In at least one version of the Wager, Pascal argues that a (prudentially) rational person will reason as follows.
V(Believing in God) = p(oo+) + (1-p)(F-) = oo+
V(Not Believing) = p(oo-) + (1-p)(F+) = oo-
So, it seems obvious that the value of believing in God exceeds the value of not believing. Whatever the evidential status of the proposition that God exists (assuming it does not express an impossibility, I suppose) it is prudent to believe.
I don’t want to rehearse the famliar objections to Pascal’s Wager but I do want to worry about some parodic reasoning. So, let’s agree that believing in God has an infinite positive payoff and not believing in God has an infinite negative payoff. Should I believe that God exists? I think the answer is no, and I think you think the answer is no.
What generates the problem I’m concerned with are imaginable worlds which have a non-zero (epistemic) probability of obtaining and which have a very high disvalue associated with them or a very high value associated with them. Because there are such imaginable worlds, it is not difficult to develop Pascalian-like situation in which one outome is the actualization of this imagined world. Notice that the worlds might be impossible, as is one world in Pascal’s Wager: viz., the world in which God exists is impossible or the world in which God does not exist is impossible. But each gets assigned some epistemic probability of being actual. To see the problem, consider this case. There is some non-zero probabiliy that some student has wired my car with an explosive. There is some non-zero probability that the explosive will be detonated when I open the car door. If there is such an explosive, then several lives will be lost, including mine. Should I check under the car? Let p be the probability that there is an explosive and (1 – p) the probability that there isn’t an explosive under the car.
V(Check) = p(F1+) + (1-p)(F2-) = F+
V(Don’t Check) = p(F3-)+ (1-p)(F4+) = F-
It seems clear that the value of checking is positive. There is some small cost in having to look under the car, but there’s a great benefit in checking and finding the explosive before opening the door. It seems clear that the value of not-checking is negative. If I don’t check and there is an explosive, then the worst outcome occurs, including the loss of lives. If I don’t check and there is no explosive, then there is a gain in convenience. Yet it is clearly nonsense to conclude that I should be checking for explosives every time I go to my car. Once you see the problem, you’ll notice all sorts of partitions that will include some extremely bad outcome with a non-zero probability of occurring. But none of these outcomes make it rational to act in hyper-cautious ways.
My conclusion: if you don’t already believe that God exists or think there’s a decent chance that God exists, then Pascal’s Wager gives you a good reason to believe God exists iff. it gives you a good reason to check under your car for explosives.
I have little direct intuition about P(I|T). I have to actually calculate. Start with:
P(I|T) = P(I|E&T)P(E|T).
Now, what is P(E|T)? The kind of evil we have the best reason to expect a priori is bad free choices by significantly free persons (SFPs), understood in the libertarian sense.
Let N be a very large integer. (Ideally, N would be infinity, but this is a parable.) There are two games which are offered to Fred, the P and Q games, and he must play the one or the other. In each, a fair coin is tossed N times, whether or not Fred plays, and each time it comes up heads, Fred gets something of positive value +V, and each time it comes up tails, Fred gets something of negative value -V. In the P game, the coin tossed is a penny, and in the Q game, it’s a quarter. To an ordinary person, the rational choice which game to play would be entirely arbitrary. Moreover, as it happens (based on priors and other evidence), the following three hypotheses are live, and have, let us suppose, equal probability 1/3:
(H1) Fred is perfectly self-interested, and knows exactly how each toss of each coin would go. When two outcomes are equally good, he chooses randomly. (Two sub-hypotheses that I won’t distinguish: (a) The coins are indeterministic, and Fred has middle knowledge; (b) The coins are deterministic, and Fred can predict perfectly.)
(H2) Fred is perfectly self-hating (i.e., tries to minimize his own utility), and knows exactly how each toss of each coin would go. When two outcomes are equally good, he chooses randomly.
(H3) Fred has no knowledge of how future coin tosses will go and chooses which game to play at random.
You now observe that Fred chooses the P game. You also observe the first toss of the P game, and see that it’s tails, while the first toss of the Q game is heads, and so Fred gets -V, but would have got +V had he played the Q game (suppose that the outcomes in the Q game aren’t affected by whether Fred plays or not). You don’t get to observe any further steps in the game.
Question: How should your observation affect your probabilities of the three hypotheses?
The qualitative intuitive answer is easy. Your observation does not affect the probability of H3 at all. It increases the probability of H2 by exactly the amount by which it decreases the probability of H1. If, however, H1 were initially more probable than H2, as typically it would be, then the probabilities of both H2 and H3 would be increased.
But the interesting question is as to the details: Just how much do the probabilities change?
Let’s grant that if sceptical theism is true, then for any evil E, we have no reason to think that the prevention of E will lead to an overall better result than letting E happen, so the fact that we do not see God preventing E is not evidence against the existence of God, since we have no more reason to think that God would prevent E than that he would not. The standard objection is that then we have no reason to prevent E, since we have no reason to think that the overall result will be better if we prevent E.
This objection is mistaken. Suppose I offer you a choice of two games–you must play the one or the other. Each game lasts forever. In Game A, you get pricked in the foot with a thorn on the first move. In Game B, nothing happens to you on the first turn. And that’s all you know. (You don’t know if God exists or anything like that.) Which game should you choose?
You should probably say you have the same probability of doing better by playing Game A as by playing Game B. Why? Well, let Games A* and B* be Games A and B minus their first steps. You know nothing about Games A* and B*. (You don’t know if the first step is a sign of what it is to come, or maybe the sign of the opposite of what is to come, or completely uncorrelated with what comes later.) Now given a pair of infinitely long games about which you know nothing, the overall difference in outcome utility can be minus infinity, plus infinity, undefined, or finite. The likelihood that this difference would be within a pinprick is zero. But if the difference is not within a pinprick, then A is better than B iff A* is better than B*, and B is better than A iff B* is better than A*. Since we know nothing about A* and B*, we should not say that it’s more likely that A* is better than B*, nor the other way around. So, the probability that A would give a better result than B is the same as the probability that B would give a better result than A. (This calculation assumes Molinism. Without Molinism, it only works for deterministic games, or as an intuition-generator.)
Now if the reasoning in the anti-sceptical-theism argument is sound, you have no reason to choose Game B over Game A, since you have equal probabilities of doing better with A as with B. But in fact, despite this equality, you should choose Game B. For since you know nothing about what Games A* and B* are like, the expected value of Games A* and B* should be the same–even if it’s infinite, or even if it’s undefined. (Think of doing this with non-standard arithmetic.) So you have two options: first a pinprick, and then something with a certain (perhaps undefined) expected value; or just something with that same (perhaps undefined) expected value. And of course you should choose the latter–you should avoid the pinprick. The lesson here is that while beliefs are guided by probabilities, action is guided by expected values.
Suppose that a Molinist God creates a world where there is a sequence of 1000 indeterministic throws of a fair coin, and suppose that middle knowledge extends to stochastic non-agential events. (My argument will also apply in the case of Thomist God who determines indeterministic events.) Suppose 514 of the coin throws, let us suppose, are heads and 486 are tails. Consider the fact p that approximately half of the throws landed heads. A standard scientific explanation of p would involve the following facts:
- The coin was fair: heads and tails each had probability 1/2.
- The individual throws of the coin were independent of one another.
- If (1) and (2) hold, then by an appropriate version of the Law of Large Numbers, it is likely that a sequence of 1000 throws of the coin would have approximately half of them be heads.
Fact (3) is a mathematical fact. Facts (1) and (2) are concrete facts about the situation at hand, and both are essential. If (1) is false, we might well expect a different heads-to-tails ratio. If (2) is false, then the Law of Large Numbers need not apply.
But this scientific explanation is unlikely to be correct if Molinism holds. For if Molinism holds, then God in effect controls what sequences of throws come up, by choosing the antecedents of counterfactuals. God makes the choice of sequence based on global providential considerations. Since the sequence is chosen on the basis of considerations of the sequence as a whole, it seems unlikely that the items in the sequence will be independent.
Suppose we say, as I suggested in the previous thread in response to Mike’s related concern, that God deliberately chooses a sequence of events that is statistically apparently random. Then p will still be true–about half of the throws will land heads. However, (2) will not be true, at least not if we condition on God’s choosing a sequence of events that is statistically apparently random. For, if (1) and (2), hold we have a non-zero probability that all the throws will be heads. But conditionally on of (1) and the claim that God chose a sequence of events that was statistically apparently random, we get a zero probability that all the throws will be heads, since if all the throws were heads, the sequence could not be statistically apparently random.
Perhaps we shouldn’t condition on God’s choosing a sequence of events that is statistically apparently random. But if we don’t condition on that, then to check whether (1) and (2) we need to compute the probabilities of all the possible choices God could have made. And we have little reason to think (1) and (2) will hold then.