It’s sometimes said that probability theory is the logic of partial belief, meaning by that that a person whose credences do not conform to the probability calculus are incoherent in just the same way that a person whose beliefs are logically inconsistent are incoherent. (We’re setting aside for purposes of this post whether logical inconsistency is a major, or even a minor, epistemic failing. The issue is whether not being a probabilist is like being an inconsistent person, however good or bad that is.)
It seems to me that this can’t be right. In particular, it seems that Dutch Book arguments for this cannot succeed. The most we can show by Dutch Book arguments is that the non-probabilist will evaluate sets of bets in such a way that leads to them giving positive evaluation to some bets that provably have negative net value taken collectively. But this is compatible with the agent having no mistaken logical views.
Note that there are lots of kinds of beliefs an agent could have that could not be true that are not logical errors. Here are three interesting categories of such belief.
- Metaphysical – An agent who believes that water is atomic doesn’t (necessarily) make any logical mistakes even though their belief cannot be true.
- Philosophical – An agent who believes that it is impossible to do a good deed that doesn’t maximise the excess of pleasure over pain has a belief that is a priori false, but it isn’t a logical mistake.
- Mathematical – An agent who believes that 37 + 27 is 54 (say because they didn’t carry the 1 when adding) does not make any logical errors, save on strongish forms of logicism.
Which of these errors is the non-probabilist making. It is hard to see how it could be more than the last. Consider someone who engages in the following reasoning.
What should my credence in p v q be? Well, my credence in p is 0.37. And my credence in q is 0.27. And p and q are logically incompatible. So my credence in p v q should be 0.37 plus 0.27, that is, 0.54. That’s is, my credence in p v q is 0.54.
There are no logical errors in this bit of reasoning, just a mathematical error. So having non-probabilistic credences doesn’t imply making any logical mistakes, at most it implies a mathematical mistake.
That’s not to say there are no logical constraints on credences. I think there are, but they are all framed in terms of comparative probabilities, not numerical probabilities. For instance, I think the following is (akin to) a logical constraint, unlike the constraint that credences be probabilities. (C here is the credence function.)
C(p | q) > C(s)
C(p | ~q) > C(s)
So, C(p) > C(s)
A credence function that doesn’t satisfy this constraint is flawed in just the way that inconsistent beliefs are flawed. But this isn’t the way that non-probabilistic credences are flawed.