How does one argue for the rationality of assigning subjective probabilities in the context of Pascal’s Wager?

Question

As a follow up from this discussion on a previous question I asked, I’m wondering how defenders of Pascal’s Wager/“strong” atheists who hold that the probability of God’s existence is zero justify their positions. In order to defend the Wager, many argue that the probability for God’s existence is defined and positive. Atheists, on the other hand, argue that the probability for God’s existence is either zero or infinitesimal (if there is a difference between the two). How are either of these positions justified? It seems that, in general, three strategies can be applied to argue for these conclusions:

1. Bayesian strategies - Some hold that they can rationally assign any probability to God’s existence as a Bayesian prior. For the atheist, this may well be zero. For a defender of the Wager, this may be any positive value. While an interesting perspective, it is certainly does not carry any power for those who do not accept the priors, making this argument somewhat weak in my eyes.
2. Deductive arguments - Atheists may argue that the very concept of God is impossible. Let us also not discuss this option - for while it is worthy of consideration, it often seems that a adjustment of definitions is all that is needed to circumvent this objection. For example, many omnipotence paradoxes are raised in objection to God. While it may be problematic on the classical definition of omnipotence, simply adding that omnipotence does not include the power to do the logically impossible is usually enough to get around the paradox. Some stronger objections, such as the problem of evil, may also be used here, but I think that this is out of the scope of this question.
3. Appeals to the principle of indifference - This seems to be the strongest way to argue in favor of the Wager/atheism here. To put it simply, the principle of indifference assigns a 1/n probability to n events when no other information is available. While this seems intuitive enough, I don’t think that it can be used here. I can think of two major problems to this approach. First, is the principle justified? Wikipedia states, among other objections,

It is not trivial to justify the principle of indifference except in the simplest and most idealized of cases (an extension of the problem limited definition). Coins are not truly symmetric. Can we assign equal probabilities to each side? Can we assign equal probabilities to any real world experience?

It also seems to be a trend in the philosophical community to abandon this interpretation, and instead accept imprecise credences - an interval, rather than precise number, of rational probability assignments. For this case, an assignment of [0, 1] would certainly seem to be in order. Regardless, for the sake of argument, let us assume that the principle of indifference is valid. We run into a second problem: Bertrand’s paradox speaks to the inapplicability of the principle when the probability space is not clear - and it certainly cannot be in this context. As an example, a defender of the Wager could say that the probability of God’s existence is 1/2 - either he exists or he doesn’t, and we have no evidence to sway us in either direction. On the other hand, an atheist could argue that out of infinitely many other Gods, Pascal’s God is just a tiny subset, making the probability for it infinitesimal. One assignment does not clearly seem better than the other. It is also not clear how we can say an infinity of Pascal’s God can be less numerous than an infinity of other Gods. If we map each of these Gods to an interval in the real numbers, we certainly cannot say that one interval is more numerous than another (more about this here).

Taking all of this into account, how do atheists and defenders of Pascal’s Wager respond to these objections? Do they abandon option (3) altogether, or make some statement about the “irreducibility” of the principle of indifference, and then simply choose a probability space that seems most reasonable to them? Or do they opt for options (1) or (2) instead?

Answer 1

I suspect the underlying problem you are concerned about is whether it's justifiable to assign probabilities to claims with insufficient evidence, and if so, how can such probabilities be assigned? For some responses to Pascal's Wager, this comes in the form of assigning a probability to the existence of God.

• Often, it is justifiable to assign probabilities. This can be seen by examining other scenarios, such as when flipping an untested coin, it's reasonable to initially assume a 1/2 chance of landing heads. This is based on prior experience with coins and by examining the coin's physical properties. It has two sides, and though it is possible to flip a coin so that it lands and stays on edge for a few seconds, it will eventually fall to one side or the other.

• For cases when there is no prior basis for the assignment of a probability. One can assign a tentative probability that is further tested. In An Intuitive Explanation of Solomonoff Induction, a scenario is described in which a soldier updates the probability he is facing a sniper based on how many times in a row his helmet gets hit with a bullet.

Suppose someone were to wonder whether there is someone on SE with the username "Josh". It would be reasonable for someone who has encountered evidence of this user's existence to assign a fairly high probability. Afterall, someone or something wrote this question. But if we were to ask people on the street who don't even know this site exists, what probability should they assign?

• Should it be 1/2 – the user either exists or doesn't. What about 1/12000000, the approximate number of users on StackOverflow? What about 1/4.4 billion, the number of internet users? Or 1/7.7 billion, the current world population?

  What about a user named "God"? Or a user named "Clark Kent"? Would it be reasonable for someone to just shrug their shoulders and say, "I don't know. Let's Google it."

So is it reasonable to assign a probability to the existence of "God"? I agree with your implication that it isn't.

• We don't have any evidence or appropriate analysis upon which to base assigning a probability.

• This isn't a scenario in which a tentative probability can be assigned and updated. If an ultimate being going by "God" does exist, it's using it's power to hide its existence.

However, it is reasonable, to suppose that we could assign a probability and analyze the possible outcomes. Such analysis would not be limited to specific values, like 0 or 1/2, but would consider various reasons for assigning a range of values from 0 to 1. This seems to be what many descriptions of Pascal's Wager attempt to do.

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Other comments:

• Some might claim that the many gods objection and principle of indifference could be used to assign a probability of 1/number-of-gods, but that's analogous to assigning a probability of 1/number-of-users to the probability of a user "God" existing on this site. The existence of the user (God) is independent of how many others have been identified. Zeus not existing doesn't make Yahweh any more likely to exist.

• The many gods objection may not be useful to assign existence probabilities, but it is useful as a counter-example or argument. If Pascal's Wager is valid, it should carry equal weight when reformulated with Zeus as it does with Yahweh or any other god.

Answer 2

Subjective probabilities ultimately come down to your behavior, your feelings, or your thoughts and dispositions towards a certain proposition. But if philosophy is about ascertaining truth about the world, why should any of this matter? One can bet an infinite amount on the earth being flat. This says nothing, at all, about whether or not it actually is.

A further, second issue is deeper. In what sense would it be meaningful to assign a probability to God anyways? What does it mean to say that God has a 50% chance of existing? It is not as if there is a fair coin being tossed deciding this outcome. God either does or does not exist. The best we can arguably seem to do is marshal arguments for or against it. One can do this without assigning a numerical measure to it.