On Rescher On Pascal's Wager (1990)

Graham Oppy

 

[This article was originally published as "On Rescher On PascalÕs Wager", International Journal For Philosophy Of Religion, 30, 1990, pp.159-168]

In Pascal's Wager: A Study Of Practical Reasoning In Philosophical Theology,[1] Nicholas Rescher aims to show that, contrary to received philosophical opinion, Pascal's Wager argument is "the vehicle of a fruitful and valuable insight--one which not only represents a milestone in the development of an historically important tradition of thought but can still be seen as making an instructive contribution to philosophical theology".[2] In particular, Rescher argues that one only needs to adopt a correct perspective in order to see that Pascal's Wager argument is a good argument. Moreover, there seems to be a certain amount of contemporary support for Rescher's claim that Pascal's Wager argument can be seen to be a good argument when properly construed.[3] However, despite this recent trend to adopt a more sympathetic stance towards Pascal's Wager argument, I propose to defend the traditional view that Pascal's Wager argument is almost entirely worthless--at least from the theological standpoint. (No doubt, it has historical significance from the standpoint of decision theory; but that's a separate matter.)

This paper is divided into two sections. I begin, in section I, by outlining the defence of Pascal's Wager argument which is given by Rescher in Pascal's Wager. Then, in section II, I explain why this defence fails.

I

As Rescher emphasises, Pascal's Wager argument unfolds against a background of scepticism about the powers of theoretical reason. "Pascal holds, with the Renaissance skeptics, that our human resources for securing knowledge by inquiring reason are wholly inadequate to the demands of apologetics".[4] Indeed--and this is a point which will be important later on--Rescher is at pains to note that Pascal insists that theoretical reason is "utterly impotent" in the context of the assessment of the question of God's existence. Consequently, the Wager argument is forced to appeal to a further species of "reasons", viz: practical reasons.

According to Rescher, the core of Pascal's Wager argument--at least from a logical point of view--lies in decision theory (i.e. in the theory of decision-making in situations of uncertainty about outcomes). We can illuminate this basic argument by referring to the following table:

 

Options


Returns to the chooser



God does not exist


God Exists



(Prob=1-p)


(Prob=p)


Bet on God


-B


(-B)


Bet against God


0


little or nothing
(perhap negative)


The idea is that, in deciding whether or not to bet on God--i.e. in deciding whether or not to believe in God--what we do is to consider the expected return for each of these options. Moreover, we suppose: (i) that the cost of making the bet is B; (ii) that the subjective probability which we assign to the claim that God exists is p; and (iii) that the returns on our decision are as outlined in the above table. (Note that, if we bet on God, and it turns out that God exists, then our return is infinite--for the value of -B is just .) But now we note that: (a) the expected return of a bet on God is .p + (-B).(1-p) which is provided that p is finite; and (b) that the expected return of a bet against God is (little or nothing).p + 0.(1-p) which is little or nothing (and quite likely negative). So--according to Pascal's Wager argument--it is reasonable to conclude that our expected return is infinite in the case that we bet on God, and little or nothing in the case that we do not bet on God; and hence it is reasonable to conclude that one ought to bet on God.

Now, of course, there are a number of assumptions which are required by this argument. It is immediately apparent that, in order to reach the conclusion that one ought to bet on God, one needs to defend the following claims:

 

1. It is reasonable to suppose that the value of a bet on God, in the case in which God exists, is .
2. It is reasonable to assign a finite value to p (i.e. to the probability that God exists).
3. It is reasonable to suppose that there are only two options between which one is to choose, viz: (i) a universe which includes the traditional Christian God; and (ii) a universe without deities (or transcendent beings) of any sort.
4. It is reasonable always to act so as to maximise expected utility.
5. It is reasonable to suppose that there are no theoretical considerations which can decide the question whether God exists.

Consequently, it is plain that there are a number of types of persons who will be untouched by this argument. Among these--as Rescher notes--are the following: (i) convinced atheists, who set the value of p to zero; (ii) those, such as out-and-out hedonists who live for the pleasure of the moment alone, who set the value of future benefits to zero; (iii) those, such as all-trusting disbelievers who hold that if God did exist He would be bound to be all-forgiving, who maintain that one would get the future benefits whether or not one believed; (iv) radical sceptics who hold that both knowledge and reasonable conviction are impossible; (v) non-calculating sceptics who, for one reason or another, are not prepared to perform the sort of self-interested calculation which the argument requires; and (vi) non-Christian theists (e.g. Zoroastrians) who have very different ideas about the rewards of belief.[5]

However, Rescher does not suppose that the fact that the argument will not touch some people points to a defect in the argument. Indeed, it is one of the main themes of Rescher's book that Pascal's Wager has been persistently misunderstood by philosophers who (he claims) have supposed that the argument is meant to provide reasons which would persuade any rational person that s/he ought to believe in God. Not so, according to Rescher. Rather, what the argument is intended to do--in Rescher's view--is to persuade any person who thinks that there is a finite probability that the Christian God exists, who holds "standard views" about the nature of that God, and who is always prepared to make decisions on the basis of considerations of maximum expected utility: "Pascal's discussion is directed at l'homme moyen sensuel, the ordinary, self-centred, "man of the world" preoccupied with his own well-being and his own prudential interests. Pascal does not address the already converted, but the glib worldly cynic--the free-thinking libertin of his day, the sort of persons who populated the social circle in which Pascal himself moved prior to his conversion."[6]

So, Rescher's assessment of the worth of Pascal's Wager argument is this: The argument is intended to be directed towards people who already hold certain assumptions, and to convince these people that they ought to believe in God. Moreover, given this intention, it is clear that the argument is a good one--i.e. it is clear that people who hold the assumptions in question ought to be persuaded by the argument to believe in God.

II

There is a logical (or perhaps mathematical) error in Rescher's version of Pascal's Wager argument which--on the face of it--completely invalidates the argument. This error stems from the fact that Rescher throughout supposes that there are only two types of values which the probability of the proposition God exists can take: zero, or finite.

(To see that Rescher does repeatedly make this assumption, consider the following extracts from his book:

All that matters for [Pascal's] reasoning is that [the probability of God's existence] is non-zero. As long as there is a finite chance of God's existence--no matter how small--the expectation of the "bet-and-believe" alternative outweighs that of its rival.[7]

Pascal's reasoning is only in a position to persuade someone who believes that God may exist.[8]

A gamble is advantageous on the basis of [the] expected-value standard whenever:

 

chance of winning



cost of stake



potential loss



>



=



chance of losing



size of prize



potential gain


And if the potential gain is infinite, this standard favours the gamble as long as the chance of winning is non-zero.[9]

[The argument] will certainly fail to touch the convinced atheist. Someone who sets the probability of God's existence at zero will obviously not arrive at the argument's conclusion.[10]

Pascal's argument is simply addressed to those who see the existence of the Christian God as a real possibility to which they are prepared to accord a nonzero probability (however "imponderable" they may deem this quantity to be in other regards). Apart from this the numerical status of this probability--even its having a definite and stable value--is quite irrelevant.[11]

Everything turns on what we are (responsibly) prepared to deem to be real possibilities, those having probabilities at any rate greater than zero (no matter how low we may think them to be).[12]

It is obvious from these remarks that Rescher supposes that either the probability that God exists is zero (in which case it is impossible that God exists), or else that the probability that God exists is finite.)

However, it is incorrect to suppose that "non-zero" and "finite" are co-extensive; for it is epistemically possible that the probability that God exists is infinitesimal. Moreover, when this is recognised, it is also immediately apparent that the calculation of the expected return of a bet on God is no longer as straightforward as the initial argument suggested. For, of course, the value of .p + (-B).(1-p) can take any value greater than (-B).(1-p) depending upon the exact values of and p.[13] And so, plainly, the argument from expected utility does not go through in all cases; it depends upon the particular values of and p which are involved.[14]

Now, to this, it may be objected that the essence of Rescher's version of Pascal's Wager argument is untouched. True, Rescher does overlook the fact that "non-zero" and "finite" are not coextensive--but all this shows is that the class of people who will be untouched by the argument must be extended to include: (vii) cautious sceptics who are not prepared to assign a finite value to the probabilty that the Christian God exists. That is, can't Rescher reply that--although his own presentation of the argument is slightly flawed--it is nonetheless true that the argument can still succeed in achieving its Apologetic aim? After all, if we grant that the piece of reasoning which we presented initially is cogent, then it is still true that we must grant that anyone who accepts the assumptions 1.-5. listed above is required to accept the conclusion that God exists.

This reply will not do. The claim that Pascal's motives are entirely Apologetic--i.e. that he only seeks to address people who already hold certain views--is plainly not as significant as Rescher supposes. For, while it might be true that "Pascal's Wager" is best interpretted as a sort of consistency argument ("if you believe those things, then you must believe in God, on pain of inconsistency"), it is clear that what one of Pascal's targets ought to do, on learning of his/her (apparent) inconsistency, is to examine all of the beliefs which s/he has which are relevant to the generation of the inconsistency. But then s/he will be faced with the question of whether s/he ought to accept the (apparently) dubitable assumptions which are central to the argument. Perhaps we can grant that anyone who accepts the assumptions 1.-5. is required to accept the conclusion that God exists--but ought anyone to accept all of these assumptions?

(Here is a parallel case: Suppose that I know someone who (somewhat confusedly) believes (i) that if there is no God, then morality has no objective foundation; (ii) that morality has an objective foundation; and (iii) that it is quite unclear whether or not there is a God. I can point out to this person that it follows logically from (i) and (ii) that there is a God--but, of course, it doesn't follow from this that they have good reason to believe that there is a God! For it might be that they have very good independent reasons for believing (iii), and very weak reasons for believing (i) or (ii) (or both). The most that a consistency argument can do is to show that I need to revise some of my beliefs--but it alone cannot tell me which beliefs need to be adjusted.)

The point which I have just made can be summarised in the claim that it is important to distinguish two senses in which Rescher could be taken to be saying that Pascal's Wager is successful. (i) On the one hand, it could be that Rescher is saying that the argument is successful because it shows that any persons who accept 1.-5. ought to believe in the Christian God, on pain of logical contradiction. But, on this construal, there is the underlying assumption that persons who accept 1.-5., and who learn that they are thus committed to belief in the existence of God, won't then choose to modify themselves in such a way that they no longer accept one (or more) of 1.-5. (ii) On the other hand, it could be that Rescher is saying that the argument is successful because it shows that any persons who accept 1.-5. ought to believe in the Christian God, because that is overall the most rational thing for them to do. And, on this construal, it would no longer be true that there is an underlying assumption that persons who accept 1.-5. won't choose to modify themselves in such a way that they no longer accept one (or more) of 1.-5.

Perhaps is can be granted that the Wager argument is a successful piece of Apologetic argumentation in sense (i). (Of course, it is crucial to this concession that the people to whom it is directed assign a finite probability to the claim that God exists.) However, it is important to note how weak this claim is--for it says nothing about the overall rationality of the situation in which these people are placed. Surely l'homme moyen sensual ought, upon being informed of Pascal's argument and the nature of his own beliefs, to ask whether it is reasonable to continue to accept each of 1.-5. But if it is quite clear that it is not at all rational to subscribe to all of 1.-5., then it is hard to see that there is any great merit in the argument, even as an instrument of Apologetics. (What's so great about getting very stupid people to believe in God? It's probably easier to do that by telling them stories about hellfire and damnation.) What would be an achievement would be to produce an argument which was successful in sense (ii)--but it remains to be argued that Pascal's Wager argument even comes close to achieving this.

So far, then, I have suggested that Rescher's appeal to the nature of the Apologetic task for which Pascal's Wager argument is designed does not manage to evade the question of the soundness of that argument. What Rescher suggests is that the validity of the argument suffices to show that it is a good argument when directed to people who accept all of the premises of that argument. However, it is important to note that, from this perspective, any valid argument which has the conclusion that one ought to believe that God exists will be just as good as Pascal's Wager argument. In order to determine whether there is a more substantial sense in which Pascal's Wager argument is a good argument, we need to consider the question whether it is reasonable to accept all of the premises of the argument. So let's see.

We have already noted that it is a crucial assumption of the argument that there are really only two alternatives between which reason is impotent to decide, viz: (a) that there is a God who rewards all those who believe in Him with eternal and infinite happiness; and (b) that life ends with death. But it seems to me that there are infinitely many other alternatives between which Pascal's homme moyen sensual ought to say that reason is equally impotent to decide. Among these possibilities are: (c) that there is a God who has predetermined our final destiny, so that what we do in this life has no consequences for our fortune in the next life--in which case it seems that we ought to do all that we can to make this life as pleasant as possible; (d) that there is a God who looks with great favour on agnostics and atheists because they have proportioned their belief to the evidence, and have not engaged in a mercenary manipulation of their own understanding ; (e) that there is a God who is not satisfied by mere belief in his existence, and who only rewards those who subscribe to the principles of the one true faith (be it Catholicism, Methodism, Anglicanism, Presbyterianism, Congregationalism, Greek Orthodoxy, Mormonism, Seventh Day Adventism, Jim-And-Tammy-Bakkerism, Judaeism, Islam, Hinduism, or the worship of Kali or Odin); (f) that there is a committee of gods who have different desiderata for permission to enter the next life, and who alternate in making decisions (so that there is no consistent basis upon which decisions are made); and so on.[15]

Now, perhaps it will be objected that, although this argument does show that there are further possibilities which Pascal ought not to have discounted, nonetheless it does not serve to establish that there is really an infinite set of possible deities. No matter; we can establish this quite directly. For consider the following:

 

1. For each natural number n there is the deity Sn who is much like the traditional Christian God, except that s/he rewards all and only those people who live for exactly n years (rounded down to the nearest whole year).
2. For each natural number n there is the deity Tn who is much like the traditional Christian God, except that s/he rewards all and only those who are among the first n people to die.
3. For each natural number n there is the deity Un who is much like the traditional Christian God, except that s/he rewards all and only those who are not among the first n people to die.

Apart from these rather arbitrary (not to say whimsical) deities, there are also some other deities (in other worlds) who are guided by slightly more substantial considerations, e.g:

 

4. For each natural number n, there is a world Wn in which there are n deities (all much like the Christian God) who reward all and only those people who believe that there are n deities who are much like the Christian God.

And so on. (It doesn't require much imagination to multiply these examples. The way in which these possible deities differ from the traditional Christian God--by which I mean, at a minimum, a being who satisfies the description "the unique omipotent, omniscient, eternal, and wholly good creator of the world ex nihilo"--is in their moral qualities. But, while these beings are not wholly good, I do not see that this fact provides any reason to suppose that the existence of these beings is somehow more improbable than the existence of the more traditional Christian God.)

I would be prepared to argue that the available evidence does not support the existence of any one of the deities which I have mentioned in the two preceding paragraphs to a greater extent than each of the other deities mentioned there. Moreover, I would also be prepared to argue that anyone who is even moderately sceptical about the abilities of theoretical reason to reach conclusions about the nature and existence of transcendent entities will agree in this judgement. However--as an anonymous reviewer of this paper pointed out to me--for the purposes of the present argument it is sufficient to note that one who holds that theoretical reason is "utterly impotent" when it comes to transcendental questions can hardly have good reason to suppose that some of these hypotheses are more likely than others. In other words: Pascal's l'homme moyen sensual ought to hold that there is an infinite range of possible deities about whose existence he acknowledges that his reason is impotent to decide. But, if this is so, then surely l'homme moyen sensual ought to assign no more than an infinitesimal value to the subjective probability that any one of these deities exists. And so it seems that there is good reason for l'homme moyen sensual to claim that a throrough scepticism about the ability of theoretical reason to decide the question of the existence of God will lead to the conclusion that it is not reasonable to assign a finite value to the probability that the standard Christian God exists--and also that it is not reasonable to suppose that there are only two alternatives between which one must choose.

To sum up the argument: Rescher's version of Pascal's Wager argument has a very peculiar status. For, on the one hand, it is supposed to be directed towards people who are very sceptical about the ability of theoretical reason to establish the existence of "transcendent entities". Yet, on the other hand, it also presupposes that the people to whom it is directed will concede that there are only two live options (either the Christian God exists, or no transcendent beings exist), and that these people will assign a finite probability to the claim that the Christian God exists. But these people--if there are any such people--are in a very uncomfortable epistemic situation; for their alleged scepticism about the abilities of theoretical reason seems to be in considerable tension with the further claims which they wish to make. But, given this fact, what these people need to do is to examine the beliefs which they already have in order to modify (or perhaps discard) some of the discordant beliefs. Moreover, it is certainly not the case that these people can reasonably use the whole set of these discordant beliefs in order to arrive at further beliefs.

Finally, it should be noted that it could be argued that there is something repugnant about the idea that one might make use of Rescher's version of Pascal's Wager argument in the service of apologetics. The reason for this claim is that, in order to use the argument as a tool of apologetics, we do not need to suppose that it is a good argument in the second of the two senses distinguished earlier in this paper. If the point is just to get people to believe in God, then it doesn't matter whether it is overall most reasonable for there people to believe in God--and so we could, quite cynically, make full use of the Wager argument against not terribly bright people in full knowledge of the fact that the argument is defective (i.e. in full knowledge of the fact that it is not reasonable to accept all of the premises of the argument). However, if we care about what it is most rational for people to believe (in the light of the evidence which they currently possess, and in light of the cognitive abilities which they enjoy), then it would be irresponsible (and indeed immoral) for us to use the Wager argument on the sorts of people in whom it could reasonably be expected to bring about belief. (If we think that there are independent means of showing that God exists, then we should appeal to those means. If we think there are no such independent arguments, then perhaps we should question our own belief that God exists.)


[1] University Of Notre Dame Press, Notre Dame, 1985.

[2] ibid., p.x.

[3] Cf. Kay, C.D. "Review Of N. Rescher's Pascal's Wager" IJPR, 1987, Vol.22, pp.112-113; and Brown, G. "A Defence Of Pascal's Wager" RS, 1985, Vol.20, pp.465-479.

[4] ibid., p.5.

[5] ibid, pp.24-25.

[6] ibid., pp.26-27.

[7] ibid., p.15.

[8] ibid., p.16.

[9] ibid., p.16-17.

[10] ibid., p.24.

[11] ibid., p.35.

[12] ibid., p.93.

[13] I have mentioned the "value of " here, because--as Cantor first noted--there are actual many different infinities, of differing magnitudes. I shan't worry about the complications to which this distinction gives rise in this paper.

[14] Note, by the way, that it is no objection to observe that infinitesimals are somewhat dubious entities. The same point could be made in the language of measure theory--i.e. in the mathematical theory which is appropriate for dealing with probabilities in the case in which there are infinitely many options. The set of worlds in which the Christian God exists may have measure zero, and yet be non-empty.

[15] Cf. J. L. Mackie The Miracle Of Theism, Oxford University Press, Oxford, 1982, p.203.


"On Rescher On Pascal's Wager" is copyright © 1990 by Graham Oppy. All rights reserved.
The electronic version is copyright © 1998 by Internet Infidels with the written permission of Graham Oppy. All rights reserved.

Top