These are the laws of probability, which we have proved to be necessarily true of any consistent set of degrees of belief. Any definite set of degrees of belief which broke them would be inconsistent in the sense that it violated the laws of preference between options … If anyone’s mental condition violated these laws, his choice would depend on the precise form in which the options were offered him, which would be absurd. He could have a book made against him by a cunning better and would then stand to lose in any event.
We find, therefore, that a precise account of the nature of partial belief reveals that the laws of probability are laws of consistency, an extension to partial beliefs of formal logic, the logic of consistency.
The Foundations of Mathematics and Other Logical Essays, Volume 5
In this post, I’m going to describe one of the more famous arguments for Bayesianism.
These arguments are about how different types of epistemological frameworks will handle different series of wagers. Let me just lay out clearly what exactly we mean by a wager, so as to remove any ambiguity.
A wager on proposition A is a betting opportunity. It involves a payoff amount S and a buy-in quantity. In general, the amount that the buy-in costs will be some fraction f of the payoff amount, so we’ll write it as fS. If you bet on A and it turns out true, then you get the payout S, but still lost the initial buy-in. And if you bet on A and it turns out false, then you get no payout and lose the fS you already spent.
S – fS
From this payout table, you can calculate that an agent will find the wager to be favorable to them exactly in the case that P(A) is greater than f. That is, the agent will want to take the bet whenever the chance of a payout is greater than the proportion of the payout that is required to buy into the bet.
Now, imagine that somebody has a credence of 52% in a proposition A and a credence of 52% in the proposition ~A. How will they evaluate the following set of bets?
B1: pays out $100 if A is true, buy-in of $51
B2: pays out $100 if ~A is true, buy-in of $51
B3: pays out a guaranteed $100, buy-in of $102
They will see both B1 and B2 as favorable bets, since the buy-in is a smaller fraction of the payout than the chance of payout. And they will see B3 as an unfavorable bet, since clearly the buy-in is a larger proportion of the payout than the chance of a payout.
But B3 is just the same as the combination of bets B1 and B2!
Why? Well, if you bet on both B1 and B2, then you are guaranteed to win exactly one of the two (since A and ~A cannot both be true, but one of the two must be). Then you will have paid in a net sum of $102, and gotten back only $100.
A similar argument can be made for any levels of credence C(A) and C(~A) that don’t sum up to 100%. And all of the usual axioms of probability theory can be argued for in the same way. Such arguments are called Dutch book arguments.
Dutch book arguments are standardly presented as revealing that if one does not form beliefs according to the laws of probability theory, then they will be able to be juiced for money by clever bookies.
This is true; somebody with beliefs like those described above can be endlessly exploited for profit. But it is much less impressive than the real conclusion of the Dutch book argument.
Recall that our agent above was found to believing a logical contradiction as a result of not having their beliefs align with probability theory (they had to believe that a bet was simultaneously favorable to them and not favorable to them)
Said another way, an agent not following the probability calculus may evaluate the same proposition differently if presented in a different form.
This is what Dutch book arguments really say: if you want your beliefs to be logically consistent, then you are required to reason according to probability theory!