If Pascal's wager can be applied to belief in anything, then what is the fallacy in Pascal's wager?

I want to know the specific flaw in Pascal's wager that allows it to motivate belief in other things than God.

For example, a similar argument to the original wager can be used to say that it is prudent is to believe in the devil, or that black magic works, or, really, any unlikely belief that might have some potential benefit.

If the wager justifies many cases, doesn't this weaken the argument, in the case of God? If so, what is the flaw in Pascal's wager, that makes it seem like a proof of something, but cannot be, since it leads to contradiction? Is there a named fallacy, to describe this example?

• I believe Conifold is correct that you're thinking of "proving too much", but I'm not sure I understand your assertion that the Wager itself can actually be modified in the way you describe. Is this an argument you've read other people making, and if so, would you mind providing a link? If it's something you've come up with yourself, could you please make one or more of your "similar argument" examples more explicit? Commented Jul 20, 2017 at 21:45
• As an interesting counterexample to Pascal's Wager, I'd like to point you to Chtulhu's Wager, which proves, using the irrefutable power of Pascal's probablility calculations, that the only logical thing to dedicate your life to is to worship Cthulhu through self-flagellation and ritual sacrifice: uncyclopedia.wikia.com/wiki/Cthulhu#Cthulhu.27s_Wager Q.E.D. Commented Jul 21, 2017 at 14:46
• Pascal's wager argues that you should believe in one of an infinite number of poential Gods, with no way of knowing which is true, and damnation if you guess wrong. Commented Jul 25, 2017 at 20:12

This is not the only issue with Pascal's Wager, but what is described in the question is called the fallacy of proving too much. It happens when an argument is structured in such a way that the reasoning can be extended to reach "absurd" conclusions. This gives an effective strategy for refuting arguments on the arguer's own terms by exposing the fact they prove too much. Silvestre describes it in On the Logical Formalization of Anselm’s Ontological Argument:

"Traditionally, a counter-argument in this sense is an argument that shares the same logical structure of another argument, has true or reasonable premises, but an absurd or patently false conclusion. In this way, since we cannot accept the conclusion of the counter-argument, we cannot accept the conclusion of the original argument either: despite its apparent soundness, there must be something wrong with the argument (although this method of refutation does not say what is wrong)."

A famous example of this is Gaunilo's counter-argument to Anselm's ontological argument for the existence of God, discussed in the paper. Very briefly, Anselm argues that God can be conceived as perfect, and therefore must also exist in reality, because otherwise he would be imperfect. Gaunilo objected that if that worked then the "perfect island" must also exist (and by extension any other conceivable object with "perfect" attached to it).

However, the validity of the reasoning's extension, and the judgement of the "absurdity" of the conclusions can both be questioned. For instance, one could argue that transplanting Pascal's Wager onto the devil does not work because with the devil there is no guarantee of "infinite reward": after all, he is a deceiver. Silvestre discusses diffusing Gaunilo's objection along this route. And sometimes proponents of an argument are committed to it enough to be willing to accept the seemingly "absurd" conclusions of its extensions - this is called biting the bullet.

Other issues with Pascal's Wager include illicit use of probability concepts and either equivocation on "God" or a false dichotomy, if the Christian God is presumed. For instance, if the Christian God does not exist but some other one does then Pascal's inference that one avoids "infinite losses" by betting on God is invalid, see Bendz's webpage on the Wager:

"The main problem with Pascal's wager is that it suffers from the fallacy of bifurcation. It only calculates with two options when there are, in fact, at least four alternatives: the Christian God and afterlife, some other god and afterlife, atheism with afterlife, and atheism without afterlife... Because Pascal's wager fails to tell us which god is likely to be the right one, you have a great probability that you picked the wrong religion and go to some other religion's version of hell. This is referred to as the "avoiding the wrong hell problem"... Believing in the wrong god has one additional problem. Most religions assure you that blasphemers will be more severely punished than un-believers. Once again, if we calculate with the rest of the possible gods, the chance of you being wrong is P=1-1/n so you both run a bigger risk than the atheist of being punished and risk the greater punishment."

And the list of problems goes on. But I believe it is the false dichotomy that exposes the Wager to proving (or rather motivating, since it was not intended as a proof) too much. There are just too many "unlikely benefactors" with potentially "infinite" rewards and losses. To run cost/benefit analysis on one of them we must run it on them all, but we can not bet on them all together because many are mutually exclusive. As a result, there is no viable alternative to not betting at all, contrary to what Pascal assumes. The Wager would work only if one of the alternatives (or several compatible ones, e.g. Christian God for Pascal) can be shown to be likely on other grounds. But it should be said that Pascal himself never intended the Wager as a rational argument. Indeed, he writes:

"“God is, or He is not”. But to which side shall we incline? Reason can decide nothing here... Your reason is no more shocked in choosing one rather than the other, since you must of necessity choose... But your happiness? Let us weigh the gain and the loss in wagering that God is... If you gain, you gain all; if you lose, you lose nothing. Wager, then, without hesitation that He is." [emphasis added]

For more on objections to and defenses of the Wager see IEP and SEP articles.

• My favorite argument against Anslem's ontological argument for the existence of God just replaces all instances of the word "God" in the argument with the word "sandwich". Commented Jul 20, 2017 at 15:27
• @Shufflepants That has nothing on "Canadian Boy/Girlfriend".
– Yakk
Commented Jul 21, 2017 at 17:31
• @Shufflepants Sure, if you define "sandwich" as "that which is such that nothing greater can be thought than it". To each his own.
– sgf
Commented Jul 22, 2017 at 17:34
• @sgf: I think no, it merely relies on persuading Pascal to accept a non-zero probability that they are, just as he expects us to accept a non-zero probability of his God. And hence there are many mutually conflicting courses of action leading to unspecified-but-non-zero probabilities of winning or losing, and you can't choose between them. I suppose Pascal might come back with a better argument, which deals with what appears to be a Burridan's Ass problem, but nevertheless the original argument has been dispatched. Commented Jul 23, 2017 at 1:27
• @sgf: and anyway it doesn't really matter whether they're the same God or not: all that matters is whether the different means of reverance all promise infinite reward or not. If religions X and Y each assert that the followers of the other are not heaven-bound, the reasoning of the wager tells you that you must follow religion X (if considered alone) and also that you must follow religion Y (if considered alone). Which may not be possible. Whether X and Y are correct about this simply doesn't enter the argument. Commented Jul 23, 2017 at 1:32

I don't think we can categorize it under a single fallacy. Furthermore...why would we want to? We'd have to explain our reasoning anyway.

What we do instead is simply look for the premises of the argument and attack them.

Pascal's Wager makes the following problematic premises:

1. There either exists a god C or there don't exist any god.
2. We either believe in a god C or we don't.
3. If there exists a god C and we believe in this god then we benefit infinitely.
4. If there exists a god C and we don't believe in this god then we suffer infinitely.

A similar argument to the original Wager can be used to say that it is prudent is to believe in the devil (ofcourse not in conjunction with a Christian God) or that black magic works or really any unlikely belief that might have some potential benefit. When it justifies many cases does it not weaken the argument in case of God?

It does. Your argument basically denies (1). We could also just change the conception of a single god for (3) or (4) with a similar idea.

Pascal's Wager presupposes ignorance - god can't be proven or disproven through reason . Pascal's Wager excludes proof due to reason. Therefore we've no reason to think that (1) is true, and we can assume any number of possibilities. Hence we can't make a sensible choice anymore. And so on.

If someone posits that premise (1) just says that, "There exists a god or there doesn't.", then the firstly leaves us the possibility of adding a premise for other gods and secondly doesn't directly lead to (3) or (4).

Now if someone objects that the other conceptions are just impossible then that person tries to include reason. This breaks with the assumption of ignorance. So then we can just argue with other arguments against the conception that the person holds. So this at least makes Pascal's Wager unsucessful, because we need the condition of ignorance. The soundness is dependent on whether we consider only one god C possible, but that is outside the argument.

edit: edited for a significant flaw, as I intended for slightly different premises but then wrote other premises. Added explanation when the latter option - the one I used beforehand - of premises is used.

• Indeed this is the main point. The results from the line of reasoning without it are catastrophic. Commented Jul 19, 2017 at 20:59

The questioner believes that Pascal must be committing a fallacy since he reasons that Pascal's reasoning can be applied to other cases, e.g., to show the existence of Black Magic or Satan. The questioner reasons as follows:

If the use value is the reason for believing in God, then surely, one should believe in any ideas insofar as believing in their existence has use value.

It could be that Pascal's own argument is fallacious, but the questioner concludes fault in the Pascal's reasoning not by examining Pascal's own argument, but by employing Pascal' method of reasoning to other cases. For this reason, the fallacy actually lies not in Pascal's reasoning, but in the questioner's reasoning, as he misapplies the reasoning method to other cases. The name of the fallacy the questioner commits is called faulty analogy.

The fallacy happens when one attributes certain quality to two things that are relevantly different. For instance, the properties we attribute to God is very different from those of Satan. Some dissimilarities between God and Satan might be that God is boundlessly good and forgiving, while Satan is revengeful and tyrannical. The quality of after-life envisioned by the God and by the Satan is also very different. Pascal is especially moved by the calculation of the eternal salvation as against the eternal damnation aspect of the use value. If the questioner wants to protest that Satan and God share all the properties equally, then by the Leibniz's law of identity, the two are the same entity. Insofar as Satan and God are two different entities, Pascal's reason for believing in God cannot be applied to the case for the Satan.

• Your analogy of his analogy is worse than the original analogy... On another topic, suggesting that Leibniz's law has anything to do with this is a major strawman, because it suggests that Pascal's argument applies exclusively to one fully specified definition of "god", yet that is evidently untrue. Commented Jul 19, 2017 at 1:53
• Please explain your claim since I cannot understand your criticism. I did not make any analogy: I explained why the question was based on faulty analogy. The questioner's analogy actually is really smart, and it took me a while to block the conclusion the questioner was making. I do not think you understand the context I was using Leibniz's law. Commented Jul 19, 2017 at 2:49
• Your image is an attempt at an analogy of the original analogy, that attempts to give context to the fallacy you claim it makes. That's what I criticised in my first sentence. Commented Jul 19, 2017 at 4:48
• @NanheeByrnesPhD If you can't address intelligent criticism of your argument, perhaps you shouldn't put so much faith in your argument's validity. Commented Jul 19, 2017 at 12:52
• "I need to block the intelligent criticism that you make in the first place since, if I concede the criticism, then I cannot show that the analogy by the questioner is faulty" ... I can't believe I just read that. Wow. "If I answer the question reasonably, honestly and intelligently, I wouldn't be able to give the answer I wanted, so I didn't do those things." Wow. If you have to be stupid and ignore all context to show someone is wrong, then they aren't wrong. Commented Jul 19, 2017 at 15:17

Since I do not think Pascal's argument is fallacious, and since I detest a long post, I offer this second post, as an appendix.

Pascal's question is whether he should believe in God (the paradigm relating to the Christian God: e.g., Soul lives after one's death; Believing in God is necessary for the eternal salvation of the soul, which is the most important thing one can hope for). These are essential assumptions of Pascal:

Pascal 1. Either God exist or does not exist.

Pascal 2. Either I should believe in God or should not believe in God.

Pascal 3. Based on the Christian paradigm, the following payoff table obtains:

Pascal's decision is based on the Bayesian decision theory which states that the rational course of action is the one that leads to the maximal expected utility. Pascal thus concludes that he should believe in God since believing in God offers infinite utility over not believing in God. I do not see any fallacy in his reasoning. If one rejects Pascal's conclusion, she must be regarding any of the above assumptions as unwarranted. But believing that the assumptions are false does not mean that the reasoning is fallacious. That is, if we accept the paradigm of the Christian God, and if accept all the assumptions (including the Bayesian method), one must accept the conclusion.

With this, I respond to the claim that Pascal's argument is fallacious, following Conifold's suggestion, due to committing the fallacy of bifurcation. So the fallacy must lie in either Pascal 1 or Pascal 2. Pascal 1 is a tautology. So the fallacy should be in Pascal 2. Indeed Conifold seems to assert that Pascal 2 is a false dichotomy since there are other possible gods that Pascal neglects to consider to believe in. But considering other possible gods are irrelevant to Pascal's question, which is whether he should or should not believe in this particular God presented to him. The situation is similar to the case whether I should or should not love this woman. If Conifold points out to me that I neglected to consider all other possible women I could love, then surely Conifold is missing the question that I am asking. The existence of other women is irrelevant to my question since my question is whether I should or should not love this particular woman. For this reason, Pascal 2 does not commit the fallacy.

• Slight change: God loathes those who believe on him just based on Bayesian decision theory, and sends them straight to hell. Where are you now? Commented Jul 19, 2017 at 8:11
• gnasher729: Haha. Marc H: Some might question your claim that a man can love multiple women, but not multiple gods. Greeks and Romans believed in poly-gods:, and they loved them all! Pascal did not question whether he should believe in God or Buddha or Confucius. He questioned whether he should or should not believe in the God. Commented Jul 19, 2017 at 13:44
• I think Pascal's logic would hold if the payoff table were correct, but the existence of other gods could change the potential payoffs for believing/disbelieving in a non-existent Christian god from -1/+1 to -infinity/+infinity. Commented Jul 19, 2017 at 14:29
• Pascal is of course entitled to assume the Christian paradigm, but his wager is presumably addressed to those who doubt or even reject it. If so he is not entitled to a payoff table "based on the Christian paradigm", or he is begging the question. And without it his payoff table does not obtain. Commented Jul 20, 2017 at 0:30
• " The core assumption is that Pascal is not trying to persuade a Buddhist, but an atheist brought up in the Christian paradigm (so the atheist knows Christian doctrines) to believe in the Christian God." I was raised "in the Christian paradigm", but I still knew about plenty of other religions. I mean, other religions are talked about extensively in the Christian Bible, so it's unrealistic to pretend that other (non-Christian) religions don't exist and aren't relevant to the discussion... Commented Jul 20, 2017 at 17:37

Pascal's Wager sounds convincing at first, but if you think clearly about it, the wager quickly reveals itself as deeply flawed logic.

In short, the wager says that

1. If the Christian god is real, then you either receive eternal reward (+infinity) or eternal punishment (-infinity) based on belief.
2. If the Christian god is not real, then you do not stand to gain or lose much by (not) believing in the Christian god (say, +1/-1).

The wager concludes that not believing in the Christian god is too risky for the modest potential gains.

However, the second point is not at all true. There are infinitely many possibilities that could completely change the expected value of the wager. Here are a few examples:

• There exists a god that punishes only Christians and rewards everyone else.
• There exists a god that punishes people who believe in him based on probabilities (such as Pascal's Wager).
• There exists a god who punishes everyone who is not an atheist, and rewards atheists for not believing in him.

For Pascal's Wager to really hold up logically, one has to show that the Christian god is more likely than any of the infinite other possibilities. There is no a priori reason to expect that. The reason that Pascal's Wager sounds so convincing at first is that the Christian paradigm was common among Pascal's peers (and still is many places), hence the Christian god seemed plausible whereas other possibilities were not considered equally plausible.

Pascal's Wager uses the following logic (excerpts from Pensées, part III, §233):

1. God is, or God is not. Reason cannot decide between the two alternatives.
2. A Game is being played... where heads or tails will turn up.
3. You must wager (it is not optional).
4. Let us weigh the gain and the loss in wagering that God is. Let us estimate these two chances. If you gain, you gain all; if you lose, you lose nothing. Wager, then, without hesitation that He is. (...) There is here an infinity of an infinitely happy life to gain, a chance of gain against a finite number of chances of loss, and what you stake is finite. And so our proposition is of infinite force, when there is the finite to stake in a game where there are equal risks of gain and of loss, and the infinite to gain.
5. But some cannot believe. They should then 'at least learn your inability to believe...' and 'Endeavour then to convince' themselves.

The first proposition is that reason cannot decide whether God exists or not. Here there is an assumption that the only deity that possibly exists is the Christian one (whereas there could be an infinite number, or a different deity than the one posited by Christians). This appears to be a quantification fallacy.

The second proposition is that it's a game of head or tails – the tossing of a coin rather than a dice, or one of those multifaceted dice used in D&D games. This repeats the quantification fallacy in the first proposition.

The third proposition states that you must wager. (Is this an argument from ignorance, "assuming that a claim is true because it has not been or cannot be proven false, or vice versa."?)

The next proposition is weighing the infinite gain of the eternity of heaven against a finite loss of a life spent believing in something that may or may not exist. That one looks like an argument from silence, "where the conclusion is based on the absence of evidence, rather than the existence of evidence".

I have always thought that there was something obnoxious about Pascal's Wager; perhaps it's the logical flaws, or perhaps it's the fact that any deity worth its salt would reject such cold calculations anyway. Or the fact that the posited deity cares about whether you believe in and worship it or not, rather than whether you try to live a good life.

The following quote attributed to Marcus Aurelius provides a more wholesome option:

“Live a good life. If there are gods and they are just, then they will not care how devout you have been, but will welcome you based on the virtues you have lived by. If there are gods, but unjust, then you should not want to worship them. If there are no gods, then you will be gone, but will have lived a noble life that will live on in the memories of your loved ones.”

The primary fallacy in Pascal's Wager is that faith is not just a matter of convenience. Simply mouthing the words does not qualify as belief.

However there is a secondary fallacy in Pascal's Wager, which your argument exposes. If your method of worship of your deity requires you to commit evil acts, then not only is this immoral, but a benevolent deity would commit you to Hell in spite of these acts being committed in Its name. This is solved by the Atheist's Wager, which directly refutes Pascal's Wager.

• You may live a good life and believe in a god, and a benevolent god exists, in which case you go to heaven: your gain is infinite.
• You may live a good life without believing in a god, and a benevolent god exists, in which case you go to heaven: your gain is infinite.
• You may live a good life and believe in a god, but no benevolent god exists, in which case you leave a positive legacy to the world; your gain is finite.
• You may live a good life without believing in a god, and no benevolent god exists, in which case you leave a positive legacy to the world; your gain is finite.
• You may live an evil life and believe in a god, and a benevolent god exists, in which case you go to hell: your loss is infinite.
• You may live an evil life without believing in a god, and a benevolent god exists, in which case you go to hell: your loss is infinite.
• You may live an evil life and believe in a god, but no benevolent god exists, in which case you leave a negative legacy to the world; your loss is finite.
• You may live an evil life without believing in a god, and no benevolent god exists, in which case you leave a negative legacy to the world; your loss is finite.

This also demonstrates the fallacy of morality being derived from religion. Every religion, and every holy text, is created by humans and is therefore fallible. Only by checking whether actions are truly moral from first principles, based on harm caused to others, is it possible to live a good and moral life.

• Benevolent humans may disagree on whether an entire life was good or evil. But whether an individual action is good or evil, I don't believe benevolent humans can disagree on that unless their criteria for judgement are based on dogma (religious or otherwise) and not on the benefit/harm to other people. That's basically the definition of "benevolent". And if a god puts more weight on belief than on actions, again by definition they fail the test of "benevolent". They may be a deity, but they aren't a good deity. Commented Jul 20, 2017 at 12:41
• @Nathan It's certainly a common scenario that there is no way for everyone to win. That's why moral judgements balance the greater good of the greater number, or balance levels of benefit for one against harm for another, or require minimised levels of harm wherever possible even for animals. Benevolence does not require that harm is eliminated, it requires that it is minimised. Commented Jul 20, 2017 at 13:59
• @Nathan A god may well choose to reward people who behave like that. But then the god is by definition malevolent towards humanity. People may well choose to serve a malevolent god because of the rewards it promises. It may even deliver on its promises - who knows? But you cannot then pretend that you are doing good deeds by following it. You are actively choosing evil, in return for those promises. Commented Jul 20, 2017 at 14:03
• @Graham - the Atheist's Wager is essentially a re-statement of Marcus Aurelius: `“Live a good life. If there are gods and they are just, then they will not care how devout you have been, but will welcome you based on the virtues you have lived by. If there are gods, but unjust, then you should not want to worship them. If there are no gods, then you will be gone, but will have lived a noble life that will live on in the memories of your loved ones.”` Commented Jul 21, 2017 at 15:24
• @YvonneAburrow True enough. It's not a particularly surprising concept. I'd worked it out myself independently while I was still at school, well before Michael Martin published his book. But he wrote a book about it, and I didn't. Partly because I'm not a philosopher. But also because I was about 14 at the time, and it seemed too obvious that someone would have thought of this already. :) Commented Jul 27, 2017 at 10:47

Um, seems to me that the problem is of assumptions. The critical assumption is that "belief" alone, will bring benefits (eternal life, not eternal damnation). The "Christianity" that Pascal refers to requires more than belief - it requires the following of rules and therefore a cost which is not accounted for in the wager. (i.e. the following of those rules may bring finite benefits - or deficits, during life.) Usually Christian churches require you to "believe with your heart" not simply say the words because you think it accrues benefits. Replacing that "god" singular with another "god" doesn't change much, unless it happens to be a God who grants benefits if all that a "believer" does is believe.

Testing Pascal's Wager by replacing the Christian god with the Christian devil only fails if you believe in the Christian god (and take "his" word for the qualities of the "devil"). Of course, if there is only one god, then all gods are the same god, which is problematic given that they all usually make strict demands before endowing benefits - and especially problematic if they don't provide any benefits.

It seems fallacious to say Pascal's Wager can only be applied to a Christian god when the rationale is to determine the greatest benefits (and minimal losses) until you've proven there are no other gods or that they cannot provide benefits to their believers. A poor analogy would be to say that otherwise it's like arguing over how many angels can fit on the head of a pin while assuming that nothing else can/will be also on the head of the pin, and that angels can be packed in tightly.

• On the other hand, some faiths (focusing on Christian, but I doubt it's unique to them) just require that you are e.g. baptized or confess before death to be saved, so there isn't the requirement of living your whole life as a believer. Commented Jul 19, 2017 at 10:34

I think the answer is simpler than any given so far.

To wit:

To gets things rolling Anselm starts out with a question,

Is it possible there is a Being of which no greater than can be conceived?

Nice question but one not too amenable to a finite mind.

Therefore, replacing God with the Devil,

Is it possible there exists a being of which no baser can be conceived?

Is all rather academic/uninteresting.

And I consider an uninteresting philosophical question to be ipso facto a fallacy.

Thus any sort of wager on the Infinite by the finite overestimates the finite mind and underestimates what it is to be God.

In short, Pascal's Wager is is a cowardly shortcut that only works if one is fool enough to believe a God worthy of being worshiped can be gotten the best of with end run around the requirement of faith.

Potentially relevant asides:

• the biblical text spends no time arguing for the existence of God, though it does on occasion lampoon belief in any of the many gods worshiped among other cultures.

• the biblical text does spend a fair amount of time lamenting God's apparent disinterest in temporal justice.

The apostle Paul says in 1 Corinthians 15:19: "If in this life only we have hope in Christ, we are of all men the most pitiable".

In other words if Pascal were to lose his wager, Paul would regard him as a very great loser. His one life would have been wasted in pointless service and sacrifice of a God that doesn't exist. In a (hypothetical) universe where God doesn't exist, surely this is as close to hell as it gets.

Isn't it the case that in making a decision as to whether to worship and serve any god, you need some pretty hard evidence to back you up? This is not something to bet on. Any truly consistent Christian would argue that his/her belief is based on more than mere probability but is supported by conclusive evidence.