such a picture and such an account of times might very well be true.

Of course, the above depended on one particular way to construct times out of simultaneity. And it depended on a simultaneity, a somewhat fishy relation. But still, it's suggestive.

I think there is a way of seeing the above remarks as a reductio of the relationalist program. That's how I saw the observation when I started writing this post. And maybe that's right, but it's not clear to me that that's right.

Spiritual experiences

The naturalist has to say that spiritual experiences are illusory. It is bad enough that the naturalist has to say this about such a large class of human experiences. But these experiences are central among the experiences that give life its savor, they are among the deepest and most significant of human experiences. Indeed, all of the deepest and most significant of human experiences include an aspect of the spiritual: the person I have encountered is seen clothed in a a significance that organic chemistry could never have, the vista stretching out before one in the night sky bespeaks a mystery beyond the merely puzzle, and so on. The naturalist has to say of the deepest and most significant of human experiences that they are illusions. And that is surely a problem.

Reference magnetism and anti-reductionism

According to reference magnetism, the meanings of our terms are constituted by requiring the optimization of desiderata that include the naturalness of referents (or, more generally, by making the joints in language correspond to joints in the world, as much as possible) and something like charity (making as many real-world uses as possible be correct).

Suppose we measure naturalness by the complexity of expression in fundamental terms—terms that correspond to perfectly natural things. (In particular, we can't talk of what cannot be expressed in fundamental terms, since reference magnetism would presumably not permit reference to what is infinitely unnatural.) Consider the reductionist thesis that the vocabulary of microphysics is the only fundamental vocabulary about the natural world. If this thesis is true, then our ordinary terms like "conscious" or "intention" or "wrong" are going to be cashed out in terms of extremely complex sentences, often of a functional sort. But I suspect that once these expressions are sufficiently complex, then there will be many non-equivalent variants of them that will fit our actual uses about as well and are about as complex. Consequently, we should expect that the meaning of terms terms like "conscious", "intention" and "wrong" to be highly underdetermined.

If we have reason to resist this underdetermination, we need to embrace an anti-reductionism on which the terms of microphysics are not the only fundamental ones, or else have another measure of naturalness.

Another argument for universal love

A part of the phenomenology of healthy full-blown love is that one sees that the beloved is such that one would have been remiss not to have recognized her lovability by loving her. The phenomology of healthy full-blown love is not misleading. But it is possible to have a healthy full-blown love for any person. So one should love everyone. For consider some person, say Sam. If one did have the healthy full-blown love for Sam, one would have correctly seen that one would be remiss in not loving Sam. But whether one would be remiss in not loving Sam doesn't depend on whether one in fact loves Sam. So, it is true that one would be remiss in not loving Sam.

In my previous post, I started the argument by noting that if you have full-blown love, you should continue loving, and yet I concluded that the conditional can be dropped—you should love (and continue loving) everyone. But why is it that the conditional had a special plausibility? I think it's because of the above phenomology of love. It's not that only the people you love are such that you should love them. But it's that by loving them that you best come to see that you should love them. Healthy love isn't blind: it sees our neighbor as she really is.

An argument for universal love

If you have full-blown love (not just be slightly fond of, but really love) someone, you should continue to love her. It is a serious moral defect to be open to discontinuing one's full-blown love. This can be discerned from the phenomenology of full-blown love.

But a failure to continue loving someone shouldn't get one out of the obligation to love her. It would be "too convenient" if simply by doing the wrong of ceasing to love one were to get out of the obligation to love our beloved.[note 1] So our principle that if you have a full-blown love then you should continue to love can be strengthened:

1. If you had a full-blown love for someone, you should love her.

But why is (1) true? I propose that the best explanation for (1) is:

2. You should love everyone you can love.

The best alternate explanation of (1) is that love is relevantly like a promise: by acquiring full-blown love for someone one commits to an obligation to love. But this view is not plausible. Think of the way that children come to deeply love their siblings. This love can grow on them early, before they have the kind of moral responsibility that would make them fit subjects for undertaking lifelong commitments.

Now, we could stick with (2) as the conclusion. But everyone is in principle lovable. But perhaps not lovable by me? But an inability to love someone who is in principle lovable is a moral defect in me, though perhaps not one that I am culpable for. And moral defects shouldn't get one out of moral obligations. So:

3. You should love everyone.

And that completes the argument. Definitely not a knockdown argument, but still something that should give some credence to the conclusion.

Popper functions, uniform distributions and infinite sequences of heads

Paper forthcoming in the Journal of Philosophical Logic, now posted. I argue that Popper functions don't solve the problems of uniform probabilities in infinite spaces. Yet another in a series of highly technical papers.

Regular probability comparisons imply the Banach-Tarski Paradox

Paper posted here (forthcoming in Synthese). Among goodies in the paper is a proof that the order extension principle (even in a weak form) implies the Banach-Tarski paradox, and a new argument against commensurability in decision theory. This is a very technical paper, so reader beware.