1. The epistemology of disagreement: concepts and modelling assumptions

I shall not give an overview of the epistemology of disagreement debate (for that, see the references listed in fn 3). In this section, I only want to define a term and provide some notation.

The term is “epistemic peer”, first coined by Gary Gutting (Reference Gutting1982). Intuitively, if I believe that Elizabeth Woodville was the wife of Henry VI, and my 8-year-old cousin disagrees with me about this, I need not lose confidence in my belief – for I am much more likely than he is to be correct about this historical fact. On the other hand, when a professor of British history tells me that I am wrong, I certainly must lose confidence. But what is the rational response when someone as likely as I am to be correct about Elizabeth Woodville was the wife of Henry VI disagrees with me?Footnote ⁵ Most epistemologists believe that this the interesting case – disagreement with an epistemic peer – and it has been the focus of the debate.

Kelly defines the term thus:

1. (i) they are equals with respect to their familiarity with the evidence and arguments which bear on that question, and

2. (ii) they are equals with respect to general epistemic virtues such as intelligence, thoughtfulness, and freedom from bias. (Kelly Reference Kelly, Gendler and Hawthorne2005: 174–5)

This is a typical definition (cf. Gelfert Reference Gelfert2011; Matheson Reference Matheson2015), and it will suffice for our purposes.

Of course, that disagreement with peers is of epistemic interest does not imply that disagreement with non-peers is not. Yet little attention has been paid to how one should revise one's beliefs in light of disagreement from an epistemic superior or an epistemic inferior. And the attention that has been paid to those cases (e.g. Zagzebski Reference Zagzebski2012) tends to focus on special contexts, like morality. The implicit assumption is that belief revision is warranted when one comes into contact with an epistemic superior, and unwarranted when one comes into contact with an epistemic inferior.

This will not do. Set aside that it is a rare thing to interact with people whom we can honestly say are pure epistemic peers. All our epistemic interactions take place within a complicated nexus of inferiors, peers, and superiors. The “smartest” person we know may believe X; two marginally less smart people may believe ~X; two peers may believe X while one believes ~X; and all the while four inferiors believe ~X. What to do, epistemically? As things stand now, no guidance is forthcoming.

What we, epistemologists interested in disagreement, should like to have is an approach to disagreement that incorporates not only gradations in confidence, which our norms already do, but also gradations in competence, which our norms do not.

I define some terms. The goal of the epistemology of disagreement debate is to identify the correct disagreement norm. We shall be considering a finite set of people, V = {v ₁, v ₂, …, v _n}, who have opinions about some proposition X. Although philosophers typically consider only the special case of n = 2,Footnote ⁶ we shall not so limit ourselves.

Associated with each v _i is a confidence, c _i ∈ (0, 1). We interpret a person's confidence in X as follows: As c _i approaches 1 (0), v _i approaches certainty that X is true (false). Although within the philosophical literature one more commonly sees c _i ∈ [0, 1], it is better to use the open interval. This becomes relevant for technical reasons later on, but it also makes more sense from the Bayesian point-of-view. To say that one has a confidence of 1 (0) is to say that no future evidence could shake one's belief that the proposition at issue is true (false). That is wrong even in the strongest real-world contexts. Even our beliefs about putative necessary truths might one day be undermined. Sometimes we discover that a mathematical “proof” we thought was rigorous is in fact subtly flawed.

Let us suppose, without loss of generality, that v ₁ is the one trying to decide whether or not to modify her belief in X in light of disagreement.

2. Bayesian belief revision

It is surprising to me that little effort has been devoted to tackling the core problem of the epistemology of disagreement – how to revise one's belief given the beliefs of others – with tools from Bayesian analysis. I am aware of only two broad attempts along these lines in the philosophical literature. Typically, the relevant papers argue, often convincingly, that some disagreement norm is incompatible with a Bayesian principle or its overarching philosophy. The goal of this paper, in contrast, is to provide a general Bayesian solution to the disagreement problem.

First, there are a number of persuasive arguments that conciliationism, usually understood in its “equal weight” form (the idea, roughly, that when two epistemic peers disagree, the rational thing for them to do is to “meet in the middle”), is incompatible with Bayesianism – because, for example, it violates conditionalization. (Cf. Jehle and Fitelson Reference Jehle and Fitelson2009; Lasonen-Aarnio Reference Lasonen-Aarnio2013; Levinstein Reference Levinstein2015; Shogenji Reference ShogenjiMs.)

Second, there is Easwaran et al.’s (Reference Easwaran, Fenton-Glynn, Hitchcock and Velasco2016) derivation of a disagreement norm they call “Upco” (“Updating on the credences of others”). This is the best-developed Bayesian approach to disagreement in the epistemological literature, and I shall consider it in some detail in the next section.

As we shall see, not only does a proper Bayesian strategy provide a means for updating confidences given others’ confidences (indeed, this is Bayesianism's raison d'être), it satisfies the three desiderata mentioned in the Introduction: It deals with multiple epistemic interlocutors; provides guidance for updating beliefs given disagreement from interlocutors of whatever competences; and it ensures that facts about epistemic dependence influence beliefs appropriately.

Moreover, the Bayesian strategy provides steadfast norms in those cases in which, intuitively, it is rational to “stick to your guns”. And it provides conciliatory norms for those cases in which belief revision seems right. For a good Bayesian, sometimes steadfasters like Kelly are correct; other times, conciliationists like Christensen are on the right side of things. We should not impose either norm on scenarios for which it is inappropriate – even though philosophers often try to do just that.

Now, some epistemologists, like Richard Feldman (Reference Feldman2009), have argued informally against a “one size fits all” approach to disagreement, often as part of a “total evidence” approach:

And Kelly (Reference Kelly, Feldman and Warfield2010) discusses coming to terms with multiple epistemic interlocutors and dependence through a total evidence approach. For adherents of this approach, perhaps this paper, and the Bayesian paradigm more broadly, can provide a useful formal framework for determining how one's total evidence should bear on a given hypothesis.

Our key move will be for v ₁ to regard her interlocutors’ judgments as random variables, the values of which – namely, c ₂, …, c _n – are revealed to her by v ₂, …, v _n. Then, v ₁ treats c ₂, …, c _n as data relevant to X on which she can update c ₁. Obviously this is a very different methodology than the typical, a priori approach described in the Introduction. But it is also different than Easwaran et al.’s Upco, which involves no probabilistic modelling at all, but is, rather, a function which falls out as a special case of Bayes's Law (under the assumption that epistemic interlocutors are independent conditional on X).Footnote ⁷

We begin by noting that confidence is typically interpreted as subjective probability (cf. §1):

(2.1)$$\matrix{ {c_{\rm i} = {\rm P}{\rm r}_{\rm i}( X ) \;} \cr} $$

Note that since X and ~X are mutually exclusive and jointly exhaustive of the sample space (the proposition is either true or it is false, and not both), (1 – c _i) is v _i’s subjective probability that ~X.

Next, consider the special case of n = 2.Footnote ⁸ v ₁ and her interlocutor, v ₂, are considering X (e.g. the defendant is guilty). v ₂ reports a confidence of c ₂ in X. Denoting v ₁’s post-disagreement confidence by c′ ₁, Bayes's Law provides unambiguous guidance to v ₁ about how, rationally, she should proceed:Footnote ⁹

(2.2)$$\matrix{ {{{c}^{\prime}}_1 = {\rm Pr}( {X\; \vert \; c_2} ) = c_{1\;} \times \; \displaystyle{{\Pr ( {c_2\; \vert \; X} ) } \over {c_{1\;} \times {\rm \;Pr}( {c_{2\;} \vert \; X} ) + ( {1-c_1} ) {\rm \;} \times {\rm \;Pr}( {c_2\; \vert \; \sim X} ) }}\;} \cr} $$

Notice how the disagreement problem reduces to specification of the likelihoods Pr(c ₂ | X) and Pr(c ₂ | ~X). That is, to reach a maximally justified belief, v ₁ must answer two questions: (1) “What is the probability that my interlocutor would say what he did (viz. c ₂) if the state of the world is such that X (e.g. the defendant is in fact guilty)?” And (2) “What is the probability that my interlocutor would say what he did if the state of the world is such that ~X (the defendant is innocent)?”

One can see immediately how the Bayesian approach satisfies the desideratum of gradations in competence (which, again, dominant epistemological approaches do not); these are incorporated into the likelihood functions themselves.

For example, take the special case in which v ₁’s interlocutor is not only her epistemic superior but is epistemically infallible (and v ₁ knows this): v ₂ reports c ₂ = 0 if X is false, c ₂ = 1 if X is true, and nothing else. Then if v ₁ hears “0” from v ₂, c′ ₁ = 0. (Because Pr(c ₂ = 0 | X) = 0.) If v ₁ hears “1” from v ₂, then c′ ₁ = 1. (Because the second term on the RHS of equation (2.2) becomes ${1 \over {c_1}}$.) Of course, in general v ₁ will have to specify likelihood functions that cover the entire domain of c ₂ – from 0 to 1. But the principle is the same.

Another alluring feature of this approach is that it satisfies the second desideratum: It easily generalizes to n of arbitrary size. Again, by Bayes's Law:

(2.3)$$\eqalign{{c}^{\prime}_1 & = {\rm Pr} \left( {X\; \vert \; {c_2}, \ldots , {c_{\rm n}}} \right) \cr & = {c_{1\; } \times \; \displaystyle{{\Pr \left( {c_2, \ldots , c_{\rm n}\; \vert \; X} \right)} \over {c_{1\; } \times {\rm \; Pr}\left( {c_2, \ldots, \! c_{\rm n}\; \vert\; X} \right) + \left( {1-c_1} \right){\rm \; } \times {\rm \; Pr}\left( {c_2,\ldots,\; c_{\rm n}\; \vert \; \sim X} \right)}}}} $$

The likelihoods can be put into more manageable form. By the definition of conditional probability, Pr(A, B | C) = Pr(A | B, C)× Pr(B | C). Thus,

(2.4)$$\matrix{ {\; {\rm Pr}{\rm (}c_2, \ldots , \; c_{\rm n}\; \vert \; X) = {\rm Pr}{\rm (}c_{\rm n}\; \vert \; c_2, \ldots , \! c_{{\rm n}-1},\; X)\; \times \; {\rm Pr}{\rm (}c_2, \ldots , \; c_{{\rm n}-1}\; \vert \; X)} \cr} $$

and

(2.5)$$\matrix{ {\; {\rm Pr}{\rm (}c_2, \ldots , \; c_{\rm n}\; \vert \sim X) = {\rm Pr}{\rm (}c_{\rm n}\; \vert \; c_2, \ldots , \! c_{{\rm n}-1}, \sim X)\; \times \; {\rm Pr}{\rm (}c_2, \ldots , \; c_{{\rm n}-1}\; \vert \; \sim X)} \cr} $$

The second terms on the right-hand sides of equations (2.4) and (2.5) can be expanded in a similar way. Doing that, and substituting into equation (2.3), yields:

(2.6)$$\eqalign{&{{c}^{\prime}}_1 = c_{1} \cr & \times {\displaystyle{{\mathop \prod \nolimits_{{\rm i} = 2}^{\rm n} {\rm Pr} \left( {c_{{\rm i}\; }\vert\; c_2, \ldots ,\; c_{{\rm i}-1},\; X} \right)} \over {c_{1\; } \times \mathop \prod \nolimits_{{\rm i} = 2}^{\rm n} {\rm Pr}\left( {c_{\rm i}\; \vert\; c_2, \ldots , c_{{\rm i}-1},\; X} \right) + \left( {1-c_1} \right){\rm \; } \times \mathop \prod \nolimits_{{\rm i} = 2}^{\rm n} {\rm Pr}\left( {c_{\rm i}\; \vert\; c_2, \ldots , c_{{\rm i}-1},\; \sim X} \right)}}}} $$

Again the disagreement problem is one of specifying likelihoods. Now, especially when it comes to multiple interlocutors, this may be an onerous task. It requires that v ₁ detail her interlocutors’ intelligence, susceptibility to bias, and other epistemic features. As a result, in the 1970s and 1980s, decision theorists proposed a number of coarse-grained models for real-world use.Footnote ¹⁰ It seems that philosophers are unaware of this body of work, despite the relevance for the epistemology of disagreement suggested by some of its titles (e.g. French's (Reference French1980) “Updating of Belief in the Light of Someone Else's Opinion”). I want to describe one such model here, to illustrate the applicability of the Bayesian modelling approach to our contemporary disagreement debate.Footnote ¹¹

The first model was developed by Peter Morris (Reference Morris1983) and Robert Winkler (Reference Winkler1968). I have chosen it because it is simple and because it incorporates the two desiderata just described.

The idea underlying the model is to treat individuals’ beliefs about some event, like a defendant's guilt, as Beta-distributed random variables. The reported c _is are regarded as the means of those distributions. The Beta distribution is appropriate because we are seeking to represent a distribution of probabilities. And it yields a new, post-disagreement distribution by summing over parameters that define the individual distributions. The mean of that new distribution may then be adopted as the post-disagreement confidence.

This provides the following disagreement norm (I omit the derivation here, it may be found in the cited work):

(2.7)$$\matrix{ {{{c}^{\prime}}_1 = \; \mathop \sum \limits_{{\rm i} = 1}^{\rm n} w_{\rm i}c_{\rm i}} \cr} \; $$

where

(2.8)$$\matrix{ {\mathop \sum \limits_{{\rm i} = 1}^{\rm n} w_{\rm i} = 1\;} \cr} $$

This is a simple weighted average, where the opinions of the v _is are granted influence in accordance with v ₁’s view of their relative competence. In the special case in which v ₁ regards them all as epistemic peers, weights are set to ${1 \over {\rm n}}$.

Note two things. First, our two desiderata are incorporated – the w _is provide for differences in competence, and as many interlocutors as v ₁ likes may offer their opinions on X for v ₁’s consideration. Second, this norm is essentially the same as conciliationism's equal weight view, albeit more general.Footnote ¹²

An example of the norm in action may be helpful. Consider the “complicated nexus” problem described in §1: v ₁ is trying to form a maximally justified belief about X in light of disagreement from 10 epistemic interlocutors – some peers, some superiors, and some inferiors.

Under Morris and Winkler's model, v ₁ ought to do two things: (1) obtain reports from v ₂, …, v ₁₁ regarding their confidences in X; and (2) assess the relative competences of v ₁, …, v ₁₁. Suppose that this yields:

Then c′ ₁ = 0.48. The computation is straightforward, as is the solicitation of confidence information from v ₂, …, v ₁₁. The only challenge for v ₁ is assessing relative competence.

Now this norm has a serious drawback, a drawback which plagues theories in the epistemology of disagreement but which has hardly been attended to in the disagreement literature.Footnote ¹³ Namely, it implicitly endorses the idea that if there is no disagreement to begin with (i.e. if all epistemic interlocutors share the same confidence), then the post-disagreement confidence should simply equal the shared, pre-disagreement confidence. This is sometimes known as the “unanimity condition”, but it is a bug, not a feature, of a theory, even in the special case of agreement with epistemic peers.

Whether the unanimity condition should hold or not turns on whether there exists dependence between (1) our epistemic agent and her interlocutors, and (2) the interlocutors themselves. For reasons I shall now give, any viable disagreement norm must be capable of modelling such dependence.

3. Disagreement and dependence

v ₁, v ₂, and v ₃ are professional horseplayers, each trying to form a maximally justified belief in Judy's Lightning will win the race. They regard each other as epistemic peers, and they have good evidence that they are in fact peers: They've been betting on races for a long time, and have had the same success in picking winners.

But v ₁, v ₂, and v ₃ are not identical. In particular, v ₁ and v ₂ share the same handicapping methodology: They rely on how horses appear the morning of the race. v ₃, on the other hand, has developed a mathematical system for predicting winners on the basis of diverse historical data. Nevertheless, these two methodologies appear equally good; v ₁, v ₂, and v ₃ win with equal frequency. Naturally, v ₁ and v ₂ tend to win together, because they share the same methodology. In contrast, v ₃ sometimes wins when v ₁ and v ₂ lose (and vice versa).

Suppose that, before this race, each reports the same confidence, γ, in Judy's Lightning will win the race. According to standard disagreement theory, v ₁ should not change her confidence in this proposition (obviously γ is unmodified under steadfastness, and it implicitly stays the same under most variants of conciliationism, too – the arithmetic average of {γ, γ, γ} is γ).Footnote ¹⁴

Two questions to consider: (1) Should v ₁’s confidence in Judy's Lightning will win the race be unchanged by her interaction with v ₂ and v ₃, given that the three do not disagree about the probability of this event? (2) If v ₁’s post-“disagreement” confidence should not remain the same, should v ₂ and v ₃ exert the same epistemic influence on v ₁ when she modifies her judgment?

The answer to (1) is, pace current theory, “no”. Even though v ₃ is an epistemic peer, equally good at getting to the truth of Judy's Lightning will win the race, v ₃ is different from v ₁. And that difference means that their assessments are at least partially independent. If she is rational, v ₁ will recognize that independence and use it appropriately to modify her confidence. For example, if γ = 0.8, then v ₁’s post-“disagreement” confidence will be greater than 0.8. The knowledge that a different handicapping methodology, even if no better than your own, is also highly confident that it has picked a winner provides you with greater reason to believe that you have got things right.

As far as (2) is concerned, v ₂ and v ₃ should certainly not exert the same epistemic influence over v ₁. Indeed, because v ₂ is more-or-less a copy of v ₁, c ₂ is not a useful datum when c ₁ = c ₂ = 0.8. That is precisely what v ₁ expects to hear prior to her interaction with v ₂, and so conditionalizing upon it should not affect her prior judgment. Because there is perfect dependence between v ₁’s judgment and v ₂’s judgment, and v ₁ knows this, there is nothing to be gained epistemically through interaction with v ₂ in this case.

The possibility of dependence between epistemic agents illuminates the limitations of Easwaran et al.’s Upco:

(3.1)$$\matrix{ {{{c}^{\prime}}_1 = \displaystyle{{c_1\; \times \; c_2} \over {c_1\; \times \; c_2 + ( {1-c_1} ) \; \times \; ( {1-c_2} ) }}\;} \cr} $$

While this norm does allow for violations of unanimity (what Easwaran et al. (Reference Easwaran, Fenton-Glynn, Hitchcock and Velasco2016) call “synergy”) – as it sometimes should – it fails to account for dependence, as in the example just given. It would allow v ₂ and v ₃ to exert the same epistemic influence over v ₁. And that, as we have seen, would be a mistake.

Under Upco, if v ₁ and v ₂ interact when both have a confidence of γ = 0.8, then the post-“disagreement” confidence is 0.94. The same result is reached if v ₁ and v ₃ interact. But such a high post-“disagreement” confidence is only plausible in the latter case. Because v ₂ provides no independent insight, v ₁ should maintain a confidence of 0.8 after interacting with v ₂ alone. Upco fails to account for this important difference.

To their credit, Easwaran et al. recognize that Upco is limited in this way. But they do not grapple with “the general question of how to deal with peer update when we think there are correlations between one's peers” (Reference Easwaran, Fenton-Glynn, Hitchcock and Velasco2016: 31), suggesting, instead, that Upco's use be restricted to scenarios of disagreement involving perfect independence between epistemic agents. But I stress that this is not a minor limitation; it is a loss of generality which renders the norm useless for real-world use. It is a struggle to imagine a real-world scenario in which perfect independence holds. Two philosophers disagree about the morality of some new law? Consider all the common training they receive. Two jurors disagree about a defendant's guilt? Think of the common evidence, presented at trial, on which their judgments rely. Two weathermen disagree about whether it will rain tomorrow? Note that both base their judgments on the same radar data.

Easwaran et al. (Reference Easwaran, Fenton-Glynn, Hitchcock and Velasco2016) do offer a conjecture about how Upco might handle dependence – but I do not think that it will work. They suggest that we assign to each term in Upco an exponent representing the weight of that interlocutor's opinion, where the weight assigned to the c ₁ terms is set to 1, and the weight of “fully independent peers” is likewise 1. Then, if peers’ opinions are correlated, we reduce the weights assigned to those peers. For example, if two peers are perfectly correlated, then they should be treated by Upco as one “fully independent peer” by assigning each a weight of one-half.

At the same time, the exponents are supposed to handle gradations in competence; indeed, this is why they are introduced in the first place. (Again, Easwaran et al. explicitly avoid in-depth analysis of dependence in their paper.) For example, “if we raise [c _i] to the power of 2, we are treating [v _i’s] report … as equivalent to the report of two independent peers with weight 1 reporting that credence” (Easwaran et al. Reference Easwaran, Fenton-Glynn, Hitchcock and Velasco2016: 30).

Here's the problem. Suppose that I think some proposition is false. My epistemic superior thinks that it is true with confidence c ₂. Whatever else we might want to say about my post-disagreement confidence, it should not be greater than c ₂. But we can choose values for this Upco variant that delivers such a result:

(3.2)$$\matrix{ {{{c}^{\prime}}_1 = \displaystyle{{{0.4}^1 \times {0.7}^2} \over {{0.4}^1 \times {0.7}^2 + {( {1-0.4} ) }^1 \times {( {1-0.7} ) }^2}} = 0.78} \cr} $$

To deal with dependence, I suggest we look elsewhere.

4. A better way

We ought to model disagreement in a way that explicitly takes into account correlation between epistemic interlocutors. Our models should not hew to unanimity; they should yield a post-disagreement confidence which turns in part on pre-disagreement dependence.

An excellent starting point would be the work of Christian Genest and Mark Schervish (Reference Genest and Schervish1985), who recognize that even though ideally rational agents should employ equation (2.3), that formula is unhelpful for real-world use. This is because it requires that v ₁ specify $\Pr ( {c_2,\;. \;. \;., \; c_{\rm n}\; \vert \; X} ) $ and ${\rm Pr}( {c_2,\;. \;. \;., \; c_{\rm n}\; \vert \; \sim X} ) $, each of which are joint distributions over n−1 random variables.

Genest and Schervish show that so long as (1) v ₁ can specify the means of the marginal distributions and (2) plausible consistency conditions hold (more on this below), then a maximally justified belief is given by:

(4.1)$$\matrix{ {{{c}^{\prime}}_1 = c_1 + \mathop \sum \limits_{{\rm i} = 2}^{\rm n} { \lambda} _{\rm i}( {c_{\rm i}-\mu_{\rm i}} ) } \cr} $$

where μ _i is the mean of i’s confidence distribution (as specified by v ₁), and the λ _is are the coefficients of linear regression of X on the vector (c ₂, …, c _n).

To reiterate: c ₁ is v ₁’s pre-disagreement confidence in the proposition at issue. c _i is the stated confidence of v _i in the proposition. v ₁ says to herself, for each of her epistemic interlocutors, “My interlocutor might report many possible confidences – from very close to 0 to very close to 1. Some values are more likely than others. What is the mean confidence that I expect to hear?” That is μ _i.

Note that if v ₁’s interlocutor reports what she expects him to (that is, if c _i = μ _i), then her confidence is unchanged by their interaction. This makes sense from the Bayesian point-of-view; the interlocutor's judgment was already baked into c ₁.

Suppose, for example, that the government announces a tax hike on the rich. I have a confidence of c ₁ = 0.8 that this policy is just. My colleague down the hall is a conservative, so I expect he'll find the policy to be unjust. Perhaps I think he's most likely to report a confidence of 0.2 (if c ₂, qua random variable, is Normal, the mean equals the mode). Now if I ask him what he thinks about the policy, and he tells me, as I expect he will, that it is unjust, my confidence should be little affected. I already knew that. But if this conservative agrees with me that the tax hike is just – well, that is surprising. It is genuinely new and useful information, and so it gives me grounds to increase my confidence that the policy is a just one.

Of course, technically c ₂ cannot be Normal – because the support of the Normal distribution is ( − ∞, ∞) and c ₂ requires a support of (0, 1). One can truncate the Normal distribution to (0, 1), but then it is no longer generally true that its mean will equal its mode. Nevertheless, for the purpose of on-the-fly belief revision, v ₁ would not go far wrong by imagining c ₂ distributed “normalish” on (0, 1), and taking its peak to be μ ₂. And, if v ₁ desires a precise answer, she can calculate the actual mean for her chosen Truncated Normal on (0, 1).

Note that it will frequently be sensible for v ₁ to set μ _i = c ₁. That is, v ₁ can presume that the mean realization of c _i (qua random variable) is what she, v ₁, already believes. This is not the case in the above example, because there I know that my opinion about the policy and my colleague's opinion are likely to be opposed given our political differences. But often μ _i = c ₁ is sensible. Example: I'm walking down the street, the sky is looking gloomy, and I think there's a 60% chance of rain. I ask a passerby what he thinks. It's highly unlikely that he'll say “60%”, exactly, but he's more likely to say “60%” than anything else. But if there were some evidence of bias – if he were wearing a t-shirt that said, “I Hate Rain” – then I would be justified in taking μ ₂ ≠ c ₁.

The λ _is satisfy certain inequalities to ensure that ${c}^{\prime}_1$ is a bona fide probability measure, and can be interpreted as indicators of how much independent insight v _i provides, above and beyond what was provided by {v ₁, …, v _i-1}. Namely:

(4.2)$$\matrix{ {\max \left\{ {\mathop \sum \limits_{{\rm i} = 2}^{\rm n} \displaystyle{{ \lambda_{\rm i}\mu_{\rm i}} \over {c_1}},\mathop \sum \limits_{{\rm i} = 2}^{\rm n} \displaystyle{{\lambda_{\rm i}( {1-\mu_{\rm i}} ) } \over {(1-c_1)}}} \right\} \le 1} \cr} $$

(N.B. here we assume that the λ _is are positive. For the most general cases, which could include negative λ _is, there are 2ⁿ inequalities.) The intuition is that as some λ _i gets close to 1, c ₁ must be kept close to μ _i. It does not make sense for v ₁ to believe, pre-disagreement, that both (1) X is very improbable, and (2) v _i, who has great insight into the truth of X, likely believes that X is very probable.

To illustrate equation (4.1) in action, let us consider the horse racing case, with c ₁ = c ₂ = c ₃ = 0.80, as above. We have three epistemic peers who report the same confidence in the proposition. Here, v ₁ should certainly take μ ₂ = c ₁ = 0.8 (they are copies). μ ₃, in contrast, will be somewhat less than this – say, 0.4. Of course, the distribution of c ₃, in v ₁’s eyes, will depend on many things. But surely its mean will be less than v ₁’s 0.80, which is extraordinarily high in the context of horse racing. (A horse that goes off at 0.3 to win is considered a heavy favourite.)

v ₁ might then evaluate λ ₂ = 0.01 and λ ₃ = 0.30, representing little possible epistemic help from v ₂ and significant possible help from v ₃. These values satisfy the necessary inequalities, and they yield a disagreement norm under which (1) v ₂ exerts no epistemic influence over v ₁, and, in contrast, (2) v ₃’s judgment does provide reason for v ₁ to become more confident in her judgment – even though, I stress, v ₁ and v ₃ do not disagree about the proposition at dispute. In particular, for the values given, v ₁’s confidence rises from 0.80 to 0.92.

This model also prevents the bad result of equation (3.2). Again, Upco with dependence fails for c ₁ = 0.4 and c ₂ = 0.7. But here, with μ ₂ = c ₁ = 0.4, we get:

(4.3)$$\matrix{ {{{c}^{\prime}}_1 = 0.4 + \lambda _2( {0.3} ) } \cr} $$

One may see that the troublesome inequality, ${c}^{\prime}_1 > c_2$, would arise if λ ₂ >1. But consistency conditions require that

(4.4)$$\matrix{ {\max \left\{ {\displaystyle{{0.4} \over {0.4-1}},\displaystyle{{0.4-1} \over {0.4}}} \right\} \le \lambda _2 \le \min \left\{ {\displaystyle{{0.4} \over {0.4}},\displaystyle{{1-0.4} \over {1-0.4}}} \right\}\;} \cr} $$

The latter inequality ensures that ${c}^{\prime}_1 \le c_2$.

Notice how this approach satisfies our three desiderata. First, multiple epistemic interlocutors are generally admissible, and the oft-considered scenarios in which n = 2 are simply dealt with as special cases.

Second, we incorporate differences in competence via the λ _is. To take the extreme cases, suppose v ₁ believes that v ₂ is epistemically infallible. Then v ₁ will set λ ₂ = 1, because v ₂’s judgment is perfectly correlated with the truth. Then ${c}^{\prime}_1 = c_1 + ( {c_2-\mu_2} ).$ Since c _i, μ _i ∈ (0, 1),Footnote ¹⁵ consistency conditions require that c ₁ = μ ₂. Therefore, ${c}^{\prime}_1 = c_2$. v ₁ entirely abrogates her judgment and adopts v ₂’s opinion, as one would expect.

Next, suppose that v ₁ believes that v ₂ is epistemically useless; v ₂’s judgment is utterly uncorrelated with the truth. Then λ ₂ = 0 and ${c}^{\prime}_1 = c_1$. v ₁ maintains her judgment in the face of disagreement from this interlocutor. And less extreme cases of epistemic superiority and inferiority are dealt with accordingly.Footnote ¹⁶

The λ _is also allow us to incorporate the third desideratum: the possibility of dependence. We have already seen examples of this, but I would like to point out here two possible sources of dependence and how each gets handled.

First, there may be dependence between v ₁ and her interlocutor(s). One imagines two weathermen (§3), each of whom makes a prediction about rain on the basis of, and only on the basis of, weather data which they both possess. Recall the interpretation of λ _i as how much independent insight v _i provides, above and beyond what was provided by {v ₁, …, v _i-1}. In this case, even if v ₂ is v ₁’s epistemic superior (being better when it comes to “general epistemic virtues such as intelligence, thoughtfulness, and freedom from bias” – §1), v ₂ provides no independent insight, and so λ ₂ is set equal to 0, and so ${c}^{\prime}_1 = c_1$.Footnote ¹⁷

Second, there may be dependence between interlocutors. Suppose, as a variant on the horse racing case, that it is v ₂ and v ₃, not v ₁ and v ₂, who use the same handicapping methodology. Then λ ₂ will be positive, because it is useful for v ₁ to know that one of v ₂ and v ₃ agree with her about the winner. But it is not useful (given that knowledge) to know that both v ₂ and v ₃ agree with her. So λ ₃ will be set to 0. v ₁ will modify her belief only in light of one interlocutor's opinion.