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.

A curious thing about infinite sequences of coin tosses

Suppose that at locations ...,−3,−2,−1,0,1,2,3,... (in time or one spatial dimension) a relevantly similar independent fair coin is tossed. Let L_n be the event that we have heads at all locations k≤n. Let R_n be the event that we have heads at all locations k≥n. Let A≤B mean that event B is at least as likely as A, and suppose all our events L_n and R_n are comparable (i.e., A≤B or B≤A whenever A and B are among the events L_n and R_n). Write A<B when A≤B but not B≤A. Assume ≤ is transitive and reflexive. Then, intuitively, we also have:

1. L_n+1<L_n
2. R_n<R_n+1.

After all, L_n+1 and R_n require one more heads result than L_n and R_n+1, respectively.

Now here is a surprising consequence of the above assumptions. Say that n is a switch-over point provided that L_n≥R_n but L_n+1<R_n+1. When there is a switch-over point, it's unique (since for m≤n, we will have L_m≥L_n≥R_n≥R_m, and for m>n, we will have L_m≤L_n+1<R_n+1≤R_m). Then:

3. Under the above assumptions, either (a) there is a switch-over point, or (b) L_n<R_m for all n and m, or (c) R_n<L_m for all n and m.

But each of these options is really rather absurd. Option (a) says that the probability distribution of our sequence of relevantly similar independent fair coins has a distinguished switch-over point. Option (b) implies that infinite sequences of heads stretching leftward (if we're talking about a spatial arrangement; backward, if temporal) are always less likely than infinite sequences of heads stretching rightward (forward, if temporal). And option (c) is just as absurd as (b). And of course in each of the three cases we have a violation of very plausible symmetry conditions on the story: in case (a), we have a violation of symmetry under shifts, and cases (b) and (c) we have a violation of symmetry under flips.

So something is wrong with the assumptions. Classical probability theory says that what's wrong are (1) and (2): in fact, all of the L_n and R_n events are equally likely, i.e., have probability 0. This seems a very plausible diagnosis to me.

Why does (3) hold? Well, suppose that (b) and (c) do not hold. Thus, L_n<R_m for some n and m. If m≥n, then we L_m≤L_n<R_m, and if m<n, then we have L_n<R_m<R_n. In either case, there is an b such that L_b<R_b. By the same reasoning, by the falsity of (c), there is an a such that L_a>R_a. As n moves from a to b, then, L_n decreases while R_n increases, and we start with L_n bigger than R_n at n=a and end with L_n smaller than R_n at n=b. This guarantees the existence of a switch-over point.

This result is basically a generalization of an observation about Popper functions for such infinite sequences in a forthcoming paper of mine.

Theism and scientific non-realism

One of the major arguments for scientific realism is that the best explanation for why our best scientific theories are predictively successful is that they are literally true or at least literally approximately true. After all, wouldn't it be incredible if things behaved observationally as if the theories were true, but the theories weren't true?

While this is a pretty good argument, it's worth noting that theists have an alternate explanation: In order that intelligent beings be able to make successful predictions of a sort that lets them exhibit appropriate stewardship over the world, God makes the world exhibit patterns of the sort that human science is capable of finding, patterns that can be subsumed under theories that are sufficiently simple for us to find. And one sort of pattern is of the as-if sort: things behave as if there were photons, which lets us organize the behavior of macroscopic things into patterns by supposing (in a way that need not carry ontological commitment) photons.

That said, there is a value to science over and beyond its helping us exercise stewardship over the world—understanding of the world is valuable for its own sake—so even given theism, a scientifically realist theistic explanation seems better than a scientifically non-realist one.

But even if realism is in general the right policy, maybe theism could provide a tenable Plan B if there turn out to be cases where scientific realism is not tenable. For instance, one might think (incorrectly, I suspect) that there is no metaphysically tenable and scientifically plausible version of quantum mechanics. Then, one might retreat to a theistic explanation of why the world behaves as if the metaphysically untenable theory were true. Or one might think (because of Zeno's paradoxes, say) that it is impossible for spacetime to be adequately modeled by a manifold of the sort that mathematics studies (one locally homeomorphic to a power of the real number line). But why do things behave as if spacetime were such a manifold? Maybe God made them behave so because this lets us organize the world in convenient ways.

If casual sex is permissible, so is polygamy

If casual sex is permissible, so is premarital sex. Now, on a view on which premarital sex is permissible, marriage is a complex normative institution that removes the right to have sex with others and confers on the spouses various duties—such as of loving, cherishing, honoring and caring for—to the other, as well as makes for at least a ceteris paribus commitment that the couple will strive to have a sexual relationship. (If premarital sex is impermissible, then marriage has one more normative component: it confers a permission for sex.)

But there need not be anything morally wrong with x's promising y that she will not have sex with anyone other than y and z. Promises of loving, cherishing, honoring and caring for another are a great thing when taken really seriously, and are simply a higher and deeper form of the commitment that we all have anyway to our friends, and there seems nothing wrong with making such promises to multiple people, as long as there are implicit or explicit rules on how apparent conflicts of love and care are to be resolved (a problem that is already anyway present in the case of a monogamous marriage, since it can come up with respect to duties to spouse and to children, since these can be in tension). The only component possibly problematic in the normative complex is the ceteris paribus commitment to a sexual relationship with multiple people. But it is hard to see what is wrong with that if casual sex is permissible. If it would be permissible for Jane to have sex with Sid and Roman on alternate days, why would it not be permissible for her to make a ceteris paribus promise to do so? This is particularly unproblematic if one thinks of marriage as permissibly dissoluble, as most people who think casual sex is permissible do.

So it seems that if casual sex is permissible, then the normative complex of commitments that constitutes marriage can be permissibly modified to a plural form. One may ask whether the modified version would still count as a marriage. If not, then polygamy is misnamed: it's not a plural marriage (poly-gamy) but a plural marriage-like relationship. But either way, we get the conclusion: If casual sex is permissible, so is polygamy.

An interesting question is whether we can prove the stronger claim that if premarital sex is permissible, so is polygamy. Probably not, since someone could think that premarital sex is permissible only in the context of a relationship with an exclusive commitment to one person. But if one thinks that something weaker than an exclusive commitment is sufficient for permissibility, maybe love, or maybe mutual respect (Martha Nussbaum), then one may still get the conclusion that polygamy is permissible.

Of course, the right conclusion to draw is that casual sex is impermissible.

Self-inflicted sufferings, Maimonedes and anomaly

Suppose I know that if I go kayaking on a sunny day for two delightful hours, I will have mild muscle pains the next day. I judge that the price is well worth paying. I go kayaking and I then suffer the mild muscle pains the next day.

My suffering is not deserved. After all, suffering is something you come to deserve by wrongdoing, and I haven't done anything wrong. But it's also awkward to call it "undeserved". I guess it's non-deserved suffering.

It would be very implausible to run an argument from evil based on a case like this. And it's not hard to come up with a theodicy for it. God is under no obligation to make it possible for me to go kayaking on a sunny day and a fortiori he is under no obligation to make it possible for me to do so while avoiding subsequent pain. It is not difficult to think that the good of uniformity of nature justifies God's non-interference.

How far can a theodicy of this sort be made to go? Well, it extends to other cases where the suffering is a predictable lawlike consequence of one's optional activities. This will include cases where the optional activities are good, neutral or bad. Maimonedes, no doubt speaking from medical experience, talks of the last case at length:

The third class of evils comprises those which every one causes to himself by his own action. This is the largest class, and is far more numerous than the second class. It is especially of these evils that all men complain,only few men are found that do not sin against themselves by this kind of evil. Those that are afflicted with it are therefore justly blamed .... This class of evils originates in man's vices, such as excessive desire for eating, drinking, and love; indulgence in these things in undue measure, or in improper manner, or partaking of bad food. (Guide for the Perplexed, XII)

Maimonedes divides evils into three classes:

1. evils caused by embodiment,
2. evils inflicted by us on one another, and
3. self-inflicted evils.

In the third class he only lists self-inflicted evils that are inflicted by bad activity, but we can extend the class as above. He insists that evils in the first and second classes are "very few and rare" and says that "no notice should be taken of exceptional cases".

The last remark is quite interesting. It goes against the grain of us analytic philosophers—exceptions are our bread and butter, it seems. But Maimonedes' insight, which mirrors Aristotle's remarks about precision in ethics, is deep and important. It suggests that the evils for which there is a plausible "problem of evil", namely the evils of the first and second classes, are an anomaly, and should be handled as such (for a development of this idea, see this paper by Dougherty and Pruss, in Oxford Studies).

Death and the Fall

It is an evil that we die. The badness of death is constituted by the cessation of the good of life. But not every cessation of a good is an evil. If I have a good conversation for several hours with a friend and then we go our separate ways, the cessation of the conversation isn't an evil. Only the cessation of a due good is an evil.

But how is it due to us not to die? Is it not a part of our very nature as human beings that we die?

Here's an argument:

1. If the empirical manifestation of our nature matches our real nature, what we are supposed to be, then death as such is not an evil, just a cessation of a good.
2. Death as such is an evil.
3. So, the empirical manifestation of our nature does not match our real nature, what we are supposed to be.

Claim (3) is already on its own a kind of doctrine of the Fall. And it calls out for explanation. The story of the Fall of Humankind provides such an explanation. The naturalist, on the other hand, cannot provide an explanation for (3). I think the naturalist should perhaps deny (2), but that is quite an implausible move.

Knowing that you can't do otherwise

Suppose as is very plausible (except for dubious interpretations of "can do otherwise") that you know that

1. Determinism implies that you cannot ever do otherwise than you in fact do.

Suppose you also know that

2. You will in fact do A,

say by induction from what you've done in similar circumstances. Finally, suppose that you know that

3. Determinism holds.

Then you know premises sufficient to conclude that you cannot do otherwise than A. So, plausibly, you are in a position to know that

4. You cannot do otherwise than A.

This is interesting. For while determinism does not by itself guarantee the possibility of knowledge of how you are determined to act, it turns out that with a bit of induction and reflection, if you know determinism to be true, you are in a position to know what you are determined to do.

It is also plausible that:

5. When you know you cannot do otherwise than A, then you are not freely choosing A.

For take Locke's locked room example. You're having great fun at the party, and don't want to leave, but unbeknownst to you, the door is locked so you can't leave. Then maybe Locke is right that you're freely staying at the party. But as soon as you find out that the door is locked, surely you're no longer freely choosing to stay at the party. The same is plausible in more sophisticated Frankfurt cases. Note that (5) can be accepted by a compatibilist.

But now we get the interesting conclusion that if you know determinism to be true, that knowledge could very well undercut some of our freedom. For it could boost knowledge of what we will in fact do to knowledge of what we will have to do.

Objection 1: Knowledge of what we will in fact do does take away freedom, so knowing that we will have to do it doesn't take away any freedom that wouldn't already have been taken away.

Response: I know I will eat lunch today, but that doesn't take away my freedom.

Objection 2: Claim (5) is no more plausible than the disjunction of the following two principles:

6. When you have a belief with knowledge-level justification that you cannot do otherwise than A and you think you know that you cannot do otherwise than A, then you are not freely choosing to do A
7. When you cannot do otherwise than A, then you are not freely choosing to do A.

For the work in (5) is either done by the justified belief or by the factiveness of knowledge—it surely isn't done by anti-Gettier conditions or even by a combination of the constituents of knowledge. Now (7) begs the question against those determinists who grant (1), while (6) is false. Here's a counterexample to (6). You have knowledge-level justification that you cannot resist some temptation, and you think you know this. But being a fallibilist about knowledge you decide to try anyway, since you can try to do even what you know is impossible. And you succeed, because you didn't in fact know. So, the reason to accept (5) is a disjunction of two claims, one of which has been shown false and the other is dialectically unacceptable, so (5) is dialectically unacceptable.

Response: Maybe. But maybe the right way to reason is this. Clearly (5) is true. Now, there are two initially plausible explanations for (5), namely (6) and (7). Since (6) is false, that leaves (7). So we have an inference to best explanation from (6) to (7). And so, even though previously I was only arguing for the interesting conclusion that knowledge of determinism could take away some freedom, we have arrived at an argument for incompatibilism. The argument starts with (5), concludes to (7) by inference to best explanation, then adds (1), and concludes that freedom is incompatible with determinism.

Responsibility and desires

Consider four cases. In each case, you know that Jones, an innocent person, is drowning and will survive if and only if you throw her a life preserver in the next two minutes. But in each of the four cases there are further facts that you know:

1. The life preserver is locked down with a mind-reading device that will open if and only if you have a desire to eat a tarantula. You lack that desire and your character is such that you are unable to form that desire in two minutes.
2. The life preserver is locked down with a mind-reading device that will open if and only if you have a desire to eat a tarantula. You lack that desire, as well as lacking a desire to rescue Jones, and your character is such that you are unable to form either desire in two minutes.
3. Same as 2, but the the mind-reading device will open if and only if you have a desire to rescue Jones. You lack that desire and your character is such that you are unable to form that desire in two minutes.
4. The life preserver is not tied down, but your character is such that you can only rescue Jones if you desire to rescue Jones. You lack that desire and your character is such that you are unable to form that desire in two minutes.

In case (1) you are not being directly responsible for failing to rescue Jones. You might, of course, be derivatively responsible, if, say, you had foreseen that the case would arise sufficiently early in the game you had foreseen that the case would come up and failed to make reasonable efforts to self-induce a desire to eat a tarantula. Such efforts could have involved reflection on the bragging rights one would gain from eating a tarantula, but it would take more than two minutes to succeed—it's too late now, anyway. With such a back story, you would be derivatively responsible for faiing to rescue Jones on the basis of your responsibility for being unable to have a desire to eat a tarantula. The case is no different from the life preserver being locked down with an ordinary lock that you have no key for and are unable to smash or pick. You have no direct responsibility, though you might have derivative responsibility if you were responsible for locking down the life preserver.

Now, in case (2), we will want to blame you. You wouldn't have rescued Jones even if you could. But while that does imply a defect of character, it is not a case of direct responsibility for failing to rescue Jones. Again, you may have derivative responsibility if you are responsible for having failed to get started earlier at self-inducing a desire to eat a tarantula. But if you're not responsible for your inability to have a desire to eat a tarantula over the next two minutes, you're not responsible for failing to rescue Jones. Though you might be responsible for failing to want to rescue Jones.

Case (3) isn't significantly different from case (2). If the mind-reading device requires you to have a desire that you are unable to form over the next two minutes, you're not directly responsible for failing to rescue, though again you may be derivatively responsible if you are responsible for your inability to have that desire.

But now consider case (4). Again, this is a case where you are unable to rescue Jones unless you form a certain desire to rescue her in two minutes, and you are unable to form that desire. The same thing as above should be true: you are at most derivatively responsible for failing to rescue Jones. And derivative responsibility requires that you be antecedently responsible for something else, in this case your inability to have over the next two minutes a desire to rescue Jones.

We need one more reflection. If you are not directly responsible in case (4) when you know the facts about your character that are given in (4), you are also not directly responsible in case (4) when he is ignorant of these facts. (You might be responsible for failing to try to induce a desire, but not for failing to induce it or for failing to rescue.[note 1])

There is a lesson here. If you are unable to do something because you're unable to have a mental state, then you're at most going to be derivatively responsible for failing to do it. Moreover this principle should not be limited to failure but needs to be applied to positive action as well: if refraining from an action would take a mental state that you are unable to gain in the time required, you're at most going to be derivatively responsible. But derivative responsibility must ultimately come from direct, non-derivative responsibility. However, if compatibilism is true, then all the things we are responsible for are determined by our motivational states. In no case like that, though, can we have non-derivative responsibility. That was the lesson of the above cases. So if compatibilism is true, there is no non-derivative responsibility, and hence there is no responsibility.

Induction, naturalness and physicalism

Something is grue provided that it is now before the year 3000 and it is green or it's the year 3000 or later and it's blue. From:

1. All observed emeralds were grue

we should not infer that all emeralds will be grue. But from

2. All observed emeralds were green

we should infer that all emeralds will be green. A standard thought (e.g., Sider in his Book book) is that the relevant difference between (1) and (2) is that "green" carves reality more at the joints, is more natural, than "grue".

Suppose that we understand naturalness in a Lewisian way: a concept is more unnatural the longer its expression in a language whose bits refer to perfectly natural stuff. And suppose we think that among the sciences only the terms of fundamental physics refer to perfectly natural stuff. Now consider:

3. All observed electrons were nesitively charged

where an object is nesitively charged provided it's negatively charged and it's before the year 3000 or it's positively charged and it's 3000 or later. We had better not infer that all electrons will be nesitively charged. But "nesitively charged" is an order of magnitude more natural than "green". Consider this beginning of an account of "green":

4. in electromagnetic radiation of the 484-789 THz range, reflecting or transmitting primarily that in the 526-606 THz range.

And this account is not finished. To make this be in terms of the perfectly natural stuff, we'd need to specify the units (terahertz) in microphysical terms, presumably in terms of Planck times or something like that, and we'll get quite messy numbers. Moreover, we need an account of reflection and transmission. I suspect that we can more easily give an account of nesitive charge: "positive" and "negative charge" seem to already be perfectly natural or close to it; the year 3000 is a bit tricky, but we can count it (or maybe just some other "neater" date) in Planck times from the Big Bang.

If naturalness then correlates with brevity of microphysical expression, "green" is not more natural, and probably is less natural, than "nesitive charge". And so we had better not base induction on naturalness.

I think the lesson of this is that we either shouldn't think of degrees of unnaturalness as distance from the perfectly natural, or we shouldn't limit the perfectly natural (even in the concrete realm) to the microphysical. The latter gives us reason to accept some kind of antireductionism about the special sciences and ordinary language.

The neural prosthetic argument against naturalism

While it is unclear whether my mental functioning could survive my getting getting a prosthetic brain, surely it could survive my getting a prosthetic brain part:

1. For any 0.5 centimeter cube in my brain and any machine that functions in exactly the same way with respect to inputs and outputs on the cube boundaries as the neural matter did, it is possible that replacing the cube with the machine would not change my mental functioning.

Claim (1) strengthened by removing "it is possible that" is in fact a key argument for functionalism: roughly, one repeats application of the strengthened claim until the whole brain has been replaced by a functional isomorph. So claim (1) certainly doesn't beg the question against functionalism. And it's pretty plausible.

Yesterday I argued that if functionalism is true, basic mental states are perfectly natural. In comments, Brian Cutter offered some excellent criticisms (though I responded back), but even if Cutter's criticisms are right, we still have:

2. If functionalism is true, the realizers of basic mental states have to be at least fairly natural.

But if we replace a cube of neural matter whose state is a part of the functional realizer of a basic mental state M by a sufficiently complex prosthetic while keeping fixed edge interaction, we can make the corresponding realizer as messy as we like. By (1), mental functioning could be unchanged by this, while (2) tells us that if functionalism is true, we'd have to lose mental state M. So we've argued that

3. If (1) and (2) are true, functionalism is false.

Now, it is actually pretty plausible that:

4. If naturalism is true, functionalism is true.

The naturalistic alternatives to functionalism just don't seem great. So, we have an argument against naturalism based on the possibility of neural prostheses.

Anyway, probably any naturalistic alternative to functionalism will be heavily biological in nature. It will tie mental functioning to organic rather than functional features of our brains. And in so doing, it is apt to violate (1) as well. Or at least it will violate a strengthened version of (1) which says that (1) necessarily holds for any mental being whose cognitive organs have the same kind of functional density that our brains have. For the replacement of a cube by a prosthetic need not change functional density, and then one could do a second replacement, and continue. Finally, by S4 one would conclude that it is possible that mental functioning could continue after total prosthetization of the brain, which would violate the organicity of our naturalistic alternative to functionalism.

So, surprisingly, gradual replacement considerations may favor dualism, not functionalism.

Functionalism, biological antireductionism and dualism

According to functionalism, a mental state such as a pain is characterized by its causal roles. But if one physical state plays the causal role of pain, so do many others and so the characterization fails. For instance, if neural state N plays the causal role of pain in me, so does the conjunction of N with my having blue eyes. One could require minimality of the state, but that won't help. First, plausibly, there is no minimal state that plays the role: if a state plays it, so does that state minus a particle. Second, even if there is one, it is very unlikely to be unique. There is likely to be redundancy, and there will be many ways of getting rid of redundancy.

The solution to this problem in the spirit of Lewisian functionalism is to restrict one's quantifiers to natural states. There are two ways of doing this. First, we could restrict the quantifiers to states which are sufficiently natural, whose degree of unnaturalness is below some threshold. (An obvious way to measure unnaturalness is to measure the length of the shortest linguistic expression taht expresses the state in terms that are perfectly natural.) But this is unlikely to work. If mental states have degreed unnaturalness, presumably there will be a lot of variation in the degree of unnaturalness. Some mental states will, for instance fall far below the threshold. Those states could then be made slightly more complicated while still staying below the threshold, so once again we would have a problem.

So we better restrict quantifiers to perfectly natural states, at least in the case of the basic mental states (or maybe protomental states—I won't distinguish these) out of which more complex ones are built. Thus we have our first conclusion:

1. If functionalism is true, basic mental states are perfectly natural.

This has an interesting corollary. Presumably no macroscopic state of a purely physical computer is perfectly natural. Thus:

2. If functionalism is true, a purely physical computer has no basic mental states, and hence no mental states.

Thus, the only way a computer could have mental states is if it wasn't purely physical (Richard Swinburne once suggested to me that if a computer had the right functional complexity, God could create a soul for it.)

What about organisms? Well, if organisms are purely physical, then their mental states will be biological states (subject to evolution and the like). So:

2. If functionalism is true, then some of the biological states of a minded purely physical organism are perfectly natural.

This is an antireductionist conclusion. Thus,

3. Functionalism implies that all minded organisms have non-physical states (dualism) or some minded organisms have perfectly natural biological states (antireductionism) (or both).

Moreover, our best account of naturalness is that it is fundamentality. If that is the right account, then our antireductionism is pretty strong: it says that some biological states are fundamental.

Moreover, functionalism is the only tenable version of physicalism (I say). Thus:

4. Physicalism implies biological antireductionism.

Deep Thoughts XXXVIII

If you don't try, you won't fail.