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Let me illustrate this question with an example. Imagine you were to compare your credence or your belief of you winning the lottery twice with your belief in the devil’s existence.

Some argue that me winning a lottery two times has a defined probability whereas we don’t know if a devil exists. Hence, we can’t assign any probability to the devil’s existence and thus we can’t say that one should have a higher credence in my double lottery win than the devil.

However, let’s change the devil to the event where I toss a fair coin and have it land on heads 5,000,000 times. Now, it seems obvious that I should place higher credence in winning the lottery twice than tossing a fair coin and having it land on heads that many times. Even though the first is improbable, the latter is much moreso.

But here is where I see a problem that my mind is having a hard time wrapping my head around. The event of me landing a coin on heads is extremely improbable, yet it is possible. On the other hand, we can’t say that the devil is possible since the devil may not exist. In a sense, we have evidence for the coin event to be possible but not the devil.

But then this creates a scenario where I’m ultimately putting higher credence in an event with no evidence (I.e. the devil) than an event with evidence (I.e. the coin event) when comparing it to the lottery event. Even if I claim that my credence in the devil should be unknown, I am de facto giving it higher credence than the coin event since when compared to the lottery event, I don’t say that my credence in the devil is lower than it. This doesn’t seem right.

What’s going on here and how should I navigate through this?

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  • Instead of asking a question with a very weird title linked to a very specific example, why not ask a more general version of the question that captures as many similar examples as possible at once? For example, something along the lines of "Are epistemic probability and empirical/statistical probability comparable?". Commented Mar 28, 2024 at 12:58
  • @Mark Good point, made the edit! Commented Mar 28, 2024 at 13:00
  • I didn't mean that my title was the ideal title, it was just a suggestion (feel free to improve upon it), but thanks. Commented Mar 28, 2024 at 13:01
  • An interesting example to ponder: what's more probable, the devil or abiogenesis? We have never observed abiogenesis take place empirically, so the statistics associated are essentially 0. However, many people have reported encounters with the devil. Commented Mar 28, 2024 at 13:06
  • @Mark What’s tough with these examples is that there’s no correct probability: in the real world, devils either exist or don’t. Abiogenesis is either true or not. It seems as if one must just make a bet? Commented Mar 28, 2024 at 14:30

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Let's call the two orderings at play, the 'arithmetic/empirical probabilistic' one and the 'epistemic/credence' one, (A, <') and (C, <'), respectively, so that the question

Are epistemic probability and empirical probability comparable?

becomes one about existence of monotone (non-decreasing) functions in either direction

Now, it seems reasonable that < is a linear order in A, [we may well take A to be the unit interval with the usual ordering, or at least the rational points] while <' may be neither antisymmetric nor linear in C, so even if we (somewhat artificially) passed to a quotient to obtain antisymmetry, the existence of a monotone f: C -> A would exactly mean linearity of <'; besides that, one of your points is that there's no way to meaningfully asign an(y) aritmetic probability to a certain d in C, which here just means that "there is no total function f: C -> A whatsoever, much less a monotone one", so in a sense 'comparability in one of the directions' is really doomed to fail

On the other direction, we may construct an f: A -> C (canonical in a sense) by taking a belief (?) like "throwing a dice with n faces will result in a face with value at most m < n" to be f(m/n), so that it makes sense to speak of winning lotteries ('l') and tossing coins ('c') in both contexts. Notice also that f cannot be surjective, by the previous paragraph, but it doesn't forbid one from having more credence in s = "the Sun will rise tomorrow" than in l, so that l <' s makes sense, even if there's no f-¹(s) to assert l < f-¹(s)

So where's the devil in these details? In

But then this creates a scenario where I’m ultimately putting higher credence in an event with no evidence (I.e. the devil) than an event with evidence (I.e. the coin event) when comparing it to the lottery event.

you worried that the previous points, not(d <' l) and (c < l) - and hence also (c <'l) - , somehow would imply that (c <' d), but it should be clear now that such points are not sufficient to conclude so

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