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gedymin comments on Skepticism about Probability - Less Wrong Discussion
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Belief in the axioms of probability theory is justified by the fact that someone with inconsistent beliefs can be Dutch-booked.
If you're willing to put money on your beliefs (i.e. bet on them), then you ought to believe in the axioms in the first place, otherwise your opponent will always be able to come up with a combination of bets that will cause you to lose money.
This fact was proved by Bruno de Finetti in 1930-ties. See e.g. AI: A Modern Approach for an easily approachable technical discussion.
I think De Finetti's justification is fine as far as it goes, but it doesn't go quite as far as people think it does. Here's a couple dialogues to illustrate my point.
Dialogue 1
A: I have secretly flipped a fair coin and looked at the result. What's your probability that the coin came up heads?
B: I guess it's 50%.
A: Great! Will you accept a bet against me that the coin came up heads, at 1:1 odds?
B: Hmm, no, that doesn't seem fair because you already know the outcome of the coinflip and chose the bet accordingly.
A: So rational agents shouldn't necessarily accept either side of a bet according to their stated beliefs?
B: I suppose so.
Dialogue 2
A: I believe the sky is green with probability 90% and also blue with probability 90%.
B: Great! I can Dutch book you now. Here's a bet I want to make with you.
A: No, I don't wanna accept that bet. The theory doesn't force me to, as we learned in Dialogue 1.
In Dialogue 1 I adjust my probability estimate as the bet is offered, no?
That's a reasonable thing to do, but can you obtain something like De Finetti's justification of probability via Dutch books that way?
The obvious steelman of dialogue participant A would keep the coin hidden but ready to inspect, so that A can offer bets having credible ignorance of the outcomes and B isn't justified in updating on A offering the bet.
Russel & Norvig:
"One common objection to de Finetti’s theorem is that this betting game is rather contrived. For example, what if one refuses to bet? Does that end the argument? The answer is that the betting game is an abstract model for the decision-making situation in which every agent is unavoidably involved at every moment. Every action (including inaction) is a kind of bet, and every outcome can be seen as a payoff of the bet. Refusing to bet is like refusing to allow time to pass."
I think a fair bet presupposes that both opponents will have access to the same amount of information, which is not the case in Dialogue 1. The bets in life are not always fair, but that has nothing to do with belief in probability axioms.
That Russell & Norvig quote doesn't appear to be a very good response to the objection it's addressing. De Finetti's argument is supposed to be a pragmatic argument for probabilism. In response to someone asking "Why should my beliefs obey the probability calculus?", de Finetti says "If you don't, you'll end up getting screwed (by being susceptible to dutch books)."
The response to de Finetti that Russell & Norvig are considering is "There are ways to get around susceptibility to dutch books other than accepting probabilism. For instance, I could formulate a policy of refusing to accept bets. Why is probabilism the right way to deal with susceptibility to dutch books?" Russell & Norvig are saying "Well, this is a thought experiment situation in which you are forced to bet."
OK, but that completely ruins the pragmatic appeal of de Finetti's theorem. I can feel the attraction of probabilism if it's the only way I'm protected against being screwed in reality. But not if it's the only way I'm protected against being screwed in an abstract model that doesn't match reality. Why should I care about getting screwed in a thought experiment?
That people often raise these objections is the reason why I prefer Savage's theorem as a decision-making foundation for probability.
This makes assumptions such as the existence of the world and the existence of bets which a global sceptic would not believe.
If you don't believe in the existence of the external world, then you shouldn't be worrying about the "notion of probability having a correlation with reality" in the first place. The OP presupposes the existence of the world. Do not shift the goalposts.
I never said I disbelieved in it- I'm postulating, not accepting a position. It is also worth noting that in my postulated position I was neither accepting nor rejecting the existence of said external world.
As I mentioned, this argument started out on the basis of trying to figure out if something could be known without assumptions that could not themselves be justified. If assumptions are necessary to know anything, it means that we effectively believe on religious-style faith.
I was arguing against a rhetorical "you", identified with the sceptic, not you personally.
That said, an extreme skepticism is altogether a different game compared to a skepticism about probabilities. The latter is reasonable; the former, although it cannot be falsified, is useless. Reality does not go away when one stops believing into it.
Logical skepticism, on the other hand, is self-defeating. To make a logical argument against the possibility of logical arguments, against the value of reasoning - that is self-contradictory.
The logical sceptic could argue that they are showing that logic is self-defeating- that when logic is taken to its ultimate conclusion it is shown to be false, therefore logically logic should be rejected. This is precisely what I would argue.
As for the matter of reality- if it exists, then of course it doesn't go away when we stop believing it. But how do we know that?
If logic is pennies, and blue is a sock, then why are cats punctuation marks? Yes, because your mask is Maryland, and WiFi smokes bananas.
(This is what happens when you live in a world without logic, and is the only response you should have to someone who is a logic skeptic).
See my below response gedymin.
Logically, "self-defeating" is not equal to "not self-defeating". If the skeptic rejects logic, then he should accept that "self-defeating" is equal to "not self-defeating". Therefore, if logic is self-defeating, then logic is also not self-defeating.
As for the second point - the epistemic perspective is more important than the ontological one. Seriously, read the conclusion of the "Simple truth".
This debate is getting silly, I'm out of here.
On the first point, if you get to a conclusion within logic which marks it as "self-defeating", then from a logical perspective logic doesn't work. Non-logic doesn't matter for those who aren't logical, but for a logical person logic matters.
On the second point, once you start postulating actually viable alternatives to the world not existing, and considering the Evil Demon Argument, there is nothing in there which is actually dealt with.