# Richard Foley: Belief, Degrees of Belief and the Lockean Thesis

Here, finally, is the first of a series of posts on the substantive chapters of Degrees of Belief. First up is Richard Foley’s chapter on “Belief, Degrees of Belief, and the Lockean Thesis”. This is the first of three chapters on the relationship between degrees of belief and “full” or “categorical” belief.

This paper is pretty short, and really serves as an introduction to the later two papers in this first part. As such, I will use it as an excuse to introduce the problem with the degrees/categorical relationship and the “Lockean thesis”.

It seems there are two ways we use the term “belief”. Sometimes we talk about beliefs as being graded: “I believe $X$ more than I believe $Y$.” This way of talking lends itself to a representation in terms of degrees of belief. This will typically mean something like probability theory. In what follows I will assume this. The other way of using “belief” is just as a flat out, categorical concept: “I believe $X$.” This lends itself more to a logical formalisation, like doxastic or epistemic logic.

How are these two ways of thinking about belief related? A standard view is to take categorical and graded belief to be related by what is known as the “Lockean thesis”. This is the idea that categorical belief is just sufficiently strong graded belief. That is, there is some kind of threshold and if a proposition is believed more strongly than this threshold, then it counts as a categorical belief.

The preface and lottery paradoxes show that at least the naïve versions of this thesis are faulty. I’ll just show the lottery paradox, but the preface works in much the same way. There is a lottery with 100 tickets. Exactly one ticket will win. You believe to degree $0.99$ that ticket $T_1$ will not win. Let’s say that this is a strong enough belief that it surpasses the threshold. (If it doesn’t, just make the lottery bigger). Likewise for any other ticket $T_i$, you believe (categorically) that it won’t win. A plausible norm for categorical beliefs is that they should be closed under conjunction. That is, if you believe $X$ and you believe $Y$, then you believe $X \land Y$. Thus, you believe the conjunction of all the propositions that claim that a single ticket will not win. That is, you believe that no ticket will win. But this is in contradiction with another belief, namely that one ticket will win.

We made no particular assumptions about the threshold for full belief in the above argument, except that the threshold is less than 1. This is certainly plausible.

So, something has to give. Foley suggests a number of responses to this problem. First, we could just give up on the epistemology of categorical belief and do everything in terms of degrees. Then there is no problem. Foley dislikes this solution for reasons that I shall return to in a moment. We could deny that the Lockean thesis is the right “bridge law” to connect categorical and graded belief. This is an awkward claim to make, since the thesis seems pretty intuitive. Finally, Foley’s preferred response is to deny that categorical belief should be closed under conjunction.

There is a worry with Foley’s preferred escape route, and one that he acknowledges. It seems that conjunction-closure is important to deductive reasoning. Unless you want to end up in some kind of “What the tortoise said to Achilles” type regress, you had better grant conjunction-closure. Foley accepts this, but argues that the relevant mental states for deductive reasoning are not beliefs, but “presuming, positing, assuming, supposing and hypothesizing” (p. 41). So belief’s not being closed under conjunction is no problem for deductive reasoning.

What of the solution in terms of just doing away with categorical belief? Can’t we just do everything with degrees of belief? Foley argues that this does violence to many of our uses of belief. For example, Foley suggests that for any kind of model of graded belief to get off the ground, various background assumptions have to be taken as (categorically) believed. That is, if we have a set up with certain numbers of coloured marbles in labelled urns, to have graded beliefs about the colour of the next marble drawn requires that we take some facts about the set-up as believed without qualification. This is a very interesting point, but I wonder whether it is conflating two different things: there are the consciously held (graded) beliefs, and then there is the perhaps subconsciously held (possibly categorical?) background information. We all know about reference classes and that probabilities only really make sense relative to some particular algebra of events. But this kind of background information is different in kind from the graded beliefs that are consciously reasoned about. I’d argue that this distinction lessens the strength of Foley’s argument here.

Continuing in this theme would be to stray from the point, but I do hope to come back to this idea at some stage: I will need to flesh out this idea for my thesis.

Foley also argues that if every opinion were graded, things would be very complicated. Categorical beliefs are easier to assimilate and reason with. He says:

[In] expository books and articles, in reports, in financial statements, in documentaries, and in most other material designed to transfer information, we want much of the information delivered in black-and-white fashion. We want definite yes or no statements. (p. 45)

This may be true, but I’m not sure it speaks to the point. We may want things to be easy, but that doesn’t mean that the easy things are somehow privileged. I may want machine-readable PDFs with proper metadata attached, but that doesn’t mean I need a separate epistemology for those kinds of PDFs.

In summary, Foley’s paper outlines the problem of two kinds of beliefs. He introduces the Lockean thesis as a way to bridge the gap; he discusses the lottery and preface paradoxes; and he outlines his preferred solution to the paradoxes. I prefer the “give up on categorical belief” option, since I find Foley’s arguments against this way out unconvincing.