It occurs to me that there is a mistake in my previous post, but it can be patched up.
To review: Considerations of inductive or ampliative risk can make the difference between it being appropriate to believe something and it being inappropriate. If the stakes are high, then you might demand more evidence than if the stakes are low.
Schematically, what’s relevant are conditional values: the benefit of believing P if it is true, the cost of believing P if it is false, the cost of not believing P if it is true, and the benefit of not believing P if it is false.
In the previous post, I reconstructed this as a payoff matrix like this:
|ampliative risk||P is true.||P is False.|
|You believe that P.||+ A||– B|
|You do not believe that P.||– C||+ D|
The problem is that the lefthand column is not well-defined.
A decade ago, in a paper on Thomas Reid’s reply to scepticism, I made this point against Peter Baumann’s attempt to construct a payoff matrix of just this kind. A propos Reid, let P be there is an external world. What would be the value of believing P or behaving as if P, on the assumption that not-P? That is, what’s the value of B in the payoff matrix? There is not way to fill in a meaningful value! Just assuming not-P isn’t enough to determine what any consequences would be. In the paper, I said that it’s like filling in a cell of the matrix with a number divided by zero. You can’t reason about utilities like that.
In philosophy of science, the point is often made by saying that there is no well-defined catchall hypothesis. One can reckon with determinate hypotheses, but the open-ended none of those, but something else isn’t a workable hypothesis.
This means that ampliative risk cannot, in general, be expressed as a decision-theoretic problem. It’s still fine to interpret James as appealing to considerations of ampliative risk, though, for at least two reasons:
First, James never wrote it in terms of a payoff matrix. The problem here is with a formal reconstruction of his argument, rather than with his argument itself.
Second, James has resources to rule out a general catchall hypothesis. In “The Will to Believe”, he only considers options between live hypotheses— viz., those that occur to one as at least potentially credible. It’s a personal, psychological matter which hypotheses are live for any given person, but the fact that a catchall isn’t a real hypothesis means that it’s not a live hypothesis for anyone.