An analogy for divine infinity

Here’s an analogy I’ve been thinking about. God’s value is related to other infinities like (except with a reversal of order) zero is related other infinitesimals. Just as zero is infinitely many times smaller than any other infinitesimal (technically, zero is an infinitesimal—an infinitesimal being a quantity x such that |x| < 1/n for every natural number n), and in an important sense is radically different from them, so too the infinity of God’s value is infinitely many times greater than any other infinity, and in an important sense is radically different from them.

Suppose we think with the medievals that value and being are correlative. Then zero value corresponds to complete non-being. There isn’t anything that has that. Between ordinary non-divine things like people and oak trees and non-being we have a radical ontological difference: there are people and oak trees, but there is no non-being. Suppose we push the analogy on the side of God. Then between ordinary non-divine things like people and oak trees and God we will have a radical ontological difference, too. Some theologians have infamously tried to mark this difference by saying that people and oak trees are but God is not. That way of marking the difference is misleading by making God seem like non-being instead of like its opposite. A better way to mark the difference is to say that in an important sense God is and people and oak trees are not (compare what Jesus is said to have have said to St Catherine of Siena: “I am who I am and you are she who is not”). In any case, the gap between God’s “is” and our “is” is at least as radical as the gap between our “is” and the “is not” of non-being.

In fact, I think the gap is more radical: we and all other creatures are closer to non-being than to God. So the analogy I’ve been thinking about, that God’s value is related to other infinities like zero to other infinitesimals (but in reverse order) is misleading: God’s value is in a sense further from other infinities than zero is from other infinitesimals. (And not just because all infinitesimals are infinitesimally close to zero. The relevant scale should not be arithmetic but logarithmic, so that the gap between zero and anything—even an infinitesimal—bigger than zero is in an important sense infinite.)

Don’t take this too seriously. Remember this.

The spectrum of values

Let’s do some rough and ready thinking about the spectrum of possible values of objects on a classical theistic view.

1. God has his value essentially.

2. Necessarily, God is more valuable than everything other than God.

3. Necessarily, everything that exists exists by participation in the good God and hence has positive value.

The spectrum of possible values thus has an upper bound: God’s value. Moreover, it follows from (2) that God is infinitely many times more valuable than anything else. For consider some object x other than God, and imagine a world (perhaps a multiverse) where x is duplicated some number n of times. By (2), God will be more valuable than the duplicates of x, and hence God is more than n times valuable than x. Since n is arbitrary, it follows that God is infinitely many times more valuable than x.

Thus, our spectrum of values has God at the top, then an infinite gap, and below that possible creatures.

What does the lower end of the spectrum of possible values of objects look like? Well, by (3), all the values are positive. So the lower end of the spectrum lies above zero. I suspect that it asymptotically approaches zero. For consider an object x and now imagine an object y which has exactly one essential causal power, that of producing x with a probability of 1/2. Intuitively, y has something like half the value of x. So it is plausible that the lower end of the spectrum of possible values approaches zero but does not reach it.

But now suppose that y has only an infinitesimal probability of producing x (imagine y has an internal spinner and it produces x whenever that spinner lands exactly at ninety degrees). Then x seems like it would be infinitely more valuable than y. If this is right, then for every value in the spectrum, there is a value that is infinitely many times smaller than it.

The spectrum of values has a top (God) but no bottom. For any value on the spectrum of values, there is a value infinitely many times smaller than it. And for any value on the spectrum of values other than God, there is a value infinitely many times greater than it.

There is thus a very natural sense in which everything is relatively infinite in value: everything is infinitely many times more valuable than something else. But only God is absolutely infinite in value: God is infinitely many times more valuable than everything else.

Incommensurability complicates things, though.

Megavalue

In my previous post, I glibly talked of the infinite value of persons. I forgot that such talk was discredited by this argument. Instead, one should talk of relatively infinite value: being infinitely more times valuable than.

I think the argument of that post can be rescued. And while I am at it, I can modify the argument to avoid another objection, that higher animals like dogs and dolphins are not infinitely less valuable than persons. I do not know if the objection is sound, but it won't matter.

1. Definition: A thing has megavalue if and only if it is infinitely more times valuable than every portion of non-living reality in the universe.

2. The sum total of life in the universe has megavalue.

3. Nothing can cause something that has infinitely more value than itself.

4. If the sum total of life in the universe has a cause and that cause is wholly within the universe, then the cause is a portion of the non-living reality in the universe.

5. There is a cause of the sum total of life in the universe.

6. A cause of the sum total of life in the universe is not wholly within the universe.

Infinite value

1. All persons have infinite value.
2. Nothing in the universe other than persons has infinite value.
3. There was a first person in the universe.
4. Everything in the universe has a cause.
5. The first person in the universe wasn’t caused by a person in the universe.
6. Nothing with finite value can cause something with infinite value.
7. So, something not in the universe and with infinite value caused the first person in the universe.

I am not confident of premise (1).

Meaning

1. A thing with no meaning cannot cause a thing with meaning.
2. There was a first meaningful thing (e.g., a thought) in the physical universe.
3. Every thing in the physical universe has a cause.
4. So, there was a meaningful thing not in the physical universe that caused the first meaningful thing in the physical universe.

Meaning and beauty

1. Only intelligent beings and things produced by them have objective meaning.
2. Something that is objectively meaningless is not objectively beautiful.
3. The earth is objectively beautiful.
4. The earth is not intelligent.
5. So, the earth is produced by an intelligent being.

Probability for truly fair infinite lotteries

Long ago, in correspondence with Plantinga and me, Peter van Inwagen suggested that the only way to model a countably infinite fair lottery is by assigning probability zero to every finite set of tickets, probability one to every co-finite set of tickets (a subset A of a set B is co-finite [relative to B] provided that the set of members of B that are not in A is finite), and an undefined probability to every other subset.

Van Inwagen’s proposal has been growing on me. In a truly fair lottery, the ordering of ticket numbers is irrelevant. Therefore, if Ω is the set of tickets and π is any permutation of Ω, the probability of A should be the same as that of πA, with each defined if and only if the other is. In other words the probability function should be permutation-invariant.

Proposition. There are only two finitely-additive real-valued probabilities invariant under all permutations of a countably infinite set Ω: the trivial probability that assigns 0 to the empty set, 1 to Ω and is undefined for all other subsets, and van Inwagen’s probability that assigns 0 to every finite set, 1 to every co-finite set and is undefined for all other subsets.

There is something very appealing about van Inwagen’s proposal: it’s the only finitely-additive real-valued probability that really captures the idea of a countably infinite fair lottery. I can't remember if van Inwagen had the above proposition in the correspondence, but he might have.

Proof of Proposition: For any two subsets X and Y that are neither finite nor co-finite, there is a permutation of Ω mapping X onto Y. Thus, by permutation invariance, either all sets that are neither finite nor co-finite have a probability or none do. Suppose first that all do. In that case, they all have equal probability. Let A be the evens, B be the odds, C the numbers equal to 0 modulo 4 and D the numbers equal to 2 modulo 4. They all must have equal probability. But A is the disjoint union of C and D, so by finite additivity, if all three have equal probability, all three must have probability zero. And so does B. Thus, P(Ω) = P(A) + P(B) = 0, a contradiction.

So, only sets that are neither finite nor co-finite have a probability. If the only subsets that have a probability are ⌀ and Ω, we are done. Suppose some other subset has a probability. If that subset is co-finite, its complement will have to have a probability too, so in either case there is a finite non-empty subset A that has a probability. Let A′ be a finite non-empty subset that has the same cardinality as A but intersects A in only one element. By permutation invariance, A′ has a probability. Thus, so does the intersection of A and A′. Hence, at least one singleton has a probability. Hence by permutation invariance all singletons have a probability. By finite additivity, that probability must be zero. It follows that all finite sets have probability, and that probability is zero, and all co-finite sets have probability, and that probability is one.

Remark 1: Suppose that we allow the probabilities to take values in some non-Archimedean ordered field. Then there are more possibilities. Specifically, for any positive infinitesimal α, we can define a probability that assigns to every finite set the probability nα where n is the set’s cardinality and to every co-finite set the probability 1 − nα where n is the cardinality of the set’s complement. And these are the only extra possibilities.

Remark 2: If we drop the countability condition on Ω, and assume the Axiom of Choice, then in the setting of the Proposition we can prove that P(A) is 0 or 1 for every subset A for which P(A) is defined.

What is philosophy?

I just came across this in Shturman and Tiktin’s delightful anthology of Soviet era jokes. (I don't know if it exists in translation.)

Question: What is philosophy?

Answer: It’s a hunt for a black cat in a dark room. Marxist philosophy is distinguished by the fact that there is no cat in the room, and Marxist-Leninist by the fact that one of the hunters in fact yells that he caught the cat.

I wonder what other philosophies it can be applied to.

In defense of a changing beatific vision

It is widely taken in the Thomistic tradition that:

1. Different people in heaven have the beatific vision to different degrees, corresponding to the saints’ different levels of holiness.

2. The beatific vision does not change with time for a given individual.

I think there is a tension between these two claims which is best resolved by dropping the no-change thesis (2). Dropping the difference thesis (1) is not an option for Catholics at least, since it’s a dogma taught by the Council of Florence.

To see the tension, note that the fact that different saints have holiness to different degrees implies that those saints who have a lesser holiness have not maxed out what human nature makes possible. And holiness is attractive to the holy, and infectious. If one saint is less holy than another, it seems likely that given a sufficient amount of time, we would expect the second saint’s greater holiness to inspire the first to even greater holiness. And then we would expect the beatific vision to increase.

We also have one New Testament case where it seems likely that a person’s level of beatific vision has increased. In 2 Corinthians, Paul writes of knowing someone who, fourteen years ago, was caught up to the third heaven. It is common to take that to be a modest reference to Paul himself, and the “third heaven” to be a reference to the beatific vision. Now, eventually Paul died and experienced the beatific vision again. It seems very implausible to think that the significant number of years between Paul’s first experience of heaven and his final experience of heaven did not result in Christian maturation and growth in virtue. Thus, it seems quite plausible that Paul had greater holiness when he died than when he was first caught up to heaven, and hence by the correspondence thesis (1), he had a greater degree of beatific vision at death than at the earlier incident.

Note, too, that a Catholic cannot say that the level of holiness is fixed at the time of death, since then purgatory wouldn’t make sense. And, intuitively, we would expect heaven to be inspiring of growth in holiness!

Now, one could insist that the level of holiness is fixed at the time of entry to heaven. But if so, then we couldn’t really say that the death of a saint is always something to rejoice at. Imagine that Paul had died at the time of his first experience of the beatific vision. Then on the no-change view of the beatific vision, he would eternally have had a lesser beatific vision than in actual world where he continued to grow in holiness for over decades more.

A picture of continual growth in holiness and the beatific vision fits better with our temporality. One may worry, however, that it takes away from the picture of resting in God. However, rest is compatible with change. One of the best ways to rest is to read a good book. But as one reads the book, one grows in knowledge of its content. And if one worries that the thought that one will come to have a greater happiness should induce in one a present sorrow of longing, I think it is plausible that with perfect virtue one would no more find the expectation of greater future happiness to be a source of sorrow than a lesser saint would find the observation of greater saints a source of envy. And, coming back to the book analogy, when one reads a good book, there need be no unhappiness at the fact that there is more of the book yet to come—on the contrary, one can rejoice that there is more to come. (In some cases, there may be a weak negative emotion as one longs for the author to reveal something—say, the solution of a mystery. But not every genre will generate that.)

Furthermore, there is good reason to think that change is not incompatible with rest. Since we will have bodies in heaven, and we will flourish in body and soul, while bodily flourishing involves change, heavenly rest must be compatible with change. And plausibly some of the bodily activities we will engage in will involve a variation in the level of happiness at least in some respects. Thus, eating is an episodic joy, and music, I take it, involves much in the way of anticipation and change.

Greedy strategies for Wordle

Today, I wanted to see how well the following greedy strategy works on Wordle. Given a set H of words, a word w partitions H into equivalence classes or “cells”, where two words count as equivalent if they give the same evaluation if w is played. We can measure the fineness of a partition by the size of its largest cell. Then, the greedy strategy is this:

1. Start with the 12792 five-letter Scrabble words of the original Wordle (things will be a bit different for the bowdlerized New York Times version).

2. Guess a word that results in a finest partition of the remaining words, with ties broken by favoring guesses that are themselves one of the remaining words, and remaining ties broken alphabetically.

3. If not guessed correctly, repeat 2–3 with the remaining words replaced by the cell of the partition that fits the guess.

A quick and dirty python (I recommend pypy) implementation is here.

Note that this greedy strategy does not cheat by using the secret list of answer words, but only uses the publicly available list of five-letter Scrabble words.

Here some observations:

• The greedy strategy succeeds within six guesses for all but one of the 2315 answer words in the original Wordle. The remaining word needs seven guesses.

• A human might be able to guess the answer word that the algorithm fails for, because a real human would not be partitioning all the five-letter Scrabble words, but only the more common words that are a better bet for being on the answer list.

• The greedy algorithm’s first guess word is “serai”, and the maximum cell size in the resulting partition is only 697 words.

• While only one answer list word remained that the greedy strategy fails for, it looks like Wardle may have deliberately removed from the answer list some other common, clean and ordinary English words that the greedy strategy does not guess in six guesses.

• It would be a cheat, but one can optimize the algorithm by starting with only the 2315 answer words. Then indeed the greedy strategy guesses every answer word in six guesses, and all but two can be done in five.

• A human can do something in-between, by having some idea of what words an ordinary person is likely to know.

Jesus's "unknown years"

Here is something that struck me about Jesus’s “unknown years”—the years not expressly described in the Gospel. Jesus was a Jew under the Torah. The Torah suffuses all aspects of life. Thus, in virtue of the fact that Jesus was sinless and hence perfectly fulfilled the Torah, we can know a lot more about his daily life than we know about the daily life of any ordinary historical figure. In Jesus' case, "ought" implied "is".

The law enforcement model of war

When an army invades a country, the invaders break a massive number of that country’s laws. They kill, commit assault and battery with deadly weapons, recklessly endanger lives, and destroy property, and do all this as part of a concerted conspiracy. And they break all sorts of more minor laws, by carrying unlicensed weapons, trespassing on government property, littering, and presumably violating traffic laws all the time (I assume one can’t slow down the progress of a column of tanks by putting up a stop sign).

This means that there is an intermediate position between just war theory and complete non-violence. This intermediate position holds that law enforcement can appropriately make use of violent means, including lethal violence when this is proportionate, and that when one’s country suffers an unjust invasion, one may (and maybe often should) legitimately engage in appropriately violent law enforcement activities against the enemy. Call this the law enforcement model of war.

I do not advocate the law enforcement model, but it is interesting to think just how it would differ from the two more standard options.

The difference from complete non-violence is clear: on the law enforcement model, violent action in defense of the country’s laws is permitted whenever proportionate. Thus, it would be permissible to destroy a tank because it is a part of a conspiracy to commit murder and destruction of property, but not because it is simply refusing to stop at a stop sign. (And discretion being the better part of valor, issuing traffic tickets in the latter case is probably not a good idea.)

The differences from just war theory are also significant, though more nuanced. While in most cases where traditional just war theory permits a defensive war, the law enforcement model would also permit defensive violence, there would be significant limitations on offensive wars, due to the fact that law enforcement is bound by significant limitations of jurisdiction. While on traditional just war theory, the fact that Elbonia is violently persecuting a Kneebonian ethnic minority on Elbonian soil might be a sufficient just cause for a war, on the law enforcement model, execrable as such persection is, it is likely to be outside of the proper jurisdiction of Kneebonian state law enforcement. Indeed, there may have to be significant limits to extraterritorial defensive operations, though some version of the doctrine of “hot pursuit” may be helpful here.

Interestingly, the law enforcement model in one respect seems to point to greater violence. The presumption in law enforcement is that criminals are not only stopped but also punished. This suggests that on a law enforcement model, we would have a presumption in favor of putting all captured invading soldiers on trial. However, even in ordinary law enforcement, punishment is only a presumption, and it can be waived for the sake of significant public goods. In the special case of defending against invaders, having a general waiver, with exceptions tailored to mitigate the worst of evils (say, attacks on defenseless populations), in order to encourage the enemy to surrender would be prudent.

In regard to the waiver of criminal penalties, it is interesting to note one difference between how we feel about war and about ordinary crime. In the case of ordinary private crime, we do not feel that it is qualitatively worse when the criminal murders a civilian rather than a police officer—indeed, we tend to feel that there is something particularly bad about murdering a police officer. In the case of war, however, we do feel that it is much worse to kill civilians. On the law enforcement model, this difference in attitudes does not seem right. But the difference can still be defended within the law enforcement model. In typical cases, soldiers fighting an unjust war are subject to incessant propaganda that they are fighting for justice. There is thus a significant probability that they are not culpable for killing enemy soldiers, because they are rationally convinced that justice requires enemy soldiers to be stopped with lethal force. But it is much harder to come to a rational conviction that justice requires enemy civilians to be stopped with lethal force. Therefore, even if in an unjust war the intrinsic wrong of killing soldiers is just as great as that of killing civilians, there is a significant difference in likely culpability.

As I said, I do not endorse the law enforcement model. But I think it is an interesting model. And I think it presents a significant challenge to those pacifists who think that law enforcement violence is sometimes justified but that violence in war never is.

What I think is wrong with the proof of the Barcan formula

The Barcan formula says:

0. ∀x□ϕ → □∀xϕ.

The Barcan formula is dubious. Suppose, for instance, that the only things in existence are a, b and c, and let ϕ(x) say that x = a ∨ x = b ∨ x = c. Then the left-hand-side of (0) is true, since necessarily a = a, b = b and c = c. However the right-hand-side is not true, since it’s false that necessarily everything is one of a, b and c: even if there are only three things in existence, there could be more.

The Barcan formula can be proved in the Simplest Quantified Modal Logic (SQML) with S5.

Recently, a correspondent asked what I do about the fact that I accept S5 and yet presumably reject the Barcan formula. This gnawed at me for a bit, and I thought about the proof of the Barcan formula as presented by Menzel. I think I now have a pretty firm idea of where I get off the boat in the proof, and it has nothing to do with S5.

The first two steps of the proof are:

1. ∀x□ϕ → □ϕ (quantifier axiom)

2. □(∀x□ϕ→□ϕ) (from (1) by Necessitation).

Claim (1) is hard to dispute. But claim (2) isn’t right. Let ϕ be the formula D(x), where D(x) says that x is divine. Then (2) says:

2. □(∀x□D(x)→□D(x)).

By Generalization, which I think is hard to dispute, we get:

3. ∀x□(∀x□D(x)→□D(x)).

But (3) is false. For let a be me. Then (3) says the following about me:

3. □(∀x□D(x)→□D(a)),

i.e., that in the possible worlds where everything is necessarily divine, I am necessarily divine. But that’s just false. For I don’t exist in possible worlds where everything is necessarily divine. Only God exists in those worlds.

So I think the problem lies with Necessitation, which is the rule that says that theorems are necessary and yields (2) from (1). Here is my story as to what the problem with Necessitation is. Some logics have presuppositions. We can, for instance, imagine a theological logic that presupposes the existence of God. If a logic has presuppositions, then unless we have established that the presuppositions are themselves necessary truths, we are not entitled to assume that the theorems of that logic are themselves necessary. Instead, all that we are entitled to assume that the theorems of that logic necessarily follow from the presuppositions.

Now, infamously, classical logic has an existential presupposition: all the names and terms are names and terms for existing things. Because it has an existential presupposition, unless we have established the necessity of the existential presupposition, all we can say about theorems is that they necessarily follow from the existential presupposition, not that they are actually necessary.

Assuming we have a name for me in the language, it is indeed a theorem of classical logic that if everything is necessarily divine, then I am necessarily divine. But we cannot conclude that it is necessary that if everything is necessarily divine, then I am necessarily divine. For that would imply that in the world where only God exists, I would exist as well and be God. Rather, all we can conclude is that:

4. It is necessary that: if I and all the other things whose existence is presupposed exist, then if everything is necessarily divine, I am necessarily divine.

And that is trivially true, because in the worlds where everything is necessarily divine, I don’t exist.

The probability of success condition for a just war

Traditional just war theory holds that a necessary condition for a just war is not just the proportionality condition that the expected benefits exceed the expected harms, but that success is likely.

In typical cases, where the success condition fails, the proportionality condition fails as well. However, there are some hightly hypothetical cases where the success condition fails but the proportionality condition is satisfied. And in those cases I think war is justified. Thus, we should drop the success condition, and simply insist on proportionality, while being clear that proportionality includes a probabilistic assessment.

Case one. Kneebonia has exactly one missile and no weapons other than that missile. They declare war and shoot that missile at a gorgeous cathedral in the Elbonian capital that took centuries to build. They offer the Elbonia the following terms of surrender: Elbonia will become a province of Kneebonia and all books in the Elbonian language will be burned and permanently banned. Elbonia has one soldier. They parachute her onto the roof of the Kneebonian missile control building, and task her with penetrating to the computer room in order to redirect the missile into the sea. However, they know that the chance of success in this mission is 1%, because she is likely to be captured. At the same time, because the Kneebonian soldiers have no weapon other than the missile, one can be pretty confident that even if the mission fails, the Elbonian soldier will survive.

The Elbonians reciprocate the declaration of war and send their one soldier in. Proportionality may well be met: the danger of one soldier being non-lethally captured is proportionate to a 1% chance of saving a precious cultural artifact that took centuries to build. But the chance of success in this war is 1%. But if there is no success, there will be likely very little harm (one soldier captured alive).

Granted, this is a defensive case. But there are offensive cases that can be imagined as well.

Case two. A regional branch of the Elbonian army is perpetrating genocide on local Kneebonian minorities. Kneebonia has only one missile, and it can shoot it at the headquarters of that branch. Intelligence data shows that if the missile strike is successful, Elbonia will surrender and agree to end the genocide. However, the missile is wonky. There is a 99% chance that instead of hitting the headquarters, it will veer off-course and explode unseen in Elbonian coastal waters, and there is a 1% chance of success. Intelligence data shows that in case of a miss Elbonia can simply withdraw its declaration of war and the war will end, with the Kneebonians slightly puzzled as to why no hostile action apparently occurred.

Again, the probability of success is 1%. Yet it seems that war is justified. Again, if there is no success, there will be no harm.

All that said, the probability of success condition is a useful heuristic. For in typical wars, where there is insufficient probability of success, the expected harms will outweigh the expected benefits.

Inscrutable probabilities and the replay argument

Suppose that we have two hypotheses about a long sequence X of zeroes and/or ones:

• S: The sequence has the underlying structure of a sequence of independent and identically distributed (iid) probabilistic phenomena with possible outcomes zero or one.

• U: The sequence is completely probabilistically unstructured, being completely probabilistically inscrutable.

What can we say about the evidential import of X on S and U?

Here is one thought. We have a picture of how a repeated sequence of independent and identically distributed binary events should look. Much as van Inwagen says in his discussion of the replay argument, we expect to have the proportion of ones to oscillate but then converge to some value. So if X looks like that, then we have evidential support for S, and if X doesn’t look like that—maybe there is wilder oscillation and no convergence—then we do not have evidential support for S.

But does the fact that the sequence X has a “nice gradual convergence” in frequencies support the structure hypothesis S? Van Inwagen in his replay argument against indeterministic free will seems to assume so.

Actually, convergence of frequencies does not support S (or U, for that matter). Here is why not. Let’s model hypothesis S as follows. There is some unknown real dispositional probability p between 0 and 1, and then our sequence X is taken by observing a sequence of iid results with the probability of the result being 1 being equal to p. Given p, any sequence of n events whose proportion of ones is m/n has probability p^m(1−p)^n − m, regardless of whether the sequence shows any “nice gradual convergence” of the frequency of ones to m/n or is just a seqence of n − m zeroes followed by m ones.

Of course, hypothesis S doesn’t say what the probability p of getting a one on a given experiment is. So, we need to suppose some probability distribution Q over the possible values of p in the interval [0,1], and then our probability of the sequence X of m ones and n − m zeroes will be ∫p^m(1−p)^n − mdQ(p). But it’s still true that this probability does not depend on the order between the ones and zeroes. The sequence could “look random”, or it could look as fishy as you like, and the probability of it on S would be exactly the same if the number of ones is the same.

On the other hand, on the hypothesis U, all possible sequences X of length n are probabilistically inscrutable, and hence we cannot say anything about any sequence being more likely than another—we might as well represent the probability of any sequence X of observations as just an interval-valued probability of [0,1].

So, no facts about the order of events in X make any difference between the hypotheses S and U. In particular, van Inwagen’s idea that an appearance of convergence is evidence for S is false.

Now, let’s say a little more about the Bayesian import of the observation X. Let’s suppose that S and I are our only two hypotheses, and that both have serious prior probabilities, say somewhere between 0.1 and 0.9. Further, let’s suppose that that the sequence X is long and has a decent amount of variation in it—for instance, let’s suppose that it has at least 50 zeroes and at least 50 ones. Because of this, P(X|S) will be some astronomically small positive number α. Indeed, we can prove that α is at most 1/2¹⁰⁰ ≈ 10⁻³⁰, for any distribution of the unknown probability p of getting a one.

On the other hand, P(X|U) will be completely probabilistically inscrutable, and hence can be represented as the full interval [0,1].

Here’s what follows from Bayes’s theorem combined with a natural way of handling inscrutable probabilities as a range of probability assignments: The posterior probability of S will be a range of probabilities that starts with something within an order of magnitude of α and ends with 1. Hence, our observation of X does not support S: upon observing X, we move from S having some moderate probability between 0.1 and 0.9, to a nearly completely inscrutable probability in a range starting with something astronomically small and ending at one. The upper end of the range is higher than what S started with but the lower end of the range is lower.

Thus, if we were to actually do the experiment that van Inwagen describes, and get the results that he thinks we would get (namely, a sequence converging to some probability), that would not support the hypothesis S that van Inwagen thinks it would support.