Michael Martin Wager

[Editor’s Note: This article was originally published in Religious Studies 19 (1983): 57-64. The page numbers below show the position of the text within that pagination scheme.]

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Pascal’s Wager as an Argument for Not Believing in God (1984)

Michael Martin

Can Pascal’s wager for the existence of God be turned against the religious believer and used as an argument for not believing in God? Although such an argument has been very briefly sketched by others[1] its details have remained undeveloped. In this paper this argument is worked out in detail in the context of decision theory and is defended against objections. The result is a plausible argument for atheism.

THE WAGER IN SIMPLE FORM

The Pascalian argument can be put briefly in this way: If one believes in God and God exists, then one gains infinite bliss after death. If, on the other hand, one believes in God and God does not exist, one has lost very little. However, if one does not believe in God, and God does exist, one receives infinite torment in Hell after death. But if he does not exist and one does not believe in him, one has gained very little. Clearly one has infinity to gain and little to lose by believing, and infinity to lose and little to gain by not believing. Hence one should believe that God exists.[2]

It is important to have a clear understanding of the status of Pascal’s argument. His argument purports to give good reasons for believing that God exists; in particular it purports to provide good reasons for a change of belief horn agnosticism or atheism to Christian theism. However, the reasons that the argument purports to give are not of the usual sort. These reasons do not make the existence of God any more likely or probable.

Let us call a reason for believing that something is the case because it is probable or likely an evidential reason. Pascal’s argument does not provide an evidential reason for believing in God. Rather Pascal’s argument purports to show that it is beneficial to believe in God and because of this one should


[1] See Anthony Flew, God and Philosophy (London: Hutchinson, 1966), chapter 9. This argument is alluded to by R. G. Swinburne in ‘The Christian Wager’, Religious Studies IV (1969), 217-28.

[2] As many scholars have pointed out, Pascal was not so naive as to suppose that one could believe in God by an act of will. However, he did think that belief could be developed by acting in certain religious ways, for example by attending mass and taking the sacraments. So the two choices of the non-believer were really to act or not to act in religious ways. See Ian Hacking, ‘The Logic of Pascal’s Wager’, APQ IX (l971), 186-92; James Cargile, ‘Pascal’s Wager’, Philosophy XLI (1966), 250-7; Michael Martin, ‘On Four Critiques of Pascal’s Wager’, Sophia XIV (1975), 1-11.

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believe in him. Let us call such a reason a beneficial reason. As we shall see in a moment, Pascal’s argument can be construed as purporting to show that practically all evidential reasons for not believing in God are outweighed by beneficial reasons for believing in God.

THE WAGER AS A PROBLEM IN DECISION THEORY

It is helpful for our purposes to interpret Pascal’s wager as a problem in decision theory.[1] Let us suppose for the moment that we do not know the probability of God’s existence. So construed the argument can be put in the following matrix:


  God exists God does not exist
Believe in God Infinite utility Finite disutility
Don’t believe in God Infinite disutility Finite utility

In this form Pascal’s wager is a problem in decision making ‘under uncertainty’, a decision in which the outcome is uncertain and the probabilities of the outcome are unknown. Using either of the two well-known rules of decision making under uncertainty–the maximax or minimax rule–one should believe in God.[2] On the maximax rule one should choose that course of action with an outcome having the most value; on the minimax rule one should pick that course of action with an outcome having the least disvalue.)

Let us suppose now that although one does not know the exact probability, one knows that God’s existence has some finite probability p, however small. the problem assumes that God’s existence is not logically impossible.) Then the problem can be construed in decision theoretical terms as a decision problem ‘under risk’ and can be formulated in the following matrix (assuming x and y are finite values):


  God exists God does not exist
Believe in God ∞ × p -x × (1 – p)
Don’t believe in God -∞ × p y × (1 – p)

In this case one would follow the famous Bayesian decision rule: Follow that course of action with the most expected value (value × probability). However, since ∞ × p + (- x × (1 – p)) = ∞ the expected value for believing


[1] For a more detailed discussion of Pascal’s wager from a decision-theoretical standpoint see Hacking, op. cit..

[2] For a discussion of these and other rules see D.C. Luce and H. Raiffa, Games and Decisions (New York: John Wiley, 1957), chapters 1-4.

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in God is ∞. Furthermore, since ( – ∞ × p) + (y × (1 – p)) = -∞ the expected value for not believing in God is -∞.[1]

Construed in the terms set forth above there is a stronger beneficial reason for the belief that God exists than for the belief that God does not exist in both decision making under risk and decision making under uncertainty, despite the fact that the evidential reason for believing in God may be less strong than the evidential reason for not believing in God. In decision making under risk a higher value is associated with belief than non-belief. In this case no probabilities (evidential reasons) are involved; the higher value provides beneficial reasons for the belief that God exists. In decision making under uncertainty there is a higher expected value associated with belief than non-belief. (This value provides the beneficial reason.) This is true despite the fact that the probability that God exists (the evidential reason) may be much lower than the probability that God does not exist.

THE WAGER REFUTED

The basic trouble with Pascal’s wager is that there are other possibilities that Pascal did not consider.[2] There are more possibilities than that God exists or does not exist.[3] Consider the following possibility. Suppose there is a Supernatural Being–we will call him the Perverse Master–who punishes with infinite torment after death anyone who believes in God or any other Supernatural Being (including himself) and rewards anyone who believes in no Supernatural Being with infinite bliss after death. One assumes that since such a Being is not logically impossible, his existence is finitely probable. Put in a matrix form as a problem of decision making under risk it would look like this (assuming p1 + p2 + p3 = 1 ):


  God exists The Perverse Master exists Neither exists
Believe that God exists ∞ × p1 -∞ × p2 zW × p3
Believe that the Perverse Master exists -∞ × p1 -∞ × p2 x × p3
Do not believe that either exists -∞ × p1 ∞ × p2 y × p3

[1] Clearly then the view is mistaken that one must assume that the probability is 1/2 that God exists and 1/2 that he does not in order to make the argument work. For this mistaken criticism see Monroe Beardsley and Elizabeth Bearsdley, Philosophical Thinking: An Introduction (New York: Harcourt, Brace, and World, p. 140). Hacking (op. cit, p. 189) claims that Pascal is committed to this assumption in one formulation of his argument. However, this assumption is not made in the version of the argument considered here.

[2] See Hacking, op. cit.; Cargile, op. cit.

[3] Whether there is infinite number of possibilities, as Flew has maintained, is another question. In any case an infinite number of possibilities is not needed to refute the argument, although such an assumption is useful in providing reasons for non-belief. See Anthony Flew ‘Is Pascal’s Wager the Only Sage Bet?” Rationalist Annual (1960), pp. 21-5.

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Now construed in this way, believing that the Perverse Master exists would be the worst choice. The expected value would be -∞. Belief in God would be the next best. The expected value would be –z × p3 (where –z is some finite disutility and p3 is some finite probability). But believing in neither would be the best choice since the expected value would be y + p3, (where y is some finite utility and p3, some finite probability).

It should be noted that belief in the Perverse Master would be no worse or better than belief in God given the possibility that another Supernatural Being exists who gives infinite reward for belief in the Perverse Master and himself and no rewards for anything else. Let us call such a being the Anti-Perverse Master. The matrix will be (assuming p1 + p2 + p3 + p4 = 1 ):


  God exists PM exists Anti-PM exists Neither exists
Believe that God exists ∞ × p1 -∞ × p2 0 -y × p4
Believe that the Perverse Master exists -∞ × p1 -∞ × p2 ∞ × p3 -x × p4
Belief that the anti-Perverse Master exists -∞ × p1 -∞ × p2 ∞ × p3 -w × p4
Do not believe that either God or the PM or the Anti-PM exists -∞ × p1 ∞ × p2 0 z × p4

In this case, all infinite expected values would cancel each other out and the choice would turn on the finite expected values in the last column. Since all of these values are negative except belief in none of the possible Supernatural Beings, the best choice is belief in none of the Supernatural Being.

The Argument Generalized

It should be clear that these considerations not only refute Pascal’s argument but provide reasons for believing in neither God, the Perverse Master nor the Anti-Perverse Master. The general strategy in my argument can be generalized and the result remains the same. No matter what other logical possibilities one can conceive of in terms of a Supernatural Being with infinite rewards and punishments, another Supernatural Being can be conceived with infinite rewards and punishments that tend to cancel out the rewards and punishments of the other

Thus, suppose it to be possible that there is a Supernatural Being who gives an infinite reward to everyone after death no matter what they believe. Then, presumably there is also the possibility of a Supernatural Being who gives an infinite punishment to everyone after death no matter what they believe.

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Put in matrix form, the infinite rewards and punishments cancel each other out and the decision turns on the finite utilities involved when no Supernatural Being exists. Thus, given all the possibilities for various Supernatural beings with infinite rewards and punishments and the canceling out effect,, the matrix reduces to this form:


  Either Supernatural

Being exists[1] or

Supernatural Being2

exists or …

or Supernatural

Beingn exists
No Supernatural

Being exists
Believe that Supernatural Being1 exists 0 x1 × p
Believe that Supernatural Being2 exists 0 x2 × p
.

.

.

.

.

.

.

.

.

Believe that Supernatural Beingn exists 0 xn × p
Do not believe that any Supernatural Being exists 0 y × p

The result is that for any matrix in which the expected value in belief in some Supernatural Being is infinite there exists a more inclusive matrix in which the infinite values cancel out. In this more inclusive matrix, the only values that remain are the finite values involved in believing or not believing if Supernatural Beings do not exist. But in this case, non-belief has the greatest utility. Consequently, in terms of beneficial reasons one should not believe that God exists or, indeed, that any other Supernatural Being exists.[1]

OBJECTIONS ANSWERED

It may be objected that I assume without argument that on the assumption that no Supernatural Beings exist it is better not to believe that any Supernatural Being exists than that some Supernatural Being does exist. However, it may be argued that this is not obvious. For it may be the case that some people would be better off believing that God exists even if he does not. Such belief might, for example, give them hope in time of crisis.


[1] The above argument will not work in exactly the same way against belief in Supernatural Beings with finite rewards and punishments. However, with slight modification the argument can proceed as before. Space does not permit me to show this extension here.

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First of all, if we take the utilities as given in the original argument, this objection does not hold. For as Pascal constructs the problem, there is more to be lost than gained in believing in God if he does not exist. The extension of Pascal’s argument to other Supernatural Beings would not change matters. Further, the utilities involved in believing in some possible Supernatural Beings seem clearly negative. Suppose there is a Supernatural Being who is believed to harm and not help people in time of crisis. Belief in such a Being in time of crisis would be cold comfort. At best, belief in some Supernatural Beings, e.g. the Christian God, would give comfort. But this comfort, considered as a positive value, would have to be weighed against negative values, e.g. the effort and trouble involved in worshipping this Being.[1] It is certainly not obvious that Pascal was wrong to suppose that this belief in a Supernatural Being (e.g. God) has a negative value if this Being does not exist when the negative values of belief are weighed against the positive values of belief.

Second, Pascal’s construal aside, one may grant for the sake of the argument that belief in a Supernatural Being has some positive utility if this Being does not exist. The crucial question is whether the positive utilities of not believing in any Supernatural Being outweigh this value in case no Supernatural Being exists. Put in the terms we introduced earlier: Are there better beneficial reasons for believing that no Supernatural Beings exist, if none do exist, than for believing that some do exist If none do exist? The answer, I believe, is yes.

( I ) There are practical values in not believing. As atheists and humanists have often pointed out, belief that God and other Supernatural Beings do not exist puts responsibility for humanity’s problems on humans, forcing them to come to grips with their own problems.

(2) There are psychological values in not believing. People’s hope that God or some other Supernatural Being will help them, if these Beings do not exist, is immature and childish.

(3) There are epistemic values in not believing in Supernatural Beings, if these beings do not exist. There is an epistemic disvalue in believing a falsehood.[2]

(4) Although belief in a Supernatural Being, if none exists, may give some comfort in time of crisis, if no Supernatural Beings exist, such comfort must be short-lived for perceptive believers. Their hopes will be frustrated, their expectations disappointed. These believers must be unhappy and feel forsaken.


[1] When worship is translated into actual religious practice, worship can be time consuming, expensive, and troublesome. The long hours spent in religious rituals, the extreme penances required by some sects, the tithe required by some churches, and the asceticism required in others are some of the more obvious examples of the negative value of religious worship.

[2] For a discussion of epistemic values in decision theory see Isaac Levy, Gambling with Truth (Nor York: Alfred Knopf. 1967): Carl Hempel, ‘Inductive Inconsistencies’, Aspects of Scientific Explanation (New York: Free Press. 1965) esp. pp. 73-9.

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But what about a Supernatural Being who people believe does not help them in time of crisis but who in fact wants his creatures to be self-reliant? Belief in this Being would not have the practical or psychological disvalue mentioned above. But it would still be an epistemic disvalue to believe in such a Being if he did not exist. Further, there would be no comfort gained by relying on such a Being in a time of crisis. The advantage would still be with the non-believer.

It might also be argued that the expected value of believing that no Supernatural Being does exist would be very small given the very low probability that no Supernatural Being exists. But in the first place, whether the expected value is low or not, the crucial point is only that the expected value of not believing is higher than the expected value of believing . This is all that is needed to have a beneficial reason for not believing. In the second place, as I have argued elsewhere,[1] there is good inductive reason to believe that God does not exist. So the probability that God does not exist is by no means small. This probability, combined with the sort of values sketched above, yields moderate expected value. Thus it would provide good beneficial reason for not believing in God.

Whether there is good inductive reason for believing in no Supernatural Being is another issue, one that is impossible to pursue here in detail. But I believe there is some reason to suppose that there is. The hypothesis that there are no Supernatural Beings is a better explanation of the available evidence than its negation. Consequently, it is inductively a better-justified hypothesis than its negation.[2]

Another objection that may be raised is this. We have shown that given any matrix that specifies finite or infinite expected values associated with the existence of a Supernatural Being A, a more inclusive matrix can be constructed which postulates a further Supernatural Being B with certain values associated with the existence of B that cancels the expected values associated with A. But it may be maintained that the above argument cuts both ways. A still more inclusive matrix can be postulated with a Supernatural Being C where not all values are cancelled. Thus the victory of the non-believer is short-lived.

There are two responses to this objection. First, if one admits a potential


[1] Michael Martin, ‘Is Evil Evidence Against The Existence of God?’ Mind (1978), pp. 429- 32.

[2] There seem to me to be three main lines of argument to show that the hypothesis that no Supernatural Beings exist is a better explanation of the available evidence that the hypothesis that some Supernatural Beings exist. First, other things being equal, hypotheses that postulate Supernatural Beings are less simple than hypotheses that do not. Second, supernatural hypotheses tend to be less testable than naturalistic hypotheses. Third, from a historical point of view, hypotheses postulating supernatural entities have in realm after realm been replaced by hypotheses that don’t; on inductive grounds one can expect this replacement to continue. Consequently the hypothesis that no Supernatural Beings exist is likely to be a better explanation of the available evidence than the hypothesis that some Supernatural Beings do exist; consequently the hypothesis that no Supernatural Beings exist is likely to be inductively better justified than its negation. See Gilbert Harmon, ‘Inference to the Best Explanation’, Philosophical Review LXXIV, 1965.

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infinite of possible Supernatural Beings, the above manoeuvre would provide no comfort for the believer. His or her advantage would be cancelled out in the next step by the non-believer. It does seem possible to go on forever constructing larger and larger matrices in which values that are associated with possible Supernatural beings are continually cancelled by values associated with other possible Supernatural Beings. In such a potentially infinite matrix the expected value of believing in some Supernatural Being would depend on the expected value associated with there being no Supernatural Being since all other values would cancel out over the long run.

Second, even if the potential infinity of possible Supernatural Beings is denied, the argument still has a dialectic point. Any Pascalian type argument can be used dialectically to establish the superiority of non-belief if the non-believer gets the believer to admit the existence of a possible Supernatural Being with certain expected values associated with the existence of this Being. The believer is trapped by his or her own argument. If the believer has the ingenuity to come back with a further move, this ingenious non-believer can counter if he or she gets the believer to admit the possibility of still another Supernatural Being with the requisite expected value associated with his existence. The argument in this dialectic model is ‘won’ by the person who fails to convince the other that a particular Supernatural Being is finitely probable.

Although this dialectic use of the argument by the believer is possible, it is hard to see that it would have much use except as a way to block the non-believer’s arguments. The believer is interested in giving beneficial reasons for believing in one particular Supernatural Being (most notably God). Postulating some strange Supernatural Being to counter the nonbeliever’s argument would have little appeal.