A false trilemma

Kant writes:

only three [forms of sovereignty] are possible: namely, either only one, or some in association, or all those together who constitute the civil society possess sovereign power (autocracy, aristocracy, and _democracy …).

The division of the sovereigns into “one”, “some” or “all” members of a society sure sounds like an exhaustive division, at least assuming that there is a sovereign rather than anarchy, which is a fair assumption, since an anarchy probably doesn’t count as a “civil society”.

But not quite! For what if there is a sovereign, but the sovereign is not a member of the civil society? I can think of at least two possibilities like that.

For, a civil society is a mutually interacting body of persons. Thus, first, we could have a personal sovereign with a one-way relationship with the civil society, where the sovereign rules but is in no way affected by what happens in the society. Since causal interaction among embodied beings is always bidirectional, the sovereign would need to be a supernatural being, such as God.

Second, we could have a sovereign that is not a person—a robot overlord. Since only persons can be members of society, such a sovereign would not be a member of society.

One might object that the robot overlord isn’t really a sovereign, since it doesn’t have a will of its own. But it is evident that one could have a fairly functional civil society run by the dictates of a robot overlord, enforced by human or robotic minions, or simply by people’s confidence that the robot overlord’s plan is a good one. If so, and if a robot overlord is not a sovereign, we have a counterexample to the thesis that anarchy is not a civil society.

Making the happiness of others our end

Kant says that a virtuous persons has the happiness of others as an end.

Note that this is not an end in the sense of a goal which we try to achieve. For the goal can’t be that all others are happy. If that were the goal, we might as well not even try: we’re not going to achieve it. Nor can the goal be that some others are happy. For that’s already so, and so nothing need be done. Could the goal be that many others are happy? I suppose that my actions could make a difference between not-many and many being happy. But it seems rather unlikely that they could. And if a careful examination of the world were to show that independently of my actions many others are happy, again such a goal would lead to my not having to do anything.

Maybe the Kantian end of happiness is distributive. For each person distinct from yourself, you aim at this person’s being happy. So you have a large number of goals, one for each person distinct from yourself. But that’s not right, either. For suppose that Alice is miserable, and you know you can’t make her happy. But you can make her happier. Surely that’s a part of what the virtuous person aims at. So is it, perhaps, that in the case of each person distinct from yourself, you aim to make them happier than they currently are? However, imagine that Alice is miserable, but you know that her condition will at least slightly improve over time, no matter what you do. Then you don’t need to do anything to make her happier, and so a goal of making others happier doesn’t make you do anything for Alice—even if you could make Alice much happier.

Perhaps, though, the end of happiness is doubly distributive: it distributes over persons and over what you might call “possible pieces of happiness”. For each person x and each possible piece H of happiness, you aim at x having H. But now the virtuous person’s aims are vast, since there are many, many possible pieces of happiness—maybe even infinitely many. Maybe this is right, but it seems implausible.

All this makes me think that when Kant talks of the happiness of others as our end, he is not talking of an end as a set of specific goals to be achieved. Maybe he is saying that in general a virtuous person has a tendency to perform actions with specific goals of the form: Make Alice happier with respect to her toothache. The “end” of happiness isn’t an end, or a multiplicity of ends even, but a kind of architectonic pattern in our goals.

Self-locating evidence and bearers of epistemic good

In the case of non-epistemic goods, it’s an obvious feature of life that someone there is a choice to be made by an individual between their own first-order good and the first-order good of the community—each requires the sacrifice of the other. In the case of epistemic goods, this is less obvious.

In the pragmatic case, the typical reason for such competition between goods is due to limited resources. This, of course, also happens in the epistemic sphere. Suppose Alice is much more intellectually talented than Bob, but only Bob has the money to go to university. If Bob spends the money on himself, he will gain private epistemic goods, but will contribute little epistemically to society as a whole. But if he gives the money to Alice, she may become a brilliant scholar or scientist, significantly contributing to society’s knowledge.

More interesting than these, however, are cases of competition between private and communal epistemic goods that are not due to epistemic resources. I find it interesting that some cases of self-locating evidence appear to be such.

Suppose there are ten billion people in the world, currently isolated from one another. A device produced by a mad scientist has a 99.9% chance at noon today of triggering a death ray that randomly kills 99.9999% of the world population. Noon has just passed. You are still alive. Should you think the device worked? Sleeping Beauty style arguments say “No”. This time I want to think about this in terms of individual epistemic goods. In N runs of the device, 0.001N runs will have you survive because the device doesn’t trigger and 0.999 ⋅ 0.000001N runs will have you survive despite the device triggering. Thus, the vast majority of the runs where you survive are runs where the device didn’t trigger. Hence, it’s best for you individually to adopt the epistemic policy of thinking the device didn’t trigger.

But on the other hand, suppose we all adopt the epistemic policy of thinking the device didn’t trigger. Then 99.9% of the time, we are unanimously collectively wrong. And if we all adopt the epistemic policy of thinking the device did trigger, then 99.9% of the time, we are unanimously collectively right. It seems thus that if we look at the epistemic goods of society, then a policy of thinking the device did trigger is best.

If this is right, it points to a potential diagnosis of why the problems about self-locating evidence (doomsday, multiverses, Sleeping Beauty, etc.) are so difficult. For there may be different bearers of epistemic goods at play—say, society vs. the individual—and it could be that different answers are appropriate depending on whose goods we are pursuing. Maybe.

Is there some sort of a probability problem with a humongous but finite universe?

It’s easy to generate probabilistic paradoxes in a universe (or multiverse) with infinitely many people (e.g., if infinitely many people roll a die, equal numbers of people get 1 as get more than 1, so why think it’s more likely to get more than 1?). But what about a very large but finite universe? I used to think: “The only relevant difference is between finite and infinite. Really big but finite—no problem.” Now I am not so sure.

Paul Heyl measured the gravitational constant G as 6.670 × 10⁻¹¹ m³ kg⁻¹ s⁻², and denote the latter quantity by G₀. Consider two theories:

• H₁: The gravitational constant is between 6.665 × 10⁻¹¹ m³ kg⁻¹ s⁻² and 6.675 × 10⁻¹¹ m³ kg⁻¹ s⁻².

• H₂: The gravitational constant is between 7.676 × 10⁻¹¹ m³ kg⁻¹ s⁻² and 7.686 × 10⁻¹¹ m³ kg⁻¹ s⁻².

It seems obvious that:

1. Heyl’s measurement strongly supports H₁ but does not completely rule out H₂.

But let’s think this through. Suppose Heyl’s evidence is the proposition E which he would express as “I measured G to be G₀.” But, very plausibly, it is an essential property of a human being that they exist in a world with such-and-such a gravitational constant. One way of getting to this conclusion is to say that the forces of gravity are part of our causal history, and then to apply the essentiality of origins. Another is to say that we couldn’t have been made of completely different matter, but the forces exerted by the matter in our bodies are an essential property of that matter.

Given this essentiality of gravitational constant assumption, it follows that at least one of H₁ and H₂ is incompatible with Heyl’s existence. Now, to get (1), we need prior probabilities on which P(H₁|E) > P(H₂|E) > 0. Such prior probabilities will assign a non-zero value to H₁E and to H₂E. But at least one of these two claims is impossible since E entails Heyl’s existence, and a probability assignment that assigns a non-zero value to something impossible is screwed up, and we should be quite suspicious of what we get from it.

We might try to avoid this by using self-locating evidence. But my colleague Yoaav Isaacs has this great paper that gives a pretty strong argument that there isn’t a good way to working with self-locating evidence. So suppose we put this option aside.

Or we might make a distinction between logical impossibility and metaphysical impossibility. I find that suspicious, too.

So, what’s left? Well, here’s one remaining suggestion. Heyl’s evidence is equivalent to the proposition that Heyl measured G to be G₀, a proposition that rigidly refers to Heyl, and hence won’t be compatible with both H₁ and H₂. But we can weaken Heyl’s evidence to something that is compatible with H₁ and H₂, something purely qualitative, like:

• E_Q: A physicist named “Paul Heyl”, who married someone named “Lucy Daugherty”, and who …, measured G to be G₀.

Here, “…” is all the other purely qualitative stuff we know about Paul Heyl, so that E_Q is compatible with both H₁ and H₂.

But now here is a problem. Suppose we live in a vast but finite universe with, say, 10¹⁰¹⁰ people. In such a universe, we might well expect large numbers of people named “Paul Heyl” who satisfy all the conditions in E_Q, including the measurement of G to be G₀, even if in fact G is in the range indicated in H₁ (measurement error!). Thus, P(E_Q|H₂) is close to 1 as is P(E_Q|H₁). Granted, we do have P(E_Q|H₁) > P(E_Q|H₂) > 0. But because the two probabilities are so close to each other, the support E_Q gives to H₁ over H₂ is very slight, and hence we no longer have (1).

It follows that unless we can find some other way of solving the problem that the essentiality of the laws of nature to humans poses for Bayesian reasoning, a fair amount of fundamental physics research would be undercut by a large enough—even if finite—universe.

Of course, maybe we can find some other way of solving it. But maybe we can’t. And if we can’t, then the E_Q solution might be our best bet—and it’ll work just fine in a universe that isn’t too vast.

Consent to deceitful experiments

1. In order to give valid consent, the subjects in an experiment need to be informed about all the harms that the experimenter plans to impose on them.

2. Being deceived is a harm.

3. Therefore, in order to give valid consent, the subjects in an experiment where the experimenter plans to use deceit need to be informed that deceit will be imposed.

Often, consent conditions are phrased in terms of risks, and it is stipulated that only non-minimal risks need to be disclosed, where minimality is measured relative to the risks in ordinary life. Being deceived about a minor matter might be thought to be a minimal risk even when the probability of deceit is nearly 100%, since in ordinary life people routinely suffer deception, and sometimes don’t mind much if at all (see here for discussion).

However, I think there may be a difference between disclosing risks and disclosing planned harms. For instance, minor pains are a daily occurrence for people. But deliberately imposing a minor pain on an experimental subject who did not consent to such imposition—apart from special cases such as pushing someone away from danger—would seem to be a morally impermissible assault.

Knowledge and induction

Assume that in fact all ravens are black. Suppose you are sequentiallly observing ravens, and noting each one to be black. After observing n ravens, your evidence that the next raven is black will typically be significantly better than the evidence that all ravens are black. Now at some point, say after observing n_A ravens, your evidence that all ravens are black will rise to the level of knowledge. Thus, plausibly, at an earlier point in the sequence, call it n_N, your evidence that the next raven is black will have risen to the level of knowledge.

Suppose now you have observed n_A − 1 ravens, and you have been handed a raven in an opaque box, which you are certain you are about to open. Since n_N < n_A, at this point you have reached n_N. Hence:

1. You do not know that all ravens are black.

2. You do know that the next raven is black.

3. You know that when you observe the next raven, you will have sufficient evidence for knowledge that all ravens are black.

But note that while you know you will have sufficient evidence for knowledge that all ravens are black, you don’t know that you will know that all ravens are black. There is nothing deeply surprising about this distinction. We might well say about someone who has been subjected to misleading or Gettiered evidence that they have sufficient evidence to know something but nonetheless they don’t know, though the case at hand feels different.

One interesting thing about this case, as I read it, is that it contradicts the thesis that K = E, i.e., that knowledge is evidence. For if knowledge is evidence, and you know that the next raven is black, then you already have the evidence you will gain by observing the next raven, and hence you are already in the position to know that all ravens are black.

Another interesting thing is that it shows that you can know something and nonetheless it be rational for you to investigate it. For you know that the next raven is black, but it’s worth investigating further, since it is only upon observation that your knowledge of the next raven’s blackness turns into the kind of evidence that gives you knowledge that all ravens are black.

All this might make one think that I have misconstrued the epistemic facts, and it is false that there can be a point n_N prior to n_A at which you know that the next raven is black. Here is one way to back up my intuition that there can be such a point n_N < n_A. Suppose that we know for sure we live in a world where the color distributions of birds are always uncorrelated between the males and the females of the species, so that information about the color of members of one sex are irrelevant to the color of members of the other sex. Also assume that you know for sure that ravens have equal numbers of each sex, that you are observing ravens in an alternative female-male-female-male-… sequence, and that your priors for the color distributions of the two sexes of ravens are the same. Then if p_M and p_F are the probabilities that all male ravens are black and all female ravens are black, and p_A is the probability that all ravens are black, then at any given point in the observation sequence p_A = p_Mp_F. Let n_M and n_F be the points in the sequence where you know that all male and all female ravens are black, respectively. Then, n_M < n_A and n_F < n_A, since p_A = p_Mp_F is significantly smaller than either p_M and p_F at all points in the sequence except when we’ve observed all the ravens of one sex, and since p_M and p_F rise fairly gradually as we go through the sequence. Thus, at the point n_A − 1, we will have already reached knowledge that all the male ravens are black and the knowledge that all the female ravens are black. In particular, then, we know that the next raven is black, since if n_A is even, the n_Ath raven is male and we know all male ravens are black, and if n_A is odd, then the n_Ath raven is female, and we know that all female ravens are black.

Epistemic rationality and Pascal's Wager

Pascal’s Wager is an argument that it is prudentially rational to engage in theistic belief promotion practices (TBPP), namely practices apt to promote one’s belief in God.

My interest this post is the standard epistemic rationality objection to the Wager, that engaging in TBPPs is irrational—a kind of brainwashing of oneself. Let’s think about the objection with a bit more care. Consider a specific TBPP Q, say Pascal’s example of going to Mass. If Q is indeed a TBPP, one expects engagement in TBPP to make it more likely that one believes in God. But how does one expect Q to achieve that goal? There are two possibilities. Either Q is expected to achieve that goal by providing one with evidence for theism or in some non-evidential way (or a combination of the two).

Suppose Q is expected to work evidentially. Then we already have the expectation of a higher credence given Q. This expectation is either rational or not. If it is not rational, then we don’t actually have good reason to engage in Q. If it is rational, however, then we should rationally raise our credence in theism right now, without having to bother engaging in Q, and for reasons having nothing to do with any wager. But if Q promotes belief in God non-rationally, then we should not engage in Q for the sake of such promotion of belief—we should not aim to non-rationally promote beliefs.

Let me make the first horn of the dilemma—namely, that the expectation of a higher credence is rational—a bit more precise. We can distinguish two (not mutually exclusive) ways in which a practice rationally increases one’s credence in a hypothesis H. One way is purely epistemic, by uncovering facts about reality. This is the usual way. But if that’s the way we expect to increase our credence in theism by engaging in Q, then we already have evidence that there are such theism-indicating facts to be discovered, and so we should already increase our credence in Q. The other way is practical, by promoting the hypothesis H in a way that shows up to us. The second way is a bit unusual, but here is an example: one way to increase your credence that you will not die of heart disease is to live a healthy life. For if you live a healthy life, you are less likely to die of heart disease, and since you will notice signs of improved cardiac health (e.g., lower resting heart rate, less huffing and puffing on stairs, etc.), your credence that you won’t die of heart diseases will also increase. But this practical way of increasing credence is utterly irrelevant in the case of theism, since nothing we can do can make God more or less likely to exist! So the only rational way that remains is the evidence-based way, and evidence-of-evidence is already evidence.

I used to be quite impressed by the worry that Pascal’s Wager leads to self-deception. I am less impressed. Here is why. There is a serious technical flaw in the argument for the first horn of the dilemma. A simple model for the relation of credence and belief is that you believe a proposition if and only if your credence is above some threshold β. This model might be false, but an analogue of what I will say should apply on more sophisticated models as well.

Here is the point. Consider a case where one is thinking about observing (and suppose this is a simple non-Newcombian observation that does not affect the hypothesis) whether some event E evidentially relevant to H has obtained. Then one’s expected posterior credence is:

1. C(E)C(H∣E) + C(∼E)C(H∣∼E),

where C is one’s credence function. But if one is a good Bayesian reasoner, then by total probability the value in (1) is simply equal to one’s prior C(H). Thus the value of one’s credence has no expectation of change upon observation when one is being rational. This seems to support the idea that if you expect your credence to go up, you should already raise it.

But in fact it’s not so simple. For even though the expected posterior credence equals one’s current credence, it could well be that it is more likely that the expected posterior credence exceeds the threshold β if you make the observation than if you don’t. Indeed, cases are obvious. Suppose the belief threshold β is 0.9, and you tossed a coin out of my sight. Suppose I have a prior credence 0.5 that this coin is fair and a prior credence 0.5 that it is double-headed. Then currently I don’t believe (or disbelieve) that the coin is fair. But if I look at the coin and I see tails, I will believe that it is fair—indeed, I will have posterior credence 1 in its fairness. But if I don’t look at the coin, I am not going to get any evidence, and I will continue not to believe that the coin is fair. If I look at the coin, the probability that I will see tails is 0.25 (I have credence 0.5 that it’s fair, and if it’s fair, the chance of tails is 0.5), and so the probability that I will believe if I look is 0.25 (since if I look and see tails, my posterior will be 1 which is bigger than β = 0.9), and the probability that I will believe if I don’t look is 0. Of course, if I don’t see tails, my credence that the coin is fair will go down. But while it will go down, it won’t affect whether I believe that the coin is fair—for I already don’t believe it (in the sense of not-believe, rather than in the sense of believe-not). And there is no irrationality of any sort in looking at the coin in this case.

In other words, the point is that while one’s rational credence has no positive or negative rational expectation of change upon observation, whether one’s rational credence is above a threshold certainly can have a positive or negative rational expection of change.

How could this work in a Pascal’s Wager situation? Let’s talk through one possibility. Take Pascal’s example of going regularly to Mass. Suppose, as Pascal says, your current credence in God is 0.5. You might think that if God exists, going to Mass has a decent chance, say 0.2, of resulting in an evident radical transformation E of your life, so evident and radical that updating your credence in theism on E will push your credence in God to above the threshold β. Of course, you might go to Mass it might not produce any such evident radical transformation (this is true even if God always improves the hearts of people who go to Mass, since he might do so more gradually or less evidently), and in that case your rational credence in God will go below 0.5. But going below 0.5 won’t affect whether you believe in God, since 0.5 is already, I assume, far below the belief threshold β. On the other hand, maybe if you don’t go to Mass, the probability that you will get evidence that will push your credence in theism above β is pretty small—smaller than 0.2. Very likely, your credence will just oscillate a little around 0.5 in the non-Mass-going case. Thus if there is a payoff you get for your credence exceeding the threshold β, it will be worth going to Mass, without there being any epistemic irrationality in the reasoning.

Thought of in this way, we get some practical guidance as to which TBPPs the agnostic or atheist should engage in. They should look for practices that, if God exists, have a decent chance of producing evidence for theism sufficient to push them above β.

I am a bit doubtful that Pascal meant us to think in the above way. He may well have been recommending TBPPs on the grounds of their non-rational effect on belief. My argument above does not defend that.