Sadistic Angels and Knowledge

by Brian on September 17, 2004

“Daniel”: has been worrying about sadistic angels with infinitary choices. But we can get puzzles in the ballpark of probability 0 problems without worrying about infinity. Just thinking about bets on things you (take yourself to) know gets the troubles started.

I take it that most of us here know that Margaret Thatcher was, at some time or other, British Prime Minister. (Some of us may not like to be reminded, but we know this is true.) The sadistic angel now offers you the following bet.

bq. The wager is this; if you accept the wager, and Margaret Thatcher was never British Prime Minister, then every human being currently alive on the planet earth will be horribly tortured for the next ninety million trillion years and then killed. If you accept the wager and Margaret Thatcher was once British Prime Minister, then a small, relatively undeserving child somewhere, will be given a lollipop.

Let’s stipulate that the child getting the lollipop is a Good Thing. (If you doubt that, change the example to fit in some other small Good Thing.)

Now it seems to me like this is a dreadful bet, one that you certainly shouldn’t take. But it also seems to me that the following claims are true.

# You know that if you take the bet, a Good Thing will happen.
# If you know that if you take the bet, a Good Thing will happen, you should take the bet.

Why is 1 true? Well, because the following are true.

* You know that Margaret Thatcher was once British Prime Minister.
* You know that if Margaret Thatcher was once British Prime Minister, then if you take the bet a Good Thing will happen.

And that seems to be sufficient to conclude 1. But still I think you shouldn’t take the bet. As far as I can tell though, one of the following four claims has to be true.

# You should take the bet.
# There are bets such that you shouldn’t take them even though you know that if you do, a Good Thing will happen.
# If you were offered this bet, you wouldn’t know that Margaret Thatcher was once British Prime Minister, even though you now actually know that Margaret Thatcher was once British Prime Minister.
# You don’t now actually know that Margaret Thatcher was once British Prime Minister.

Frankly none of these seem at all appealing. I think option 3 is the least bad of the bunch, and a few philosophers (esp. John Hawthorne and Jason Stanley) have been defending a line like that recently, but none of the options here are great.

Why does this link to Daniel’s post? Well, if you think that things you know get probability 1 (as say Timothy Williamson has been urging) then this puzzle has exactly the same probabilistic structure as Daniel’s. But I think it’s worth noting that you can get the puzzle without any worries about infinity, without any worries about assessing who has won the bet (beyond at least the worries that always attach to bet winning) and without even talking about probabilities. So I’m not sure that tinkering with how the probabilities work really fixes the problem.



dsquared 09.18.04 at 12:47 am

I think the issue here is that this ought to weaken our belief (those of us who are Bayesians) that probabilities (in the mathematical sense, or more generally in the sense that you would use when getting any expected utility theory off the ground) map exactly onto degrees of belief. So one might say that something you know gets a degree of belief of 1, but deny that this maps in any straightforward way onto probabilities as defined by measure theory.

(btw, I can reconstruct my example using equipment no more complicated than a bicycle wheel, it just occurred to me this afternoon. The angel just has to say:

“Some time in the last few minutes, God set a bicycle wheel spinning, and it is currently rotating at 200rpm in a horizontal plane. Consider the point P, which is the centre of gravity of the valve on that bicycle wheel, and the circle C described by P as the wheel rotates.

Some time in the next ten seconds, tap your finger on the table. If P is at the most northerly point of C at the instant when your finger comes into contact with the table, everyone dies, otherwise the kid gets the lollipop.”

Given that you don’t know when God set the wheel spinning, then you only need the Principle of Insufficient Reason to turn this into a random draw from a uniform variate, without any worries about infinite decimal expansions.)


Ralph 09.18.04 at 1:02 am

Is this puzzle about belief or about rational choice? In other words, is the answer supposed to depend on whether the subject believes the story being presented, and so believes that God set the wheel spinning, or believes that Margaret Thatcher was Prime Minister, or that “every human being currently alive on the planet earth will be horribly tortured”, and so forth?

Because I don’t see how it’s a useful puzzle if it depends on what the subject believes, or about some kind of psychology.

I originally assumed that earlier versions of this question were about multiplying an arbitrarily small probability of being wrong by an arbitrarily large punishment for being wrong.

Now I can’t seem to figure out at all what it is supposed to be about.


Giles 09.18.04 at 1:11 am

Unfortunately I think a phsycist will tell you that unless completely level the spinning of the wheel is uneven – faster when the valve is down, slower when the valve is up. Also assume so there is a point that the valve is moving most slowly is north . The “tapping” points are therefore not drawn from an even distribution – although the measure of P is still zero. The question this then raises is that if we let this distribution of the rotation tend to say one where P is north with probability 1 – then obviously the bets bad. So the distribution must be entirely independent, which the wheel cant be.


perianwyr 09.18.04 at 1:46 am

If X is true, humankind gets teleportation technology, which will solve all human ills. If X is false, demons will come through the teleporter from Hell and enslave all of humanity forever. Furthermore, the Marines don’t appear.


John Quiggin 09.18.04 at 2:08 am

This reminds me of the Monty Python sketch about Inspector Rene ‘Doubty’ Descartes. Assigned as security for a US state visit he found himself paralysed by doubt about the presidentness of the Nixon-like person. He was relieved from duty after using an ashtray to refute the hypothesis (which he had found increasingly appealing) of a hologram.

To be serious, doesn’t your example just show that “know” means “believe with high probability”? Consider for example the hypothesis that the real Margaret Thatcher (not as I recall a notable devotee of the market when she started out) was secretly replaced by an agent of the Mont Pelerin society sometime in the 1970s. If Alan Greenspan offered me a billion to one against this, I’d happily put down a dollar to back it – stranger things have happened. If this counts as a loss on your bet, then I’d refuse the bet.


Rob 09.18.04 at 2:29 am

For all we know they could be talking about the Maggie Thatcher living in Nova Scotia.


Lindsay Beyerstein 09.18.04 at 3:55 am

I believe that Margaret Thatcher was a British Prime Minister. That belief is about as well-founded as any belief I have about the external world.

An externalist would say that I know it. But if I’m trying to figure out whether to take the bet, I have to ask whether I know that I know it. Since I don’t have any transcendental arguments to vindicate my belief in an external world, I claim to know that I know that I know that Margaret Thatcher was Prime Minster.

This isn’t just capitulating to skepticism. If externalism is true, then I really do know it (without necessarily knowing how or why I know it). On the other hand, from the first person perspective, I don’t feel sufficiently confident in my estimations of my own knowledge to risk enslaving all of humanity.


Blar 09.18.04 at 6:11 am

This new version is a lot easier than the original. Of course you should pass on the bet, because your claim #2 is true (unless you want to use a super-stringent definition of “know” that makes #4 the true claim).

The problem is that people have two intuitions about knowing:
1. Knowledge means certainty. So, if you know X, then X is certainly, absolutely, 100% true.
2. People know things.

These can’t both be true, because it’s at least possible that you’re, say, a brain in a vat or a computer that is a subject in some kind of futuristic research on consciousness or delusional. So if we want to be able to keep using the word “know” (which seems like a good idea, since it really is a nice word to have around) it had better mean something other than certainty. Usually we can pretend that knowledge means certainty, because there’s generally not enough of a difference between the two to matter, but if the stakes get high enough (like, say, the end of the world), then you had better be able to admit that even the simple facts that you know best are just very very very likely.


s_bethy 09.18.04 at 6:17 am

I’m sorely tempted to throw in Rumsfeld’s famous formulation, “There are known knowns. These are things we know that we know. There are known unknowns. That is to say, there are things that we know we don’t know. But there are also unknown unknowns. There are things we don’t know we don’t know.”, but that would be impertinent, so I won’t.


mv 09.18.04 at 7:38 am

I know that Margaret Thatcher was the British Prime Minister because I have received that piece of information (from multiple sources). It’s an abstract fact to me.

Now I have a new piece of information. That is, I know that God is willing to make this large, lopsided wager. By definition, God knows some things I don’t. The fact that God is willing to make this wager is enough to make me question my belief that Margaret Thatcher was the British Prime Minister. I could say that I don’t know it anymore.

Same idea on a smaller scale: Kryptonite brand bike locks are supposed to be hard to pick. If I knew this, and a friend of mine bet me $10 that he could pick one with a ballpoint pen, I might suspect that he knew something I didn’t. If he bet me ten cents to a thousand dollars, I would be pretty sure he knew something I didn’t. (If he bet me a thousand dollars to ten cents, I would be even more sure.)


Aleph One 09.18.04 at 7:38 am

I agree with John. If “know” means “this is absolutely, without a shadow of a doubt true”…then I don’t know anything–not even that gravity exists (Hey, it could just be coincidence that things fall down).
Given that, I wouldn’t take the bet, since I think the likelihood I’m mistaken about Thatcher having been PM is greater than (lollipop goodness)/(excessive amounts of torture), so the expected utility of the bet is negative.

And s_bethy, let me rebut Rumsfeld with Piet Hein:
“Knowing what
thou knowest not
is in a sense


s_bethy 09.18.04 at 7:58 am

“Knowing what
thou knowest not
is in a sense

Isn’t that exactly what Rumsfeld was claiming?

“Naive you are
if you believe
life favours those
who aren’t naive.”


digamma 09.18.04 at 8:12 am

This thread is much clearer than the other, and I agree with most of the posters – our knowledge of Mrs. Thatcher’s service isn’t enough to make that wager.

The lesson, I think, is that most of what we know we know with a certainty that approaches but never equals 100%.


s_bethy 09.18.04 at 8:18 am

Using the poetic stylings of Donald Rumsfeld and Piet Hein to explain sadistic angels and Margaret Thatcher is bound to be less impenetrable than maths that involve infinity.


bad Jim 09.18.04 at 9:14 am

The subtext of this post appears to be that forgetting Thatcher might result in ninety million trillion years of universal torture, while remembering her gives someone a lollipop. Somehow I would have expected the reverse.


abb1 09.18.04 at 9:46 am

I got bored yesterday at work and started arguing with a bunch of wingnuts. One of them (who appropriately named himself “Warmongering Lunatic”) posted this:

Under the Charter of the United Nations, the staus [sic] of a permanent member of of [sic] the Security Council can only be changed by amendment of the Charter, and no country gan [sic] gain permenent [sic] Security Council membership without such an ammendment [sic]. No such amendment has ever been enacted, and accordingly all changes in its membership, form a Charter point of view, are illegal.

Since the Republic of China and the Soviet Union have been stripped of their permanent seats, and the People’s Republic of China and Russian Federation have both been granted a permanent seat in violation of the Charter, the so-called Security Council accordingly is an illegal body, and any action it takes is illegal.

Accordingly, Kofi Annan is not legally the Secretary General of the United Nations, his selection having been made by a body other than the legal Security Council of the United Nations as set forth in the Charter.

So, you see, everything is a matter of opinion, subject to interpretation. Please don’t take the Thatcher bet, fellas, better safe than sorry.


dsquared 09.18.04 at 12:24 pm

WC Fields’ advice to his son:

“Son, one day you will meet a man with long fingers and a shiny suit who will produce a pack of cards and offer to bet you five dollars that without touching the cards, he can make the jack of diamonds rise out of the deck and spit vinegar in your ear. Son, do not take that bet, because if you do, you will certainly end up with an earful of vinegar.”


Brian Weatherson 09.18.04 at 12:48 pm

Are you sure that’s WC Fields? I always saw it attributed to Damon Runyon.

I actually think that kind of thing is relevant to the problem. There’s not much literature (in philosophy at least) on what the evidential effects of learning someone is offering you a bet should be. It’s often assumed to have no effect at all, which is not a smart way to live at the very least.


John Quiggin 09.18.04 at 1:12 pm

“here’s not much literature (in philosophy at least) on what the evidential effects of learning someone is offering you a bet should be.”

In economics, this is the “lemons” principle (if a newish car is offered for sale, it’s probably a lemon), which got Akerlof his Nobel.

DD, it’s cider. The advice is to Sky Masterson from his father


Tom T. 09.18.04 at 6:00 pm

So the Conservatives were right all along, when they said that if Margaret Thatcher became PM, good things would happen?


Mats 09.18.04 at 7:10 pm

Brian, are you really trying to save the puzzle from being related to poor handling of infinities? Why are you then using the outcome of “every human being currently alive on the planet earth will be horribly tortured for the next ninety million trillion years and then killed”.

If the ratio of the plight of this outcome to the bad of the child not getting the lollipop is known to be finite: take the bet! Otherwise it requires some deeper analysis, just because of the infiniteness of the ratio of the values of the outcomes.


anthony 09.18.04 at 7:56 pm


Intrigued by this “wingnut”s claim, I checked out the charter. Chapter V Article 23 does indeed describe the 5 permanent members explicitly, including the currently nonexistent Republic of China and USSR, suggesting that there are presently no representatives of those two permanent members, and that rectifying this would take an amendment of that article (I’m not sure if this requirement is stipulated anywhere). However, they appear to cover themselves in Article 28 when they say “The Security Council shall be so organized as to be able to function continuously. Each member of the Security Council shall for this purpose be represented at all times at the seat of the Organization.” Obviously, this is more relevant in the case that the official representative is unable to attend a vote, but I think the idea that it must be so organized as to function continuously is a broad enough requirement that it allows for them to ad lib when something on the order of a permanent member ceasing to exist occurs. Furthermore, Article 30 says “The Security Council shall adopt its own rules of procedure, including the method of selecting its President.” Whatever you want to say about the composition of the body itself, you can’t really ascertain the legality of it’s actions, as the only arbiter in this case is the Council itself.

I guess that’s what comes from following up on the claims of wingnuts.


W. Kiernan 09.18.04 at 9:06 pm

Listen, this has nothing to do with mathematics or logic. You’re talking about betting against gods. History, I mean mythology, clearly shows nothing good can come from people associating with gods in any way; everyone knows that gods are dishonest bastards who cheat like they breathe.

For example, take Actaeon; he was a hunter; it was his job to be out in the woods. Of course he peeked, I would have too. Now if Diana was so modest and virginal, what was she doing strutting around all starkers like that? But noooo, it’s all Actaeon’s fault. And whatever did Job do to deserve such insensate abuse from Jahweh? And didn’t Arachne weave just as well as Athena, and what good did it do her? she ended up as a bug anyway.

Fact is, gods cheat; gods suck. I tell you, if you hang out with gods, Richard, it will not make them any better; no, if you hang out with gods you too will turn into a bum.


Matt Weiner 09.18.04 at 10:21 pm

About infinity and gods–I think you can strip those out and still have Brian’s problem. Let’s say it’s a bar bet between you and an eccentric but always truthful dictator, who if Maggie was never PM will torture everyone in his country. That outcome probably doesn’t have a utility of negative infinity, but it’s still bad enough that I wouldn’t take the bet.


pwax 09.19.04 at 12:50 am

Of course you realize this is all nonsense? If you cannot trust your knowledge of Margeret Thatcher’s being prime minister you cannot trust your reasoning about the bet.


bad Jim 09.19.04 at 8:44 am

We spent the entire twentieth century demonstrating that substituting dictators for gods did not improve results.

So much for humans.


Nick Simmonds 09.20.04 at 5:55 am

My anser hasn’t changed an iota, except to take note of the stipulation that the lollipop is a Good Thing.

First, while it may be impossible that Margaret Thatcher was not prime minister, it is *not* impossible that the listener is misunderstanding the terms of the bet. In bets where the absolute utility of the Bad Thing reaches the value of the probability that the listener understands perfectly (which, depending on the listener, is likely to be very high but never 100%, no matter how many times it’s repeated) one should always decline the bet.

Second, while for the sake of argument it may be impossible (zero probability) that our knowledge of the past is false, it is also infinitely bad that everyone on the Earth should be tortured to death. An infinity multiplied by zero is a indeterminate but real finite number. The balance of probability is that an indeterminate finite number is greater (rather, in this case, lesser) than a known finite number, and so the bet is a bad one.

Finally, the bet *still* hinges on the assumption that God is not playing silly buggers, and I don’t think it can be taken as read that God is a benevolent (or even neutral) being, especially if He has such sadistic angels.


Kenny Easwaran 09.20.04 at 7:50 am

Where does everyone get infinite negative utility from everyone on earth being tortured for 90 million billion years or whatever? That sounds pretty finite to me.

Brian – as for not taking into account the information conveyed by the fact that a bet is being offered, I remember something like this coming up in the sleeping beauty stuff at FEW this past spring. This seemed to me the most compelling argument Mike Titelbaum gave to support the Dutch book supporting Elga rather than the one supporting Lewis.

Anyway, it seems that some sort of regularity principle in one’s subjective probabilities will prevent one from making any bets like this one. Though they’ll make Bayesian updating more difficult.


Nick Simmonds 09.20.04 at 11:14 am

Where does everyone get infinite negative utility from everyone on earth being tortured for 90 million billion years or whatever? That sounds pretty finite to me.

It’s not that–the torture is mostly irrelevant in terms of evaluating the utility. The fact that everyone will be killed after the torture is what brings in the infinite blah blah blah.

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