Priests and conjurors are of the same trade.
~ Thomas Paine
Justin Barrett, Christian cognitive scientist, argues that belief in god is an indication that (the Christian) god exists. I respond that belief cannot be evidence for its own validity. Belief depends on evidence that is different from the belief itself. Without respect for the distinction between belief and evidence, one ends up with magical thinking. Magical thinking may be fun and it may be soothing, but it is irrational and illogical. When magical thinking is used to browbeat others, it can be dangerous because it makes it difficult to consider the possibility that others may be right.
Suppose there is an association between the strength of human belief in god and god's existence. In the interest of simplicity I will ignore the question of whether the belief is held by an individual or by a collective. According to this argument, the presence of strong belief suggests that god exists and that the presence of weak belief suggests that god does not exist.
How might this work? Consider Newcomb's problem (Nozick, 1969). A person is faced with a choice between the option of opening two boxes placed before him and the option of opening just one box. He knows that the box on the left contains $1,000. The box on the right is either empty or it contains $1M. A nearly omniscient being (a demon of sorts) is in the business of predicting what people will do in this situation. The demon virtually never makes mistakes. If he predicts that the person will open both boxes, he leaves the right box empty. If he predicts that the person opens only the right box, he places $1M inside. Knowing all this, what should the person do?
One view is that the person should open both boxes. Whatever the demon predicted, the person is better off by $1,000. This analysis implies that the person can exercise free will in the sense that he can uncouple his choice from the demon's prediction. Another response is that whatever the person "chooses" to do remains statistically linked to the demon's prediction. Choosing to open only the right box does not presume to influence the demon's action retroactively in time; it only plays on the statistical link between own choice and the demon's choice.
Newcomb's problem means that you can get rich if you forsake your belief in free will (if only Newcomb's game were offered with someone paying the bill!). How is it then that the demon can be so accurate in his predictions? The answer is that neither the person's choice nor the demon's behavior is free. The actions of both depend on a third variable, a common cause. Let's call it Big C. Big C in its inscrutable wisdom has ordained that the person will open one box or both boxes and that the demon will predict it. You can think of Big C is a shortcut term for the aggregate causal force of the universe that makes events what they are (e.g., the logos of Heraclitus, Spinoza's nature, or the Dao).
Now reconsider Barrett's argument that the belief in and the existence of god are linked. To play Newcomb's problem you have to surrender your belief in free will before trying to rescue your belief in god. So you "choose" to believe in god, hoping that this means that god exists. This would work if only there were a Big C that simultaneously makes you choose god and brings god into existence. The obvious problem is that your chosen god can no longer be the Christian (Muslim, Jewish) god, because that god is defined as the ultimate uncaused cause. Now it appears that Big C is the real creator god. How can you make his existence probable by believing in him? You can do that by assuming the existence of a Super Big C that causes you to believe in Big C and also causes Big C to exist. And so on ad infinitium and nauseam. As they say in New England: "You can't get there from here!"
One more time: Strong religious belief says nothing about the probability of god's existence, and therefore, any increase in belief (e.g., over historical time) in the existence of a creator god cannot indicate—let alone cause—an increased probability that such a god exists.
Let's take another look at the Reverend Bayes's formula. A theist of Barrett's school wishes that p(G|B) > p(G|~B), that is, the probability that god exists given that I (or we) believe is greater than the probability that god exists given that I (or we) do not believe. In ratio form, Bayes's theorem is this:
p(G|B) / p(G|~B) = p(B|G) / p(~B|G) x p(~B) / p(B).
Notice that the probability of god's existence, p(G), is already canceled out. It cannot be raised by more fervent belief. What does happen, though, if the overall probability of belief, p(B), goes up, is that that the ratio of p(G|B) / p(G|~B) goes down. This is not what a theist would want.
If classical probability theory yields disappointing results, perhaps there is hope in quantum probability theory.
Nozick, R. (1969). Newcomb's problem and two principles of choice. In. N. Rescher (Ed.), Essays in honour of Carl G. Hempel (pp. 114-146). Dordrecht, Holland: Reidel.