Are credences beliefs about probability?
There's a lively literature on how graded beliefs (credences) relate to ungraded beliefs. Some hold that ungraded beliefs are reducible to credences, others that credences are reducible to ungraded beliefs; yet others hold that both are real and irreducible to one another. I find this debate confusing. By which of course I mean: confused. I'll try to explain why. I'll also comment on one particular proposal, that credences are beliefs about probability; an interestingly new version of this has just been defended in Buchanan and Dogramaci 2026.
First, why I find the debate confusing. Imagine a parallel debate about desire. Desires patently vary in strength: we desire some things more than others. In decision theory, an agent's desires are represented by a utility (or "desirability") function that assigns numbers to propositions; the higher the number, the stronger the desire. Now imagine someone asks how these graded desires relate to ungraded desires: are graded desires reducible to ungraded ones, or the other way round?
I hope it's clear why this would be confused. There are no ungraded desires! Every desire has a (comparative) strength; the utility function represents this strength.
We might still ask how the ungraded concept (or word) 'desire' relates to the graded concept of a utility function. In particular, can we analyse simple desire reports in terms of the subject's utility function? If so how? These are good questions. (They are explored, for example, in Phillips-Brown 2021 and Blumberg and Hawthorne 2022.)
You might say that the case of belief is importantly different: while desires vary in strength, beliefs do not. But is that true? It seems false to me. Surely it can't be taken for granted.
So we shouldn't assume that there is any such thing as ungraded belief. What we have is an ungraded concept (or word) 'belief'. We may ask how this concept relates to the concept of credence. Can we analyse belief reports in terms of the subject's credence function, or the other way round?
So that's a meaningful question. Buchanan and Dogramaci 2026 suggest that credence reports can indeed be analysed in terms of (ungraded) belief reports. Their proposal follows an earlier proposal in Moon and Jackson 2020, according to which 'S gives credence x to p' means 'S believes that the evidential probability of p conditional on S's evidence equals x'. To avoid some counterintuitive consequences of this proposal, Buchanan and Dogramaci suggest that we shouldn't fix the evidence to be the subject's evidence at the relevant time. We shouldn't fix the evidence at all. According to Buchanan and Dogramaci, 'S gives credence x to p' means 'S believes that the probability of p is x', where 'the probability of p is x' expresses an "assessment-sensitive proposition" (in the sense of MacFarlane 2014) whose truth-value varies not just from world to world, but from point of assessment to point of assessment. More precisely, what matters about points of assessment is the evidence available at them. Formally, 'p has probability x' is true relative to a world w and evidence E iff the evidential probability (at w) of p conditional on E is x.
This may look strange at first, but the basic idea is well-known in the semantics of epistemic modals. Consider 'might'. Whether something might be the case depends on the available information; this information parameter can be shifted by operators. Thus 'S believes that might p' expresses (roughly) that p is compatible with S's information (not: with the speaker's information). To formalise this, we have to make the truth-value of 'might p' sensitive to a (shiftable) information parameter. (Assessment-sensitive accounts add that when the parameter isn't shifted, it is generally fixed by the context of assessment, rather than the context of speech.) Since 'probably' is also an epistemic modal, it is natural to extend this treatment to 'the probability of p is x'.
But let's return to that information parameter. Buchanan and Dogramaci suggest that the information parameter can be identified with an evidence proposition: 'the probability of p is x' is true relative to E (and a world w) iff the evidential probability (at w) of p conditional on E equals x. (Whether evidential probability is world-relative is an interesting question that we don't need to get into.) For the project of analysing credence statements, this leads to a fatal problem.
The Bayesian concept of credence is supposed to measure an agent's comparative confidence, showing up, for example, in betting behaviour and other choices. How this quantity relates to evidential probability is an open question on which the founding fathers of Bayesianism were divided. Carnap (and arguably Laplace) thought there is a connection: ideally, an agent's credence in p should equal the extent to which p is supported by their evidence. Ramsey, de Finetti, Savage and Jeffrey disagreed. But even for Carnap, the link between credence and evidential probability is normative. No Bayesian ever thought that it is analytic. But that's what the proposals in Moon and Jackson 2020 and Buchanan and Dogramaci 2026 would entail.
We can bring out this problem by thinking of cases that put pressure on the normative link between credence and probability. In Schwarz 2025, I describe scenarios in which, I claim, a rational agent should give high credence to some proposition p even though they know that the proposition has low evidential probability (conditional on their evidence). It doesn't matter if I'm right. What matters is that the position I'm defending is intelligible. We can imagine an agent whose credence in p is high (as revealed by their verbal and non-verbal behaviour: they accept bets on p, they say that they're confident in p, etc.), but who also judges that p has low evidential probability (conditional on their evidence and any other relevant evidence). If there was an analytic connection between credence and beliefs about evidential probability, this kind of scenario would be unintelligible.
There's a lesson here for the semantics of epistemic modals. In the scenarios I describe, the agent might well say 'probably p'. So the information parameter relative to which epistemic modals like 'probably' are evaluated can't be identified with an evidence proposition. It has to be represented by a full probability measure (or a set of such measures). This is, in fact, a fairly standard assumption in the semantics of epistemic modals. (See, for example, Yalcin 2007 or Stalnaker 2014.)
With this change, we can rescue the proposal in Buchanan and Dogramaci 2026. Recall that, according to B&D, 'p has probability x' is true or false relative to (a world and) an evidence proposition: 'p has probability x' is true relative to E iff the evidential probability of p given E is x. Let's drop the reference to evidence and evidential probability. We can say that 'p has probability x' is true relative to a probability measure iff that measure assigns x to p.
So the following version of B&D's proposal survives the problem I've raised: 'S gives credence x to p' means 'S believes that the probability of p is x', where 'the probability of p is x' is true relative to a probability measure iff that measure assigns x to p.
I have some quibbles about the suggested equivalence, but I'm willing to grant it for the sake of the argument. So we can translate credence reports into belief-about-probability reports. But have we thereby reduced the concept of credence to an ungraded concept of belief?
Buchanan and Dogramaci assume that we have. They assume that the reduction of credence to belief succeeds as long as we can translate credence talk into belief talk. But what if the relevant belief talk can only be understood in terms of credence?
I've granted that 'S gives credence x to p' is equivalent to 'S believes that the probability of p is x'. But what is it to believe that the probability of p is x? What kind of state is this? It's not an ordinary belief. Ordinary beliefs represent the world as being one way or another. This one doesn't. What if the state we attribute with 'S believes that the probability of p is x' can only be understood with the concept of credence?
I have two more complaints.
Remember that 'credence' has a well-established use in decision theory and Bayesian epistemology. The question is whether we can analyse this concept in terms of ungraded belief. A positive answer would lead to a strange view.
Ungraded beliefs play no role in Bayesian decision theory. What an agent should do is a matter of their credences and utilities. If these credences are a sub-class of beliefs (namely, beliefs about probability), we get a picture on which only these beliefs (beliefs about probability) figure in the explanation of rational behaviour; beliefs that are not about probability are idle wheels in an agent's cognitive architecture. This looks unappealing. But if we give other ungraded beliefs a role in decision theory, we have to rework decision theory; the beliefs-about-probability won't play the standard decision-theoretic role of credence. That is, the concept of credence that will be reduced to that of belief won't be the concept we find in standard decision theory.
My last complaint. Credence functions assign numbers to propositions, but what they represent – an agent's credal state – isn't essentially numerical. That we measure credal states with numbers between 0 and 1 is obviously a matter of convention. We could just as well use numbers from 0 to 100. We could also use a non-probabilistic representation; for many purposes, log-probabilities (or negative log-probabilities) are actually more convenient than probabilities. In any case, all these numerical representations measure something that isn't intrinsically numerical. What is that something?
An attractive idea is that credence functions measure comparative confidence (much as utility functions measure comparative desire). This approach is nicely defended in another new paper, Bowen 2026. Here we take as given a relation of "more confident than". If this relation satisfies certain formal constraints, it is representable by a unique probability measure, which we call the agent's 'credence'. Personally, I think we need to say more about what "more confident than" means, ultimately bringing in dispositions to act. But never mind those details. The point is that this whole idea makes no sense if credences are beliefs about probability. The theory of measurement explains how certain qualitative, relational facts can be represented by numerical functions. It doesn't explain how such facts could be represented by ungraded beliefs about a numerical function.
So no, credences are not beliefs about probability.