The interpersonal Satan's Apple

Consider a moral interpersonal version of Satan’s Apple: infinitely many people independently choose whether to give a yummy apple to a (different) hungry child, and if infinitely many choose to do so, some calamity happens to everyone, a calamity outweighing the hunger the child suffers. You’re one of the potential apple-givers and you’re not hungry yourself. The disaster strikes if and only if infinitely many people other than you give an apple. Your giving an apple makes no difference whatsoever. So it seems like you should give the apple to the child. After all, you relieve one child’s hunger, and that’s good whether or not the calamity happens.

Now, we deontologists are used to situations where a disaster happens because one did the right thing. That’s because consequences are not the only thing that counts morally, we say. But in the moral interpersonal Satan’s Apple, there seems to be no deontology in play. It seems weird to imagine that disaster could strike because everyone did what was consequentialistically right.

One way out is causal finitism: Satan’s Apple is impossible, because the disaster would have infinitely many causes.

More on discounting small probabilities

In yesterday’s post, I argued that there is something problematic about the idea of discounting small probabilities, given that in a large enough lottery every possibility with has a small probability. I then offered a way of making sense of the idea by “trimming” the utility function at the top and bottom.

This morning, however, I noticed that one can also take the idea of discounting small probabilities more literally and still get the exact same results as by trimming utility functions. Specifically, given a probability function P and a probability discount threshold ϵ, we form a credence function P_ϵ by letting P_ϵ(A) = P(A) if ϵ ≤ P(A) ≤ 1 − ϵ, P_ϵ(A) = 0 if P(A) < ϵ and P_ϵ(A) = 1 if P(A) > 1 − ϵ. This discounts close-to-zero probabilities to zero and raises close-to-one probabilities to one. (We shouldn’t forget the second or things won't work well.)

Of course, P_ϵ is not in general a probability, but it does satisfy the Zero, Non-Negativity, Normalization and Monotonicity axioms, and we can now use LSI^↑ level-set integral to calculate utilities with P_ϵ.

If U_ϵ is the “trimmed” utility function from my previous post, then LSI^↑_Pϵ(U) = E(U_2ϵ), so the two approaches are equivalent.

One can also do the same thing within Buchak’s REU theory, since that theory is equivalent to applying LSI^↑ with a probability transformed by a monotonic map of [0,1] to [0,1] keeping endpoints fixed, which is exactly what I did when moving from P to P_ϵ.

How to discount small probabilities

A very intuitive solution to a variety of problems in infinite decision theory is that “for possibilities that have very small probabilities of occurring, we should discount those probabilities down to zero” when making decisions (Monton).

Suppose throughout this post that ϵ > 0 counts as our threshold of “very small probabilities”. No doubt ϵ < 1/100.

In this post I want to offer a precise and friendly amendment to the solution of neglecting small probabilities. But first why we need an amendment. Consider a game where an integer K is randomly chosen between  − 1 and N for some large fixed positive N, so large that 1/(2+N) < ϵ, and you get K dollars. The game is clearly worth playing. But if you discount “possibilities that have very small probabilities”, you are left with nothing: every possibility has a very small probability!

Perhaps this is uncharitable. Maybe the idea is not that we discount to zero all possibilities with small probabilities, but that we discount such possibilities until the total discount hits the threshold ϵ. But while this sounds like a charitable interpretation of the suggestion, it leaves the theory radically underdetermined. For which possibilities do we discount? In my lottery case, do we start by discounting the possibilities at the low end ( − 1, 0, 1, ...) until we have hit the threshold? Or do we start at the high end (N, N − 1, N − 2, ...) or somewhere in the middle?

Here is my friendly proposal. Let U be the utility function we want to evaluate the value of. Let T be the smallest value such that P(U>T) ≤ ϵ/2. (This exists: T = inf {λ : P(U>λ) ≤ ϵ/2}.) Let t be the largest value such that P(U<t) ≤ ϵ/2 (i.e., t = sup {λ : P(U<λ) ≤ ϵ/2}). Take U and replace any values bigger than T with T and any values smaller than t with t, and call the resulting utility function U_ϵ. We now replace U with U_ϵ in our expected value calculations. (In the lottery example, we will be trimming from both ends at the same time.)

The result is a precise theory (given the mysterious threshold ϵ). It doesn’t neglect all possibilities with small probabilities, but rather it trims low-probability outliers. The trimming procedure respects the fact that often utility functions are defined up to positive affine transformations.

Moreover, the trimming procedure can yield an answer to what I think is the biggest objection to small-probability discounting, namely that in a long enough run—and everyone should think there is a non-negligible chance of eternal life—even small probabilities can add up. If you are regularly offered the same small chance of a gigantic benefit during an eternal future, and you turn it down each time because the chance is negligible, you’re almost surely missing out on an infinite amount of value. But we can apply the trimming procedure at the level of choice of policies rather than of individual decisions. Then if small chances are offered often enough, they won’t all be trimmed away.

A principle about infinite sequences of decisions

There are many paradoxes of infinite sequences of decisions where the sequence of individual decisions that maximize expected utility is unfortunate. Perhaps the most vivid is Satan’s Apple, where a delicious apple is sliced into infinitely many pieces, and Eve chooses which pieces to eat. But if she greedily takes infinitely many, she is kicked out of paradise, an outcome so bad that the whole apple does not outweigh it. For any set of pieces Eve eats, another piece is only a plus. So she eats them all, and is damned.

Here is a plausible principle:

1. If at each time you are choosing between a finite number of betting portfolios fixed in advance, with the betting portfolio in each decision being tied to a set of events wholly independent of all the later or earlier events or decisions, with the overall outcome being just the sum or aggregation of the outcomes of the betting portfolios, and with the utility of each portfolio well-defined given your information, then you should at each time maximize utility.

In Satan’s Apple, for instance, the overall outcome is not just the sum of the outcomes of the individual decisions to eat or not to eat, and so Satan’s Apple is not a counterexample to (1). In fact, few of the paradoxes of infinite sequences of decisions are counterexamples to (1).

However, my unbounded expected utility maximization paradox is.

I don’t know if there is something particularly significant about a paradox violating (1). I think there is, but I can’t quite put my finger on it. On the other hand, (1) is such a complex principle that it may just seem ad hoc.

Must we accept free stuff?

Suppose someone offers you, at no cost whatsoever, something of specified positive value. However small that value, it seems irrational to refuse it.

But what if someone offers you a random amount of positive value for free. Strict dominance principles say it’s irrational to refuse it. But I am not completely sure.

Imagine a lottery where some positive integer n is picked at random, with all numbers equally likely, and if n is picked, then you get 1/n units of value. Should you play this lottery for free?

The expected value of the lottery is zero with respect to any finitely-additive real-valued probability measure that fits the description (i.e., assign equal probablity to each number). And for any positive number x, the probability that you will get less than x is one. It’s not clear to me that it’s worth going for this.

If you like infinitesimals, you might say that the expected value of the lottery is infinitesimal and the probability of getting less than some positive number x is 1 − α for an infinitesimal α. That makes it sound like a better deal, but it’s not all that clear.

Of course, infinite fair lotteries are dubious. So I don’t set much store by this example.

Two different ways of non-instrumentally pursuing a good

Suppose Alice is blind to the intrinsic value of friendship and Bob can see the intrinsic value of friendship. Bob then told Alice that friendship is intrinsically valuable. Alice justifiedly trusts Bob in moral matters, and so Alice concludes that friendship has intrinsic value, even though she can’t “see” it. Alice and Bob then both pursue friendship for its own sake.

But there is a difference: Bob pursues friendship because of the particular ineffable “thick” kind of value that friendship has. Alice doesn’t know what “thick” kind of value friendship has, but on the basis of Bob’s testimony, she knows that it has some such value or other, and that it is a great and significant value. As long as Alice knows what kinds of actions friendship requires, she can pursue friendship without that knowledge, though it’s probably more difficult for her, perhaps in the way that it is more difficult for a tone-deaf person to play the piano, though in practice the tone-deaf person could learn what kinds of finger movements result in aesthetically valuable music without grasping that aesthetic value.

The Aristotelian tradition makes the grasp of the particular thick kind of value involved in a virtuous activity be a part of the full possession of that virtue. On that view, Alice cannot have the full virtue of friendship. There is something she is missing out on, just as the tone-deaf pianist is missing out on something. But she is not, I think, less praiseworthy than Bob. In fact Alice’s pursuit of friendship involves the exercise of a virtue which Bob’s does not: the virtue of faith, as exhibited in Alice’s trust in Bob’s testimony about the value of friendship.

Pursuing a thing for its own sake

Suppose you pursue truth for its own sake. As we learn from Aristotle, it does not follow that you don’t pursue truth for the sake of something else. For the most valuable things are both intrinsically and instrumentally valuable, and so they are typically pursued both for their own sake and for the sake of something else.

What if you pursue something, but not for the sake of something else. Does it follow that you pursue the thing for its own sake? Maybe, but it’s not as clear as it might seem. Imagine that you eat fiber for the sake of preventing colon cancer. Then you hear a study that says that fiber doesn’t prevent colon cancer. But you continue to eat fiber, out of a kind of volitional inertia, without any reason to do so. Then you are pursuing the consumption of fiber not for the sake of anything else. But merely losing the instrumental reason for eating fiber doesn’t give you a non-instrumentally reason. Rather, you are now eating fiber irrationally, for no reason.

Perhaps it is impossible to do something for no reason. But even if it is impossible to do something for no reason, it is incorrect to define pursuing something for its own sake as pursuing it not for the sake of something else. For that you pursue something for its own sake states something positive about your pursuit, while that you don’t pursue it for the sake of anything else states something negative about your pursuit. There is a kind of valuing of the thing for its own sake that is needed to pursue the thing for its own sake.

It is tempting to say that you pursue a thing for its own sake provided that you pursue it because of the intrinsic value you take it to have. But that, too, is incorrect. For suppose that a rich benefactor tells you that they will give you a ton of money if you gain something of intrinsic value today. You know that truth is valuable for its own sake, so you find out something. In doing so, you find out the truth because the truth is intrinsically valuable. But your pursuit of that truth is entirely instrumental, despite your reason being the intrinsic value.

Hence, to pursue a thing for its own sake is not the same as to pursue it because it has intrinsic value. Nor is it to pursue it not for the sake of something else.

I suspect that pursuing a thing for its own sake is a primitive concept.

Human worth and materialism

1. A typical human being has much more intrinsic value than any 80 kg arrangement of atoms.

2. If materialism is true, a typical human being is an 80 kg arrangement of atoms.

3. So, materialism is not true.

Transsubstantiation and magnets

On Thomistic accounts of transsubstantiation, the accidents of bread and wine continue to exist even when the substance no longer does (having been turned into the substance of Christ’s body and blood). This seems problematic.

Here is an analogy that occurred to me. Consider a magnet. It’s not crazy to think of the magnet’s magnetic field as an accident of the magnet. But the magnetic field extends spatially beyond the magnet. Thus, it exists in places where the magnet does not.

Now, according to four-dimensionalism, time is rather like space. If so, then an accident existing when its substance does not is rather like an accident existing where its substance does not. Hence to the four-dimensionalist, the magnet analogy should be quite helpful.

Actually, if we throw relativity into the mix, then we can get an even closer analogy, assuming still that a magnet’s field is an accident of the magnet. Imagine that the magnet is annihilated. The magnetic field disappears, but gradually, starting near the magnet, because all effects propagate at most at the speed of light. Thus, even when the magnet is destroyed, for a short period its magnetic field still exists.

That said, I don’t know if the magnet’s field is an accident of it. (Rob Koons in conversation suggested it might be.) But it’s comprehensible to think of it as such, and hence the analogy makes Thomistic transsubtantiaton comprehensible, I think.

Does our ignorance always grow when we learn?

Here is an odd thesis:

1. Whenever you gain a true belief, you gain a false belief.

This follows from:

2. Whenever you gain a belief, you gain a false belief.

The argument for (2) is:

3. You always have at least one false belief.

4. You believe a conjunction if and only if you believe the conjuncts.

5. Suppose you just gained a belief p.

6. There is now some false belief q that you have. (By (3))

7. Before you gained the belief p you didn’t believe the conjunction of p and q. (By (4))

8. So, you just gained the belief in the conjunction of p and q. (By (5) and (7))

9. The conjunction of p and q is false. (By (6))

10. So, you just gained a false belief. (By (8) and (9))

I am not sure I accept (4), though.

“Accuracy, probabilism and Bayesian update in infinite domains”

The paper has just come out online in Synthese.

Abstract: Scoring rules measure the accuracy or epistemic utility of a credence assignment. A significant literature uses plausible conditions on scoring rules on finite sample spaces to argue for both probabilism—the doctrine that credences ought to satisfy the axioms of probabilism—and for the optimality of Bayesian update as a response to evidence. I prove a number of formal results regarding scoring rules on infinite sample spaces that impact the extension of these arguments to infinite sample spaces. A common condition in the arguments for probabilism and Bayesian update is strict propriety: that according to each probabilistic credence, the expected accuracy of any other credence is worse. Much of the discussion needs to divide depending on whether we require finite or countable additivity of our probabilities. I show that in a number of natural infinite finitely additive cases, there simply do not exist strictly proper scoring rules, and the prospects for arguments for probabilism and Bayesian update are limited. In many natural infinite countably additive cases, on the other hand, there do exist strictly proper scoring rules that are continuous on the probabilities, and which support arguments for Bayesian update, but which do not support arguments for probabilism. There may be more hope for accuracy-based arguments if we drop the assumption that scores are extended-real-valued. I sketch a framework for scoring rules whose values are nets of extended reals, and show the existence of a strictly proper net-valued scoring rules in all infinite cases, both for f.a. and c.a. probabilities. These can be used in an argument for Bayesian update, but it is not at present known what is to be said about probabilism in this case.

Choices on a spectrum

My usual story about how to reconcile libertarianism with the Principle of Sufficient Reason is that when we choose, we choose on the basis of incommensurable reasons, some of which favor the choice we made and others favor other choices. Moreover, this is a kind of constrastive explanation.

This story, though it has some difficulties, is designed for choices between options that promote significantly different goods—say, whether to read a book or go for a walk or write a paper.

But a different kind of situation comes up for choices of a point on a spectrum. For instance, suppose I am deciding how much homework to assign, how hard a question to ask on an exam, or how long a walk to go for. What is going on there?

Well, here is a model that applies to a number of cases. There are two incommensurable goods one better served as one goes in one direction in the spectrum and the other better served as one goes in the other direction in the spectrum. Let’s say that we can quantify the spectrum as one from less to more with respect to some quantity Q (amount of homework, difficulty of a question or length of a walk), and good A is promoted by less of Q and incommensurable good B is promoted by more of Q. For instance, with homework, A is the student’s having time for other classes and for non-academic pursuits and B is the student’s learning more about the subject at hand. With exam difficulty, A may be avoiding frustration and B is giving a worthy challenge. With a walk, A is reducing fatigue and B is increasing health benefits. (Note that the claim that A is promoted by less Q and B is promoted by more Q may only be correct within a certain range of Q. A walk that is too long leads to injury rather than health.)

So, now, suppose we choose Q = Q₁. Why did one choose that? It is odd to say that one chose Q on account of reasons A and B that are opposed to each other—that sounds inconsistent.

Here is one suggestion. Take the choice to make Q equal to Q₁ to be the conjunction of two (implicit?) choices:

1. Make Q at most Q₁

2. Make Q at least Q₁.

Now, we can explain choice (a) in terms of (a) serving good A better than the alternative, which would be to make Q be bigger than Q₁. And we can explain (b) in terms of (b) serving good B better than the alternative of making Q be smaller.

Here is a variant suggestion. Partition the set of options into two ranges R₁, consisting of options where Q < Q₁ and R₂, where Q > Q₁. Why did I choose Q = Q₁? Well, I chose Q over all the choices in R₁ because Q better promotes B than anything in R₁, and I chose Q over all the choices in R₂ because Q better promotes A than anything in R₁.

On both approaches, the apparent inconsistency of citing opposed goods disappears because they are cited to explain different contrasts.

Note that nothing in the above explanatory stories requires any commitment to there being some sort of third good, a good of balance or compromise between A and B. There is no commitment to Q₁ being the best way to position Q.

Simplicity and gravity

I like to illustrate the evidential force of simplicity by noting that for about two hundred years people justifiably believed that the force of gravity was Gm₁m₂/r² even though Gm₁m₂/r^2 + ϵ fit the observational data better if a small enough but non-zero ϵ. A minor point about this struck me yesterday. There is doubtless some p ≠ 2 such that Gm₁m₂/r^p would have fit the observational data better. For in general when you make sufficiently high precision measurements, you never find exactly the correct value. So if someone bothered to collate all the observational data and figure out exactly which p is the best fit (e.g., which one is exactly in the middle of the normal distribution that best fits all the observations), the chance that that number would be 2 up to the requisite number of significant figures would be vanishingly small, even if in fact the true value is p = 2. So simplicity is not merely a tie-breaker.

Note that our preference for simplicity here is actually infinite. For if we were to collate the data, there would not just be one real number that fits the data better than 2 does, but a range J of real numbers that fits the data better than 2. And J contains uncountably many real numbers. Yet we rightly think that 2 is more likely than the claim that the true exponent is in J, so 2 must be infinitely more likely than most of the numbers in J.

Bayesian reasoning isn't our duty

Ought implies can. Most people can’t do Bayesian reasoning correctly. So Bayesian reasoning is not how they ought to reason. In particular, a reduction of epistemic ought to the kinds of probability fcts that are involved in Bayesian reasoning fails.

I suppose the main worry with this argument is that perhaps only an ought governing voluntary activity implies can. But the epistemic life is in large part involuntary. An eye ought to transmit visual information, but some eyes cannot—and that is not a problem because seeing is involuntary.

However, it is implausible to think that we humans ought to do something that nobody has been able to do until recently and even now only a few can do, and only in limited cases, even if the something is involuntary.

If Bayesian reasoning isn’t how we ought to reason, what’s the point of it? I am inclined to think it is a useful tool for figuring out the truth in those particular cases to which it is well suited. There are different tools for reasoning in different situations.

Probabilistic trolleys

Suppose a trolley is heading towards five people, and you can redirect it towards one. But the trolley needs to go up a hill before it can roll down it to hit the five people, and your best estimate of its probability of making it up the hill is 1/4. On the other hand, if you redirect it, it’s a straight path to the one person, who is certain to be killed. Do you redirect? Expected utilities:  − 1.25 lives for not redirecting and  − 1 lives for redirecting.

Or suppose you are driving a fire truck to a place where five people are about to die in a fire, and you know that you have a 1/4 chance of putting out the fire and saving them if you get there in time. Moreover, there is a person sleeping on the road in front of the only road to the fire, and if you stop to remove the person from the road, it will be too late for the five. Do you brake? Expected utilities:  − 5 lives for braking and  − 1 − 3.75 =  − 4.75 lives for continuing to the fire and running over the person on the road.

I think you shouldn’t redirect and you should brake. There is something morally obnoxious about certainly causing death for a highly uncertain benefit when the expected values are close. This complicates the proportionality condition in the Principle of Double Effect even more, and provides further evidence against expected-value utilitarianism.