Thursday, December 3, 2015

Coin guessing using past data, once again

In my previous post, I showed that given a backwards infinite sequence of coin tosses, there is a simple strategy leveraging data about infinitely many past coin flips that guarantees that you guessed correctly infinitely often. I then suggested that this supports the idea that one can't leverage an infinite amount of past data, and that in turn supports causal finitism--the denial of the possibility of infinite causal histories. But there is a gap in that argument: Maybe there is some strategy that guarantees infinitely many correct guesses that doesn't require the guesser to make use of data about infinitely many past coin flips. If so, then the paradox doesn't have much to do with infinite amounts of data.

Fortunately for me, that gap can be filled (modulo the Axiom of Choice). Given the Axiom of Choice, it's a theorem that there is no strategy leveraging merely a finite amount of past data at each step that guarantees getting any guesses right. In other words, for every strategy that leverages a finite amount of past data, there is a sequence of coin flips such that that sequence would result in the guesser getting every guess wrong. The proof uses the Compactness Theorem for First-Order Logic.

17 comments:

Angra Mainyu said...

I have a question:

Are you arguing (or suggesting at least) that it's metaphysically impossible for any agent (human or not) to use an infinite amount of data, even if it's not in the form of an infinite causal chain? (e.g., data from infinitely many galaxies).
Or are you placing limits on the kind of agents or the origin of the infinite amount of data, etc.?

Alexander R Pruss said...

An irreducibly infinite amount of data about contingent events can't be used by anyone, I think.

Alexander R Pruss said...

I managed to re-prove the result mentioned in this post without using the Axiom of Choice.

Angra Mainyu said...

That's interesting. I trust your proof is correct, but I'm not convinced that it establishes the result you intend about using data (e.g., I don't find the strategy surprising in the other thread; maybe here there are other things I wouldn't find surprising).

However, I think if you're right, that seems to have interesting consequences for the compatibility of infinitely many galaxies, people, etc., with Catholicism, Christianity, and even theism.

For example, how about the following exchange? ;-)

Alice: I think probably infinitely many people have gone to Purgatory.
Bob: I disagree. Either there are only finitely many galaxies, or only finitely many people. But if there are infinitely many, only finitely many have gone to Purgatory.
Alice: I wish we could find a way to tell.
Bob: We can ask God, can't we?
Alice: You mean, like by praying?
Bob: Sure. Why not? He'll probably not answer. But we can at least try.
Alice: I think trying would be a sin.
Bob: I disagree. But if you think it's a sin, I can try myself. In fact, I will.
Alice: I advice against it.
Bob: Noted...
Bob: God, could you please let us know whether infinitely many have gone to Purgatory?
[Something funny happens in the sky]
Loud voice coming from the sky, apparently: Hi, Bob. I'm God. You shouldn't have asked. But I can't tell you whether infinitely many people have gone to Purgatory.
Bob: God! I'm sorry. I didn't mean to ask. Please, forgive me!
Voice: Okay, you're forgiven. But don't do it ever again.
Bob: Yes, yes, I promise. I won't do it again. Just one question, please. I need to know: Why can't you tell me?
Voice: Because I don't have the power to answer your question about Purgatory.
Bob: Excuse me? If you're God, you're omnipotent. Of course, God would have the power. But you either don't, or you're lying to me. God wouldn't lie, either. So, you're not God. Who are you? Lucifer, trying to trick us?
Voice: No, no, no. I'm God. So, I'm omnipotent. But not being able to do what's metaphysically impossible doesn't preclude omnipotence, and it's metaphysically impossible to use an irreducibly infinite amount of data about contingent events. Whether infinitely many people have gone to Purgatory is as irreducibly infinite as whether infinitely coin tosses landed heads, and whether a human or relevantly similar person goes to Purgatory is a contingent matter, like a coin toss.
Bob: If it turns out that it's metaphysically impossible to send info faster than light, are you going to tell me that a being incapable of allowing an astronaut on Mars to talk to her spouse on Earth with less than a 1-second delay might still be omnipotent? Surely, that doesn't capture the meaning of "omnipotent".
Voice: But...but...
Bob: Besides, if you can't use data from infinitely many contingent events, you couldn't have told me that you can't tell me whether infinitely many people have gone to Purgatory, because implicitly you're using the info that there are infinitely many people, and that's an irreducibly infinite amount of data about contingent events.
Voice: Hmm...oops, I guess I was just guessing, not really using the data, because it's impossible. Or something. Hmm...
Bob: So, you just guessed, when you thought you were giving me good info? You're fallible. So, you're not God.
Voice: Hmm...I always thought I was God. But I guess I'm just Yahweh, the most powerful entity described in the Bible. But the contingency argument proves God exists. So, God is someone else. And since it's impossible to use an irreducibly infinite amount of data from contingent events, it seems to me infinitely many people are metaphysically impossible.

Alexander R Pruss said...

"Besides, if you can't use data from infinitely many contingent events, you couldn't have told me that you can't tell me whether infinitely many people have gone to Purgatory, because implicitly you're using the info that there are infinitely many people, and that's an irreducibly infinite amount of data about contingent events."

Very clever, but there might be only a single datapoint being used, viz., that an omnipotent being intended to create infinitely many people. (Given divine simplicity there may be some difficulties here, but at first blush at least it seems to work.)

Alexander R Pruss said...

There are questions that for logical reasons a perfect being can't correctly answer. Here's an example (based on a Rescher example): "Is your response negative?"

Angra Mainyu said...

"Very clever, but there might be only a single datapoint being used, viz., that an omnipotent being intended to create infinitely many people. (Given divine simplicity there may be some difficulties here, but at first blush at least it seems to work.)"

Fair enough. I was thinking about the doctrine of special creation, which would seem to require infinitely many data points. But I guess the Voice could just reply that Catholicism is mistaken about that.
But in any case, that does not block the conclusion that the voice is not omnipotent.

Alternative scenario:

Alice: I think there probably are infinitely many fallen people.

Etc.

"There are questions that for logical reasons a perfect being can't correctly answer. Here's an example (based on a Rescher example): "Is your response negative?"

I'm afraid I don't understand the example (could you be more specific with the reference?).
But what I'm saying is that under any conception of "omnipotence" that matches usage, an omnipotent being would be able to answer that question. Let me put it this way: we're talking about a being who knows that the answer is X, has the capability to speak in general, but is unable to simply utter "X".

entirelyuseless said...

"Is your response to this question negative?"

If someone says yes or no to that, his response is wrong. That is true even for an omniscient being. So presumably he won't answer.

But you can turn this into an argument against an omniscient being existing at all:

"God thinks that this sentence is false."

This is not a liar paradox. It does not say that the statement is false itself. So is it true or not? If it is true, then God is wrong. And if it is false, then God doesn't know something (namely that it is false.)

You could respond that God doesn't "think" that any particular thing is false, since he is simple.

But then you can modify the sentence:

"God thinks, in the relevantly analogous sense, that this sentence is false."

If God thinks this is true, then it is false, and if he thinks it is false, it is true, and either way is not omniscient. If God does not think in any relevantly analogous way at all, the you should not call him omniscient, but adopt an apophatic theology instead.

Alexander R Pruss said...

Angra:

The doctrine of special creation does not say that God has created infinitely many people. Nor that those he created were created by separate decisions.

"we're talking about a being who knows that the answer is X, has the capability to speak in general, but is unable to simply utter 'X'"

He's able to utter 'X', but not in a way that makes makes it be the answer to the question.

Another fun case. Presumably there are infinitely many truths expressible in English which God never utters. Question: "What is the first (in lexicographic ordering) sentence of English that is both true and never uttered?" God knows the answer to the question is X, he has the capability of uttering X, but he cannot answer the question.

Angra Mainyu said...

Alex,

Given entirelyuseless's reply, I suspect your question based on a Rescher example was a self-referential one. Is that correct?
If so (if not, please clarify) I don't think the situations is analogous at all. An omnipotent, omniscient being would have the knowledge and power to respond and explain what's wrong with the question, but in any case, there is no correct answer. So, an omnimax being could just reply: "There is no correct answer to that question, and I only answer correctly".
On the other hand, in the case of whether infinitely many people have fallen and/or infinitely many people who went to Purgatory, etc., there is a correct answer in the scenario, but any agent that exists either does not know the correct answer (and so, he or she is not omniscient), or does know the answer, but is incapable of uttering it (or otherwise use the info), so he or she is not omnipotent.
Either way, it seems to me that if you're right, infinitely many galaxies with people (or without people; we can just modify the scenario a little bit) are incompatible with theism.

Angra Mainyu said...

"He's able to utter 'X', but not in a way that makes makes it be the answer to the question."

I don't think that works. Let's simplify:

Let's say that there are infinitely many galaxies, and infinitely many fallen people:

Bob: God. Are there infinitely many fallen people?
Voice: I don't have the power to answer the question, because no one can use an irreducibly infinite amount of data about contingent events.
Bob: But there is a correct answer, and it's either "Yes, there are infinitely many fallen people" or "No, there are not infinitely many fallen people". If you don't know the answer, you're not omniscient. If you do, then either you can utter "Yes, there are infinitely many fallen people" or "No, there are not infinitely many fallen people", or you're not omnipotent. Surely, on any plausible construction of the term "omnipotence", if you know that answer but you cannot even utter it now, you are not omnipotent.
Voice: But there are questions no being can answer correctly. There are some paradoxes involved, since they would entail a contradiction.
Bob: Even if true, that is not related to the matter at hand. If you can't even utter "Yes, there are infinitely many fallen people" or "No, there are not infinitely many fallen people" (whichever is correct), then you're not omnipotent.

I think Bob won that argument, hands down.

"Presumably there are infinitely many truths expressible in English which God never utters. Question: "What is the first (in lexicographic ordering) sentence of English that is both true and never uttered?" God knows the answer to the question is X, he has the capability of uttering X, but he cannot answer the question."

God can answer: The question is ill-formed, because whether a sentence is true depends on the context of utterance.

But let's say you ask: "What is the first (in lexicographic ordering) sentence of English that is both necessarily true and never uttered?"

I don't know that there is such sentence. But if God existed, he would know, and if there is no such sentence, he would be able to answer: There is no such sentence.

If there is such sentence, God could answer: That's sentence #83457282482 (or whatever)
Or he could answer (for example): it's the sentence composed of n words such that the first word is , the second word is , ..., the n-1-th word is , and the last word is spelled , and so on.

At any rate, I don't see any analogy between paradoxes and an incapability for saying whether there are infinitely many fallen people, whether infinitely many people decided to have children, etc.

Alexander R Pruss said...

You can modify the question: "What is the first (in lexicographic ordering) sentence of English that is both necessarily true and never uttered, or otherwise explicitly indicated?" :-)

The more serious issue is: "I don't see any analogy between paradoxes and an incapability for saying whether there are infinitely many fallen people, whether infinitely many people decided to have children, etc."

I think that a general ability to answer questions like "Have infinitely many people decided to have children?" leads to all sorts of paradoxes at least in special cases. For instance, suppose that the answer is "No" to this question but "Yes" to "Are there infinitely many people?". What probability should I now assign to the proposition that some friend of mine--who doesn't yet have a child--has decided to have a child? On the one hand, intuitively, it is very unlikely that he is one of the finite number who has decided to have a child. On the other hand, the information that only finitely many people decided to have children is logically equivalent to the information that only finitely many people other than my friend decided to have children, and that seems to tell me little about my friend (at least given a fair amount of independence between free decisions).

Alexander R Pruss said...

By the way, I am very grateful for all these comments. In parallel with responding to them, I am thinking and writing about this stuff. It's a delightful collaborative approach. Angra: If you want to be properly acknowledged, email me your real name.

Angra Mainyu said...


You can modify the question: "What is the first (in lexicographic ordering) sentence of English that is both necessarily true and never uttered, or otherwise explicitly indicated?" :-)

Okay, but that's another one. :-)

So, let's assume for the sake of the argument that it's possible that there is one such sentence. Surely, it's not necessary. So, here's how God could answer:

God: The question assumes that there is one such first sentence. But that's a contingent matter. It's possible that there is, and it's possible that there is not. Moreover, when there is a first sentence of English that is both necessarily true and has never been uttered, or otherwise explicitly indicated, uttering, etc., it would bring it about that a different sentence (or none) is the first.
Now, you're not asking about the first never utteredetc. so far, but also in the future. But any answer to that question, would be false. It's strictly logically impossible that I tell you the correct answer, because of the self-referential way the question has been constructed. But of course, that is no more of a challenge to my omnipotence than saying I can't create a boulder I can't lift, while remaining omnipotent.

"I think that a general ability to answer questions like "Have infinitely many people decided to have children?" leads to all sorts of paradoxes at least in special cases. For instance, suppose that the answer is "No" to this question but "Yes" to "Are there infinitely many people?". What probability should I now assign to the proposition that some friend of mine--who doesn't yet have a child--has decided to have a child?"

That's a question about how to assign epistemic probability under the assumption that something whose epistemic probability is near zero has happened. I would say it might get weird (but not in this case; see below), but I don't think this leads to paradoxes.
In reality, I would say that if a being were to tell us that, we should assign very low probability to the hypothesis that the being is telling the truth. But for that matter, I think we should always assign negligibly low probability to the claim that a being claiming to be God is God (sorry, but I have no other way to explain my take on that :-)).

Still, under the assumption we know those things:

"On the one hand, intuitively, it is very unlikely that he is one of the finite number who has decided to have a child. On the other hand, the information that only finitely many people decided to have children is logically equivalent to the information that only finitely many people other than my friend decided to have children, and that seems to tell me little about my friend (at least given a fair amount of independence between free decisions)."

I would say that the right answer is to ignore the answers to those questions, or more precisely, to give them only negligible weight (which is the same for all intents and purposes) when it comes to making an assessment about your friend. The reason is not the equivalence you mention, I think, but what we already know about how things work on Earth. For example, if one were to ponder "what's the probability that at least one person on Earth has decided to have children in the past seven day", one should still assign close to 1. Given the info available to us, the weirdness clearly seem to be happening on other planets, realms or whatever.

But the more serious issue is this: let's say that you're correct that the ability for saying some such things is for whatever reason problematic. Then, I would say that actually counts against the compatibility of theism and infinitely many people, galaxies, etc., given the strength of the assessment (by conceptual analysis) that assessment that a being who can't answer those questions (i.e., in my examples) isn't omniscient and omnipotent.

Angra Mainyu said...

Alex

You're welcome. It's fun for me too. :-)

Also, thanks for the offer. I sent you an email.

entirelyuseless said...

I would agree that if we make a strong argument that God cannot say whether or not an infinite number of people have made such and such a choice, this is an argument that the existence of God is not compatible with the existence of an infinite number of people.

But paradoxes regarding infinities result in all sorts of ways, and one of the simplest ways to explain them all at once is to say that no such infinity is actually possible, even the ones we have not yet proved to be impossible. So an argument like that would be a much stronger argument against the existence of an infinite number of people, than against the existence of God.

Angra Mainyu said...

entirelyuseless,

I'm not arguing against the existence of God, though I admit that I'm indirectly hinting at an argument against the epistemic justification of belief in God assuming that Alex's point against using an infinite amount of data holds.
However, I think in the context of current discussions in philosophy of religion, a successful argument for incompatibility between theism and infinitely many people, galaxies, etc., would be interesting on its own, independently of the justification of theistic belief.
That said, I don't claim that the argument is successful. I only claim that it's successful under the assumption that no one can use an irreducibly infinite amount of data from contingent events.