Monday, 6 February 2017

What is justified credence?

Aafira and Halim are both 90% confident that it will be sunny tomorrow. Aafira bases her credence on her observation of the weather today and her past experience of the weather on days that follow days like today -- around nine out of ten of them have been sunny. Halim bases his credence on wishful thinking -- he's arranged a garden party for tomorrow and he desperately wants the weather to be pleasant. Aafira, it seems, is justified in her credence, while Halim is not. Just as one of your full or categorical beliefs might be justified if it is based on visual perception under good conditions, or on memories of recent important events, or on testimony from experts, so might one of your credences be; and just as one of your full beliefs might be unjustified if it is based on wishful thinking, or biased stereotypical associations, or testimony from ideologically driven news outlets, so might your credences be. In this post, I'm looking for an account of justified credence -- in particular, I seek necessary and sufficient conditions for a credence to be justified. Our account will be reliabilist.

Reliabilism about justified beliefs comes in two varieties: process reliabilism and indicator reliabilism. Roughly, process reliabilism says that a belief is justified if it is formed by a reliable process, while indicator reliabilism says that a belief is justified if it is based on a ground that renders it likely. Reliabilism about justified credence also comes in two varieties; indeed, it comes in the same two varieties. And, indeed, of the two existing proposals, Jeff Dunn's is a version of process reliabilism (paper) while Weng Hong Tang offers a version of indicator reliabilism (paper). As we will see, both face the same objection. If they are right about what justification is, it is mysterious why we care about justification, for neither of the accounts connects justification to a source of epistemic value.  We will call this the Connection Problem.

I begin by describing Dunn's process reliabilism and Tang's indicator reliabilism. I argue that, understood correctly, they are, in fact, extensionally equivalent. That is, Dunn and Tang reach the top of the same mountain, albeit by different routes. However, I argue that both face the Connection Problem. In response, I offer my own version of reliabilism, which is both process and indicator, and I argue that it solves that problem. Furthermore, I show that it is also extensionally equivalent to Dunn's reliabilism and Tang's.

Reliabilism and Dunn on reliable credence


Let us begin with Dunn's process reliabilism for justified credences. Now, to be clear, Dunn takes himself only to be providing an account of reliability for credence-forming processes. He doesn't necessarily endorse the other two conjuncts of reliabilism, which say that a credence is justified if it is reliable, and that a credence is reliable if formed by a reliable process. Instead, Dunn speculates that perhaps being reliably formed is but one of the epistemic virtues, and he wonders whether all of the epistemic virtues are required for justification. Nonetheless, I will consider a version of reliabilism for justified credences that is based on Dunn's account of reliable credence. For reasons that will become clear, I will call this the calibrationist version of process reliabilism for justified credence. Dunn rejects it based on what I will call below the Graining Problem. As we will see, I think we can answer that objection.

For Dunn, a credence-forming process is perfectly reliable if it is well calibrated. Here's what it means for a process $\rho$ to be well calibrated:
  • First, we construct a set of all and only the outputs of the process $\rho$ in the actual world and in nearby counterfactual scenarios. An output of $\rho$ consists of a credence $x$ in a proposition $X$ at a particular time $t$ in a particular possible world $w$ -- so we represent it by the tuple $(x, X, w, t)$. If $w$ is a nearby world and $t$ a nearby time, we call $(x, X, w, t)$ a nearby output. Let $O_\rho$ be the set of nearby outputs -- that is, the set of tuples $(x, X, w, t)$, where $w$ is a nearby world, $t$ is a nearby time, and $\rho$ assigns credence $x$ to proposition $X$ in world $w$ at time $t$.
  • Second, we say that the truth-ratio of $\rho$ for credence $x$ is the proportion of nearby outputs $(x, X, w, t)$ in $O_\rho$ such that $X$ is true at $w$ and $t$.
  • Finally, we say that $\rho$ is well calibrated (or nearly so) if, for each credence $x$ that $\rho$ assigns, $x$ is equal to (or approximately equal to) the truth-ratio of $\rho$ for $x$.
For instance, suppose a process only ever assigns credence 0.6 or 0.7. And suppose that, 60% of the time that it assigns 0.6 in the actual world or a nearby world it assigns it to a proposition that is true; and 70% of the time it assigns 0.7 it assigns it to a true proposition. If, on the other hand, 59% of the time that it assigns 0.6 in the actual world or a nearby world it assigns it to a proposition that is true, while 71% of the time it assigns 0.7 it assigns it to a true proposition, then that process is not well calibrated, but it is nearly well calibrated. But if 23% of the time that it assigns 0.6 in the actual world or a nearby world it assigns it to a proposition that is true, while 95% of the time it assigns 0.7 it assigns it to a true proposition, then that process is not even nearly well calibrated.

This, then, is Dunn's calibrationist account of the reliability of a credence-forming process. Any version of reliabilism about justified credences that is based on it requires two further ingredients. First, we must use the account to say when an individual credence is reliable; second, we must add the claim that a credence is justified iff it is reliable. Both of these moves creates problems. We will address them below. But first it will be useful to present Tang's version of indicator reliabilism for justified credence. It will provide an important clue that helps us solve one of the problems that Dunn's account faces. And, having it in hand, it will be easier to see how these two accounts end up coinciding.

Tang's indicator reliabilism for justified credence


According to indicator reliabilism for justified belief, a belief is justified if the ground on which it is based is a good indicator of the truth of that belief. Thus, beliefs formed on the basis of visual experiences tend to be justified because the fact that the agent had the visual experience in question makes it likely that the belief they based on it is true. Wishful thinking, on the other hand, usually does not give rise to justified belief because the fact that an agent hopes that a particular proposition will be true -- which in this case is the ground of their belief -- does not make it likely that the proposition is true.

Tang seeks to extend this account of justified belief to the case of credence. Here is his first attempt at an account:

Tang's Indicator Reliabilism for Justified Credence (first pass)  A credence of $x$ in $X$ by an agent $S$ is justified iff
(TIC1-$\alpha$) $S$ has ground $g$;
(TIC2-$\alpha$) the credence $x$ in $X$ by $S$ is based on ground $E$;
(TIC3-$\alpha$) the objective probability of $X$ given that the agent has ground $g$ approximates or equals $x$ -- we write this $P(X | \mbox{$S$ has $g$}) \approx x$.

Thus, just as an agent's full belief in a proposition is justified if its ground makes the objective probability of that proposition close to 1, a credence $x$ in a proposition is justified if its ground makes the objective probability of that proposition close to $x$. There is a substantial problem here in identifying exactly to which notion of objective probability Tang wishes to appeal. But we will leave that aside for the moment, other than to say that he conceives of it along the lines of hypothetical frequentism -- that is, the objective probability of $X$ given $Y$ is the hypothetical frequency with which propositions like $X$ are true when propositions like $Y$ are true.

However, as Tang notes, as stated, his version of indicator reliabilism  faces a problem. Suppose I am presented with an empty urn. I watch as it is filled with 100 balls, numbered 1 to 100, half of which are white, and half of which are black. I shake the urn vigorously and extract a ball. It's number 73 and it's white. I look at its colour and the numeral printed on it. I have a visual experience of a white ball with '73' on it. On the basis of my visual experience of the numeral alone, I assign credence 0.5 to the proposition that ball 73 is white. According to Wang's first version of indicator reliabilism for justified credence, my credence is justified. My ground is the visual experience of the number on the ball; I have that ground; I base my credence on that ground; and the objective probability that ball 73 is white given that I have a visual experience of the numeral '73' printed on it is 50% -- after all, half the balls are white. Of course, the problem is that I have not used my total evidence -- or, in the language of grounds, I have not based my belief on my most inclusive ground. I had the visual experience of the numeral on the ball as a ground; but I also had the visual experience of the numeral on the ball and the colour of the ball as a ground. The resulting credence is unjustified because the objective probability that ball 73 is white given I have the more inclusive ground is not 0.5 -- it is close to 1, since my visual system is so reliable. This leads Tang to amend his account of justified credence as follows:

Tang's Indicator Reliabilism for Justified Credence  A credence of $x$ in $X$ by an agent $S$ is justified iff
(TIC1) $S$ has ground $g$;
(TIC2) the credence $x$ in $X$ by $S$ is based on ground $g$;
(TIC3) the objective probability of $X$ given that the agent has ground $g$ approximates or equals $x$ -- that is, $P(X | \mbox{$S$ has $g$}) \approx x$;
(TIC4) there is no more inclusive ground $g'$ such that (i) $S$ has $g'$ and (ii) the objective probability of $X$ given that the agent has ground $g'$ does not equal or approximate $x$ -- that is, $P(X | \mbox{$S$ has $g'$}) \not \approx x$.

This, then, is Tang's version of indicator reliabilism for justified credences.

Same mountain, different routes


Thus, we have now seen Dunn's process reliabilism and Tang's indicator reliabilism for justified credences. Is either correct? If so, which? In one sense, both are correct; in another, neither is. Less mysteriously: as we will see in this section, Dunn's process reliablism and Tang's indicator reliabilism are extensionally equivalent -- that is, the same credences are justified on both. What's more, as we will see in the final section, both are extensionally equivalent to the correct account of justified credence, which is thus a version of both process  and indicator reliabilism. However, while they get the extension right, they do so for the wrong reasons. A justified credence is not justified because it is formed by a well calibrated process; and it is not justified because it matches the objective chance given its grounds. Thus, Dunn and Tang delimit the correct extension, but they use the wrong intension. In the final section of this post, I will offer what I take to be the correct intension. But first, let's see why it is that the routes that Dunn and Tang take lead them both to the top of the same mountain.

We begin with Dunn's calibrationist account of the reliability of a credence-forming process. As we noted above, any version of reliabilism about justified credences that is based on this account requires two further ingredients. First, we must use the calibrationist account of reliable credence-forming processes to say when an individual credence is reliable. The natural answer: when it is formed by a reliable credence-forming process. But then we must be able to identify, for a given credence, the process of which it is an output. The problem is that, for any credence, there are a great many processes of which it might be the output. I have a visual experience of a piece of red cloth on my desk, and I form a high credence that there is a piece of red cloth on my desk. Is this credence the output of a process that assigns a high credence that that there is a piece of red cloth on my desk whenever I have that visual experience? Or is it the output of a process that assigns a high credence that there is a piece of red cloth on my desk whenever I have that visual experience and the lighting conditions in my office are good, while it assigns a middling credence that there is a piece of red cloth on my desk whenever I have that visual experience and the lighting conditions in my office are bad? It is easy to see that this is important. The first process is poorly calibrated, and thus unreliable on Dunn's account; the second process is better calibrated and thus more reliable on Dunn's account. This is the so-called Generality Problem, and it is a challenge that faces any version of reliabilism. I will offer a version of Juan Comesaña's solution to this problem below -- as we will see, that solution also clears the way for a natural solution to the Graining Problem, which we consider next.

Dunn provides an account of when a credence-forming process is reliable. And, once we have a solution to the Generality Problem, we can use that to say when a credence is reliable -- it is reliable when formed by a reliable credence-forming process. Finally, to complete the version of process reliablism about justified credence that we are basing on Dunn's account, we just need the claim that a credence is justified iff it is reliable. But this too faces a problem, which we call the Graining Problem. As we did above, suppose I am presented with an empty urn. I watch as it is filled with 100 balls, numbered 1 to 100, half of which are white, and half of which are black. I shake the urn vigorously and extract a ball. I look at its colour and the numeral printed on it. I have two processes at my disposal. Process 1 takes my visual experience of the numeral only, say '$n$', and assigns the credence 0.5 to the proposition that ball $n$ is white. Process 2 takes my visual experience of the numeral, '$n$', and my visual experience of the colour of the ball, and assigns credence 1 to the proposition that ball $n$ is white if my visual experience is of a white ball, and assigns credence 1 to the proposition that ball $n$ is black if my visual experience is of a black ball. Note that both processes are well calibrated (or nearly so, if we allow that my visual system is very slightly fallible). But we would usually judge the credence formed by the second to be better justified than the credence formed by the first. Indeed, we would typically say that a Process 1 credence is unjustified, while a Process 2 credence is justified. Thus, being formed by a well calibrated or nearly well calibrated process is not sufficient for justification. And, if reliability is calibration, then reliability is not justification and reliabilism fails. It is this problem that leads Dunn to reject reliabilism about justified credence. However, as we will see below, I think he is a little hasty.

Let us consider the Generality Problem first. To this problem, Juan Comesaña offers the following solution (paper). Every account of doxastic justification -- that is, every account of when a given doxastic attitude of a particular agent is justified for that agent -- must recognize that two agents may have the same doxastic attitude and the same evidence while the doxastic attitude of one is justified and the doxastic attitude of the other is not, because their doxastic attitudes are not based on the same evidence. The first might base her belief on the total evidence, for instance, whilst the second ignores that evidence and bases his belief purely on wishful thinking. Thus, Comesaña claims, every theory of justification needs a notion of the grounds or the basis of a doxastic attitude. But, once we have that, a solution to the Generality Problem is very close. Comesaña spells out the solution for process reliabilism about full beliefs:

Well-Founded Process Reliablism for Justified Full Beliefs  A belief that $X$ by an agent $S$ is justified iff
(WPB1) $S$ has ground $g$;
(WPB2) the belief that $X$ by $S$ is based on ground $g$;
(WPB3) the process producing a belief state $X$ based on ground $g$ is a reliable process.

This is easily adapted to the credal case:

Well-Founded Process Reliablism for Justified Credences  A credence of $x$ in $X$ by an agent $S$ is justified iff
(WPC1) $S$ has ground $g$;
(WPC2) the credence $x$ in $X$ by $S$ is based on ground $g$;
(WPC3) the process producing a credence of $x$ in $X$ based on ground $g$ is a reliable process.

Let us now try to apply Comesaña's solution to the Generality Problem to help Dunn's calibrationist reliabilism about justified credences. Recall: according to Dunn, a process $\rho$ is reliable if it is well calibrated (or nearly so). Consider the process producing a credence of $x$ in $X$ based on ground $g$ -- for convenience, we'll write it $\rho^g_{X,x}$. There is only one credence that it assigns, namely $x$. So it is well calibrated if that truth-ratio of $\rho^g_{X,x}$ for $x$ is equal to $x$. Now, $O_{\rho^g_{X,x}}$ is the set of tuples $(X, x, w, t)$ where $w$ is a nearby world and $t$ a nearby time where $\rho^g_{X,x}$ assigns credence $x$ to proposition $X$. But, by the definition of $\rho^g_{X,x}$, those are the nearby worlds and nearby times at which the agent has the ground $g$. Thus, the truth-ratio of $\rho^g_{X,x}$ for $x$ is the proportion of those nearby worlds and times at which the agent has the ground $g$ at which $X$ is true. And that, it seems to me, is the something like the objective probability of $X$ conditional on the agent having ground $g$, at least given the hypothetical frequentist account of objective probability of the sort that Tang favours. As above, we denote the objective probability of $X$ conditional on the agent $S$ having grounds $g$ as follows: $P(X | \mbox{$S$ has $g$})$. Thus, $P(X | \mbox{$S$ has $g$})$ is the truth-ratio of $\rho^g_{p,x}$ for $x$. And thus, a credence $x$ in $X$ based on ground $g$ is reliable iff $x$ is close to $P(X | \mbox{$S$ has $g$})$. That is,

Well-Founded Calibrationist Process Reliabilism for Justified Credences (first attempt) A credence of $x$ in $X$ by an agent $S$ is justified iff
(WCPC1) $S$ has ground $g$;
(WCPC2) the credence $x$ in $X$ by $S$ is based on ground $g$;
(WCPC3) the process producing a credence of $x$ in $X$ based on ground $g$ is a (nearly) well calibrated process -- that is, $P(X | \mbox{$S$ has $g$}) \approx x$.

But now compare Well-Founded Calibrationist Process Reliabilism, based on Dunn's account of reliable processes and Comesaña's solution to the Generality Problem, with Tang's first attempt at Indicator Reliabilism. Consider the necessary and sufficient conditions that each imposes for justification: TIC1 = WCPC1; TIC2 = WCPC2; TIC3 = WCPC3. Thus, these are the same account. However, as we saw above, Tang's first attempt to formulate indicator reliabilism for justified credence fails because it counts as justified a credence that is not based on an agent's total evidence; and we also saw that, once the Generality Problem is solved for Dunn's calibrationist process reliabilism, it faces a similar problem, namely, the Graining Problem from above. Tang amends his version of indicator reliabilism by adding the fourth condition TIC4 from above. Might we amend Dunn's calibrationist process reliabilism is a similar way?

Well-Founded Calibrationist Process Reliabilism for Justified Credences  A credence of $x$ in $X$ by an agent $S$ is justified iff
(WCPC1) $S$ has ground $g$;
(WCPC2) the credence $x$ in $X$ by $S$ is based on ground $g$;
(WCPC3) the process producing a credence of $x$ in $X$ based on ground $g$ is a (nearly) well calibrated process -- that is, $P(X | \mbox{$S$ has $g$}) \approx x$;
(WCPC4) there is no more inclusive ground $g'$ and credence $x' \not \approx x$, such that the process producing a credence of $x'$ in $X$ based on ground $g'$ is a (nearly) well calibrated process -- that is, $P(X | \mbox{$S$ has $g'$}) \approx x'$.

Since TIC4 is equivalent to WCPC4, this final version of process reliabilism for justified credences is equivalent to Tang's final version of his indicator reliabilism for justified credences. Thus, Dunn and Tang have reached the top of the same mountain, albeit by different routes
  

The third route up the mountain


Once we have addressed certain problems with the calibrationist version of process reliabilism for justified credence, we see that it agrees with the current best version of indicator reliabilism. This gives us a little hope that both have hit upon the correct account of justification. In the end, I will conclude that both have indeed hit upon the correct extension of the concept of justified credence. But that have done so for the wrong reasons, for they have not hit upon the correct intension.

There are two sorts of route you might take when pursuing an account of justification for a given sort of doxastic attitude, such as a credence or a full belief. You might look to intuitions concerning particular cases and try to discern a set of necessary and sufficient conditions that sort these cases in the same way that your intuitions do; or, you might begin with an account of epistemic value, assume that justification must be linked in some natural way to the promotion of epistemic value, and then provide an account of justification that vindicates that assumption. Dunn and Tang have each taken a route of the first sort; I will follow a route of the second sort.

I will adopt the veritist's account of epistemic value. That is, I take accuracy to be the sole fundamental source of epistemic value for a credence, where a credence in a true proposition is more accurate the higher it is; a credence in a false proposition is more accurate the lower it is. Given this account of epistemic value, what is the natural account of justification? Well, at first sight, there are two: one is process reliabilist; the other is indicator reliabilist. But, in a twist that should come as little surprise given the conclusions of the previous section, it will turn out that these two accounts coincide, and indeed coincide with the final versions of Dunn's and Tang's accounts that we reached above. Thus, I too will reach the top of the same mountain, but by yet another route.

Epistemic value version of indicator reliabilism


In the case of full beliefs, indicator reliabilism says this: a belief in $X$ by $S$ on the basis of grounds $g$ is justified iff the objective probability of $X$ given that $S$ has grounds $g$ is high --- that is, close to 1. Tang generalises this to the case of credence, but I think he generalises in the wrong direction; that is, he takes the wrong feature to be salient and uses that to formulate his indicator reliabilism for justified credence. He takes the general form of indicator reliabilism to be something like this: a doxastic attitude $s$ towards $X$ by $S$ on the basis of grounds $g$ is justified iff the attitude $s$ 'matches' the objective probability of $X$ given that $S$ has grounds $g$. And he takes the categorical attitude of belief in $X$ to 'match' high objective probability of $X$, and credence $x$ in $X$ to 'match' objective probability of $x$ that $X$. The problem with this account is that it leaves mysterious why justification is valuable. Unless we say that matching objective probabilities is somehow epistemic valuable in itself, it isn't clear why we should want to have justified doxastic attitudes in this sense.

I contend instead that the general form of indicator reliabilism is this:

Indicator reliabilism for justified doxastic attitude (epistemic value version)  Doxastic attitude $s$ towards proposition $X$ by agent $S$ is justified iff
(EIA1) $S$ has $g$;
(EIA2) $s$ in $X$ by $S$ is based on $g$;
(EIA3) if  $g' \subseteq g$ is a ground that $S$ has, then for every doxastic attitude $s'$ of the same sort as $s$, the expected epistemic value of attitude $s'$ towards $X$ given that $S$ has $g'$ is at most (or not much above) the expected epistemic value of attitude $s$ towards $X$ given that $S$ has $g'$.

Thus, attitude $s$ towards $X$ by $S$ is justified if $s$ is based on a ground $g$ that $S$ has, and $s$ is the attitude towards $X$ that has highest expected accuracy relative to the most inclusive grounds that $S$ has.

Let's consider this in the full belief case. We have:

Indicator reliabilism for justified belief (epistemic value version)  A belief in proposition $X$ by agent $S$ is justified iff
(EIB1) $S$ has $g$;
(EIB2) $s$ in $X$ by $S$ is based on $g$;
(EIB3) if  $g' \subseteq g$ is a ground that $S$ has, then
  1. the expected epistemic value of disbelief in $X$, given that $S$ has $g'$, is at most (or not much above) the expected epistemic value of belief in $X$, given that $S$ has $g'$;
  2. the expected epistemic value of suspension in $X$, given that $S$ has $g'$, is at most (or not much above) the expected epistemic value of belief in $X$, given that $S$ has $g'$.

To complete this, we need only an account of epistemic value. Here, the veritist's account of epistemic value runs as follows. There are three categorical doxastic attitudes towards a given proposition: belief, disbelief, and suspension of judgment. If the proposition is true, belief has greatest epistemic value, then suspension of judgment, then disbelief. If it is false, the order is reversed. It is natural to say that a belief in a truth and disbelief in a falsehood have the same high epistemic value -- following Kenny Easwaran (paper), we denote this $R$ (for `getting it Right'), and assume $R >0$. And it is natural to say that a disbelief in a truth and belief in a falsehood have the same low epistemic value -- again following Easwaran, we denote this $-W$ (for `getting it Wrong'), and assume $W > 0$. And finally it is natural to say that suspension of belief in a truth has the same epistemic value as suspension of belief in a falsehood, and both have epistemic value 0. We assume that $W > R$, just as Easwaran does. Now, suppose proposition $X$ has objective probability $p$. Then the expected epistemic utility of different categorical doxastic attitudes towards $X$ is given below:
  • Expected epistemic value of belief in $X$ = $p\cdot R + (1-p)\cdot(-W)$.
  • Expected epistemic value of suspension in $X$ = $p\cdot 0 + (1-p)\cdot 0$.
  • Expected epistemic value of disbelief in $X$ = $p\cdot (-W) + (1-p)\cdot R$.
Thus, belief in $X$ has greatest epistemic value amongst the possible categorical doxastic attitudes to $X$ if $p > \frac{W}{R+W}$;  disbelief in $X$ has greatest epistemic value if $p < \frac{R}{R+W}$; and suspension in $X$ has greatest value if $\frac{R}{R+W} < p < \frac{W}{R+W}$ (at $p = \frac{W}{R+W}$, belief ties with suspension; at $p = \frac{R}{R+W}$, disbelief ties with suspension). With this in hand, we have the following version of indicator reliabilism for justified beliefs:

Indicator reliabilism for justified belief (veritist version)  A belief in $X$ by agent $S$ is justified iff
(EIB1$^*$) $S$ has $g$;
(EIB2$^*$) the belief in $X$ by $S$ is based on $g$;
(EIB3$^*$) the objective probability of $X$ given that $S$ has $g$ is (nearly) greater than $\frac{W}{R+W}$;
(EIB4$^*$) there is no more inclusive ground $g'$ such that (a) $S$ has $g'$ and (b) the objective probability of $X$ given that $S$ has $g'$ is not (nearly) greater than $\frac{W}{R+W}$.

And of course this is simply a more explicit version of the standard version of indicator reliabilism. It is more explicit because it gives a particular threshold above which the objective probability of $X$ given that $S$ has $g$ counts as 'high', and above which (or not much below which) the belief in $X$ by $S$ counts as justified --- that threshold is $\frac{W}{R+W}$.

Note that this epistemic value version of indicator reliabilism for justified doxastic states also gives a straightforward account of when a suspension of judgment is justified. Simply replace (EIB3$^*$) and (EIB4$^*$) with:

(EIS3$^*$) the objective probability of $X$ given that $S$ has $g$ is (nearly) between $\frac{W}{R+W}$ and $\frac{R}{R+W}$;
(EIS4$^*$) there is no more inclusive ground $g'$ such that (a) $S$ has $g'$ and (b) the objective probability of $X$ given that $S$ has $g'$ is not (nearly) between $\frac{W}{R+W}$ and $\frac{R}{R+W}$.

And when a disbelief is justified. This time, replace (EIB3$^*$) and (EIB4$^*$)  with:

(EID3$^*$) the objective probability of $X$ given that $S$ has $g$ is (nearly) less than $\frac{R}{R+W}$;
(EID4$^*$) there is no more inclusive ground $g'$ such that (a) $S$ has $g'$ and (b) the objective probability of $X$ given that $S$ has $g'$ is not (nearly) less than $\frac{R}{R+W}$.

Next, let's turn to indicator reliabilism for justified credence. Here's the epistemic value version:

Indicator reliabilism for justified credence (epistemic value version) A credence of $x$ in proposition $X$ by agent $S$ is justified iff
(EIC1) $S$ has $g$;
(EIC2) credence $x$ in $X$ by $S$ is based on $g$;
(EIC3) if $g' \subseteq g$ is a ground that $S$ has, then for every credence $x'$, the expected epistemic value of credence $x'$ in $X$ given that $S$ has $g'$ is at most (or not much above) the expected epistemic value of credence $x$ in $X$ given that $S$ has $g'$.

Again, to complete this, we need an account of epistemic value for credences. As noted above, the veritist holds that the sole fundamental source of epistemic value for credences is their accuracy. There is a lot to be said about different potential measures of the accuracy of a credence -- see, for instance, Jim Joyce's 2009 paper 'Accuracy and Coherence', chapters 3 & 4 of my 2016 book Accuracy and the Laws of Credence, or Ben Levinstein's forthcoming paper 'A Pragmatist's Guide to Epistemic Utility'. But here I will say only this: we assume that those measures are continuous and strictly proper. That is, we assume: (i) we assume that the accuracy of a credence is a continuous function of that credence; and (ii) any probability $x$ in a proposition $X$ expects credence $x$ to be more accurate than it expects any other credence $x' \neq x$ in $X$ to be. These two assumptions are widespread in the literature on accuracy-first epistemology, and they are required for many of the central arguments in that area. Given veritism and the continuity and strict propriety of the accuracy measures, (EIC3) is provably equivalent to the conjunction of:

(EIC3$^*$) the objective probability of $X$ given that the agent has ground $g$ approximates or equals $x$ -- that is, $P(X | \mbox{$S$ has $g$}) \approx x$;
(EIC4$^*$) there is no more inclusive ground $g'$ such that (i) $S$ has $g'$ and (ii) the objective probability of $X$ given that the agent has ground $g'$ does not equal or approximate $x$ -- that is, $P(X | \mbox{$S$ has $g'$}) \not \approx x$.

But of course EIC3 = TIC3 and EIC4 = TIC4 from above. Thus, the veritist version of indicator reliabilism for justified credences is equivalent to Tang's indicator reliabilism, and thus to the calibrationist version of process reliabilism.

Epistemic value version of process reliabilism


Next, let's turn to process reliabilism. How might we give an epistemic value version of that? The mistake made by the calibrationist version of process reliabilism is of the same sort as the mistake made by Tang in his formulation of indicator reliabilism -- both generalise from the case of full beliefs in the wrong way by mistaking an accidental feature for the salient feature. For the calibrationist, a full belief is justified if it is formed by a reliable process, and a process is reliable if a high proportion of the beliefs it produces are true. Now, notice that there is a sense in which such a process is calibrated: a belief is associated with a high degree of confidence, and that matches, at least approximately, the high truth-ratio of the process. In fact, we want to say that this process is belief-reliable. For it is possible for a process to be reliable in its formation of beliefs, but not in its formation of disbeliefs. So a process is disbelief-reliable if a high proportion of the disbeliefs it produces are false. And we might say that a process is suspension-reliable if a middling proportion the suspensions it forms are true and a middling proportion are false. In each case, we think that,  corresponding to each sort of categorical doxastic attitude $s$, there is a fitting proportion $x$ such that a process is $s$-reliable if $x$ is (approximately) the proportion of truths amongst the propositions to which it assigns $s$. Applying this in the credal case gives us the calibrationist version of process reliabilism that we have already met -- a credence $x$ in $S$ is justified if it is formed by a process whose truth-ratio for a given credence is equal to that credence. However, being the product of a belief-reliable process is not the feature of a belief in virtue of which it is justified. Rather, a belief is justified if it is the product of a process that has high expected epistemic value.

Process reliabilism for justified doxastic attitude (epistemic value version)  Doxastic attitude $s$ towards proposition $X$ by agent $S$ is justified iff
(EPA1-$\beta$) $s$ is produced by a process $\rho$;
(EPA2-$\beta$) If $\rho'$ is a process that is available to $S$, then the expected epistemic value of $\rho'$ is at most (or not much more than) the expected epistemic value of $\rho$.

That is, a doxastic attitude is justified for an agent if it is the output of a process that maximizes or nearly maximizes expected epistemic value amongst all processes that are available to her. To complete this account, we must say which processes count as available to an agent. To answer this, recall Comesaña's solution to the Generality Problem. On this solution, the only processes that interest us have the form, process producing doxastic attitude $s$ towards $X$ on basis of ground $g$. Clearly, a process of this form is available to an agent exactly when the agent has ground $g$. This gives

Process Reliabilism about Justified Doxastic Attitudes (Epistemic value version) Attitude $s$ towards proposition $X$ by $S$ is justified iff
(EPA1-$\alpha$) $s$ is produced by process $\rho^g_{s, X}$;
(EPA2-$\alpha$) If  $g' \subseteq g$ is a ground that $S$ has, then for every doxastic attitude $s'$, the expected epistemic value of process $\rho^{g'}_{s', X}$ is at most (or not much more than) the expected epistemic value of process $\rho^{g}_{s, X}$.

Thus, in the case of full beliefs, we have:

Process reliabilism for justified belief (epistemic value version)  A belief in proposition $X$ by agent $S$ is justified iff
(EPB1) Belief in $X$ is produced by process $\rho^g_{\mathrm{bel}, X}$;
(EPB2) if  $g' \subseteq g$ is a ground that $S$ has, then
  1. the expected epistemic value of process $\rho^g_{\mathrm{dis}, X}$ is at most (or not much more than) the expected epistemic value of process $\rho^g_{\mathrm{bel}, X}$;
  2. the expected epistemic value of process $\rho^g_{\mathrm{sus}, X}$ is at most (or not much more than) the expected epistemic value of process $\rho^g_{\mathrm{bel}, X}$;

And it is easy to see that (EPB1) = (EIB1) + (EIB2), since belief in $X$ is produced by process $\rho^g_{\mathrm{bel}, X}$ iff $S$ has ground $g$ and a belief in $X$ by $S$ is based on $g$. Also, (EPB2) is equivalent to (EIB3). Thus, as for the epistemic version of indicator reliabilism, we get:

Indicator reliabilism for justified belief (veritist version) A belief in $X$ by agent $S$ is justified iff
(EPB1) $S$ has $g$;
(EPB2) the belief in $X$ by $S$ is based on $g$;
(EPB3) the objective probability of $X$ given that $S$ has $g$ is (nearly) greater than $\frac{W}{R+W}$;
(EPB4) there is no more inclusive ground $g'$ such that (a) $S$ has $g'$ and (b) the objective probability of $X$ given that $S$ has $g'$ is not (nearly) greater than $\frac{W}{R+W}$.

Next, consider how the epistemic value version of process reliabilism applies to credences.

Process reliabilism for justified credence (epistemic value version)  A credence of $x$ in proposition $X$ by agent $S$ is justified iff
(EPC1) the credence in $x$ is produced by process $\rho^g_{x, X}$;
(EPC2) if  $g' \subseteq g$ is a ground that $S$ and $x'$ is a credence, then the expected epistemic value of process $\rho^{g'}_{x', X}$ is at most (or not much more than) the expected epistemic value of process $\rho^g_{x, X}$.

As before, we see that (EPC1) is equivalent to (EIC1) + (EIC2). And, providing the measure of accuracy is strictly proper and continuous, we get that (EPC2) is equivalent to (EIC3). So, once again, we arrive at the same summit. The routes taken by Tang, Dunn, and the epistemic value versions of process and indicator reliabilism lead to the same spot, namely, the following account of justified credence:

Reliabilism for justified credence (epistemic value version)  A credence of $x$ in proposition $X$ by agent $S$ is justified iff
(ERC1) $S$ has $g$;
(ERC2) credence $x$ in $X$ by $S$ is based on $g$;
(ERC3) the objective probability of $X$ given that the agent has ground $g$ approximates or equals $x$ -- that is, $P(X | \mbox{$S$ has $g$}) \approx x$;
(ERC4) there is no more inclusive ground $g'$ such that (i) $S$ has $g'$ and (ii) the objective probability of $X$ given that the agent has ground $g'$ does not equal or approximate $x$ -- that is, $P(X | \mbox{$S$ has $g'$}) \not \approx x$.


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