Closing Pascal’s Box

          Nearly four centuries ago Blasie Pascal laid out his famous wager in an attempt to prove that it was rational to believe in God. The argument seems deceptively simple, either God exists or not, we have to “place a bet,” so to speak, and the only way we “win” the wager is if we bet for God, and God exists. Hidden in this attempt to justify theism is a rather complex use of probability and decision theories, voluntarism, pragmatism, and an often overlooked use of infinity.¹ This argument had implications far beyond the Philosophy of Religion, and set the stage for the continuing debate on epistemic justification, how it is rational for us to form and hold beliefs. In effect, he opened Pandora's box on this topic.

          In this paper I will show that some of the last openings for Pascal's mode of thinking have been closed off in recent years. While it is clear that there is a great number of things that can be said about his argument, I will focus on voluntarism, forming beliefs at will, and pragmatic justifications, basing beliefs on non-epistemic concerns. With these two features alone, we can remove any doubt that rational people can be epistemically justified in being convinced by the wager, and we can close Pascal's box.

The Box

          Pascal laid out three arguments in his wager, to cover potential ways of approaching the problem. The three arguments may be categorized as superdominance, expectation, and generalized expectations.² Superdominance is a decision that must be made without any possibility of the decision-maker knowing the truth of the matter, where “[r]eason cannot settle which way we should incline”.³ Expectation is when one can reasonably assign some probability to potential states of the world, and a wager.⁴ One might assign probabilities to each position on a roulette table, and place bets accordingly. Pascal's final move is to show in his generalized argument, provided that there is a non-zero likelihood of God's existence, and the infinite gain given that God does exist, that it would be irrational to not wager for God.⁵

          The superdominance argument is a decision under uncertainty, which is required in a state of ignorance, since we may not be able to fix odds to the existence of God. The basic idea is that it is rational to examine the potential benefits and choose the best possible outcome.⁶ In the case of this “game ... being played at the extremity of this infinite distance”, betting for God superdominates betting against God.⁷

	God exists	God does not exist
Bet for God	Infinite reward	No change
Bet against God	Loss	No change

          Since betting for God either gives a benefit of “all” or loses you nothing, but betting against God will either cause “misery” or no change, the clear choice is to bet for God because the only way to gain anything is by betting for God.⁸ However, it is conceivable that one might not be “at the extremity of this infinite distance”, or rather the decision might be one under risk, but not under uncertainty. One might be able to assign probability to the existence of God, and if that is the case, then one also assigns potential rewards if the wager lands in favor of the bet, specifically that there is an infinite reward for wagering if God exists.⁹

          This leads immediately to the generalized form of the argument, which can be expressed in mathematical forms. For my purposes it is only necessary to understand that the inclusion of the infinite rewards granted from a wager for God undeniably tilts the scale in favor of betting for the existence of God. That is because any non-zero probability of God's existence multiplied by the infinite reward must result in an infinite expected utility. Betting against God in the same circumstances will, at best result in a finite reward.¹⁰ The true power in this argument rests on the inclusion of infinity in a mathematical formula.

          This argument relies on a few common-sense assumptions about the nature of belief: we are able to form beliefs on pragmatic grounds, and we are able to control the content of our beliefs, or rather that we can voluntarily form beliefs at will. William Clifford seemed to hold a similar understanding about beliefs when he wrote “[i]t is wrong in all cases to believe on insufficient evidence; and where it is presumption to doubt and to investigate, there is worse than presumption to believe.”¹¹ Two hundred years after Pascal opened the box, Clifford continued to use the presumption that our beliefs are voluntary and can be settled on pragmatic grounds.

          The reason for Clifford's condemnation of hasty belief is that beliefs lead to actions, or other beliefs that lead to actions, and the complex set of beliefs interact and inform each other.¹² When one introduces the habit of not properly questioning one belief, it can lead to a chain of false beliefs, and actions which “leave its stamp upon our character.”¹³ The real danger seems to be that accepting false beliefs, or rather, not properly proving a belief is based on truth has the tendency of weakening the foundations of knowledge for all people.¹⁴ Therefore, he admonishes people to question their beliefs and to withhold judgment until matters can be settled by evidence.

          Instead of terms of wagering, Clifford speaks in “placing trust”, “honestly earning” and “stifling … doubts” in a belief, but the effect is very similar.¹⁵ While Pascal argues for pragmatic reasons to form a belief, Clifford argues for pragmatic reasons to withhold belief if there is a lack of evidence; both hold that it is possible to will belief in decisions under uncertainty or risk. If this is not the case that either pragmatic reasoning can be held as reasons to believe, of if we are not able to will belief, as we suppose we will actions, then neither of these arguments will have any power, even if they have any chance of being logically valid. It may well be that we have no real doxastic control (control over what we believe) given any level of knowledge or ignorance.

Doxastic Control

          From Pascal forward, there has developed a fairly widely discussed notion that, while one is not free, in the truest sense, to form any belief that one chooses, one may form beliefs in roundabout fashions, that is to say that “one should act in ways that are conducive to the acquisition of … belief”.¹⁶ Part of the reason that it seems that one does not have radical doxastic voluntary control is that beliefs aim at truth.¹⁷ As Bernard Williams stated:

If I could acquire a belief at will, I could acquire it whether it was true or not; moreover I would know that I could acquire it whether it was true or not. If in full consciousness I could will to acquire a “belief” irrespective of its truth, it is unclear that before the event I could seriously think of it as a belief, i.e. as something purporting to represent reality.¹⁸

          The idea is this, to be able to have doxastic control, I must know that I have that control, and if I have that control then I know that I can form beliefs with no consideration to their truth. If I can do that, then I can form beliefs that I know to be false, and that is hardly consistent to what we mean when we talk about beliefs. I cannot force myself to believe things I already know to be false, so I can not form beliefs irrespective of the truth. Knowing the truth of a matter limits what it is possible for us to believe.

          That is only a limitation on beliefs where knowledge can be rightly claimed, where empirical evidence or rationality settles the matter. One could not claim given the definition of “2”, “4”, “+”, “=”, that the expression “2+2=4” is something we can, at will, form a belief about, because the definitions of the terms entail that it is a true statement. One only needs to do the math to confirm this. In a similar matter, one only needs to look into a room to see if the lights are on or off to know the truth about the status of the lights.

The realm of potentially will-formed beliefs is pushed farther into the darkness of ignorance, like that of the truth of the existence of God at the extremity of infinite distance. Although there are more practical, daily beliefs that might be examined. If I wish to ensure the truth of my belief that the lights in my office are on, I only need to turn the lights on.¹⁹ While this might seem philosophically uninteresting, it may be a fairly common way in which we act. “Consider, for example, the countless number of people who exercise regularly and carefully monitor their diets in order to avoid being overweight. … [I]f one is overweight, then it makes it considerably more difficult to avoid believing that one is overweight.”²⁰  By directly altering the state of affairs to reflect our desired beliefs, we can change them to make them true. That does not mean that we are justified in holding the belief until the affairs resemble our desired beliefs. It also does not apply to anything that is outside of our immediate control.

          Take Thomas Kelly's example of a “nervous parent, unable to sleep because he or she does not believe that his or her child has arrived safely.”²¹ Surely, this parent desires the truth of the child's safety, in Bernard Williams' meaning of having a truth-centered motive to believe, a desire that it is true that the child is safe.²² At the same time, choosing to believe without sufficient evidence seems like a mistake as Clifford's admonishment decries. Kelly points out that parent does not simply push off these concerns, but attempts to intentionally acquire evidence for the truth of the desired belief by making a phone call.²³

          Seeking evidence for a truth-centered desired belief seems not only plausible, but also a fairly common strategy for forming beliefs where there is a current ignorance of the truth. However, this is not to say that a truth-centered desire is sufficient to be epistemically justified in holding a belief. One must find the evidence that actually does support the belief, and that is only possible if the world is so arranged that the evidence does exist and one has access to it.²⁴

          It is not that practical consideration do not plays a part of the causal history of a belief, for we might very well have reasons that start a process of investigation, but that practical reasons do not ground the belief itself.²⁵ This is because when we arrive at a point where we are able to ask ourselves “whether to believe that p”, the question collapses to “is p true” with no intermediate step, which is to say that one cannot settle the first question without also believing that they've answered the second.²⁶ Whatever rational project leads to asking that question still reaches that question. At that point, regardless of the motivation to ask, we are only left with trying to determine the truth of a statement.

          So, if we engage in a Pascalian project to believe by acting in ways that aid in our believing, we inevitably lead to asking if the desired belief is true. No matter how badly we desire, because of the strong belief-truth conceptual connection, unless we can change our environment to match the belief, we will be forced to take stock of evidence that we have for that belief. On that evidence alone will we be able to answer both “should I believe” and “is it true” because they are equivalent questions.
          If we were able to somehow do otherwise and form a belief without evidence, we'd have the type of doxastic control necessary to form beliefs at will, but that would mean that we could form beliefs that we knew were false. However, it is not possible to hold a belief that is, at the same time, known to be false. Once again, we are left with only the ability to examine the evidence to see if a potential belief can be grounded by truth.

          So, applied to Pascal's wager, one might very well be convinced that one should believe in God, that the expected utility is indeed higher if one believes and God exists, but it is not clear that we could have the type of control to will ourselves to believe. At best one could start a project of gathering the right sort of supporting evidence. It is not just that beliefs aim at the truth, it is that questions about what to believe collapse into questions about the truth, which can only be settled by evidence. Because we do not have voluntary control of over what sort of evidence exists in the world, we cannot have voluntary control over what types of beliefs we can form.

Pragmatic Considerations

          Let us turn back to the claims about pragmatic reasons to believe. It is possible to imagine a situation where it might be overwhelmingly advantageous to hold a particular belief. For the clearest example, imagine a situation where both the belief and the evidence for that belief seem to be fully under one's own control, like in Gregory Kavka's The Toxin Puzzle. You've been given the opportunity to win a significant amount of money from a billionaire.

He places before you a vial of toxin that, if you drink it, will make you painfully ill for a day, but will not threaten your life or have any lasting effect. … The billionaire will pay you one million dollars tomorrow morning if, at midnight tonight, you intend to drink the toxin tomorrow afternoon. He emphasizes that you need not drink the toxin to receive the money; in fact, the money will already be in your bank account hours before the time for drinking it arrives, if you succeed. … All you have to do is sign the agreement and then intend at midnight tonight to drink the stuff tomorrow afternoon. You are perfectly free to change your mind after receiving the money and not drink the toxin.²⁷

          In this puzzle you clearly have control of your actions, so you have control over if you're going to drink the toxin. All the evidence that might exist as to the truth of believing that you will drink the toxin (the intention to drink) you create, in a similar fashion as one might turn on a light in a room to believe the lights are on. The pragmatic reasons for forming the belief are quite compelling, and the expected gain of drinking seems high enough to rationalize the intention to drink the toxin.

	Drink the toxin	Not drink the toxin
Form the intention	Sick for a day and $1 million	$1 million
Not form the intention	Sick for a day	Status quo

          The option of forming the intention to drink is clearly the only way that you could gain any benefit from this experience. Forming the intention superdominates not forming the intention, and so it is rational to desire to form the intention. However, the money will either be in your account in the morning, or it won't be.²⁸

	The money is in the account	The money is not
Drink the toxin	Sick for a day and $1 million	Sick for a day
Not drink the toxin	$1 million	Status quo

          The problem is that when you realize that you will know that the money is in your account before you drink the toxin, the superdominance shifts toward not drinking the toxin. Drinking the toxin if the money is in the account is unnecessary therefore irrational, and more so if the money is not there. It seems that there is no way to rationally drink the toxin, making forming the intention/belief that you will drink irrational as well.
So, you have reasons to form the belief that you will, and reasons to not follow through, especially since you will either have or not have the money before you drink; having no reason to drink the toxin is inconsistent with you forming the intention that you will drink it.²⁹ What this demonstrates is that there is a difference between reasons for acting and reasons for intending/believing.³⁰

          Pascal may have well given us the strongest possible reasons for attempting to form a belief, the intention to believe. Infinite rewards are clearly the greatest utility possible. Because those are pragmatic reasons for action but not evidence of the truth, he did not give us reasons to actually believe. The type of motivation Pascal has supplied, being pragmatic reasons, clearly do have a role in the causal history of the belief.

[T]he expected consequences of holding the belief might make a difference to whether it is rational to hold it, simply because its epistemic status might be historically dependent on the expected consequences of its being held. … However, those philosophers who have defended the idea that the expected consequences of a belief can make a difference to its rationality have meant something considerably stronger than this.³¹

          They meant that practical considerations can rationalize a belief even when they have no impact on epistemic considerations, they can produce irrational beliefs.³² This seems less plausible if we hold that beliefs aim at truth. Recognizing that it would be beneficial to hold a given belief could provide reasons to attempt to gain the belief, but the gaining of the belief would be dependent on gaining enough evidence, and not on the pragmatic reasons.³³ If someone were to be convinced by Pascal, and was successful in forming a belief in God it would not be because the considerations of expected utility.³⁴ Just as it isn't clear that you could form the intent to drink the toxin based on the expected utility, any conversion or belief formation will be based on evidence of the truth because belief aims at truth.

          To separate this a bit further, consider two agnostics convinced of the expected utility of believing in God by Pascal. If one is successful while the other continues the effort, and they both discover that the actual utility is vastly different than the expected, and it was now in their favor to not hold the belief, the still-agnostic would only need to stop the effort, but the now-believer would have to undertake another effort to reverse the belief. ³⁵ The reason for the difference is that the now-believer is no longer under the sway of an expected utility argument, but the still-agnostic is. The believer has formed a belief which requires further actions to change, but the agnostic does not and a mere shift in the expected utility can cause her to give up the actions she is taking.³⁶ Belief is grounded on something more substantial than a pragmatic consideration, namely truth via evidence.

Conclusion

          Pascal may have provided us with a strong argument that if God exists and rewards people that wager that is so, then the only rational action is to make the wager for God. If by “wagering” Pascal intended that we should form a rational belief, it seems that his argument is irrelevant to us. Because our understanding of truth requires that any question of whether we should believe, is identical to whether the belief is true, and that truth is grounded on evidence. If we lack evidence then we can not rationally form a belief. The nature of belief requires that we cannot knowingly believe false things, and because of that we cannot knowingly choose to believe at will. We simply lack the doxastic voluntarism to will ourselves to believe. While pragmatic considerations, like expected utility, do have a place in the causal history of a belief, it is a mistake to conflate reasons for actions with reasons for beliefs and intentions. So, pragmatic concerns give us reasons to inquire, but not means to rationally believe, frustrating Pascal's intentions when he first opened the box.

________________________________________________

1 Hájek 2012

2 Hájek 2012, §2-4.

3 Hájek 2012, §2.

4 Hájek 2012, §3.

5 Hájek 2012, §4.

6 Hájek 2012, §1.

7 Hájek 2012, §2.

8 Hájek 2012, §2.

9 Hájek 2012, §3.

10 Hájek 2012, §4.

11 Clifford 1877, 10.

12 Clifford 1877, 3.

13 Clifford 1877, 3.

14 Clifford 1877, 4.

15 Clifford 1877, 1.

16 Kelly 2002, 6.

17 Williams 1973, 148.

18 Williams 1973, 148.

19 Feldman 2000, 671.

20 Kelly 2002, 7.

21 Kelly 2002, 9.

22 Williams 1973, 150.

23 Kelly 2002, 9.

24 Kelly 2002, 10-11.

25 Kelly 2002, 15.

26 Shah 2003, 447.

27 Kavka 1983, 33-4.

28 Kavka 1983, 34.

29 Kavka 1983, 34-6.

30 Kavka 1983, 35-6.

31 Kelly 2002, 12.

32 Kelly 2002, 12-3.

33 Kelly 2002, 15.

34 Kelly 2002, 15.

35 Kelly 2002, 18-9.

36 Kelly 2002, 18-9.

Bibliography

Clifford, William. 1877. “The Ethics of Belief.” Lectures and Essays. Ed. Leslie Stephen and Frederick Pollock. 1886. London: Macmillan and Co.

Feldman, Richard. 2000. “The Ethics of Belief.” Philosophy and Phenomenological Research. International Phenomenological Society. Vol. 60, No. 3 (May, 2000), 667-695.

Hájek, Alan. 2012. “Pascal's Wager.” Standford Encylopedia of Philosophy. Stanford: Stanford University. Winter 2012.

Kavka, Gregory. 1983. “The Toxin Puzzle.” Analysis. Oxford: Oxford University Press. Vol. 43, No. 1 (Jan. 1983). 33-36.

Kelly, Thomas. 2002. “The Rationality of Belief and Some Other Propositional Attitudes.” Philosophical Studies 110: 163-196.

Shah, Nishi. 2003. “How Truth Governs Belief.” The Philosophical Review. Durham, NC: Duke University Press. Vol. 112, No. 4 (October 2003) 447-482.

Williams, Bernard. 1973. “Deciding to Believe.” Problems of the Self. Cambridge: Cambridge University Press. 136-151.