What do you mean “the statement is affected by a generalisation”? What does it mean for something to be “affected by a generalisation”? What does it mean for a statement to be “affected”?
The claim is a general one. Are general claims always false? I highly doubt that. That said, this generalisation might be false, but it seems like establishing that would require more than just pointing out that the claim is general.
I think this misses the point, somewhat. There are important norms on rational action that don’t apply only in the abstract case of the perfect bayesian reasoner. For example, some kinds of nonprobabilistic “bid/ask” betting strategies can be Dutch-booked and some can’t. So even if we don’t have point-valued will-to-wager values, there are still sensible and not sensible ways to decide what bets to take.
If you weaken your will-to-wager assumption and effectively allow your agents to offer bid-ask spreads on bets (i’ll buy bets on H for x, but sell them for y) then you get “Dutch book like” arguments that show that your beliefs conform to Dempster-Shafer belief functions, or Choquet capacities, depending on what other constraints you allow.
Or, if you allow that the world is non-classical – that the function that decides which propositions are true is not a classical logic valuation function – then you get similar results.
Other arguments for having probability theory be the right representation of belief include representation theorems of various kinds, Cox’s theorem, going straight from qualitative probability orderings, gradational accuracy style arguments…
This seems to be orthogonal to the current argument. The Dutch book argument says that your will-to-wager fair betting prices for dollar stakes had better conform to the axioms of probability. Cox’s theorem says that your real-valued logic of plausible inference had better conform to the axioms of probability. So you need the extra step of saying that your betting behaviour should match up with your logic of plausible inference before the arguments support each other.
Savage’s representation theorem in Foundations of Statistics starts assuming neither. He just needs some axioms about preference over acts, some independence concepts and some pretty darn strong assumptions about the nature of events.
So it’s possible to do it without assuming a utility scale or a probability function.
I’ve had rosewater flavoured ice cream.
I bet cabbage ice cream does not taste as nice.
Sorry I’m new. I don’t understand. What do you mean?
I have lots of particular views and some general views on decision theory. I picked on decision theory posts because it’s something I know something about. I know less about some of the other things that crop up on this site…
it is clear that each party to this dispute – as to all that persist through long periods of time – is partly right and partly wrong
— Bertrand Russell History of Western Philosophy (from the introduction, again.)
Uncertainty, in the presence of vivid hopes and fears, is painful, but must be endured if we wish to live without the support of comforting fairy tales
— Bertrand Russell, History of Western Philosophy (from the introduction)
Hi. I’ll mostly be making snarky comments on decision theory related posts.
The VNM utility theorem implies there is some good we value highest? Where has this come from? I can’t see how this could be true. The utility theorem only applies once you’ve fixed what your decision problem looks like…
What the Dutch book theorem gives you are restrictions on the kinds of will-to-wager numbers you can exhibit and still avoid sure loss. It’s a big leap to claim that these numbers perfectly reflect what your degrees of belief ought to be.
But that’s not really what’s at issue. The point I was making is that even among imperfect reasoners, there are better and worse ways to reason. We’ve sorted out the perfect case now. It’s been done to death. Let’s look at what kind of imperfect reasoning is best.