February 21, 2022

Pascal’s Wager

I don’t think there is an easy answer to Pascal’s wager, Amanda-Askell addresses many of the common objections to Pascal’s wager here

But I do think the reason it sounds plausible is to do with people often overlooking seemingly minor costs and benefits that aggregate significantly. Living a pious life incurs significant costs!

  1. There are many gods you could wager for, not just one! The basic idea: there are n-many gods that reward belief. If there’s a non-zero chance of getting infinite utility if you wager for a different God, then wagering for any of these gods has infinite expected utility (EU). So the wager doesn’t give you any more reason to believe in God X over any of the alternative gods. Simple Answer: Suppose you find yourself standing at the gates of heaven. St Peter offers you one of two options: you can walk through door A and go into heaven, or you can walk through door B and have a 1 in 1,000,000,000 of getting into a heaven and a 999,999,999 in 1,000,000,000 chance of being annihilated. Now you want to get into heaven — it’s not some crummy heaven that you won’t enjoy. Do you really think it’s rational to be indifferent between these two options? If you think you should have even the slightest preference for door A, then this objection doesn’t work.
  1. Almost all actions have infinite expected utility if wagering for God has infinite expected utility. So if Pascal’s wager is true then I can do almost anything I want to. The basic idea: If the expected value of believing in god is infinite then any action that has a non-zero chance of ending with me wagering for god is also infinitely valuable. So the wager doesn’t give you any more reason to believe in God than it does to roll a dice and believing in God if 4 comes up, or pick up a beer knowing that it might end with you getting drunk and believing in God. Answer: The same probability dominance argument applies to the mixed strategies objection. Insofar as you think that rolling a dice or drinking a beer has a lower probability of producing the infinite utility outcome (heaven) than some other action does — such as simply wagering for God now — you ought, all things considered, to perform the action with the higher probability of producing the infinite utility outcome. In this case, that means wagering for god rather than employing a mixed strategy.
  1. Doesn’t the wager beg the question? The basic idea: Pascal’s wager assumes key features of the god it seeks to prove the existence of. For example, that god rewards belief and not non-belief. Answer: Firstly, the aim of the wager isn’t to prove existence of god: it’s to establish that belief in god is prudentially/morally rational. Now consider the following argument: If you think there’s a >10% chance that there’s a dish in the dishwasher that’s made of china, then you ought to check that the dishwasher is off. You think there’s >10% chance that there’s a dish in the dishwasher that’s made of china. So you ought to check that the dishwasher is off.’ Pascal’s wager doesn’t fallaciously assume characteristics of god any more than this argument fallaciously assumes characteristics of dishes.
  1. What about the atheist-loving god? The basic idea: Suppose there’s a god that sends all non-believers to heaven and all believers to hell. Given the logic of Pascal’s wager, I ought not to believe in God. Answer: If it’s rational for you to think that disbelief in God (or cars, or hands) will maximize your chance of getting into heaven, then that’s what you ought to do under PW. What’s the evidence for the belief-shunning God? Possibly: Divine hiddenness’ plus God making us capable of evidentialism. The evidence against? God making us capable of performing expected utility calculations, all the historical testimonial evidence for belief-loving Gods. I suspect the latter will outweigh the former. But if you’re making this objection you’re already on my side really: we’re now just quibbling about what God wants us to do.
  1. What about infinite utility producing scientific hypotheses?**** The basic idea: Okay, so Pascal’s wager doesn’t tell us which God to believe in, just to maximize the probability of gaining infinite utility. But what about the possibility of more naturalistic infinite utility hypotheses (singularity, lab universes, etc.)? Answer: Given the response to 1, you ought to perform whatever set of actions that has the highest probability of getting you into heaven. Given this, insofar as a belief in a supernatural being or God is consistent with actions that maximize the chance of a scientific means of gaining infinite utility, you ought to do both regardless of which is more plausible. Also, higher cardinalities of infinite utility will dominate lower cardinalities of infinite utility in EU calculations. And supernatural hypotheses may be more likely to produce higher cardinalities of utility than their empirically-grounded cousins. ****
  1. You can’t quantify the utility of heaven. Answer: The wager doesn’t start by looking at a religious text and trying to work out how good their heaven is. The argument is premised on some infinite-utility outcome being possible, such that you ought to have a non-zero credence in infinite-utility outcomes. It doesn’t matter how inconsistent that outcome is with common conceptions of heaven, as long as it’s in principle possible the argument will go through. You might want to declare that such heavens are absolutely (and not just nomologically) impossible, but it’s hard enough to defend logical omniscience, let alone no-such-thing-as-heaven omniscience.
  1. God wouldn’t reward prudentially-grounded belief. Answer: You have to take your credence that a given God would reward belief into account when calculating what to do. Suppose you are certain that only two gods are possible: A and B. Each of their heavens produce infinite utility, and they’re equally likely to exist. The only way to get into heaven is through belief, but god A might reward prudentially grounded belief while god B doesn’t (all with certainty). Clearly you ought to wager for A. Suppose god B becomes sufficiently more probably. Then perhaps you ought to try to inculcate non-prudentially-grounded belief in yourself and others!
  1. I think God’s just as likely to reward belief as to reward non-belief. Answer: Suppose that, for action A that has the potential to produce infinite utility (given all of the possible states of the world), A and ~A are just as likely to produce infinite utility. Then you would need to find a tie-breaker between the two, or flip a coin. This doesn’t undermine the argument of the wager. But it seems highly unlikely that belief and non-belief would have exactly the same rational subjective likelihood of getting you into heaven. What could be the evidential basis for this perfect symmetry?
  1. What about the problem of evil, etc? Answer: Evidential considerations for or against a certain god are obviously relevant to what you ought to do or believe, since they are relevant to the likelihood that given actions will produce infinite utility. But Pascal’s wager doesn’t solve (or aim to solve) theological problems like the problem of evil. But its conclusion still holds as long as those problems don’t warrant adopting credence 0 in there being any infinite utility outcome that’s consistent with any action we can perform. It seems unlikely that the standard objections to God’s existence are as devastating as this requires!
  1. I have credence 0 (or near enough) in God’s existence. Answer: The near enough strategy isn’t going to work, unless you add the premise that you ought to treat even extreme-utility outcomes that you have a sufficiently low credence in as though you had credence 0 in them. That seems like a bad principle. If you genuinely have credence 0 in all potentially infinite-utility producing states of the world, credence 1 that you have these credences etc. then you are indeed immune to Pascal’s wager. Would it be reasonable to have such credences? This seems implausible under a standard account of credences, since these states of the world appear to be far from impossible.
  1. But we don’t have voluntary control over our beliefs! Answer: Are you certain that doxastic voluntarism is false? If not, the chance that your voluntary belief could occur and would result in your getting into heaven ought to be taken into account when you’re trying to determine what you ought to do and belief (constructing the full decision procedure for maximizing your chance of gaining infinite utility an interesting task!). But suppose you’re certain that doxastic voluntarism is false: you still ought to try to convince others of God’s existence, give money to organizations that try to do this, etc. The argument would simply support a different set of actions.
  1. The wager ignores the disutility of believing in God and the utility of not believing in God. Answer: The wager doesn’t ignore either of these: they simply don’t affect the act or belief that it is rational for you to perform or adopt. Suppose that the annoyance of wagering for god is like continuous torture for you. And suppose the utility of not believing in god is extremely pleasurable for you. You still ought to wager for god, since infinite expected utility swamps any finite (dis)utility. Even if the utility of both is infinite (see 2), it’s still probability and not finite utility considerations that determine whether or not you ought to wager.
  1. Dammit Jim, it’s just not scientific! Answer: The wager doesn’t give evidence for god:it’s a moral/prudential argument for belief. The view that your beliefs always ought to be in accordance with your evidence is powerful and useful, but should we be certain that it’s true, and that there are never prudential reasons to hold a belief? If not, then the full force of Pascal’s wager returns, since any non-zero credence that there are prudential reasons for belief is enough to let infinite utility back in. Even if you could be rationally certain in this norm, however, it just changes the actions Pascal’s wager warrants (see 11).
  1. That’s not how the maths works. Answer: Pascal’s wager appeals to the claim that a finite, nonzero chance of getting an infinitely good outcome is better than any probability of a finitely good outcome. We can appeal to something like Bartha’s relative utility theory to get both this result and the result that a greater chance of an infinite outcome is better than a lower chance of the same outcome. It would be somewhat surprising if our accounts of infinity (e.g. hyperreals, surreals) were in conflict with either of these claims. In theories where infinities can be multiplied by finite, nonzero numbers, they tend to produce infinities.
  1. The only reason you’d believe this is because you want to believe in God anyway. Answer: There’s a class of responses to the wager that bring into question your motives for defending the argument in the first place. I don’t really think that motives like wanting to believe in god have much bearing on the efficacy of the argument, but they should probably give you reason to doubt my weighing of the arguments and evidence, etc. But I don’t have such motives. I came to this through intellectual curiosity, though I don’t think that means that I’ll end up finding the conclusions unmotivating.
  1. Doesn’t the wager promote an unethical life of belief over an ethical life of non-belief? Answer: In principle the wager could promote this. But I don’t see any reason to think that this is overwhelmingly likely. It doesn’t necessarily favor adopting a given religion unflinchingly. And if we are more confident that god is more non-malevolent than not and that we haven’t been grossly mislead about the nature of moral truth, then we have strong reasons to act morally. The wager applies to actions as well as beliefs, so even if you think you’ll be forgiven’ for a certain action it’s unlikely that under PW it’ll be worth performing an action you are confident is wrong.
  1. The wager is only valid because there are problematic features of infinities. Answer: The infinite version of Pascal’s wager relies on features of infinities: e.g. that the expected value of an infinitely good outcome is will not be finite. But the uncertainty argument will apply even if you’re pretty certain these features won’t be present in the correct account of infinities. I don’t think that our worries about the wager give us sufficient reason to reject these principles of infinities. In principle we could reformulate much of wager by simply appealing to sufficiently large finite amounts of utility, but Pascal’s wager seems to be consistent with features of infinities that we are happy with in other domains.
  1. What if we have bounded utility functions? Aren’t unbounded utility functions problematic?**** Answer: Utility functions that are bounded above and below can prevent both positive and negative infinite forms of Pascal’s wager. But there are some obvious drawbacks to this response: 1. It’s ad hoc. What other reason do we have to think we don’t have an unbounded concave utility function over happiness (that isn’t’ just a result of our inability to adequately handle large numbers)? 2. Counterintuitive results at the point where a unit of happiness has no extra value for us, 3. Might not work for non-preference forms of utilitarianism (moral PW argument) and 4. We shouldn’t be certain that out utility function is bounded.
  1. If we allow ourselves to be skeptical about mathematical and normative principles, we’ll end up skeptical about everything!**** Answer: I don’t think this is the case, for a couple of reasons. Firstly, the question is a bit misleading. The uncertainty I’ve appealed to here isn’t mathematical uncertainty (though I think we can appeal to that as well in some cases) it’s normative uncertainty. And it’s not really skepticism, it’s just taking into account that we shouldn’t be certain that a given normative principle (mentioned in 1) is true. If we end up uncertain about everything like this, I don’t think that would be a bad thing. However, I’ll try to discuss objections to this view in another post.
  1. Isn’t this just a reductio of expected utility theory? Answer: I think that the existence of fanaticism problems presents a huge worry for expected utility theorists who allow unbounded utility functions. In fact, I’m surprised people haven’t written this up as an impossibility theorem with an anti-fanaticism axiom, since it seem you have to either accept the wager, accept other problematic conclusions, or give up on some plausible aspect of unbounded expected utility theory. I don’t think the reductio worry helps people who don’t want to buy Pascal’s wager though, since it doesn’t warrant acting as if some fanaticism-avoiding decision theory were true.