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I'm glad I didn't misread your points, indeed I felt pretty good about my comprehension once I saw another of your replies (An earlier draft of my post included language like: "It seems I am stuck believing in infinite rewards and punishment" in regards to seeing that step 5 invokes scripture, and step 1 merely invokes infinite reward and punishment. It seems the trap I fell into was intended!)
The impression I get from Pascal's Wager: an a-priori argument for God for those who think it distasteful to apply that "empiricism" business to the beautiful question of theism. When deployed in that manner, it is open to the non-empirical attack of "the Atheist's God." The Thiest's retort "that seems unlikely!" amounts to cherry-picking evidence.
Your more fleshed out version of Pascal's Wager appears to be in the business of evaluating evidence. Of course, one would need evidence in order to even consider the hypothesis about infinite rewards and punishments, given that empirically, there doesn't seem to be infinite of anything around us! The police do not open a phonebook and randomly determine a suspect to investigate when they hear of a new crime. The laws of probability and what we might call "reasonable thought" obligate them to possess evidence before considering any suspect in the first place. It would be even more disturbing to learn the accused is a rival of the sheriff!
Your focus on infinite rewards and punishments is not separate from sacred texts. The reason anyone discusses Pascal's Water and infinite rewards and punishments is because of the sacred texts. So this business of "deciding what are the infinite rewards and punishments" is of course a strategic choice of starting point. It seems to me we should start with the evidence in front of us: the sacred texts. Maybe I chose that strategically? I don't have perfect access to my mind's internals. The sacred texts seem to me quite easily explainable as a lie to steer people's behavior by giving them incentives (That's what I meant by "incentive structure")
Thanks, that clarifies.
Ah, I see we relate to epistemology slightly differently. Let me argue that mine is better and more rigorous.
Have you ever read Eliezer Yudkowsky's The Sequences? I imagine, given that you're in this space, that there's some slight chance. Not that I recommend spending that time, but what follows will have some of the same ideas (though he rejects Pascal's wager, in a somewhat unprincipled manner).
Generally speaking, everything you know has a probability attached to it, according to how likely it is, from your perspective, true. That I'm typing into a computer right now? I'm quite certain of that, but there's always the possibility that Cartesian doubt is right and I'm under some variety of extreme delusion. That you're not in this room right now? I'm quite sure of that as well, though perhaps there's some remote chance that you happen to be in the area and crept in. In these examples it's kind of silly to pay attention to the tiny chances that my evaluation is wrong. There are some cases where it's more useful. If I am expecting someone to arrive soon, there's some subjective probability that someone will arrive in the next five minutes, which might be pretty relevant as to how much I need to be rushing to prepare. I said "subjective probability" there. I want to emphasize that what we are talking about is not what the probability is from some neutral world observer. I am talking about what the probability is to you. This isn't any different from what we ordinarily mean by probability: when you roll a die, you could hypothetically apply the laws of physics and work out exactly how it will land. But we still say the probability is one in six, because that is the probability according to the knowledge of the players involved. Alright, everything has probabilities. It is important to keep in mind that in the more extreme examples, you cannot dismiss that. There's no clear boundary line between a 1 in a graham's number chance of being right, and near certainty, only a sloping gradation. Everything that you can think of has a probability of being the case, somewhere between 0 and 1.
When we learn things, a key part of what is going on is that we think some facts about the world become more or less likely. This happens according to Bayesian updating (or at least, would happen if we were perfectly rational and had unlimited computation at our disposal. But it's a useful concept anyway.): that is, there is some likelihood about the world. You come across evidence. This evidence is more likely under some hypotheses than under other hypotheses. Following Bayes' rule (yes, the basic probability rule), you revise your likelihood of the former hypotheses up, and of the latter ones down. Hooray; you've now taken that piece of evidence into account, placing just the right amount of weight on it, and have new, more accurate, probabilities. One useful concept, then, is of subjective likelihoods attached to every hypothesis, and a universal prior. That is, some probability assigned to every world state or possible hypothesis, and from there, throughout our lives, with every piece of evidence, we adjust all the probabilities accordingly, giving the probabilities that would be the case of a perfectly rational agent. (This is known as Solomonoff induction.)
Such a construct, of course, does not exist. Various parts of that aren't true. We don't have probabilities at hand for every possible hypothesis. Most ideas we haven't even thought of. There are serious questions about how you would even generate the probabilities, if there is some objective way to do so (Kolmogorov complexity—that is, one measure of the amount of information in it—has been suggested, but I don't think that can apply to everything, and there is no clear way to define that neutrally, either). And we couldn't even calculate it if it did, as it is provably noncomputable. Rather, we come up with ideas, assign likelihoods to them by who knows what rule (though it has to be a somewhat reasonable one, since we're right a lot of the time), pay attention to some things and not others, and often have to realize how likelihoods of things change, not compute everything after every piece of evidence. Nevertheless, it still is a useful construct, as it shows how a perfect reasoner might work, and it is something we approximate by our own reasoning. If we build our ideas off of that better form of reasoning, they'll remain theoretically correct and rational, even if what we do only only approximates it.
Enough background. Let's go through your comment. I'll skip the first paragraph.
I'll note that Pascal's Wager isn't really an argument that God exists, it's an argument that it is instrumentally (but not necessarily epistemically) rational to wager for God. It's an argument for a course of action. That said, I don't have a problem with non-empirical arguments. There is no reason why evidence that adjusts our probabilities (as discussed above) has to be real world data; both that and realizations in our ideas will do so.
The argument isn't opposed to empiricism in general, or even in any specific instance. Apply all the empirical evidence you like; it'll only make your picture of the world better. I think my first example clearly involved empiricism, looking at the actual revealed religions. It would be absurd to argue for a religion without at least some empiricism. Arguing that it is unlikely is precisely what it is the relevant question (well, along with how large is the benefit/harm). The wager dismisses as comparatively irrelevant possibilities that do not offer any infinite benefits or harms, but it still cares about empiricism.
Why, then, reject "the Atheist's God"? I don't, actually, reject it in the same way as I ignore the finite benefits. Rather, I compare the probability of that, versus the probability of other options, consider rewards and penalties of possible courses of action, and go with the one with the best expected value. I'm just convinced that that's less likely, comparatively, to a God of some of the various large revealed religion happening to be true, and so it makes more sense to follow the latter rather than the former.
This was the main reason that I gave all that background above.
In this case, then, you talk of bringing up the hypothesis, and argue that even mentioning the possibility is something that needs justification. In general, this isn't necessary. You're always free to think up ideas, just often the probability will be low. There's nothing wrong with me considering the idea that the moon is made of cheese, and that they discovered it during the landings, but didn't reveal it after financial pressure from lobbyists in Big Cheese to prevent cheese mining. I'll just reject it out of hand as technically possible but extremely improbable, under my ordinary, somewhat inscrutable, probability assigning rules.
Then, it is false when you say you need evidence to consider the hypothesis. It is fine to consider the hypothesis that there are infinite rewards and punishments. In fact, this is entirely a rational thing to do, as discussed before: it has some probability. Feel free to think the probability low. But the argument I articulated before does not care if the probability of infinite benefits and harms is low. When the payoff is infinite, that outweighs everything else.
I think what you were saying is that you need a reason to take it seriously. Usually, things are only taken seriously when there's a reasonable likelihood of them happening, because extreme improbability usually outweighs whatever finite considerations we are considering. But here, that doesn't matter, as that infinite probability will overcome whatever finite improbability we are talking about. (Side note: the actual reason police can't start investigating random people is due to labor costs (it's just not efficient) and rules requiring reasonable cause, because we protect citizens, not that it would be impossible to assign probabilities legitimately.)
Sure, sacred texts were what first led me to look into this. But that doesn't mean that the basic Pascalian concerns would not be right, even were the sacred texts never written. I'm still convinced that, were the sacred texts never to have existed, it would still be right to realize that infinites are what matter, try to figure out what's more or less likely (in that case a considerably harder task) and devote one's life to it.
Sacred texts are first in the actual facts of my thinking about it, but that does not mean that there is not independent motivation—indeed, the most extreme possible motivation—to do so.
That is, arguments do not gain their legitimacy from whatever led one to look at them. They have their legitimacy in their own right, by their own merits. And in this case, the merits of the argument are good. Nor does the need to seek infinites depend on any sacred-text-reliant premises.
In the sense of I'm bringing this up to try to present an argument for religiosity, sure, it's strategic. But in terms of whether you should do this, no that's just what you should do. In every choice you make, whatever effects that has dominates over everything else. It would be extremely silly not to look at the thing in comparison to which everything you're ordinarily thinking about it is of infinitesimal value.
I think the authors of the scriptures believed them. Several of them endured physical suffering for it. But that aside, okay, that's possible, and would decrease how likely you are to think each of the sacred texts we're talking about are telling the truth. Fair enough. But that doesn't adjust the overall fact that it is infinites you are to look to and evaluate. That doesn't get you out of the overall question. (And if you can't find anything more reliable, you might turn to the scriptures anyway, on the off-chance that they are what they say they are, but that isn't at all necessary to the initial steps of the argument—seek out infinites, with all your might—which it sounds like is a big departure from how you've looked at life up to this point.)
Sorry to write at such length, but I though giving a better background on epistemology would help. Don't feel the need to respond to each detail.
Thanks for the reminder that Pascal's Wager is about instrumental beliefs and not epistemology. I realized that sometime in between posting this and reading your reply...
I'm not even sure I "should" think according to any mechanistic rules -- everyone notes we don't actually compute Bayes in our heads -- at least not at the high level of thoughts. Just like ethics is more about systematizing what we feel in our guts, I navelgaze because I think systematizing is fun, for example, systematizing what we actually do. I get the impression your argument is prescriptive (not that you personally are evangelizing anyone), so I would like to be up-front and honest that absolutely nothing you say would ever change how I act, except maybe cause me to think of a reply.
It's difficult for me to decouple 1) and 5). The mugging implications seem too real to me. Isn't accepting this just a vulnerability to be mugged by anyone? Upon further reflection, I don't think we even need to bring up infinities to realize that expected value has mugging problems. The mugger will just tell me that there is some amount of reward -- not infinite -- that I should accept since I don't assign anything a probability of zero. As the mugger names higher and higher values, it's true the probability doesn't (seem to) drop comparatively. Without bringing infinity into the mix, expected value seems to have some issues! So I'm not sure if a hyperreal (or whatever) analog to expected value would help me feel any better. You seem smarter than me though, so I'm assuming you already know about this though.
I get that this isn't going to convince you. My goal is mostly just to make you go, "Oh. That's a good argument. I don't really have any answer to that." Planting seeds, etc. Thanks for the honesty, though.
Yeah, I agree that at some point the probability seems to drop less than the value grows. No idea whether I'm smarter, but I've probably thought about this set of issues way more. My answer to the what about muggings question is just that that's way less of an issue when you're already centering your actions around an infinite. At that point, it's not just a finite loss to the mugger, but you're risking losing some infinite amount.
I don't think it makes sense to reject expected value because the Von Neumann-Morgenstern utility theorem says that, to be rational (under a seemingly reasonable definition of rational), your actions need to be able to be treated as following a utility function, so you end up having to act as if you have expected value.
Why do you think that the Completeness Axiom is an axiom of rationality, rather than a modelling convenience? I once checked through the great Bayesian decision theorists, e.g. Savage and Morgenstern for an argument for this axiom, but they ALSO seem to view it as a modelling convenience. As I recall, Savage explained the axiom as, "No, this isn't a requirement of rationality, but I can't do the maths in a simple [by HIS standards!] way otherwise." When I ask living great Bayesian philosophers, decision theorists, or statisticians, they ALSO view it as a modelling convenience, or change the subject from representation theorems to Dutch Book Arguments, epistemic accuracy arguments, and so on.
This isn't just a technical point, since it's not clear to me that a rational agent must assign an additive probability to their belief in Mysteries, such as the Trinity, because in a Bayesian model this also requires determining likelihoods of the deductive closure of your beliefs, over a sigma-algebra of propositions, under the assumption of the Trinity. (Otherwise you don't know whether your credences are coherent.) However, this is a problem for Trinitarian Christianity, rather than unitarian monotheisms. Again, this seems to be another case where your reasoning seems to favour Islam, rather than standard Christianity.
(By the way, I recently talked to a large number of Bayesian statisticians, all of whom were literally laughing out loud when they learned that people like Yudowsky think that you can determine credences in hypotheses like "God exists" or "This interpretation of quantum mechanics is true." That is not how someone who understood Bayesian mathematics would speak, in their view. For one thing, they brought up the problem of determining a partition.)
I actually don't hold to the (standard formulation of) the axiom of completeness. It doesn't work with infinites.
But there are reformulations for infinites that end up letting you still use all the same theorems. But you're right, even there, it's a modelling convenience; it's possible that you're preferences be stronger than the ratio between 1 and every infinite.
But then it would seem like you could dismiss the smaller ones, and only care about the commensurable ones in the largest class? (That is, with nothing incommensurably larger than them?)
I'd also start wondering whether it's possible to take this and model it with infinities anyway, but there probably wouldn't be a unique way to do that.
But it sounds like you're more technically informed on these matters. What do you mean by a sigma-algebra with regard to deductive beliefs? It seems reasonable enough to me to assign probability to some set of incoherent beliefs. Like, it might make sense to guess how subjectively likely it is that some math problem works out one way or the other—I'm certainly entitled to be surprised by it.
Could you elaborate on determining a partition? My thought would be that it would be impossible to actually do things like that for everything in practice, and that generating precise probabilities in general is hard, but in theory, it would be correct if an agent acted that way? (See the page on Solomonoff induction)
It's not so much a question of caring about the importance, but rather whether one is rationally obliged to have a preference over all of the options.
A sigma-algebra is a set that is closed under complement, (countable) intersections, and (countable) unions. For example, if A in the algebra and B is in the algebra, then A U B is in the algebra.
Deductive closure is the requirement that a set of propositions contains every implication of conjunctions from that set. This is also called the logical omniscience requirement of Bayesianism: it assumes you know all the logical relations and have updated accordingly.
Not sure what you mean here?
Agreed, but then you're going beyond the Bayesian model of belief.
There are quite a few things going on with partitions in Bayesianism, but for example, P(H) = P(H | A1) P(A1) + ... + P(H | An) P(An), where {A1, ... An} is a partition of propositions (mutually exclusive and exhaustive). The probabilities for the elements of such partitions must add up to one, by the Law of Total Probability.
To create such partitions, Bayesian epistemologists use "catch-all" hypotheses, meaning basically "The disjunction of all the possibilities that I haven't considered." Problem: how do you determine P(H | Ac), where Ac is a catch-all hypothesis? If you can't do this, then you don't know whether your probability distribution is coherent.
Bayesian decision theorists and statisticians stare at me blankly when I bring this up, because they don't do Bayesianism the way that philosophers do it. They assume that the probability distribution is over what Savage called a "small world", with a nice simple and manageable set of events (they almost all prefer that domain rather than propositions, AFAIK) that is an idealised model of some portion of the real world. That's definitely a great way to reason if you're making some practical decision or making an inference within a simplified model of some phenomenon, but it's incompatible with the high aspirations of Bayesian epistemologists, who are interested in a rational agent's reasoning, and agents don't just reason about small worlds.
Solomonoff induction is popular among some rationalists, but it has no particular status within Bayesianism: http://philsci-archive.pitt.edu/12429/
It's also controversial within Bayesianism (and even moreso statistics/decision theory/philosophy) whether people's beliefs should be representable as precise probabilities over a sigma-algebra, but that's a huge topic beyond the scope of what I have time to discuss here.
Ah, that was off the top of my head. I actually was referring to the Archimedean property, not completeness, so I didn't respond to you properly.
Since completeness is defined, at least per wikipedia, with a ≤ instead of a <, it would seem relatively hard to deny? The others are less obviously necessary.
What followed: there are inconsistent, deductively false beliefs, that nevertheless need subjective credences.
Fair enough—well, not necessarily in the sense that you're not performing updates, but in the sense that you have no universal probability function.
Ah, yes, that is a serious problem.
Nice paper, as well.
I've definitely had conversations with people—or, well, more, rants on my part—over these problems, though put in far less precise of a manner. Yes, these are serious issues.
I guess I just don't have any better, clearer way to handle things.
When we are considering any actual possibility, we are moving it out of the catchall part of the partition, and there it can behave a lot better, I think, so I don't know how much it messes things up, though I imagine still enough that there might be intractable problems.
Thanks for the precision, and the reminder that all this is a just-so story covering over a sea of infinite complexity and Humean doubt.
Anywhere I should direct myself for that last paragraph?
Suppose you have a revealed preference analysis of preference. Note that I do NOT NOT NOT mean a revealed preference theory of evidence about preferences, but the idea that observed choice behaviour is what preference is. In that case, Completeness holds trivially, provided the choices in question are in fact made.
However, if you understand preferences as mental attitudes, then it is perfectly possible that someone does not have an attitude such that (1) they prefer A to B, (2) they prefer B to A, or (3) they are indifferent (in the technical sense) between A and B. For example, Duncan Luce did experiments that found that, under some conditions, people's choices in apparently repeatable choice-situations fluctuated probabilistically. IIRC, they preferred A or B to a random choice between the two, indicating that this was not indifference. Now, it's possible that they were interpreting those choice-situations as non-repeatable, but it could also be that their preferences with respect to A and B don't form a strict ordering.
There's no basis in decision theory or mathematics for that claim, AFAIK.
There is a cool literature on imprecise probabilities you might like to look at:
https://plato.stanford.edu/entries/imprecise-probabilities/
I haven't read any applications of this approach to Pascal's Wager, but since IP is arguably a more realistic model of human psychology than maximising expected utility (which assumes unique additive probabilities) someone should definitely do that.
Me too! I don't want to dox myself, but I think that Bayesian decision theory is similar to things like General Equilibrium Theory, neoclassical capital, and other concepts in economics, in that they can be useful tools to make decisions given idealised assumptions, but they shouldn't be taken too literally. Like any scientific model, their value comes not from their approximation of truth, but because of empirical and formal properties they possess, e.g. track-records and approximations of relevant features in the world (empirical) and tractability/computability properties (formal).
For more about the topics raised in the last paragraph in my comment above, the Stanford Encyclopedia page is pretty good - written by a very good young philosopher, Seamus Bradley, who has done other work on the topic worth reading. Much of the great work in this literature, e.g. by John Maynard Keynes, Teddy Seidenfeld, Peter Walley, Clark Glymour, Henry E. Kyburg, and Isaac Levi, is extremely technical, even for decision theorists. The SEP page covers most of their ideas at a more accessible level, sometimes in its appendix. The best young guy in the field is probably Richard Pettigrew, who has done some magnificent work that has still not been incorporated into the broader Bayesian consciousness, e.g. https://philarchive.org/rec/PETWIC-2 (see this video for a relatively easy introduction to that paper - https://youtube.com/watch?v=1W_wgQpZF2A ).
I find this topic very interesting, because (like you, I think?) I see Pascal's Wager (or something like it) as the best current case for religious belief. I actually like Arthur Balfour's variation of this type of reasoning, which avoids some of the features of Pascal's arguments that are awkward, e.g. regarding infinite expected payoffs:
https://archive.org/details/adefencephiloso01balfgoog/page/n12/mode/2up?view=theater
Here's John Passmore's summary of Balfour's position, from One Hundred Years of Philosophy (1968):
Basically, the idea would be that, at least assuming a common human nature, it is prudentially rational to believe in God, because one is permitted to do so in the absence of a refutation; there is no refutation of God's existence; and one can expect better consequences from such belief.
Whether that reasoning is sound is one of the most important questions of philosophy, in my view, and it's brought me very deep into epistemology/decision theory. Balfour's starting place is Hume, but Subjective Bayesianism (either with precise or imprecise probabilities) seems very apt for such reasoning. Indeed, on a Subjective Bayesian view, I don't think there is any reason to think that theism is less rational than belief in even our most supported scientific theories.
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