Is it irrational to assign a probability to a metaphysical proposition?

Question

One of the responses to Pascal’s Wager is that it may be meaningless to assign a probability to a certain metaphysical proposition such as God’s existence.

In this post about the assignment of probability with respect to metaphysical realities, @Conifold comments,

The assignment is of course meaningless, along with "more", "less", "many other", etc., applied to something as vague as "metaphysical possibilities". See Bertrand paradox for a simple illustration of how probability values depend on arranging "possibilities" into a sample space. Even if one was willing to accept the "base assumption" of equiprobability it is impossible to make sense of what that means here.

Suppose that you are a person who thinks that it is meaningless to assign a probability to God’s existence or that you think the probability of God’s existence is unknown given the lack of experience with any supernatural entities.

Now, in the interpretation of subjective probability, which is often linked to Bayesian thinking, probability is defined as a degree of belief as such:

Your degree of belief in E is p iff p units of utility is the price at which you would buy or sell a bet that pays 1 unit of utility if E, 0 if not E

The neat thing about this definition is that it defines probability in a practical way, in the form of a bet.

Now imagine that this bet is somehow forced upon you, either by the government, or by punishment in such a way where you have no choice but to take this bet. Now, a person who is forced to take this bet where he can only pick between 0 and 1 units, must still take an action (or else, he may be murdered for example).

But if he does pick a number of units (think of this as real money), which he must, does this not mean that he undeniably has a credence in the proposition (which can be translated into a probability) even if he claims not to, making him inconsistent?

Note that if the terms of this bet still seem unrealistic, you can waive these concerns by not thinking of the bet too literally:

These problems stem largely from taking literally the notion of entering into a bet on E, with its corresponding payoffs. The problems may be avoided by identifying your degree of belief in a proposition with the betting price you regard as fair, whether or not you enter into such a bet; it corresponds to the betting odds that you believe confer no advantage or disadvantage to either side of the bet (Howson and Urbach 1993). At your fair price, you should be indifferent between taking either side.

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Disclaimer: I am not myself a Bayesian and don’t know on which side of the fence I belong here yet (unless, of course, I was forced to bet on it ;))

P.S. to the current downvoters, explain the downvotes instead of being cowardly, especially since answers by prominent posters have already been given, giving at least some level of indication that they thought the question was worth responding to. This site unfortunately has a wonky history of downvoting/closing things without explanation which is sometimes a sign that there’s no basis for them except a dislike of users.

Answer 1

For something that appears to be false, I'd simply assign a 0 probability.

As someone who doesn't use probabilistic epistemology for things for which we have no reasonable probability estimates, all such things would either have a 0 or 1 probability (or I might refrain from assigning one of those values if I just don't know).

But I'd have a non-zero confidence about those values, and that confidence increases or decreases based on how sure I am that it's true or false (although I don't have a means of putting a concrete percentage value on that).

This addition of confidence may not, strictly speaking, make it that much different than pure probability. But one key difference is that confidence values don't need to add up to 1. That may sound minor, but it's actually rather significant.

I don't need to consider every* one of the infinitely-many conceivable possibilities to try to come up with some relative percentage for each. And that's just for all possible gods. I might also need to consider all of the countless other implausible-but-not-impossible things to consider whether my time might be better spent on those instead of a god.

Rather than doing all of the above, I can instead just evaluate the evidence until I get to a point of high confidence (not a percentage value) about 1 particular claim, then reject further investigation or commitment as a waste of time, and spend my time on more productive things.

* Proponents of Pascal's wager tend to take for granted that their religion (and more specifically, their particular interpretation of it) stands head-and-shoulders above every other religion (in terms of plausibility). But this would also be a claim an atheist may not accept.

Answer 2

There is nothing inconsistent, or even irrational, about declining to assign a probability to a given proposition. Doing so just amounts to suspending belief or disbelief. Sometimes we wish to say we don't have any idea whether some proposition is true or not and that's all.

I might be compelled to take a bet on a proposition and assign a probability to it when I have little or no evidence to go on. But in the case of propositions such as whether gods exist, I don't see how that would work. How will the party compelling me to bet determine whether I have won or lost? For Pascal, the bet is entirely personal and there is no counterparty. I win or lose depending on whether I discover that my soul has survived death and I am to be rewarded or punished in the afterlife.

Answer 3

I'm not sure we can say that in a forced decision the person has credence toward different outcomes. For example, they may decide to ask their interrogator-executioner their last name -- if it has an even number of letters they pick Yes, otherwise No.

Or they may think that No is safer in that if they are right, they live, if they are wrong, they still live in the afterlife (and maybe try to do a little last minute repenting/groveling to St. Peter ;-)

We can't say they are deciding based on credence unless they are picking based on which they truly feel is the most correct. They may feel they have no idea and choose some arbitrary method to get that person an answer.

I guess a potential loophole is that I can refuse to answer since you didn't specify a timeframe in which I need to make that decision -- maybe I can say "Ok, I promise to decide -- I'll return in 80 years with my definitive answer". I'm guessing they'd say, no, decide in the next 5 seconds or you will die. Given certain death, you're better off guessing somehow.

Answer 4

But if he does pick one, which he must, does this not mean that he undeniably has a credence in the proposition (which can be translated into a probability) even if he claims not to, making him inconsistent?

Suppose it is objectively true that individual X subconsciously assigns probability P to proposition Y. How can this be? Where, within the mind of X, can this probability P be said to reside? What does it even mean for this probability P to exist?

If physicalism is true, the statement "individual X assigns probability P to proposition Y" must ultimately reduce to concrete physical facts (see the related discussion: How does physicalism interpret mathematical theorems in physicalist terms?). So, what physical facts would correspond to a probability? How are propositions represented in the brain, and how are beliefs about those propositions stored? Are neurons physically encoding a "probability" for each proposition? Does the brain compute probabilities, and if so, is this an accurate model of how the brain works? (Interesting article: Are our brains Bayesian?)

I’m willing to grant that the individual likely has some belief about proposition Y, but I wouldn’t immediately leap to the conclusion that "it can be translated into a probability." If physicalism is true, I’m unsure how one could substantiate such a claim.

Even under alternative metaphysical views, such as dualism, it’s not clear how one could establish that probability P must exist within a spiritual or immaterial mind.

Answer 5

The basis of your question is unrealistic to the point of being meaningless. How can you place a bet on the probability of an unprovable proposition, since by definition you have know way of knowing whether your bet is correct?

However, leaving that aside, your question seems to be about whether a person, for whatever reason, assigning a probability to some unprovable notion is in some way expressing a degree of credence. I think your question is tautologous. If I say I am convinced god exists, I will naturally assign a probability of 1. If I am convinced god does not exist, I will assign a probability of zero. In other words, posing the question 'what do you believe to be the probability of x?' seems to me to be no different to the question 'how strongly do you believe x' where x is some entirely unprovable notion for which there is no logical way to quantify probability.

Answer 6

Now imagine that this bet is somehow forced upon you, either by the government, or by punishment in such a way where you have no choice but to take this bet.

By ignoring the group of individuals, who given a choice between "bet or die", choose die. You are effectively cherry-picking your data. Any answer to a question based on a fallacy is tainted. For example:

By a highly improbable event (miracle) one of the players who chose "die" doesn't die and escapes.

This escapee organizes and leads underground movement to destroy this vile bet and give all the freedom of choice.

He wins and the sinister bet is illuminated and never used again.

How does this scenario factor in your betting scheme?

If out of 100 players only 3 bet and the rest chose death. How does that impact the relevance of your bet?

Finally, your assumption that the threat of death will lead to a bet choice 100% of time is a hasty generalization based on your personal bias.