Philosophical Devices: Proofs, Probabilities, Possibilities, and Sets by David Papineau

Author:David Papineau [Papineau, David]
Language: eng
Format: azw3
Publisher: OUP Oxford
Published: 2012-10-03T16:00:00+00:00

7.8 Dutch Books

I said that degrees of belief or subjective probabilities offer one way of interpreting probability statements—that is, one way of attaching numbers between 0 and 1 to propositions in such a way as to satisfy the axioms of probability.

However, as yet I haven’t really shown this, for I haven’t yet shown that degrees of belief do satisfy the axioms of probability.

And in fact there is no guarantee that they will. Nothing in psychology rules out the possibility that an agent at a time might attach a degree of belief 0.6 to the proposition it will rain and simultaneously a degree of belief 0.6 to the proposition it won’t rain, thus violating the immediate implication of the probability axioms that Pr(p) = 1 – Pr(not-p). (Maybe the agent wasn’t thinking very hard, and somehow managed to take a positive view of both these propositions at the same time.)

However, there is an argument that any rational degrees of belief must conform to the axioms of probability, even if actual degrees of belief don’t always do so.

The argument is that anybody whose degrees of belief violate the axioms of probability can have a ‘Dutch Book’ made against them. A Dutch Book is a set of bets which are guaranteed to win whatever happens.

By way of illustration, consider the person who believes it will rain to degree 0.6 and also believes it won’t rain to degree 0.6. Well, this person will happily pay 60p to win £1 on its raining, and also happily pay 60p to win £1 on its not raining. But anybody who makes this pair of bets will certainly lose whatever happens, because they will have paid out £1.20 in total and will only win £1 whether it rains or not.

It is not hard to prove that a Dutch Book can be made against you if and only if your degrees of belief fail to satisfy the axioms of probability.

(The subject in the above illustration got into trouble because of degrees of belief in p and not-p which added to more than 1. This might make it seem safe to have degrees of belief that add to less than 1. However, in that case you could be induced to bet against both p and not-p in a way that is guaranteed to lose.)

Since it seems clearly irrational to adopt attitudes that can make it certain that you will incur a loss, it follows that any rational agent will have degrees of belief that do conform to the probability calculus. (Such agents are called ‘coherent’; those whose degrees of belief violate the axioms are ‘incoherent’.) (See Box 17.)

Note that there is nothing in this ‘Dutch Book Argument’ to specify what degrees of belief you should have, beyond requiring that they must conform to the probability axioms. You can be coherent by having a subjective probability of 0.6 for it will rain and of 0.4 for it won’t rain. But you could equally achieve coherence by attaching 0.8 and 0.2 to these two propositions, or 0.

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