Credences are Beliefs about Probabilities: A Defense from Triviality

Abstract

It is often claimed that credences are not reducible to ordinary beliefs about probabilities. Such a reduction appears to be decisively ruled out by certain sorts of triviality results–analogous to those often discussed in the literature on conditionals. I show why these results do not, in fact, rule out the view. They merely give us a constraint on what such a reduction could look like. In particular they show that there is no single proposition belief in which suffices for having a particular credence, regardless of one’s evidence. But if we allow such propositions to vary with evidence–as we should–then the results do not rule out a reduction. So, at least on this count, credences might very well just be beliefs about probabilities.

1 Introduction

A small group of theorists defend or are tempted by the thesis that credences are fundamentally ordinary beliefs about probabilities (Moon & Jackson, 2020 call the view “belief-first”; see also Lance, 1995; Hawthorne & Stanley, 2008; Weisberg, 2013; Dogramaci, 2018; Lennertz, 2021):

Credences are Beliefs about Probabilities (CBP): For every proposition, p, and real number, n, every attitude of confidence, or credence, of degree n in p is, fundamentally, a belief that the probability of p is n.

For example, my 50% credence that the cafeteria will serve Johnny Marzetti is fundamentally a belief that it is 50% probable that the cafeteria will serve Johnny Marzetti.

CBP has some advantages. It seamlessly explains ascriptions of what appear to be attitudes toward complex contents–like “Sally believes it is 50% likely that the cafeteria will serve Johnny Marzetti or they’ve changed the menu.” This is because, according to CBP, such sentences do ascribe attitudes (beliefs) with complex contents, rather than requiring inventing new attitudes for each different sort of complexity (Lennertz, 2021; Moss, 2018). CBP also illuminates the role played by credences in reasoning about what to do, given a principle where knowledge is required to rely on something in reasoning (Hawthorne & Stanley, 2008; Weisberg, 2013). Finally, since CBP locates the probability in the content of the attitude, it supports a straightforward explanation of rational norms on credences as grounded in norms on belief (Moss, 2018).¹

Despite these advantages, CBP is unpopular. It faces a number of objections.² I’ll focus on one from formal triviality results (Russell & Hawthorne, 2016; Schroeder, 2018; Goldstein, 2019, though these authors don’t take the results to directly show that CBP is false). These appear to show that if CBP and other natural assumptions are true, rational people are only ever 0% or 100% confident of each proposition; they never have intermediate levels of confidence. Since this conclusion appears unacceptable, we must drop one of the assumptions.

These triviality results take as an assumption a biconditional that appears to follow from CBP. Simplifying a bit, Moon and Jackson (2020, p. 654) suggest:

Biconditional: S has a credence of degree n in p if and only if S believes that p is n-probable.

It is easy to interpret the triviality results as refuting Biconditional. Schroeder paraphrases his conclusion as follows: “there is no possible object of credence, q, such that believing it is necessary and sufficient for having a credence of n in p, where p is any object of credence and n is any value in (0, 1)” (2018, p. 302). A key assumption in Goldstein’s proof is a transparency principle like Biconditional: “an agent believes [A is n-likely] just in case her credence in A is n” (2019, p. 192). Russell and Hawthorne state their assumption: “conditional on the claim that the probability of A is x, you should assign credence x to A” (2016, p. 313). Their interpretation of “conditional on the claim that” is to issue a conditional probability, but a nearby interpretation is “if you believe that”, which yields the right-to-left direction of Biconditional.

In this paper, I uncover two relevant readings of Biconditional. I argue that the triviality results show the following: if the first reading of Biconditional is true, rational people must only have credences with extremal or trivial values (0 or 1); but the other reading appears consistent with people having the full range of strengths of credences. Given the plausible assumption that our credences aren’t merely of trivial strengths, then the triviality results investigated in this paper show that the first reading of Biconditional is false, but they leave open that the second could be true. One thing that follows from this is a constraint on the sort of proposition that is hypothesized to be believed when one has a credence. But this constraint is consistent with the most plausible way of cashing out CBP anyway.³

2 Triviality Results

I’ll discuss Schroeder’s (2018) result in detail and then relate it to Goldstein’s (2019) and Russell and Hawthorne’s (2016). None is inconsistent with CBP.

2.1 Schroeder’s Result

After giving a pictorial version of his proof, Schroeder describes it:

Let p and q be any objects of credence and n be any value in the interval (0, 1). If q is consistent with p, then there is a way of believing q—namely, by being certain of q—which entails having a credence of 1 in p, and hence which is insufficient to have a credence of n in p. And if q is consistent with ~p, then there is a way of believing q —namely, by being certain of q—which entails having a credence of 0 in p, and hence which is insufficient to have a credence of n in p. But if q is not consistent with p and q is not consistent with ~p, then q is itself inconsistent. So there is no consistent object of credence q such that belief in q is sufficient for a nonextreme credence of n in p, for any object of credence p. (Schroeder, 2018, p. 303)

Schroeder targets a claim that is stronger than Biconditional. He argues that there is no proposition of any sort, q (whether of the form the probability of p is n or not), such that a person has a credence of n in p just in case they believe q.

Schroeder’s argument is sound, but it doesn’t refute CBP. I’ll formally reconstruct the beginning of his proof in a natural deduction system, which will help us see this clearly, as it highlights the key issue: quantifier scope.

I think that the most helpful way to reconstruct Schroeder’s reasoning is by considering three possibilities for the relationship between p and q (though this isn’t precisely how he presents it in the above quotation): they are consistent, they are inconsistent though q is not a contradiction, and q is a contradiction. Formalizing the first possibility is sufficient for diagnosing why the proof doesn’t refute CBP (Appendix A outlines the full proof). I underline names to distinguish them from variables. And for a person, S, and credence function, C, I use C^S to refer to C as the credence function that represents S. To streamline, I appeal to a proposition (Consistency_a) and an inference rule (Certainty Entailment)⁴ that are not axioms of probability theory but follow from them, as well as a substantive philosophical assumption (Certainty to Belief):⁵

We can see informally how the rest of the proof goes by noting why line 21 is counterintuitive. If supposition 1 were true, the proposition belief in which would be sufficient for a credence of n in p would, itself, be consistent with p. For if it were inconsistent, then we would be able to infer that p is false from our belief in that proposition. But then we shouldn’t have a nonzero credence in p. In this manner, Schroeder’s total proof refutes the initial reductio assumption (I here ignore the possibility of rejecting the other assumptions).

Again, Schroeder’s proof is sound. But note how it proceeds, fixing q before the relevant credence function. This shows that we can’t take a proposition, p, and degree, n, and find another proposition, q, such that belief in q is necessary and sufficient for having credence of degree n in p, no matter what our rational credence function is. But it does not show that the following is impossible: fix p, n, and a credence function and then find a proposition q such that believing q is necessary and sufficient for having credence of n in p, given that credence function. That is, it doesn’t show that we can’t, if we fix the credence function first, find a proposition, belief in which is sufficient for credence n in p.

Here’s another way of putting my point: this proof shows that there is no proposition, the probability of p is n, full stop. For any such candidate proposition, belief in it doesn’t suffice for having a credence of n in p, for every credence function.⁶ But this doesn’t show that there aren’t many propositions, the probability_C of p is n, one of which is believed each time a person has a credence of n in p.⁷ Indeed, it seems eminently plausible that one can utter “the probability of p is n” in one context and express something true while uttering it in another and express something false. One who, at t₁, knows nothing other than that a two-sided coin will be tossed and says, “the probability of heads is 50%” says something true. If they learn at t₂ that the coin is biased toward heads and, again, say, “the probability of heads is 50%”, they say something false. A plausible explanation is that sentences of the form, “the probability of p is n”, are context-sensitive. The speaker at t₁ uses this sentence to express the proposition that the probability, given their evidence at t₁, of heads is 50%. At t₂ they use it to express the proposition that the probability, given their evidence at t₂, of heads is 50%. These are different propositions, since one is true and one is false. And it is possible that for each time, believing the proposition that the probability, given one’s evidence at that time, of heads is 50%, is, at that time, having a credence of 50% that the coin lands heads. Nothing about the result above rules this out.

A third way of viewing my point appreciates that Biconditional is not explicit about the binding of variables:

Biconditional: S has a credence of degree n in p if and only if S believes that p is n-probable. (Moon & Jackson, 2020)

We can distinguish (at least) two readings⁸:

Biconditional_f: For any person, S, proposition, p, and real number, n, there is a proposition, that p is n-probable, such that for any credence function, C, C^S(p) = n if and only if S believes that p is n-probable.

Biconditional_v: For any person, S, proposition, p, real number, n, and credence function, C, C^S(p) = n if and only if there is a proposition, that p is n-probable, such that S believes that p is n-probable.

The f subscript of Biconditional_f indicates that the proposition targeted for belief is fixed prior to the agent’s credence function; the v subscript of Biconditional_v indicates that that proposition can vary with the credence function. Biconditional_f implies the reductio assumption in Schroeder’s proof, if we treat p is n-probable as a name for a proposition (it is stronger than Schroeder’s reductio assumption which considers only 0 < n < 1). However, supposing that “the probability of p is n” expresses different propositions in different contexts, it is the only weaker Biconditional_v that follows from CBP:

CBP: For every proposition, p, and real number, n, every attitude of confidence, or credence, of degree n in p is, fundamentally, a belief that the probability of p is n.

So, the proof, though sound, doesn’t show that our target thesis is false (or trivialized).

Could Schroeder’s proof be salvaged with a reductio assumption implied by the weaker Biconditional_v? No. It is instructive to see the difference between Biconditional_v and Biconditional_f to see what would happen if we tried:

The original proof made a supposition for existential elimination before instantiating the credence function with C. But this isn’t possible here. There are some false starts we could take if we were to, at the next step make an assumption of C ^S(p) = n so that we could use biconditional elimination, but none leads to the result we want. So, the original proof shows that there is no proposition belief in which suffices for a credence of n in p (for 0 < n < 1), regardless of the agent’s credence function. But we cannot show that for each particular credence function, there is not a proposition belief in which suffices for having that credence function assign n to p (for 0 ≤ n ≤ 1).

A toy model illustrates how Biconditional_v is consistent with assigning non-trivial credence values to simple propositions. Take a toy domain of 3 worlds, w₁, w₂, w₃. Suppose an initial rational credence function, Cr, makes every world equiprobable: Cr(w₁) = Cr(w₂) = Cr(w₃) = 1/3. Suppose that p is true in w₁ and w₂, and false elsewhere. Now suppose that S learns some piece of evidence. Biconditional_v implies that S’s credence in p is one-half if and only if there is a proposition, that p is ½-probable, such that S believes that p is ½-probable. In this case there are two things that the evidence learned might have been that would yield that S has a credence of one-half in p. So there are two propositions which, if S came to believe them, would suffice for Cr_{S’s evidence}(p) = ½. The evidence could be the information that w₂ isn’t the actual world (E₁), in which case we have:

p is ½-probable₁ = {w₁, w₃}.

Or the evidence could be the information that w₁ isn’t the actual world (E₂), in which case we have:

p is ½-probable₂ = {w₂, w₃}.

Suppose S is rational and has gained some evidence. If S either comes to believe p is ½-probable₁ or comes to believe p is ½-probable₂, then Cr_{S’s evidence}(p) = ½. And if Cr_{S’s evidence}(p) = ½, then S will either believe p is ½-probable₁ or believe p is ½-probable₂. Thus we have a case where Biconditional_v⁹ is true, at least for simple propositions, but S has a credence in p that is neither 0 nor 1 – so, is not trivialized.

More programmatically, credences, or probabilities, evolve with evidence. We need to track this evidence in advocating for or arguing against theses about the connection between belief and credence. As evidence evolves we are either dealing with new probability functions or previous ones conditionalized on more evidence.¹⁰ Even before seeing Schroeder’s result, we should have thought it surprising to find a proposition belief in which is necessary and sufficient for a credence of n in p regardless of one’s credence function and evidence.

2.2 Other results

Goldstein proves multiple triviality results. The one that is similar to Schroeder’s relies on the following thesis (2019, p. 192):

Belief Transparency: For any proposition, p, and real number, n, where C represents an agent S’s rational credence function: S believes p is n-likely if and only if C(p) = n.

I detail Goldstein’s proof in Appendix B. One way of viewing why his result holds but doesn’t trivialize CBP is that in his quasi-logical language, “p is n-likely” expresses the same proposition in every context. But the result doesn’t hold if we drop that assumption. Here’s another way of viewing it: Belief Transparency is an alternative formulation of Biconditional, and as with Biconditional, there are two possible readings, one where p is n-likely picks out a proposition that is fixed just by p and n, Biconditional_f, and another where it varies based on the credence function of the relevant agent, Biconditional_v:

Belief Transparency_f: For any proposition, p, and real number, n, there is a proposition, p is n-likely, such that where C represents an agent S’s rational credence function: S believes p is n-likely if and only if C(p) = n.

Belief Transparency_v: For any proposition, p, and real number, n, where C represents an agent S’s rational credence function, there is a proposition, p is n-likely, such that S believes p is n-likely if and only if C(p) = n.

Like Schroeder’s, Goldstein’s result trivializes Belief Transparency_f, but not Belief Transparency_v–or the corresponding Biconditional_v. And, again, Biconditional_v is consistent with CBP.

Russell and Hawthorne prove a plethora of triviality results. The closest to our focus trivializes the following principle (2016, p. 313):

Exact Probability: For any proposition, p, real number, n, and rational credence function, C, if C(the probability of p is n) > 0, then C(p | the probability of p is n) = n.

This principle is not as obviously related to CBP, but it still causes trouble for it as follows: Suppose credence 1 is sufficient for belief. Then the consequent of Exact Probability implies that a rational person who has credence 1–and, so, believes–that the probability of p is n will have credence of degree n in p. So, if Exact Probability were trivialized, then CBP would be too for any beliefs of which a person is certain.

Exact Probability also requires the contentious assumption that there is a single object of credence expressed by the quasi-logical sentence “the probability of p is n”. As with our other results, we can distinguish different readings:

Exact Probability_f: For any proposition, p, and real number, n, where 0 < n < 1, there is a proposition, the probability of p is n, such that for any rational credence function, C, if C(the probability of p is n) > 0, then C(p | the probability of p is n) = n.

Exact Probability_v: For any proposition, p, real number, n, where 0 < n < 1, and rational credence function, C, there is a proposition, the probability of p is n, such that if C(the probability of p is n) > 0, then C(p | the probability of p is n) = n.

As discussed in Appendix C, Russell and Hawthorne’s result follows from Exact Probability_f, whereas Exact Probability_v, in which what is meant by “the probability of p is n” is not fixed, is consistent with CBP.¹¹

2.3 Taking Stock

I’ll draw three main lessons from these formal results. (1) There is no single proposition that we might call the probability of p is n, which a person believes if and only if they have credence (based on any rational credence function with any evidence) of n in p. Nonetheless, (2) CBP may still be true (non-trivially). Finally, (3) scenarios that witness the consistency of CBP and the triviality results have a particular character. For each credence of n in a proposition, p, there is some proposition, q, such that a person’s credence of n in p is their belief in q–in a way that q varies based on the person’s credence function and evidence.

This third lesson can look unsatisfying. If one has a credence of n in p, then we can find some q belief in which serves as a proxy for that credence.¹² But, as intended, q is not just a random proxy. Rather it is a proposition about the probability of p given the relevant probability function and evidence in that context. This seems exactly the sort of thing that a credence might be. More specifically, we can say that a credence of n in p is a belief that the probability of p is n, given your evidence.¹³ This allows two people to have a credence of the same strength in a given proposition, where, for each person, that credence is a belief in a different proposition. For example, consider a case where Fred knows that a fair coin was flipped and has landed, but knows nothing else. Fred’s credence of 0.5 that the coin landed heads is a belief that the probability that the coin landed heads given his current evidence is 50%. Contrast this with a case where Sally knows that a coin was flipped and there is a testifier who saw the toss and tells Sally it landed heads, but who Sally knows tells the truth exactly 50% of the time. Sally’s credence of 0.5 that the coin landed heads is a belief that the probability that the coin landed heads given her current evidence is 50%. Since Fred’s and Sally’s bodies of evidence are different, this view makes the plausible prediction that they have different beliefs that are sufficient for their credences (in contrast with Biconditional_f).¹⁴ This way of viewing the issue also makes sense of why, in intrapersonal cases, the proposition that serves as a proxy for a credence of n in p might change as a person’s evidence changes. Either the probability function that proposition is about or the evidence that the probability function is conditioned on changes. This is consistent with our ordinary conception of credence functions: when a rational person gains evidence, either the function that represents their credences changes or, if it stays the same, it is understood as updated by that evidence. It would be counterintuitive to think that belief in the same proposition would occur whenever one had a particular credence in a proposition despite those evidential changes, as in Biconditional_f.¹⁵ Updating the proposition to be about current evidence facilitates beliefs about such propositions playing the role that credences do in reasoning, decision-making, and action. In such cases, it is almost always the probability of propositions given one’s current evidence that grounds the practical relevance of our attitudes about them.

3 A Single Proposition?

In this section, I’ll entertain objections to this CBP-consistent picture of credences. These objections are to this particular picture; they are independent of the triviality results, which, as we’ve seen, don’t by themselves show that CBP is false.

My defense of CBP relies on there being many propositions about the probability of p being n, which may involve different probability functions or probability functions conditionalized on different evidence. This might appear to treat our belief states as excessively and objectionably shifty. The core of this objection can be seen from a philosophy of mind perspective and a philosophy of language perspective. I’ll discuss each in turn.

First, the philosophy of mind: when a person, say Jorge, gains some evidence, he is required, not just to change the degree to which he believes a proposition is probable, but to change the notion of probability that his beliefs are about. Earlier his belief was about what was probable given his evidence then. Now Jorge’s belief is about what is probable given his evidence now. There are two ways to conceptualize this: either his earlier and later beliefs are about different probability functions or they are about the same probability function, conditioned on different evidence. If his earlier belief is about P_E(•), his later belief is either about P_E&E'(•) or P_E(• | E').¹⁶ Either way, something beyond the strength of his thought’s degree has changed. We might think that this is an objectionably shifty way to conceive of changes in confidence.

I’ll begin my response by conceding that it would be implausible if a change in confidence required Jorge to explicitly reflect on the probability function his thought is about or to list the evidence that that function is conditionalized upon. But that is not required. Instead, Jorge might have that belief in a way that he would describe to himself as “p is n-likely given my current evidence.” Suppose that he gets some evidence but doesn’t change his confidence. He might again say to himself “p is n-likely given my current evidence.” The second time he says this to himself, the proposition expressed is a bit different. But there is no great mental energy that needs to be expended here, nor is it implausible that our thoughts evolve like this. Consider an analogy to thoughts about time. Suppose Francesca believes what she could express by saying to herself “I am sitting on the couch now” and then, ten minutes later, has a belief that she would express with the same phrase. Her beliefs at those two times seem to have different contents, but this doesn’t mean that beliefs about time are too objectionably shifty to be captured by our usual model of belief. It is not hard for us to have beliefs like this, nor is it hard for theorists to understand them. The same is true in our case, where a person’s degree of confidence changes.

Furthermore, I contend that there is nothing gained by thinking of these evolutions of mind fundamentally in terms of credences rather than beliefs, as we might if we rejected CBP. Consider Tomiko, who has some degree of credence in a proposition. The popular way of modeling her state, provided that she is rational, is as a probability function yielding that degree as its value for that proposition, as argument. But once Tomiko receives more evidence, what counts as her credence function is either a different probability function or the same function conditionalized on her updated evidence. That is, if her earlier credence function was represented by P_E(•), her later credence function will be represented by either P_E&E'(•) or P_E(• | E'). These are just the probability functions that featured in Jorge’s belief on the CBP picture. Tomiko’s state appears to have the same sort of shiftiness as Jorge’s. So, rejecting CBP gets us no further in addressing any excessive shiftiness than accepting it does. There simply is shiftiness in representing these states of mind no matter how we divide up the attitude and content. But that should convince us that the shiftiness is not objectionable.¹⁷

A different take on the shiftiness objection is associated more closely with concerns in the philosophy of language. Lewis mentions it in his original results about conditionals (1976, p. 133). It seems like two people agree if they both say, “It is 87% likely that it will rain on July 30, 2049 in central Lagos” and two people disagree if one says “It is 87% likely that it will rain on July 30, 2049 in central Lagos” and the other says “It is not 87% likely that it will rain on July 30, 2049 in central Lagos.” It is common to think that these sentences are used to express credences of 87% and something other than 87%, respectively, that it will rain on July 30, 2049 in central Lagos. It would be easy to explain this agreement and disagreement if there were a single proposition belief in which sufficed for credence of 0.87 that it will rain on July 30, 2049 in central Lagos (and one which sufficed for a credence of some other degree in that proposition). But we’ve seen that there can be no such proposition. And the agreement and disagreement don’t look easily explicable on a picture where the two people who appear to agree utter the sentence based on different evidence and, so, express different propositions (similarly for disagreement). Thus, the way we’ve developed CBP appears unable to explain agreement and disagreement.¹⁸

The debate over this issue has raged in the literature on epistemic and deontic modals in the last couple decades. I can’t fully engage the range of possible responses here. Some theorists question whether there truly is disagreement in cases like the one I’ve described,¹⁹ but I’m going to suppose there is some genuine disagreement and lay out two popular and attractive paths for answering the objection.

The first suggests that even though statements about what is likely or probable are shifty, in these cases of agreement and disagreement, the two interlocutors are focusing on a common parameter–as would be so if we both use “that” while pointing at the same object. We might, when speaking with another person about what is likely or not, presuppose that our evidence and probability functions are relevantly similar (López de Sa, 2008). My assertion that it is 87% likely, given my evidence, that it will rain on July 30, 2049 in central Lagos is inconsistent with your assertion that it is not 87% likely, given your evidence, that it will rain on July 30, 2049 in central Lagos, given the presupposition that you and I share the relevant evidence and probability function. A different proposal of this general kind holds that speakers are talking about some evidence and probability function that will be obtained before it needs to be used to solve the relevant question or problem (Yanovich, 2014). Thus, what I really assert is that it is 87% likely, given the evidence that will be obtained before we need to act on these topics, that it will rain on July 30, 2049 in central Lagos, and what you really assert is it is not 87% likely, given the evidence that will be obtained before we need to act on these topics, that it will rain on July 30, 2049 in central Lagos. These assertions are inconsistent. A very similar proposal, which yields the same kind of solution, is that these assertions are about what the shared evidence will be after the participants take on board the very utterances being made (Mandelkern, 2020). All of these suggestions explain agreement and disagreement in virtue of what is asserted (perhaps together with what is presupposed).

The other popular and attractive path for dealing with the objection concedes there is no single proposition that is asserted by the first speaker and denied by the second, as the simple version of the shifty account suggests. Nonetheless, the interlocutors might agree or disagree based on something else pragmatically conveyed by uttering those sentences in the context. For instance, a speaker might ‘put in play’, without committing to, a proposition that the hearer responds to–perhaps it is 87% likely, given the conversational group’s evidence, that it will rain on July 30, 2049 in central Lagos (von Fintel & Gillies, 2011). Or a speaker might suggest to degree 0.87 that it will rain on July 30, 2049 in central Lagos and the hearer may disagree with this act of suggestion (Montminy, 2012). Finally, it might be that a speaker expresses a stance of taking it as 87% likely in their reasoning and deliberation that it will rain on July 30, 2049 in central Lagos and the hearer refuses to take up that stance (Lennertz, 2014).

4 Conclusion

I want to close by refocusing on my goal. I aimed to show that the triviality results we’ve explored don’t refute the claim that credences are fundamentally beliefs about probabilities (CBP). In Sect. 2, I showed that these results don’t by themselves rule out CBP. Nonetheless, they do show that there is no single proposition belief in which is necessary and sufficient for having a particular credence. That leaves us with a view that appears consistent with CBP which says that as a person gains evidence, the proposition belief in which constitutes their credence changes. In Sect. 3 I explored a couple of ways of objecting to the shiftiness of this view, but I suggested that none of these objections are decisive, drawing on the large literature on these questions to chart possible ways forward. Importantly, the main goal of this paper–to show that the considered triviality results are consistent with CBP–holds independent of any particular defense from Sect. 3 of the shifty view.

Appendices

Appendix A

Here is a reconstruction of Schroeder’s entire proof. In addition to what we saw in the text, I assume an instance of a probability axiom, Additivity, and an instance of a truth derived from the axioms, which we can call Maximality:²⁰

The assumption in 39 seems plausible. Any candidate for the proposition, belief in which is necessary and sufficient for credence n in p cannot be a contradiction. If it were a contradiction, one could not rationally have credence n in p (for doing so would require believing a contradiction). And this would be so for every p and every n, where 0 < n < 1.

Appendix B

Here is a formalization of Goldstein’s result (2019) as a standard triviality proof, not a reductio. I’ll let C_p(•) = C(• | p). C_p(•) is a rational credence function if C(•) is. Again I use some standard truths of probability, like the Law of Total Probability and a couple of rules that I call CP and Negation CP:^21,22,23

32 trivializes Belief Transparency_f. If it is true, then rational people only have extreme credences. As in Schroeder’s proof, we can reject Belief Transparency_f but retain Belief Transparency_v. However, if we do so, we cannot derive the triviality result in this way.

Appendix C

Here is a formalization of Russell and Hawthorne’s result (2016). Again, I’m taking truths and inference rules based on probability and logic as basic, rather than proving them from the axioms²⁴:

Russell and Hawthorne don’t, themselves, draw the conclusion in 28. Instead they think that the problem is in the derivation to step 19. They explore a different way of thinking of contents in a dynamic framework. According it, either Inconsistency to Entailment (18) is not true–if statements like the probability of p is n count as propositions–or 19 does not express an instantiation of Inconsistency to Entailment – if they do not (see Russell & Hawthorne, 2016 for details).

This proof, like Schroeder’s and Goldstein’s, is sound. But, we cannot replicate this sort of derivation while relying on Exact Probability_v rather than Exact Probability_f, and CBP is consistent with Exact Probability_v:

Exact Probability_v: For any proposition, p, real number, n, where 0 < n < 1, and rational credence function, C, there is a proposition, the probability of p is n, such that if C(the probability of p is n) > 0, then C(p | the probability of p is n) = n.

We might think, however, that there should be more to say about which proposition of the form, the probability of p is n, is the one that makes C(p | the probability of p is n) = n true for a given C. A natural suggestion is that it is the proposition where probability just is the credence function that the person has at that time. So, credences about the probability of p are credences about that very credence function. I’ll write the principle as follows:

Exact Probability_r: For any rational credence function, C, proposition, p, and real number, n, where 0 < n < 1, if C(C(p) = n) > 0, then C(p | C(p) = n) = n.

Is this principle subject to a triviality result like the one for Exact Probability_f? This might seem particularly pressing since the way I’ve wiggled out of earlier triviality proofs involved disambiguating between the scopes of quantifiers that bind credence functions and propositions about probabilities. But there could be no such ambiguity in this principle. Unfortunately, I can’t guarantee that there is no result trivializing it. But I will give some license for optimism by showing that proceeding in similar ways to the triviality result for Exact Probability_f won’t yield such a result:

But we can’t use C as our proxy in 7, since we’ve already made assumptions about C elsewhere (namely that its assigning n to p is consistent with ~ p). We could choose a different proxy:

But now 8 doesn’t follow from 1 by universal instantiation, since in 1, the same variable over credence functions appears with wider and narrower scope. 1 makes no claims about the relationship between different credence functions. Perhaps we should reexamine our reductio assumption at step 3, focusing not on C but C_~p. By doing so, we don’t need to proceed by reductio; we can reason directly as follows:²⁵

It’s not hard to show that ~ p and C_~p(p) = n are inconsistent. But the remainder of the proof stalls when we must say that the probability of p is n is not a contradiction. When we cash this out in the way we decided in the proof, we see that it is a contradiction. It can’t be true that C_~p(p) = n. Again, this doesn’t mean that the English sentence, “The probability of p is n” is a contradiction on every interpretation, since this can be used to pick out any number of credence functions. But on the interpretation used in the proof, it is a contradiction. We should also recognize that this latest attempt to carry out the proof couldn’t be on the right track, since we didn’t use our reductio assumption in the proof. Even if we had been able to derive a contradiction, some other principle, not the reductio assumption, would be the culprit.