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.

Wronging others

Alice is about to inherit a large some of money and then she hears that she has a long-lost sibling with whom she’d have to share. So she hires an assassin to kill the sibling. Happily, the “assassin” turns out to be a police officer who promptly arrests Alice for attempted murder. Bob happens to be in exactly the same position and acts the same way. However, Bob actually has a long-lost sibling, while Alice has been misinformed.

Legally, it may be that Alice can get off on the grounds of the doctrine of impossible attempts. But morally speaking, Alice and Bob are both just as guilty as if they had successfully committed murder. And, further, they are equally guilty (barring some other differences between the cases).

However, Bob wronged and violated the rights of his long-lost sibling. This is true even though the assassination failed, because one is wronged and has one’s rights violated by an attempt on one’s life.

Alice did not wrong or violate the rights of any long-lost sibling since she does not have any such sibling.

It is correct, as a matter of the description of the situation, that Bob wronged a sibling, and Alice did not. But this fact does not add to the wrongfulness of Bob’s action. This suggests that the patient-centered concept of wronging someone does not actually carry much ethical weight.

Maybe.

But maybe what we should say is this: Bob’s action was indeed more wrong than Alice’s, but Alice is no less culpable than Bob.

If we say this, then we may want to push the reasoning further. Suppose Carl is subjectively in a situation like Alice and Bob, but (a) like Bob, Carl does have a long-lost sibling, and (b) the assassin is real and actually kills the sibling. I used to think that attempted murder was no less wrong than a successful one. But if we are to mae a distinction between Alice and Bob, perhaps we can make a similar distinction between Bob and Carl. What Carl did was more wrong than what Bob did, even though Bob is just as culpable as Carl.

I am not sure, though. Consider this. Imagine that all three malefactors are in their right minds, acting freely, with no excuses available. In that case, each one of them is fully culpable for each wrong they did. But the following principle seems pretty plausible:

1. If X and Y are fully culpable for wrong actions A and B, respectively, and B is more wrong than A, then Y is more culpable than X in regard to these respective actions.

It follows that if we think that Bob did something more wrong than what Alice did, then Bob is more culpable than Alice.

But maybe (1) is false. Or maybe we can say this. Carl is only culpable for an attempted murder, but isn’t actually culpable for the successful murder, because culpability only attaches to attempts. Neither option seems attractive. So I am back to where I was: Alice’s action is not less wrong than Bob, and Bob’s is no less wrong than Carl’s. The stuff about violating rights and wronging what one owes others, that’s all true, but it doesn’t actually affect the degree of wrongness.

The replay argument against agential control

Van Inwagen’s replay experiment is supposed to show that indeterministic free will is problematic. We imagine someone choosing betweeen A and B, then we rewind the state of the universe to just how it was before their choice, and repeat. Van Inwagen thinks that as we continue the experiments, we will get the proportion of choices that are As converging to some number, say 30%, and that tends to make us think that the choices are random rather than something in the agent’s control.

Let’s imagine we’ve actually done this replay a very large number of times. Here are the three possibilities for what we might observe:

• O₁: The proportion of As settled down to some number strictly between 0 and 1.

• O₂: All the choices were As or all the choices were Bs.

• O₃: The proportion of As changed and did not settle down.

For familiar libertarian reasons, it seems that O₂ would be evidence against the hypothesis that the choice is in the agent’s control: they both support the hypothesis that the agent is compelled or nearly compelled, either by their character or by external forces, to choose as they did.

Van Inwagen claims that observation O₁ is evidence against the hypothesis that the choice is in the agent’s control. For Bayesian reasons, given that O₁, O₂ and O₃ are exhaustive and mutually exclusive, and that O₂ is also evidence against the agent’s control, the only way this could be is O₃ favored the agential control hypothesis (unless our prior probability of O₃ is equal to 1, which it’s not).

But I don’t think observation O₃ would favor the agential control hypothesis. There are two reasons for my judgment.

First, let’s subdivide observation O₃ into two suboptions.

• O_3a: There is some discernible pattern to the As and Bs that precludes the proportion of As from settling down, e.g., ABBAAAABBBBBBBBAAAAAAAAAAAAAAAA...,

• O_3b: It’s just a complete mess and the proportion doesn’t settle.

Now, O_3a is a strange option. A pattern initially may seem like evidence of control. But once we recall that the experiment involved a rewinding of the universe’s state, we can see that any pattern that is not just all As or all Bs cannot be the result of agential control, since control of a pattern across time would require memory, and we have assumed that any memories are wiped. So, given the rewinding, we know that any pattern—other than the patterns that do not require memory, namely the patterns in options 2 and 3, is a fluke. Thus, 4a shouldn’t be evidentially different from 4b.

What about O_3b? Well, O_3b is more of a mess than O₁. A mess is, if anything, more random than a sequence with a probabilistic structure, and hence if anything less indicative of agential control. Thus, O_3b is either neutral with regard to the hypothesis of agential control or even evidence against it. And O_3b is in the boat by the above remarks. Thus, O₃ is either neutral with regard to agential control or evidence against it.

Second, the reason van Inwagen thinks O₁ goes against the agential control hypothesis is because it favors the hypothesis that we simply have a probabilistic structure of independent identically distributed random experiments: I will call this the “iid hypothesis”. Now, suppose we observe option O₃, and let’s consider the specific observed sequence. Let r be the final observed proportion of As. This is some number strictly between 0 and 1 (otherwise we would have O₂). Of course, because we are in option O₃, we didn’t actually observe convergence to r, but still there was some final proportion. Whatever that number r is, van Inwagen thinks that if we had observed convergence to r, that would have been evidence against the agential control hypothesis, because it would have indicated a probabilistic random structure. But now observe that on the iid hypothesis, if there are m of the As and n of the Bs in a sequence, and the probability of A is p, then the probability of the sequence is p^m(1−p)ⁿ. Note that this does not depend on the order of the As and Bs in the sequence. Hence when we keep fixed the proportion of As in the sequence, any particular sequence that looks like it converges to that proportion and any particular sequence that simply happens, after some swings, to end at that proportion are equally likely. Further, the main competing hypothesis with the iid hypothesis is that no meaningful probabilities can be attached to the situation. In that case, we cannot say anything about whether O₁ or O₃ is more likely. Thus, any particular sequence we could get that exhibits O₁ does not favor the iid hypothesis any more than any particular sequence we could get that exhibits O₃.

Let us recapitulate. Option O₃ either is neutral on whether there is agential control or is some evidence against it. Option O₂ opposes agential control. Thus, option O₁ must either be neutral on whether there is agential control or be some evidence for agential control. And so van Inwagen’s replay argument does not work.

Nano St Petersburg

In the St Petersburg game, you toss a fair coin until you get heads. If it took you n tosses to get to heads, you get a utility of 2ⁿ. The expected payoff is infinite (since (1/2) ⋅ 2 + (1/4) ⋅ 4 + (1/8) ⋅ 8 + ... = ∞), and paradoxes abound (e.g., this.

One standard way out is to deny the possibility of unboundedly large utilities.

Interestingly, though, it is possible to imagine St Petersburg style games without really large utilities.

One way is with tiny utilities. If it took you n tosses to get to heads, you get a utility of 2ⁿα, where α > 0 is a fixed infinitesimal. The expected payoff won’t be infinite, but the mathematical structure is the same, and so the paradoxes should all adapt.

Another way is with tiny probabilities. Let G(n) be this game: a real number is uniformly randomly chosen between zero and one, and if the number is one of 1, 1/2, 1/3, ..., 1/n, then you get a dollar. Intuitively, the utility of getting to play G(n) is proportional to n. Now our St Petersburg style game is this: you toss a coin until you get heads, and if you got heads on toss n, you get to play G(2ⁿ).

Dominance and infinite lotteries

Suppose we have an infinite fair lottery with tickets 1,2,3,…. Now consider a wager W where you get 1/n units of utility if ticket n wins. How should you value that wager?

Any value less than zero is clearly a non-starter. How about zero? Well, that would violate the dominance principle: you would be indifferent between W and getting nothing, and yet W is guaranteed to give you something positive. What about something bigger than zero? Well, any real number y bigger than zero has the following problem as a price: You are nearly certain (i.e., within an infinitesimal of certainty) that the payoff of W will be less than y/2, and hence you’ve overpaid by at least y/2.

But what about some price that isn’t a real number? By the above argument, that price would have to be bigger than zero, but must be smaller than every positive real number. In other words, it must be infinitesimal. But any such price will violation dominance as well: you would be indifferent between W and getting that price, yet it is certain that W would give you something bigger—namely one of the real numbers 1, 1/2, 1/3, ....

So it seems that no price, real numbered or other, will do.

(This argument is adapted from one that Russell and Isaacs give in the case of the St Petersburg paradox.)

One way out will be familiar to readers of my work: Reject the possibility of infinite fair lotteries, and thereby get yet another argument for causal finitism.

But for those who don’t like controversial metaphysics solutions to decision theoretic problems, there is another way: Deny the dominance principle, price W at zero, and hold that sometimes it is rational to be indifferent between two outcomes, one of which is guaranteed to be better than the other no matter what.

This may sound crazy. But consider someone who assigns the price zero to a dart tossing game where you get a dollar if the dart hits the exact center and nothing otherwise, reasoning that the classical mathematical expected value of that game for any continuous distribution of dart tosses (such as a normal distribution around the center) is zero. I think this response to an offer to play is quite rational: “I am nearly certain to lose, so what’s the point of playing?” Now, that case doesn’t violate the same dominance principle as the lottery case—it violates a stronger dominance principle that says that if one option is guaranteed to be at least as good as the other and in some possible scenario is better, then it should be preferred. But I think the dart case may soften one up for thinking this:

1. If (a) some game never has an outcome that’s negative, and (b) for any positive real, it is nearly certain that the outcome of some game will be less than that, I should value it at zero or less.

And if we do that, then we have to value W at zero. Yes, if you reject W in favor of nothing, you’ve lost something. But probably very, very little.

Here is another weakish reason to be suspicious of dominance. Dominance is too similar to conglomerability, and conglomerability should be suspicious to anyone who likes exotic probabilistic cases. (By the way, this paper connects with this.)

How not to value wagers

Given the Axiom of Choice, there is a rotationally invariant finitely additive probability measure defined for all subsets of a circle. We can use such a finitely probability measure to define an expected value Ef or integral of a bounded function f on the circle, and we might want to have a decision theory based on this expected value. Given a wager that pays f(z) at a uniformly randomly chosen location z on the circle, we are indifferent to buying the wager at price Ef, we must accept the wager at lower prices, and we must reject it at higher prices.

This procedure, however, leads to the following interesting thing: There will be bounded wagers that pay more than y no matter what, but where one is indifferent with respect to buying the wager at price y. To see this, let x be an irrational number, and as in my previous post, let u be a bounded function on the circle such that u(ρz) > u(z) for all z where ρ is rotation by x degrees. Then let f(z) = u(ρz) − u(z). Because of the additivity of integrals with respect to finitely additive measures and rotational invariance, we have Ef = ∫f(ρz)dP(z) − ∫f(z)dP(z) = 0. But f(z) > 0 for all z. So the decision theory tells us to be indifferent to the game where you get payoff f(z) at z when the game is offered for free, even though no matter what the outcome of the game, you will received a strictly positive amount.

More generally, given the Axiom of Choice, there is no finitely-additive rotationally-invariant expected value assignment for bounded utilities that respects the principle that any gamble that is sure to pay more than y ought to be accepted at price y.

Domination and uniform spinners

About a decade ago, I offered a counterexample to the following domination principle:

1. Given two wagers A and B, if in every state B is at least as good as A and in at least one state B is better than A, then one should choose B over A.

But perhaps (1) is not so compelling anyway. For it might be that it’s reasonable to completely ignore zero probability outcomes. If a uniform spinner is spun, and on A you get a dollar as long as the spinner doesn’t land at 90^∘ and on B you get a dollar no matter what, then (1) requires you to go for B, but it doesn’t seem crazy to say “It’s almost surely not going to land at 90^∘, so I’ll be indifferent between A and B.”

But now consider the following domination principle:

2. Given two wagers A and B, if in every state B is better than A, then one should choose B over A.

This seems way more reasonable. But here is a potential counterexaple. Consider a spinner which uniformly selects a point on the circumference of a circle. Assume x is any irrational number. Consider a function u such that u(z) is a real number for any z on the circumference of the circle. Imagine two wagers:

• A: After the spinner is spun and lands at z, you get u(z) units of utility

• B: After the spinner is spun, the spinner is moved exactly x degrees counterclockwise to yield a new landing point z′, and you get u(z′) units of utility.

Intuitively, it seems absurd to think that B could be preferable to A. But it turns out that given the Axiom of Choice, we can define a function u such that:

3. For any z on the circumference of the circle, if z′ is the result of rotating z by x degrees counterclockwise around the circle, then u(z′) > u(z).

And then if we take the states to be the initial landing points of the spinner, B always pays strictly better than A, and so by the domination principle (2), we should (seemingly absurdly) choose B.

Remarks:

• The proof of the existence of u requires the Axiom of Choice for collections of countable sets of reals). In my Infinity book, I argued that this version of the Axiom of Choice is true. However, arguments similar to those in the book’s Axiom of Choice chapter suggest that the causal finitist has a good way out of the paradox by denying the implementability of the function u.

• Some people don’t like unbounded utilities. But we can make sure that u is bounded if we want (if the original function u is not bounded, then replace u(z) by arctan u(z)).

• Of course the function u is Lebesgue non-measurable. To see this, replacing u by its arctangent if necessary, we may assume u is bounded. If u were measurable and bounded, it would be integrable, and its Lebesgue integral around the circle would be rotation invariant, which is impossible given (3).

It remains to prove the existence of u. Let ∼ be the relation for points on the (circumference of the circle) defined by z ∼ z′ if the angle between z and z′ is an integer multiple of x degrees. This is an equivalence relation, and hence it partitions the circle into equivalence classes. Let A be a choice set that contains exactly one element from each of the equivalence classes. For any z on the circle, let z₀ be the point in A such that z₀ ∼ z. Let u(z) be the (unique!) integer n such that rotating z₀ counterclockwise around the circle by an angle of nx degrees yields z. Then for any z, if z′ is the result of rotating z′ by x degrees around the circle, then u(z′) = u(z) + 1 > u(z) and so we have (3).

A cosmological argument from the Hume-Edwards Principle

The Hume-Edwards Principle (HEP) says:

1. If you’ve explained every item in a collection, you’ve explained the whole collection of items.

This sounds very plausible, but powerful counterexamples have been given. For instance, suppose that exactly at noon, cannonball is shot out of a cannon. The collection C of cannonball states after noon has the property that each state in C is explained by an earlier state in C (e.g., a state at 12:01:00 is explained by a state at 12:00:30). By the Hume-Edwards Principle, this would imply that C is self-explanatory. But it plainly is not: it requires the cannon being fired at noon to be explained.

But I just realized something. All of the effective counterexamples to the Hume-Edwards Principle involve either circular causation or infinite causal regresses. We can now argue:

2. HEP is necessarily true.

3. If circular causation is possible, counterexamples to HEP are possible.

4. If infinite causal regresses are possible, counterexamples to HEP are possible.

5. So, neither circular causation nor infinite causal regresses are possible.

6. If there is no first cause, there is a causal circle or an infinite causal regress.

7. So, there is a first cause.

Similarly, it is very plausible that if infinite causal regresses are impossible, then causal finitism, the thesis that nothing can have an infinite causal history, is true. So, we get an argument from HEP to causal finitism.

Dialectically, the above is very odd indeed. HEP was used by Hume and Edwards to oppose cosmological arguments. But the above turns the tables on Hume and Edwards!

Objection: Not every instance of causal regress yields a counterexample to HEP. So it could be that HEP is true, but some causal regresses are still possible.

Response: It’s hard to see how there is sufficient structural difference between the cannonball story and other regresses to allow one to deny the cannonball story, and its relatives, while allowing the kind of regresses that are involved in Hume’s response to cosmological arguments.

Final remark: What led me to the above line of thought was reflecting on scenarios like the following. Imagine a lamp with a terrible user interface: you need to press the button infinitely many times to turn the lamp on, and once you do, it stays on despite further presses. Suppose now that in an infinite past, Alice was pressing the button once a day. Then the lamp was always on. Now I find myself with two intuitions. On the one hand, it seems to me that there is no explanation in the story as to why the lamp was always on: “It’s always been like that” just isn’t an explanation. On the other hand, we have a perfectly good explanation why the lamp was on n days ago: because it was on n + 1 days ago, and another button press doesn’t turn it off. And I found the second intuition pushing back against the first one, because if every day’s light-on state has an explanation, then there should be an explanation of why the lamp was always on. And then I realized this intuition was based on somehow finding HEP plausible—despite having argued against HEP over much of my philosophical career. And then I realized that one could reconcile HEP with these arguments by embracing causal finitism.

Aquinas on drunkenness and sleep

Aquinas argues that

drunkenness is a mortal sin, because then a man willingly and knowingly deprives himself of the use of reason, whereby he performs virtuous deeds and avoids sin, and thus he sins mortally by running the risk of falling into sin.

On the other hand, Aquinas also argues that sleep suspends the use of reason:

What a man does while asleep, against the moral law, is not imputed to him as a sin; as Augustine says (Gen. ad lit. xii, 15). But this would not be the case if man, while asleep, had free use of his reason and intellect. Therefore the judgment of the intellect is hindered by suspension of the senses.

One might try to reconcile the two claims by saying that sleep is something that befalls us involuntarily, and that it would be wrong to willingly and knowingly go to sleep. But that would not fit with ordinary human practice, and would contradict Aquinas’ own rejection of the claim that it is “against virtue for a person to set himself to sleep”. Moreover, Aquinas notes without any moral warnings that sleep—like baths, contemplation of the truth and other apparently quite innocent things—assuages sorrow.

So what’s going on?

And to add a further complication, Proverbs 31:6 seems to recommend the use of alcohol as an analgesic.

I can think of three things one could say on behalf of Aquinas.

First, one might attempt a Double Effect justification. In sleep, the body rests. Aquinas certainly thinks so: the discussion of the suspension of reason during sleep presupposes that the primary effect of sleep is on the body. It is this bodily rest, rather than the suspension of reason, that is intended. One might worry that the suspension of reason is a means to rest. However, non-human animals, who lack reason in Aquinas’s sense of the word, also sleep. Presumably whatever benefits they derive from the sleep are available to us, and it seems not unlikely that many of these do not depend on the suspension of reason. Similarly, alcohol helps with pain in non-human animals, and so the mechanism by which it helps may not depend on the suspension of reason.

That said, I don’t think Aquinas would want to take this approach (though it may well work for me). For Aquinas thinks that it is stupid we cannot claim that an invariable or typical effect of something intended counts as unintended (Commentary on the Physics, Book II, Lecture 8, paragraph 214). But the suspension of reason is indeed an invariable or typical effect of sleep.

A second approach focuses on Aquinas’ response to the question of why the loss of rationality during the sexual act does not render the sexual act wrong, from which I already quoted the rejection of the claim that it’s vicious to set oneself to sleep:

it is not contrary to virtue, if the act of reason be sometimes interrupted for something that is done in accordance with reason … .

This approach does not seem to be based on Double Effect, but rather on some sort of principle that it is permissible to suspend a good for the sake of that same good. This principle applies neatly to sleep as well as to the biblical case of analgesic use of alcohol (given that reason opposes suffering the pain).

But this approach would also moderate Aquinas’s seemingly absolute rejection of drunkenness. For we can imagine cases where it seems that reason would recommend drunkenness, such as when a tyrant will kill you if you refuse to get drunk with them. And once one allows drunkenness in such extreme cases, what is to prevent allowing it in more moderate cases, such as getting drunk with one’s boss in the hope of getting a deserved promotion… or maybe just for fun? Aquinas can say that these cases are immoral and hence against reason, but that would beg the question.

A third approach would note that sleep, unlike drunken stupor, is a natural human state, and information processing in sleep is itself a part of our human rational processing. However, while this gives a neat explanation of why it’s permissible to set oneself to sleep, it doesn’t explain the permissibility of the analgesic use of alcohol or, more significantly in modern times, of the use of general anaesthesia during medical procedures.

A different approach for justifying sleep, the analgesic use of alcohol and general anaesthesia insists that temporary suspension of a good is different from willful opposition to the good. To eat in an hour rather than now does not oppose the good of food. The down side of this fourth approach is that it seems to destroy Aquinas’s argument against drunkenness as opposed to the good of reason. And it seems to let in too much: can’t one say that by torturing someone, one is merely suspending their painless state?

I think the best philosophical solution is the first, Double Effect. Aquinas alas can’t use it because his version of Double Effect is too narrow, given his view that typical effects of intended things count as intended.

It can be rational to act as if one's beliefs were more likely true than the evidence makes them out to be

Consider this toy story about belief. It’s inconvenient to store probabilities in our minds. So instead of storing the probability of a proposition p, once we have evaluated the evidence to come up with a probability r for p, we store that we believe p if r ≥ 0.95, that we disbelieve p if r ≤ 0.05, and otherwise that we are undecided. (Of course, the “0.95” is only for the sake of an example.)

Now, here is a curious thing. Suppose I come across a belief p in my mind, having long forgotten the probability it came with, and I need to make some decision to which p is relevant. What probability should I treat p as having in my decision? A natural first guess is 0.95, which is my probabilistic threshold for belief. But that is a mistake. For the average probability of my beliefs, if I follow the above practice perfectly, is bigger than 0.95. For I don’t just believe things that have probability 0.95. I also believe things that have probability 0.96, 0.97 and even 0.999999. Intuitively, however, I would expect that there are fewer and fewer propositions with higher and higher probability. So, intuitively, I would expect the average probability of a believed proposition to be a somewhat above 0.95. How far above, I don’t know. And the average probability of a believed proposition is the probability that if I pick a believed proposition out of my mental hat, it will be true.

So even though my threshold for belief is 0.95 in this toy model, I should treat my beliefs as if they had a slightly higher probability than that.

This could provide an explanation for why people can sometimes treat their beliefs as having more evidence than they do, without positing any irrationality on their part (assuming that the process of not storing probabilities but only storing disbelieve/suspend/belief is not irrational).

Objection 1: I make mistakes. So I should take into account the fact that sometimes I evaluated the evidence wrong and believed things whose actual evidential probability was less than 0.95.

Response: We can both overestimate and underestimate probabilities. Without evidence that one kind of error is more common than the other, we can just ignore this.

Objection 2: We have more fine-grained data storage than disbelieve/suspend/believe. We confidently disbelieve some things, confidently believe others, are inclined or disinclined to believe some, etc.

Response: Sure. But the point remains. Let’s say that we add “confidently disbelieve” and “confidently believe”. It’ll still be true that we should treat the things in the “believe but not confidently” bin as having slightly higher probability than the threshold for “believe”, and the things in the “confidently believe” bin as having slightly higher probability than the threshold for “confidently believe”.

Animalist functionalism

The only really plausible hope for a materialist theory of mind is functionalism. But the best theory of our identity, materialist or not, is animalism—we are animals.

Can we fit these two theories together? On its face, I think so. The thing we need to do is to make the functions defining mental life be functions of the animal, not of the brain as such. Here are three approaches:

1. Adopt a van Inwagen style ontology on which organisms exist but brains do not. If brains don’t exist, they don’t have functions.

2. Insist that some of the functions defining mental life are such that they are had by the animal as a whole and not by the brain. Probably the best bet here are the external inputs (senses) and outputs (muscles).

3. Modify functionalism by saying that mental properties are properties of an organism with such-and-such functional roles.

I think option 2 has some special difficulties, in that it is going to be difficult to define “external” in such a way that the brain’s connections to the rest of the body don’t count as external inputs and outputs and yet we allow enough multiple realizability to make very alien intelligent life possible. One way to fix these difficulties with option 2 is to move it closer to option 3 by specifying that the external inputs and outputs must be inputs and outputs of an organism.

Options 1 and 3, as well as option 2 if the above fix is used, have the consequence that strong AI is only possible if it is embedded in a synthetic organism.

All that said, animalist functionalism is in tension with an intuition I have about an odd thought experiment. Imagine that after I got too many x-rays, my kidney mutated to allow me exhibit the kinds of functions that are involved in consciousness through the kidney (if organism-external inputs and outputs are required, we can suppose that the kidney gets some external senses, such as a temperature sense, and some external outputs, maybe by producing radio waves, which help me in some way) in addition to the usual way through the brain, and without any significant interaction with the brain’s computation. So I am now doing sophisticated computation in my kidney of a sort that should yield consciousness. On animalist functionalism, I should now have two streams of consciousness: one because of how I function via the brain and another because of how I function via the mutant kidney. But my intuition is that in fact I would not be conscious via the kidney. If there were two streams of consciousness in this situation (which I am not confident of), only one would be mine. And that doesn’t fit with animalist functionalism (though it fits fine with non-animalist functionalism, as well as with animalist dualism, since the dualist can say that the kidney’s functioning is zombie-like).

Given that functionalism is the only really good hope we have right now for a materialist theory of mind, if my intuition about the mutant kidney is correct, this suggests that animalism provides evidence against materialism.

Game Boy Fiver [Wordle clone]: How to compress 12972 five-letter words to 17871 bytes

Update: You can play an updated version online here in the binjgb Game Boy emulator. This is the version with the frequency-based answer list rather than the official Wordle list, for copyright reasons.

There is a Game Boy version of Wordle, using a bloom filter, a reduced vocabulary and a reduced list of guess words, all fitting on one 32K cartridge. I decided to challenge myself and see if I could fit in the whole 12972 word Wordle vocabulary, with the whole 2315 word answer list. So the challenge is:

• Compress 12972 five-letter words (Vocabulary)

• Compress a distinguished 2315 word subset (Answers).

I managed it (download ROM here), and it works in a Game Boy emulator. There is more than one way, and what I did may be excessively complicated, but I don’t have a good feel for how fast the Game Boy runs, so I did a bit of speed optimization.

Step 0: We start with 12972 × 5 = 64860 bytes of uncompressed data.

Step 1: Divide the 12972 word list into 26 lists, based on the first letter of the word. Since in each list, the first letter is the same, we now need only store four letters per word, along with some overhead for each list. (The overhead in the final analysis will be 108 bytes.) If we stop here,

Step 2: Each four letter “word” (or tail of a word) can be stored with 5 bits per letter, thereby yielding a 20 bit unsigned integer. If we stop here, we can store each word in 2.5 bytes, for a total of 32430. That would fit on the cartridge if there was no code, but it is some progress.

Step 3: Here was my one clever idea. Each of the lists of four letter “words”, is in alphabetical order, and encoded the natural way as 20 bit numbers, the numbers will be in ascending order. Instead of storing these numbers, we need only store their arithmetical differences, starting with an initial (invalid) 0.

Step 4: Since the differences are always at least 1, we can subtract one from each difference to make the numbers slightly smaller. (This is a needless complication, but I had it, and don’t feel like removing it.)

Step 5: Store a stream of bytes encoding the difference-minus-ones. Each number is encoded as one, two or three bytes, seven-bits in each byte, with the high bit of each byte being 1 if it’s the last 7-bit sequence and 0 if it’s not. It turns out that the result is 17763 bytes, plus 108 bytes of overhead, for a total of 17871 bytes, or 28% of the original list, with very, very simple decompression.

Step 6: Now we replace each word in the alphabetically-sorted Answers list with an index into the vocabulary list. Since each index fits into 2 bytes, this would let us store the 2315 words of the Answers as 2315 × 2 = 4630 bytes.

Step 7: However, it turns out that the difference between two successive indexes is never bigger than 62. So we can re-use the trick of storing successive differences, and store the Answers in 2315 bytes. (In fact, since we only need 6 bits for the differences, we could go down to 1737 bytes, but it would complicate the code significantly.)

Result: Vocabulary plus Answers goes down to 108+17763+2315=20186 bytes. This was too big to fit on a 32K cartridge using the existing code. But it turns out that most of the existing code was library support code for gprintf(), and replacing the single gprintf() call, which was just being used to format a string containing a single-digit integer variable, with gprint(), seemed to get everything to fit in 32K.

Example of the Vocabulary compression:

• The first six words are: aahed, aalii, aargh, aarti, abaca, abaci.

• Dropping the initial “a”, we get ahed, alii, argh, arti, baca, baci.

• Encoding as 20-bit integers and adding an initial zero, we get 0, 7299, 11528, 17607, 18024, 32832, 32840.

• The differences-minus-one are 7298, 4228, 6078, 416, 14807, 7.

• Each of these fits in 14-bits (two bytes, given the high-bit usage), with the last one in 7-bits. In practice, there are a lot of differences that fit in 7-bits, so this ends up being more efficient than it looks—the first six words are not representative.

Notes:

• With the code as described above, there are 250 bytes to spare in the cartridge.

• One might wonder whether making up the compression algorithm saves much memory over using a standard general purpose compressor. Yes. gzip run on the 64860 bytes of uncompressed Vocabulary yields 30338 bytes, which is rather worse than my 17871 byte compression of the Vocabulary. Plus the decompression code would, I expect, be quite a bit more complex.

• One could save a little memory by encoding the four-letter “words” in Step 2 in base-26 instead of four 5-bit sequences. But it would save only about 0.5K of memory, and the code would be much nastier (the Game Boy uses library functions for division!).

• The Answers could be stored as a bitmap of length 12972, which would be 1622 bytes. But this would make the code for generating a random word more complicated and slower.

Sports injuries and the problem of evil

An argument from evil against the existence of God based on sports injuries would not, I think, be found very persuasive. Why not?

I take it that this is because of the retort: “The athletes freely undertook these risks.”

This free undertaking is not the whole of the explanation of why sports injuries are less troubling as examples of the problem of evil. Another part is the idea that there are significant goods at stake in sports, and real danger is a constitutive component of some of these goods. But that the risk is freely accepted is surely a major part of our lack of worry. We are much more worried about evils that befall those who did not freely undertake the relevant risks.

Note also that as a rule we do not—though there are notable exceptions—blame the athletes for freely undertaking the risks of sports injuries. We feel that, generally speaking, they are within their rights to undertake these risks for these potential benefits.

But if God exists, he is closer to us than we are to ourselves. He cares for our good more than we care for ourselves. And he has more rights over us than we have over ourselves. If this is correct, then just as we have the right (limited as it is) to accept certain serious risks, God has an even greater right to impose risks on us.

The observation that God would have more rights over us than do over ourselves by no means solves the problem of evil. But it helps.

Fixing an earlier regress argument about intentions

In an earlier post, I generated a regress from:

1. If you are responsible for x, then x is an outcome of an intentional act with an intention that you are responsible for,

where both responsibility and outcomehood are partial. But I am now sceptical of 1. It is plausible when applied to things that aren’t actions, but there is little reason to think an action I am responsible for has to be the outcome of another action of mine.

Maybe what I should say is this:

2. Any action that I responsible for has an intention I am responsible for.

3. Anything that isn’t an action that I am responsible for is an outcome of an action I am responsible for.

This still seems to generate a regress or circle. By (3), if I am responsible for anything, I am responsible for some action, say A₁. This will have an intention I₁ that I am responsible for. Now either I₁ is itself an action A₂ or an outcome of some action A₂ that I am responsible for. In both cases, I am responsible for A₂. And then A₂ will have an intention I₂ that I am responsible for. And so on.

How can we arrest this? I think there are exactly two ways out:

4. Some action A_n is identical with its intention I_n.

5. Some action A_n has its own intention I_n as an outcome of itself.

Divine hiddenness and divine command ethics

Once upon a time, there was an isolated village in the mountains. It had a large electronic billboard. Every so often, unsigned demands appeared on the billboard. Most of these demands seemed reasonable, and the villagers find themselves with an ingrained feeling that they should do what the billboard says, and typically they do so, often deferring to the billboard even when the reasonableness of its demands is less clear. There were two main theories about the billboard. Some villagers said that thousands of miles away there was an authoritative and benevolent monarch who had cameras and microphones hidden around the village, and who issued commands via the billboard. Others said that there was no monarch, but centuries ago, as a science fair project, a clever teenager wrote a machine learning program that offered good advice for the village—a program that wasn’t sophisticated enough to count as really intelligent, but nonetheless its deliverances were quite helpful—and hooked it up to the billboard, and eventually the origins of the system were largely forgotten. The evidence is such that neither group of villagers is irrational in holding to their theory.

Suppose that the monarch theory was in fact the correct one.

Question: Did the monarch’s demands constitute valid commands for the villagers who accepted the software theory?

Response: No. Anonymous demands are not valid commands even when they are issued by a genuine authority. A valid command needs to make it evident whom it comes from. When the authority chooses not to make a subordinate be aware of the demand as an authoritative command, the demand is not an authoritative command.

Objection: Given the widely ingrained feeling that the billboard should be obeyed, even the villagers who accepted the software theory had a duty to obey the billboard. That was just part of the governing structure of the village: to obey the billboard.

Response: Perhaps. But even so, the duty to obey the billboard (at least over the villagers who accept the software theory) wasn’t grounded in the monarch’s authority, but in either the authority of the individual’s conscience or the law-giving force of village custom.

Question: Did the monarch’s demands constitute valid commands for the villagers who accepted the monarch theory?

Response: I am not sure. I think a case can be made in either direction.

Intentional acts that produce their own intentions

Start with these assumption:

1. If you are at least partly responsible for x, then x is at least partly an outcome of an intentional act with an intention that you are at least partly responsible for.

2. You are at least partly responsible for something.

3. You do not have an infinite regress of intentions.

4. You do not have a circle of distinct intentions.

For brevity, let’s drop the “at least partly”. Let’s say you’re responsible for x. Then x must be an outcome of an intentional act with an intention I₁ you’re responsible for; the intention I₁ then must be an outcome of an intentional act with an intention I₂ you’re responsible for; and so on.

It seems we now have a contradiction: by (1) and (2), either you have infinitely many intentions in the list I₁, I₂, ..., and hence a regress contrary to (3), or else you come back to some intention that you already had, and hence you have a circle, contrary to (4).

But there is one more possibility, and (1)–(4) logically entail that this one more possibility must be true:

5. For some n, I_n = I_n + 1.

This is like a circle, but not quite. It is a fixed point. What we have learned is that given (1)–(5):

6. You have at least one intention that is at least partly an outcome of an intentional act with that very intention.

This seems even more absurd than a circle or regress. One wants to ask: How could an intention be its own outcome? But this question has a presupposition:

7. Any at least partial outcome of an intentional act is an at least partial outcome of the intentional act’s intention.

What we learn from (6) is that the presupposition (7) is false. (For if (7) were true, then given (6) some intention would be its own at least partial outcome, which is indeed absurd.)

But how can (7) be false? How can an outcome (again, let’s drop the partiality for brevity) of an intentional act not be an outcome of the act’s intention? I think there are two possibilities. First, the act’s intention can itself be an outcome of the act. Second, the act’s intention can be something that is parallel to the act and neither is an outcome of the other. The second view fits with acausalist views in the philosophy of intention, but it does not seem plausible to me—there needs to be a causal connection of some sort between an intentional act and its intention. And in any case the second view won’t help solve our puzzle.

So, we are led to a view on which at least sometimes the intention of an intentional act is an outcome rather than a cause of the act. If we think of an intention as explaining the rational import of an act, then in such cases the rational import of the act is retrospective in a way.

It would be neatest if every time one performed an intentional act, the intention were an outcome of the act. But we have good reason to think that sometimes the intention precedes the intentional act. For instance, when I decide on a plan of action, and then simply carry out the plan, the plan as it is found in my mind informs my sequence of acts in the way an intention does, and so it makes sense to talk of the plan as an intention. But now think about the mental act of deliberation by which I decided on the plan, including on the plan’s end. Here it makes sense to think of the plan’s end as being a part of the intention behind the mental act—the mental act is made rational by aiming at the plan’s end.

But all this is predicated on (1). And it now occurs to me that (1) is perhaps not as secure as it initially seemed. For imagine this case. I am trying to decide on what to do, so I engage in deliberation. The deliberation is an intentional mental act, whose intention is to come to a decision. But perhaps I do not need to be responsible for this intention in order to be responsible for the decision I come to. I can be simply stuck with having to come to a decision, and still be responsible for the particular decision I come to. In other words, deliberative processes could be a unique case where I am responsible for an act’s outcome without being responsible for the act’s intention. That doesn’t sound quite right to me, though. It seems that if the outcome of the deliberative processes is not what I intend, and, as is often the case, is not even what I foresee, then I am not responsible for that outcome.