An impartiality premise

In an argument that David Lewis’s account of possible worlds leads to inductive skepticism, I used this premise:

1. If knowing that x is F (where F is purely non-indexical and x is a definite description or proper name) does not epistemically justify inferring that x is G (where G is purely non-indexical), then neither does knowing x is F and that x is I (now, here, etc.: any pure indexical will do) justify inferring that x is G.

This is less clear to me now than it was then. Self-locating evidence might be a counterexample to this principle. I know that the tallest person in the world is the tallest person in the world. But suppose I now learn that I am the tallest person in the world. It doesn’t seem entirely implausible to think that at this point it becomes reasonable (or at least more reasonable) to infer that the number of people in the world is small. For on the hypothesis that the number of people is small, it seems more likely that I am the tallest than on the hypothesis that the number of people is large. (Compare: That I won some competition is evidence that the number of competitors was small.)

But I think I can fix my argument by using this premise:

2. If knowing that x is F (where F is purely non-indexical and x is a definite description or proper name) and that a uniformly randomly chosen person (or other occupied location) is x would not epistemically justify inferring that x is G (where G is purely non-indexical), then neither does knowing x is F and that x is I (now, here, etc.: any pure indexical will do) justify inferring that x is G.

There are multiple versions of (b) depending on how the random choice works, e.g., whether it is a random choice from among actual persons or from among possible persons (cf. self-sampling vs. self-indication).

It takes a bit of work to convince oneself that the rest of the argument still works.

Yet another argument for thirding in Sleeping Beauty?

Suppose that a fair coin has been flipped in my absence. If it’s heads, there is an independent 50% chance that I will be irresistably brainwashed tonight after I go to bed in a way that permanently forces my credence in heads to zero. If it’s tails, there will be no brainwashing. When I wake up tomorrow, there will be a foul taste in my mouth of the brainwashing drugs if and only if I’ve been brainwashed.

So, I wake up tomorrow, find no taste of drugs in my mouth, and I wonder what I should to my credence of heads. The obvious Bayesian approach would be to conditionalize on not being brainwashed, and lower my credence in heads to 1/3.

Next let’s evaluate epistemic policies in terms of a strictly proper scoring accuracy rule (T,F) (i.e., T(p) and F(q) are the epistemic utilities of having credence p when the hypothesis is in fact true or false respectively). Let’s say that the policy is to assign credence p upon observing that I wasn’t brainwashed. My expected epistemic utility is then (1/4)T(p) + (1/4)T(0) + (1/2)F(p). Given any strictly proper scoring rule, this is optimized at p = 1/3. So we get the same advice as before.

So far so good. Now consider a variant where instead of a 50% chance of being brainwashed, I am put in a coma for the rest of my life. I think it shouldn’t matter whether I am brainwashed or put in a coma. Either way, I am no longer an active Bayesian agent with respect to the relevant proposition (namely, whether the coin was heads). So if I find myself awake, I should assign 1/3 to heads.

Next consider a variant where instead of a coma, I’m just kept asleep for all of tomorrow. Thus, on heads, I have a 50% chance of waking up tomorrow, and on tails I am certain to wake up tomorrow. It shouldn’t make a difference whether we’re dealing with a life-long coma or a day of sleep. Again, if I find myself away, I should assign 1/3 to heads.

Now suppose that for the next 1000 days, each day on heads I have a 50% chance of waking up, and on tails I am certain to wake up, and after each day my memory of that day is wiped. Each day is the same as the one day in the previous experiment, so each day I am awake I should assign 1/3 to heads.

But by the Law of Large Numbers, this is basically an extended version of Sleeping Beauty: on heads I will wake up on approximately 500 days and on tails on 1000 days. So I should assign 1/3 to heads in Sleeping Beauty.

Mushrooms

Some people have the intuition that there is something fishy about doing standard Bayesian update on evidence E when one couldn’t have observed the absence of E. A standard case here is where the evidence E is being alive, as in firing squad or fine-tuning cases. In such cases, the intuition goes, you should just ignore the evidence.

I had a great conversation with a student who found this line of thought compelling, and came up with this pretty convincing (and probably fairly standard) case that you shouldn’t ignore evidence E like that. You’re stranded on a desert island, and the only food is mushrooms. They come in a variety of easily distinguishable species. You know that half of the species have a 99% chance of instantly killing you, and otherwise having no effect on you other than nourishment, and the other half have a 1% chance of instantly killing you, again otherwise having no effect on you other than nourishment. You don’t know which are which.

To survive until rescue, you need to eat one mushroom a day. Consider two strategies:

1. Eat a mushroom from a random species the first day. If you survive, conclude that this species is likely good, and keep on eating mushrooms of the same species.

2. Eat a mushroom from a random species every day.

The second strategy makes just as much sense as the first if your survival does not count as evidence. But we all know what will happen if you follow the second strategy: you’ll be very likely dead after a few days, as your chance of surviving n mushrooms is (1/2)ⁿ. On the other hand, if you follow the first strategy, your chance of surviving n mushrooms is slightly bigger than (1/2)(0.99)ⁿ. And the first strategy is precisely what is favored by updating on your survival: you take your survival to be evidence that the mushroom you ate was one of the safer ones, so you keep on eating mushrooms from the same species. If you want to live until rescue, the first strategy is your best bet.

Suppose you’re not yet convinced. Here’s a variant. You have a phone. You call your mom on the first day, and describe your predicament. She comforts you and tells you that rescue will come in a week. And then she tells you that she was once stuck for a week on this very island, and ate the pink lacy mushrooms. Then your battery dies. You rejoice: you will eat the pink lacy mushrooms and thus survive! But then suddenly you get worried. You don’t know when your mom was stuck on the island. If she was stuck on the island before you were conceived, then had she not survived the mushrooms, you wouldn’t have been around to hear it. And in that case, you think her evidence is worthless, because you wouldn’t have any evidence had she not survived. So now it becomes oddly epistemically relevant to you whether your mom was on the island before or after you were conceived. But it seems largely epistemically irrelevant when your mom’s visit to the island was.

Does my existence by itself confirm a multiverse?

Suppose I am considering two hypotheses, H₁ and H₂, and according to H₂ there are more people. Does the fact that I exist give me reason to prefer H₂, all other things being equal? If so, then my existence is apt to confirm the existence of a multiverse over a single universe.

Here is one reason to think this works. The probability that I exist in a given world, all other things being equal, seems proportional to the number of people in that world. Each person in that world corresponds to another opportunity for me to exist.

While this is tempting, here is a toy model that should give us pause. Suppose that I am defined by a real number parameter between 0 (inclusive) and 1 (not inclusive). According to hypothesis H₁, a single real number is picked uniformly at random in the range, and the person with that parameter is created. According to hypothesis H₂, two real numbers are picked uniformly and independently in the range, and persons corresponding to these are created. Learning that a person with my parameter is created seems to provide me with evidence for H₂, since it’s twice as likely on H₂ as on H₁.

But this is tricky. In classical probability theory, it is correct to say that my parameter is twice as likely to be generated on H₂ as on H₁, but that’s only because both probabilities are zero, and zero is twice zero, so while H₂ is twice as likely as H₁, it is also true that H₁ is twice as likely as H₂!

Perhaps, though, we want to depart from classical probability theory in some way, say by allowing non-zero infinitesimal probabilities or by an intuitive handwavy “this is twice as likely as that”. However, it is then no longer clear that on H₂ there is twice as big a chance of hitting my parameter. For there are (infinitely) many ways of picking a number between 0 and 1 uniformly randomly.

Here’s one way:

1. You write down “0.”, then roll a fair ten-sided die infinitely many times, writing down the results as the digits after the decimal point, thereby generating a decimal representation of a number. If the number ends with infinitely many nines, try again.

(The final proviso is to ensure that intuitively each number is equally likely. Without that proviso, 1/10 would be more likely than 1/3, as there would be two ways of getting 1/10, namely 0.1000... and 0.0999..., but only one way to get 1/3, namely 0.3333.....)

Here is another way:

2. You write down “0.”, then roll a fair ten-sided die infinitely many times, omitting the results of the first die throw, but writing down the results as the digits after the decimal point, thereby generating a decimal representation of a number. If the number ends with infinitely many nines, try again.

Intuitively, method B has ten times the probability of generating any given number than method A has, as long as literally the numerically same die throws occur in the two cases. For consider the number 1/3 = 0.3333.... By method A to generate it you need every die to show a three. By method B, to generate 1/3, all you need is for all the die throws other than the first one to be threes, and so there are ten times as many ways to generate the number.

Now, if the single selection of a parameter on H₁ uses method B while the double selection of a parameter on H₂ uses method A, then intuitively we are five times as likely to generate my parameter on H₁ than on H₂. Thus merely saying that on both hypotheses the parameters are generated uniformly is insufficient to determine how the comparison between the probabilities of generating my parameter goes.

We might insist that in both hypotheses the same method for generating parameters is used. But notice that in cosmological applications, this is implausible. If H₂ is some multiverse hypothesis and H₁ is a single universe hypothesis, we are unlikely to be able to count on the two hypotheses involving even the same laws of nature, much less the same selection process for the parameters of the persons. (Besides all this, it is really unclear what it even counts to say that there are two different runs of method A.)

So, here’s what I am thinking. On classical probability theory, there is no difference in the probability of my parameter getting generated on H₂ than on H₁, because both probabilities are zero. On non-classical probability theory, we can perhaps make sense of a difference between the probabilities, but cannot count on the hypothesis with more people being more likely to generate my parameter.

Given all this, there does not seem to be a way of making sense of comparing the evidential impact of my existence on the two hypotheses using probabilistic methods. Maybe all we have is intuition.

Self-locating beliefs in the Trinity

Here is a difficulty for the doctrine of the Trinity that I don’t remember coming across before:

1. The Father and the Son have the numerically same divine mind.

2. If x and y have the numerically same divine mind, then x and y have the same divine beliefs.

3. The Father has an “I am the Father” divine belief.

4. So, the Son has an “I am the Father” divine belief. (1–3)

5. An “I am the Father” divine belief in the Son would be false.

6. There are no false divine beliefs.

7. So, the Son has no “I am the Father” divine belief. (5–6)

8. Contradiction!

Here, premise (1) follows from the heuristic that what there are two of in Christ, there is one of in the Trinity: there are two minds in Christ, so one mind in the Trinity. Non-heuristically, if there are two minds in Christ—the human and the divine mind—the mind must be a function of the nature, and as there is one divine nature in the Trinity, there is one mind in the Trinity.

There is a quick way out of the paradox: Restrict premise (2) to propositional beliefs rather than de se or self-locating beliefs. The belief that would be expressed in English by “I am the Father” is a de se or self-locating belief. There are corresponding propositional beliefs, such as the belief that the Father is the Father and the Son is the Son, but these are unproblematically had in common by the Father, the Son and the Holy Spirit.

However, while this quick way gets one out of the argument, it nonetheless leaves raises the difficult question of how it is the Father knows de se that he is the Father and the Son knows de se that he is not the Father, while yet there is one mind.

The solution had better be in terms of the relations between the divine persons, for there is no difference between the persons of the Trinity except the relational. I am reminded here of Thomas’s discussion of creation and the Trinity:

And therefore to create belongs to God according to His being, that is, His essence, which is common to the three Persons. Hence to create is not proper to any one Person, but is common to the whole Trinity.

Nevertheless the divine Persons, according to the nature of their procession, have a causality respecting the creation of things. For as was said above, when treating of the knowledge and will of God, God is the cause of things by His intellect and will, just as the craftsman is cause of the things made by his craft. Now the craftsman works through the word conceived in his mind, and through the love of his will regarding some object. Hence also God the Father made the creature through His Word, which is His Son; and through His Love, which is the Holy Ghost. And so the processions of the Persons are the type of the productions of creatures inasmuch as they include the essential attributes, knowledge and will.

Thus, each divine person is fully the Creator, but is fully the Creator in a way that takes into account the relationship that defines the person in the Trinity: the Father creates in a Fatherly way, the Son as the Logos through which creation is done, and the Spirit as the Love in which creation is inspired. What makes it be the case that the Father creates in a Fatherly way is just that the Father creates and he stands in the relations that constitute him as Father; what makes it be the case that the Son creates in a Filial way is just that the Son creates and he stands in the relations that constitute him as Son; and similarly for the Holy Spirit.

We might thus imagine the following story. There is a state F of the divine mind such that the Father’s Fatherly instantiation of F constitutes F into a belief that he (de se) is the Father. The Son instantiates the numerically same state F in a Filial way. But while a Fatherly instantiation of F is correctly described in English as constituting an “I am the Father” belief, a Filial instantiation of F is not aptly so described. Perhaps, a Filial instantiation of F is aptly described as a believing of “I am the Son of the one who is the Father.” Thus, the de se beliefs of the persons of the Trinity are constituted by mental states common to the Trinity and the relations constituting the persons.