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There are many objections to the Pascal's wager, as originally stated, see for instance here. Personally, I don't think that the wager can be defended. However, I am wondering if it is possible to formulate - and successfully defend - an argument of a broadly similar type. The argument should run along the following lines:

  1. I accept that God may or may not exist. (Here, any plausible definition of the term "God" may be used). Moreover, I concede that God's existence is unlikely given the available evidence, but not extremely so - I may assign probability of about p = 1:1000000 to there being God. Note that I require the definition to be plausible - in particular, if the God is question is self-contradictory, then p = 0, and the argument is bound to fail.

  2. I accept a set of plausible assumptions about the nature of the utility function (my utility function?), conditional on there being God, or on there being no God. These could be statements such as "Assuming that there is no God, the point of life is increasing the total happiness of sapient beings" or "... the point of life is seeking joy and avoiding suffering", etc. (Note that these should be plausible: you don't get to say that "The only thing that has any utility at all is worshipping existing God").

  3. There is a certain action X such that: if God exists, X has large positive utility, if God does not exist then X has negative but small utility; and preferably X only "makes sense" if God exists. For instance, X could be believing in God, or worshipping God, or following commandments of a certain faith, etc.

  4. The relation between respective utilities and probabilities is such that it is advantegeous to do X, i.e. something like p Utility(X|God exist) + (1-p) Utility(X|God does not exist > 0. Therefore, I choose to do X.

Is there any argument like that which "works"?

  • Take into account that for example the "Christian" god, if I understand things right, wouldn't be happy with you basing your actions on that kind of argument, so your Utility(X|God exists) may be lower than you think. I'd be happy if anyone with insight into Muslim, Buddhist and other religions can comment. – gnasher729 Nov 28 '15 at 17:38
  • How does one account for the multitude of possible gods and the notion that an attempt at pleasing one of them may come at the expense of displeasing another? In the event that only one god exists, how does one choose the correct one? There is a good argument that abstaining from pleasing the wrong god has perhaps the highest probability of at least not offending the correct god. In addition, there are some unpredictable gods in history that may punish someone without any known reason. – KnightHawk Nov 30 '15 at 21:04
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The tricky part of an argument like this is the proof of Utility(X). This argument works very well in situations where the utility is obvious (I either do God's will by helping this orphan child, or I make myself feel better by helping this orphan child), but it gets more difficult to assign meaningful utility values in other situations. We humans are not very good at estimating these sorts of values (is 54627 * 183059 - 2,984,120 * 1 greater or less than 0?) This makes the wager tricky.

However, even if we are given our Utility function, its still tricky, because we have to use that function after it has been proven in order to determine if any action is good or bad. Thus our Utility function must include the utility costs of applying the formula. Otherwise we may spend forever in a chaotic loop around U(X) = 0, unsure how to proceed. The Utility function must be self-referential to avoid this issue.

The fact that you are going to use probabilities on these utility functions implies you need the basics of arithmetic. This doesn't sound like anything too important, but it's not a trivial detail, as we soon find out.

Now the final question before things get crazy is: do you have to be able to prove the validity of your results? If you are willing to take this on faith, as most if not all religious beliefs are taken, you can rest assured that this sort of logic can be made to work. However, for the faithless, who must rely on hard mathematics and provability, the concept gets squirely fast.

Why? Well, if you put all of these pieces together, you have the basis of Tarski's undefinability theorem, and whenever his name gets involved, you know you're in the really messy bits of logic and mathematics. His theorem basically states that no self-referential system which can prove all truths of mathematics (read: contains the Peano axioms of arithmetic) and contains negation can define its own semantics. In particular, it cannot define what "true" means within the system. His proof involves the diagonal lemma, which makes all sorts of things frustrating, but it ends up using the system's descriptions of arithmetic to form inconsistent results. The way this occurs is specific to the system, but Tarski proved that it must occur in every system satisfying the ruleset he was exploring (and its surprisingly easy to accidentally support that ruleset).

This means that you get stuck with the interpretation of the Utility function as being handled outside of the language of said functions (a metalanguage feature). This creates all sorts of interesting loopholes that are impossible to plug within the Utility function.

There are ways around this. I've enumerated a few in my own exploration, but the key is that you have to describe the Utility functions in a language which sidesteps Tarski's work. If you accidentally step into the languages he described in his undefinability theorem, that step seals the fate of not only the resulting Utility function, but the entire class of Utility functions created in that form.

  • You can design the system to refuse to admit the diagonalization lemma. Dan Willard has created a class of curious self-verifying theories which are intentionally too weak to support diagonalization (in particular, multiplication is not total in his theories). Wisps of this sort of thinking show up because Dan's systems start with an infinity (everything), and subtract and divide from there, rather than starting from 0 and adding/multiplying, and I've seen several belief systems which focus on being "part of everything."
  • You don't have to prove your system. This can be undervalued, but remember that nobody has been able to demonstrate a proof of their religion which is satisfactory for all other people. Faith is kind of a big deal for religions.
  • You can admit uncertainty. If your system is not complete, meaning it does not admit a proof for everything, you can avoid the provability issues Tarski addressed. The phrase "if it is God's will" comes to mind.
  • You can omit negation from logic. Negation is a key part of Tarski's proof. If you never negate anything, his work cannot disprove yours.
  • You can weaken the wager slightly. If you can demonstrate that U(X) >= 0, whether or not there is a God (i.e. it's not bad, no matter what), then it is trivial to show that U(X) cannot be less than 0. "Do what you will, may it harm none"

I'm certain there are others, but those are the ones I have found interesting.

  • I'm afraid we are talking at cross purposes here. In particular, I'm quite happy to keep things at a rather informal level, where Tarski's results will not be that much of an issue. I can hardly imagine formalising one's utility so much that these problems may arise. Surely, it is even harder to have a system that is somehow able to describe utility (which is connected to physical world, which is messy and complicated) but at the same time is too weak to include integer arithmetic. – Jakub Konieczny Nov 29 '15 at 2:13
  • If you don't formalize it, then step 4 becomes trickier, because you cannot actually compute Utility(X|God exist) + (1-p) Utility(X|God does not exist > 0, and humans are notoriously bad at estimating equations like that when small probabilities or large values are involved (which is why Casinos exist). – Cort Ammon Nov 29 '15 at 2:17
  • Note that to argue that the expected utility of a certain action is positive, there is no need even be able to evaluate utility of a single specific action. Pascal does not do it. Instead, he argues that utility of a certain action if God exists is infinite (or at least orders of magnitute larger than anything else involved) so that even rescaled by a factor like 1/1000000 it still dominates all the other terms which would appear. – Jakub Konieczny Nov 29 '15 at 2:19
  • Pascal also started from the assumption that God was real and that there was only one possible infinite God. Unfortunately, the same argument he makes is one that can be made for jackpots... the winnings are so large that even if you rescale it by the probability of winning, it must dominate... only it doesn't actually dominate in the Casino case when you do the real math. – Cort Ammon Nov 29 '15 at 2:21
  • Cort Ammon, I was writing the second comment before I saw yours, but it happens to partially answer it. I'm not asking that all terms in the expression that you mention be computed - I'm merely asking for sufficiently good estimates. – Jakub Konieczny Nov 29 '15 at 2:21
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There is a generic way of reformulating Pascal's wager that does avoid some of the standard objections. It goes like this:

If you seek to enter into a relationship with a being, such that relating to that being is both positive and possible, then, if you are successful you gain all such benefits. If you are unsuccessful, then you lose nothing*.

On the one hand, note that this general-form argument doesn't even require a divine being (it could be the President, or a new friend at school, or a pretty girl on Facebook, etc.), or that there be any ontological uncertainty about the being. However, it does have that big asterisk. You're losing the time and effort you put into pursuing the relationship, arguably, and if there are any other costs of your effort, you incur those as well.

Whether this argument is compelling to you, with regards to God, depends on if you think it's more reasonable to believe that God, if God exists, will reward any sincere effort to reach out (this is the view taken by noted Christian apologist C.S. Lewis) or that God, if God exists, will ignore and/or punish all but the approved and proscribed method of seeking God (this is typical of a "fundamentalist" view of God). In other words, you have to assume that you know some things about what God would be like, even while withholding judgment about God's existence. The more positive a conception you have of God-if-God-exists, the more compelling the argument becomes.

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I am not familiar enough with logic to state this in any technical fashion, but it seems there is a much simpler point concering time comparisons as implied in Pascal's wager.

An assumption of divine judgment is presumably an assumption also of the immortality of the soul. Thus you are comparing "utility" and "evidence" within an infinitesimally finite period of time with the possibility of outcomes over an infinite period of time. Which is how the Christian imagination actually thinks of it.

One might say that even the tiniest inkling of the infinite span dominates any probabilities of "getting away with a bit more utility" in the finite span. But in fact, this simply erases any probability calculations altogether. I am assuming, of course, moral laws and a divine judgment, in Pascal's sense, as given.

  • Sure, infinite time multiplied by even a small improvement in utility per unit of time gives infinite utility. However, to make it work you need to assume that the god favours those who pretend to believe over those who do not believe but are generally good people, which I don't think is generally believed. So, there is infinite utility hanging in the balance, but it is not sure which strategy is more likely to give you this infinite improvement. – Jakub Konieczny Nov 29 '15 at 1:52
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This might work, but needs to deal with some standard objections:

  1. Pascal's wager isn't possible:

It's not clear that it's to cause yourself to believe in God as a result of Pascal's wager. Many people think you can only believe what you take the evidence to support, and you don't (by hypothesis) take the evidence to support the existence of God. So Pascal's wager might just be an argument that you should take steps to change your cognitive dispositions, exposure yourself to biased evidence, etc. to increase your willingness to believe in God's existence. (But these steps might have significant disutility).

  1. Pragmatic belief in God has low utility

Many organized religions don't take God to reward those who believe in God for pragmatic reasons only. So it's not clear that belief in God should be assigned high utility.

  1. Widening the state space

Your presentation of Pascal's wager takes there to be two states the world could be in relevant to the utility of taking the wager: "God exists" or "God doesn't exist." But we should probably break each of these into many sub-states. "God exists" becomes "The Christian god exists"; "The Jewish god exists"; ... and many other possibilities, each of which has its own sub-possibilities. That won't sink your argument, even if some of these don't take God to reward followers. But the "God doesn't exist" state should be broken into states like "an evil demon exists who will punish people for believing in God." And depending on the credences we assign to these states attaining, Pascal's wager might have negative utility.

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