I got back yesterday from the conference on formal and traditional epistemology at Oklahoma organized by Jim Hawthorne and Wayne Riggs. It was utterly fabulous! Except that I was really sick when I talked, and had to leave early to get home (with a fever of 102!). I have very little idea what I said, except for one remarkable lapse: I forgot what my last argument was supposed to be! I don’t know what I said, but what I wanted to say, I’ll write here.
It’s about David Christensen’s DBA from his beautiful book Putting Logic in Its Place and the way in which subjectivists should understand the perspectival character of rationality. The book is easy to read, very entertaining, and the arguments quite compelling, especially the ones about the import of the Preface Paradox for deductive closure principles about rationality. But the argument that I don’t think quite succeeds is the argument for probabilistic incoherence being a defect.
Here’s how the argument goes.
The argument is intended to sustain the following result:
Simple Agent Probabilism: If a simple agent’s degrees of belief violate the probability axioms, they are defective.
A simple agent is one who values money and nothing else, and whose money preferences are positively linear. In this way, the value of each extra dollar is the same as the value of any dollar, no matter how much money the agent has or lacks. The argument begins with the following principle:
Sanctioning: A simple agent’s degrees of belief sanction as fair monetary bets at odds matching his degrees of belief.
I take these as very plausible normative judgements: any agent who values money positively and linearly, and who cares about nothing else, should evaluate bets in this way. The way in question here is the way described in Sanctioning: if your degree of belief in p is 2/3 and you are offered a bet that will pay you $1 if p is true and cost you $2 if p is false, then if you are a simple agent, you should regard this bet as fair. We can understand, then, the role of the appeal to simplicity of the agent: it is a way of controlling for interference into the assessment of rationality of degrees of belief by messiness concerning preferences. A person’s preferences might be inconsistent; they might be non-linear; they might violate transitivity; etc. When we want to connect attitudes about fairness of bets to the rationality of degrees of belief, we want to control for such insanity (if such we prefer to label it). Once we control for this by stipulating that the agent is simple, the hope is that we can read off conclusions about rational credences from information about attitudes toward fair bets.
The remainder of the argument uses the following principles:
Bet Defectiveness: For a simple agent, a set of bets that is logically guaranteed to leave him monetarily worse off is rationally defective.
Belief Defectiveness: If a simple agent’s beliefs sanction as fair each of a set of bets, and that set of bets is rationally defective, then the agent’s beliefs are rationally defective.
Dutch Book Theorem: If an agent’s degrees of belief violate the probability axioms, then there is a set of monetary bets, at odds matching those degrees of belief, that will logically guarantee the agent’s monetary loss.
So, the idea of the argument is this. First, restrict discuss to simple agents, so that money and a linear ordering on it is all we need to talk about to measure the agent’s preferences. Fair bets for such an agent are ones that match the agent’s degree of belief. So, to have rational degrees of belief, a simple agent’s degrees of belief have to match his attitudes about fair bets. Unfair bets favor the book or favor the agent. One class of bets that favors the book are those that are logically guaranteed to favor the bookie. These we say are rationally defective.
The trick of the argument is to connect this Bet Defectiveness principle with the principle that follows it, Belief Defectiveness. Here is the way Christensen argues. He first notes that Belief Defectiveness doesn’t follow from Bet Defectiveness. He considers cases of what I will call “messy preferences” to show that the second doesn’t follow from the first. In such cases, the value of a payoff for one bet may affect the value of the payoff for the next bet: I’ll value roast duck more, he says, if I don’t yet have one than if I do. But, of course, this possibility is ruled out by noting the qualification in the principles in question that the agents in question are assumed to be simple agents. So, the claim is, without messy preferences, Belief Defectiveness follows from Bet Defectiveness, since without value interference, bets that are individually acceptable will also be acceptable in combination. Once we get to Belief Defectiveness, the rest is just math! Add in the Dutch Book Theorem and you get Simple Agent Probabilism.
This argument assumes that messy preferences are the only things that can block the move from defects in bets to defects in beliefs, but they clearly are not. Here is where holism about rationality conflicts with probabilistic coherence requirements. Christensen claims that the bet described in the last paragraph is one that, given your degree of belief you should regard as fair. We should ask, however, whether this is a prima facie or ultima facie “should”. As an ultima facie claim, it won’t be true unless messy preferences are all that need to be controlled for to line up attitudes about fair bets with degrees of belief.
The point to note, however, is this: it is not only preferences that can be messy, but beliefs as well. Suppose I am a strongly puritanical Calvinist, believing wholeheartedly in the work ethic of such a view, and holding that divine anger will be visited on any who try to benefit in ways outside of honest toil. My degree of belief in p might be 2/3, but I won’t regard the bet in question as fair; in fact, I’ll regard every bet offered as an unfair bet. I will think any bet of any sort will work to my disadvantage. Or, again, suppose I’m convinced that bets are quite often finkish: that accepting them triggers a change in the prospects of winning. So, no matter what bet we’re considering, I can’t regard any bet as a fair bet, because there aren’t any such bets: fairness of a bet, I also believe, requires that no party to the bet will have a special advantage over the other party, and that viewpoint is one we can’t ever accept. Or, once more, suppose I can’t tell what my degree of belief is and I have no idea on reflection what the chances of p are. You ask me whether a bet on p that costs $2 and pays $3 if p is true is a fair bet. I have no idea. Perhaps I’ll reflect further and come to some decision in some way or other, but my reflection may lead to either answer compatible with my inaccessible degree of belief being 2/3.
In each of these cases, nothing said contravenes the assumption that the agent in question is a simple agent. What is messy here are not the preferences of the agent, but rather the beliefs of the agent. Moreover, in each case, there is no reason to view the additional features of the agent’s perspective on the world to be out of order or inadmissible or illegitimate in a way that would allow one to ignore such complexities. In each case, the point of the example is to show that a proper understanding of the agent’s perspective on the world requires more, and perhaps less, than an account of some particular degree of belief they have. Rationality is perspectival in this sense, and it is a mistake to think that the degree of belief in question is all that needs to be noted in characterizing the perspective in question.
Once these points are noted, we can only accept a prima facie version of the Sanctioning principle:
PFS (Prima Facie Sanctioning): A simple agent’s degrees of belief prima facie sanction as fair, i.e. give one a defeasible reason for regarding as fair, monetary bets at odds matching his degrees of belief.
PFS, however, is not strong enough to sustain Simple Agent Probabilism, except when the defect of suffering from probabilistic incoherence is itself taken only to be a prima facie defect.