A failed Deep Thought

I was going to post the following as Deep Thoughts XLIII, in a series of posts meant to be largely tautologous or at least trivial statements:

1. Everyone older than you was once your age.

And then I realized that this is not actually a tautology. It might not even be true.

Suppose time is discrete in an Aristotelian way, so that the intervals between successive times are not always the same. Basically, the idea is that times are aligned with the endpoints of change, and these can happen at all sorts of seemingly random times, rather than at multiples of some interval. But in that case, (1) is likely false. For it is unlikely that the random-length intervals of time in someone else’s life are so coordinated with yours that the exact length of time that you have lived equals the sum of the lengths of intervals from the beginning to some point in the life of a specific other person.

Of course, on any version of the Aristotelian theory that fits with our observations, the intervals between times are very short, and so everyone older than you was once approximately your age.

One might try to replace (1) by:

2. Everyone older than you was once younger than you are now.

But while (2) is nearly certainly true, it is still not a tautology. For if Alice has lived forever, then she’s older than you, but she was never younger than you are now! And while there probably are no individuals who are infinitely old (God is timelessly eternal), this fact is far from trivial.

Punishment, causation and time

I want to argue for this thesis:

1. For a punishment P for a fault F to be right, P must stand in a causal-like relation to P.

What is a causal-like relation? Well, causation is a causal-like relation. But there is probably one other causal-like relation, namely when because of the occurrence of a contingent event E, God knows that E occurred, and this knowledge in turn explains why God did something. This is not exactly causation, because God is not causally affected by anything, but it is very much like causation. If you don’t agree, then just remove the ``like’’ from (1).

Thesis (1) helps explain what is wrong with punishing people on purely statistical grounds, such as sending a traffic ticket to Smith on the grounds that Smith has driven 30,000 miles in the last five years and anyone who drove that amount must have committed a traffic offense.

Are there other arguments against (1)? I think so. Consider forward-looking punishment where by knowing someone’s present character you know that they will commit some crime in ten days, so you punish them now (I assume that they will commit the crime even if you do not punish them). Or, even more oddly, consider circular forward-looking punishment. Suppose Alice has such a character that it is known that if we jail her, she will escape from jail. But assume that our in society an escape from jail is itself a crime punishable by jail, and that Alice is not currently guilty of anything. We then jail her, on the grounds that she will escape from jail, for which the punishment is us now jailing her.

One may try to rule out the forward-looking cases on the grounds that instead of (1) we should hold:

2. For a punishment P for a fault F to be right, P must come after F.

But that’s not right. Simultaneous causation seems possible, and it does not seem unjust to set up a system where a shoplifter feels punitive pain at the very moment of the shoplifting, as long as the pain is caused by the shoplifting.

Or consider this kind of a case. You know that Bob will commit a crime in ten days, so you set up an automated system that will punish him at a preset future date. It does not seem to be of much significance whether the system is set to go off in nine or eleven days.

Or consider cases where Special Relativity is involved, and the punishment occurs at a location distant from the criminal. For instance, Carl, born on Earth, could be sentenced to public infamy on earth for a crime he commits around Alpha Centauri. Supposing that we have prior knowledge that he will commit the crime on such and such a date. If (2) is the right principle, when should we make him infamous on earth? Presumably after the crime. But in what reference frame? That seems a silly question. It is silly, because (2) isn’t the right principle—(1) is better.

Objection: One cannot predict what someone will freely do.

Response: One perhaps cannot predict with 100% certainty what someone will freely do, but punishment does not require 100% certainty.

Punishment, reward and theistic natural law

I’ve always found punishment and (to a lesser extent) reward puzzling. Why is it that when someone does something wrong is there moral reason to impose a harsh treatment on them, and why is it that when someone does something right—and especially supererogatory—is there moral reason to do something nice for them?

Of course, it’s easy to explain why it’s good for our species that there be a practice of reward and punishment: such a practice in obvious ways helps to maintain a cooperative society. But what makes it morally appropriate to impose a sacrifice on the individual for the good of the species in this way, whether the good of the person receiving the punishment or the good of the person giving the reward when the reward has a cost?

Punishment and reward thus fit into a schema where we would like to be able to make use of this argument form:

1. It would be good (respectively, bad) for humans if moral fact F did (did not) obtain.

2. Thus, probably, moral fact F does obtain.

(The argument form is better on the parenthetical negative version.) It would be bad for humans if we did not have distinctive moral reasons to reward and punish, since our cooperative society would be more liable to fall apart due to cheating, freeriding and neglect of others. So we have such moral reasons.

As I have said on a number of occasions, we want a metaethics on which this is a good argument. Rule-utilitarianism is such a metaethics. So is Adams’ divine command theory with a loving God. And so is theistic natural law, where God chooses which natures to exemplify because of the good features in these natures. I want to say something about this last option in our case, and why it is superior to the others.

Human nature encodes what is right and wrong for. Thus, it can encode that it is right for us to punish and reward. An answer as to why it’s right for us to reward and punish, then, is that God wanted to make cooperative creatures, and chose a nature of cooperative creatures that have moral reasons to punish and reward, since that improves the cooperation.

But there is a way that the theistic natural law solution stands out from the others: it can incorporate Boethius’ insight that it is intrinsically bad for one to get away unpunished with wrongdoing. For our nature not only encodes what is right and wrong for us to do, but also what is good or bad for us. And so it can encode that it is bad for us to get away unpunished. It is good for us that it be bad for us to get away unpunished, since its being bad for us to get away unpunished means that we have additional reason to avoid wrongdoing—if we do wrong, we either get punished or we get away unpunished, and both options are bad for us.

The rule-utilitarian and divine-command options only explain what is right and wrong, not what is good and bad, and so they don’t give us Boethius’ insight.

What is an existential quantifier?

What is an existential quantifier?

The inferentialist answer is that an existential quantifier is any symbol that has the syntactic features of a one-place quantifier and obeys the same logical rules of an existential quantifier (we can precisely specify both the syntax and logic, of course). Since Carnap, we’ve had good reason to reject this answer (see, e.g., here).

Here is a modified suggestion. Consider all possible symbols that have the syntactic features of a one-place quantifier and obeys the rules of an existential quantifier. Now say that a symbol is an existential quantifier provided that it is a symbol among these symbols that maximizes naturalness, in the David Lewis sense of “naturalness”.

Moreover, this provides the quantifier variantist or pluralist (who thinks there are multiple existential quantifiers, none of them being the existential quantifier) with an answer to a thorny problem: Why not simply disjoin all the existential quantifiers to make a truly unrestricted existential quantifier, and say that that is the existential quantifier? THe quantifier variantist can say: Go ahead and disjoin them, but a disjunction of quantifiers is less natural than its disjuncts and hence isn’t an existential quantifier.

This account also allows for quantifier variance, the possibility that there is more than one existential quantifier, as long as none of these existential quantifiers is more natural than any other. But it also fits with quantifier invariance as long as there is a unique maximizer of naturalness.

Until today, I thought that the problem of characterizing existential quantifiers was insoluble for a quantifier variantist. I was mistaken.

It is tempting to take the above to say something deep about the nature of an existential quantifier, and maybe even the nature of being. But I think it doesn’t quite. We have a characterization of existential quantifiers among all possible symbols, but this characterization doesn’t really tell us what they mean, just how they behave.

Combining epistemic utilities

Suppose that the right way to combine epistemic utilities or scores across individuals is averaging, and I am an epistemic act expected-utility utilitarian—I act for the sake of expected overall epistemic utility. Now suppose I am considering two different hypotheses:

• Many: There are many epistemic agents (e.g., because I live in a multiverse).

• Few: There are few epistemic agents (e.g., because I live in a relatively small universe).

If Many is true, given averaging my credence makes very little difference to overall epistemic utility. On Few, my credence makes much more of a difference to overall epistemic utility. So I should have a high credence for Few. For while a high credence for Few will have an unfortunate impact on overall epistemic utility if Many is true, because the impact of my credence on overall epistemic utility will be small on Many, I can largely ignore the Many hypothesis.

In other words, given epistemic act utilitarianism and averaging as a way of combining epistemic utilities, we get a strong epistemic preference for hypotheses with fewer agents. (One can make this precise with strictly proper scoring rules.) This is weird, and does not match any of the standard methods (self-sampling, self-indication, etc.) for accounting for self-locating evidence.

(I should note that I once thought I had a serious objection to the above argument, but I can't remember what it was.)

Here’s another argument against averaging epistemic utilities. It is a live hypothesis that there are infinitely many people. But on averaging, my epistemic utility makes no difference to overall epistemic utility. So I might as well believe anything on that hypothesis.

One might toy with another option. Instead of averaging epistemic utilities, we could average credences across agents, and then calculate the overall epistemic utility by applying a proper scoring rule to the average credence. This has a different problematic result. Given that there are at least billions of agents, for any of the standard scoring rules, as long as the average credence of agents other than you is neither very near zero nor very near one, your own credence’s contribution to overall score will be approximately linear. But it’s not hard to see that then to maximize expected overall epistemic utility, you will typically make your credence extreme, which isn’t right.

If not averaging, then what? Summing is the main alternative.

Closed time loop

Imagine two scenarios:

1. An infinitely long life of repetition of a session meaningful pleasure followed by a memory wipe.

2. A closed time loop involving one session of the meaningful pleasure followed by a memory wipe.

Scenario (1) involves infinitely many sessions of the meaningful pleasure. This seems better than having only one session as in (2). But subjectively, I have a hard time feeling any preference for (1). In both cases, you have your pleasure, and it’s true that you will have it again.

I suppose this is some evidence that we’re not meant to live in a closed time loop. :-)

Shuffling an infinite deck

Suppose infinitely many blindfolded people, including yourself, are uniformly randomly arranged on positions one meter apart numbered 1, 2, 3, 4, ….

Intuition: The probability that you’re on an even-numbered position is 1/2 and that you’re on a position divisible by four is 1/4.

But then, while asleep, the people are rearranged according to the following rule. The people on each even-numbered position 2n are moved to position 4n. The people on the odd numbered positions are then shifted leftward as needed to fill up the positions not divisible by 4. Thus, we have the following movements:

• 1 → 1

• 2 → 4

• 3 → 2

• 4 → 8

• 5 → 3

• 6 → 12

• 7 → 5

• 8 → 16

• 9 → 6

• and so on.

If the initial intuition was correct, then the probability that now you’re on a position that’s divisible by four is 1/2, since you’re now on a position divisible by four if and only if initially you were on a position divisible by two. Thus it seems that now people are no longer uniformly randomly arranged, since for a uniform arrangement you’d expect your probability of being in a position divisible by four to be 1/4.

This shows an interesting difference between shuffling a finite and an infinite deck of cards. If you shuffle a finite deck of cards that’s already uniformly distributed, it remains uniformly distributed no matter what algorithm you use to shuffle it, as long as you do so in a content-agnostic way (i.e., you don’t look at the faces of the cards). But if you shuffle an infinite deck of distinct cards that’s uniformly distributed in a content-agnostic way, you can destroy the uniform distribution, for instance by doubling the probability that a specific card is in a position divisible by four.

I am inclined to take this as evidence that the whole concept of a “uniformly shuffled” infinite deck of cards is confused.