It’s time for another post on Huber and Schmidt-Petri’s Degrees of Belief book! This time I am looking at James Hawthorne’s chapter: The Lockean Thesis and the Logic of Belief. Incidentally, the Degrees of belief book category will house all the posts in this series.
Last time we saw that there is a problem with the “Lockean thesis” which bridges the gap between categorical (full) belief and degrees of belief. This problem was cashed out with degrees of belief understood as being modelled by probabilities. Hawthorne takes a slightly different tack, having degrees of belief modelled by a qualitative belief ordering. That is, we define a relation between the objects of belief (events, propositions, whatever) such that $X \succeq Y$ is read as meaning “$X$ is at least as likely as $Y$”.
Hawthorne then builds up a logic of qualitative belief. There are a number of plausible sounding axioms for qualitative belief orderings, though many of these are well known in the literature. He then suggests a way of modelling acceptance within this qualitative framework. Recall that acceptance in the probabilistic framework amounted to degree of belief surpassing some threshold. Hawthorne takes the same route out of the problem with the Lockean thesis as did Foley. That is, he too rejects the conjunction rule. However, he goes further than Foley in describing, in detail, the logic of accepted beliefs that arises out of his understanding of acceptance.
The trick is basically to satisfy only a restricted version of the conjunction rule. That is if you believe each of some set of propositions, you are only required to believe $n$-fold conjunctions of them but not $n+1$-fold conjunctions. I am playing very fast and loose with what Hawthorne actually says, which is much more sophisticated, because I don’t really think it’s worth going through the details in this blog post. I refer the interested reader to Hawthorne’s paper.
There is quite a bit of research on various kinds of logic of acceptance, so I don’t really want to say much more about this in case I say something foolish.
But I do want to pick up on one aspect of Hawthorne’s presentation that I particularly liked. There are well-known theorems in the literature that prove that certain kinds of qualitative belief ordering can be represented by a unique probability measure. Savage’s Foundations of Statistics has one such theorem. Krantz et al.’s Foundations of Measurement, Volume I has another. Terence Fine’s Theories of Probability has another. They are all very similar. They all share the same two flaws, as I see it. First, the relation is assumed to be complete: that is, any two elements are comparable. I find it reasonable that you might entertain two propositions, but be unable to say anything about which of the two you consider more likely. Second, the space of propositions is required to be uncountably infinite in order to secure the uniqueness of the representation. This “richness” assumption feels artificial and unwarranted.
Hawthorne makes it very clear that for him, completeness and richness are not requirements of rational belief, but rather, requirements of extendibility. To take the case of completeness, Hawthorne says that incomplete belief orderings are perfectly reasonable. However, it would be unreasonable for an agent to have a preference that could not be extended into a complete one. He says that there being no complete extension of your belief ordering points to an implicit incoherence. That is, if $X$ and $Y$ are incomparable, and there is no complete extension of the relation, then neither of $X\succeq Y, Y \succeq X$ is even possible for you. This seems to be incoherent. So Hawthorne requires only that the relation be completeable. He does the same sort of thing for richness. This I think, is a very sensible way of thinking about the less intuitive axioms of representation theorems. The idea of completeness and richness as conditions of extendibility isn’t, I think, new to Hawthorne, but I do think his presentation of it is excellent. I think Hawthorne does a good job of putting completeness in its place.