II. The Simple Threshold View

When is it justified to have a belief? On one view, you are justified in believing a proposition if and only if (iff) you have a high enough credence in that proposition given your total evidence. And your credence is high enough iff it exceeds a certain threshold.Footnote ³ On this simple threshold view, your credences or “degrees of belief” are rational: they satisfy the Kolmogorov axioms of probability and the ratio definition of conditional probability.

The simple threshold view translates your evidence-based credences into what it deems justified beliefs. When your credence in a proposition $ A $ , given your whole body of evidence, surpasses the threshold, your belief in $ A $ is justified. Your credences, in turn, are justified by the total evidence you received, which is summarized by the strongest proposition of which you are now certain. To be clear, your credences at time t are represented by a probability function conditionalized on the total evidence you received up to time t. Typically, your total evidence leads to non-extreme credences: subjective probabilities strictly in-between 0 and 1. You would have only few justified beliefs if you were to set the threshold for them to 1—too few.

The exact threshold for justified belief, if there is any, is of course subject to debate. But there is a natural lower bound: justified belief in a proposition requires a credence in that proposition strictly above $ 1/2 $ . Suppose for reductio that the threshold were equal to $ 1/2 $ or less. Then you could be justified in believing a proposition $ A $ and its negation ¬ $ {A} $ to be true at the same time. But you are never justified in believing such a contradiction to be true. The resulting inconsistent belief state is irrational and so unjustified.

The simple threshold view can explain how your beliefs are supported by your evidence. Imagine you look out of the window and you see it raining. Based on this evidence, your credence in rain goes up so that it seems to be high enough for a justified belief in rain. Conversely, if you are justified in believing that it rains, your credence in rain must be high enough in some sense. The view is not too implausible. But there are issues.

A—if not “the”—problem for the simple threshold view is posed by statistical evidence, which supports a high credence but no justified belief. For example, suppose there are 100 prisoners in a yard under the supervision of a guard. Ninety-nine of them join a pre-planned attack to kill the guard. One prisoner clearly refrains, standing alone in a corner. We know this from a reliable video recording. However, the video footage does not allow us to discern the individual prisoners—they all wear the same uniforms and the quality is not good enough to identify faces or other characteristics. There is no other evidence. Each prisoner is tried in a court of law.Footnote ⁴

Are you justified in believing that the prisoner standing trial is guilty? On the simple threshold view, you are because your credence in his guilt is high enough. If legal proof should be tantamount to high enough credence of guilt, this would also mean that we should find the prisoner guilty. But many are not willing to endorse this consequence.Footnote ⁵ Lara Buchak, for example, says:

On this, I agree with Buchak. However, Buchak also argues that justified belief cannot be reduced to credence. And so “threshold views of the relationship between licensed court verdicts and rational credence are false.” I disagree. In what follows, I offer a threshold view on which justified belief requires high credence, but purely statistical evidence does not give rise to justified belief.

III. A Threshold View of Justified Belief

I develop now the notion of justified belief in terms of credence. As on the simple threshold view, your credences are determined by your total evidence. This means you consider all the pieces of evidence you received. If you are a fact-finder, you exclude the inadmissible pieces of evidence. But there is more. Your total evidence also determines the possibilities you consider overall. You consider all and only those possibilities your credences assign a definite positive probability value. The total evidence you received partitions the underlying set $ W $ of all logical possibilities into possibilities $ {\pi}_i $ that are assigned a positive credence $ {P}_{\Pi}\left(\left\{{\pi}_i\right\}\right)>0 $ . Such a possibility is a maximally specific way things might be with respect to the partition Π induced by your total evidence. A partition Π on $ W $ is a set of pairwise disjoint and non-empty subsets $ {\pi}_i $ of $ W $ so that $ \cup {\pi}_i=W $ . If you are a fact-finder, your initial partition includes the possibilities that the defendant is guilty and innocent.

Propositions are understood relative to a partition. Any subset of a partition Π is a proposition. A proposition $ A\hskip0.3em \subseteq \hskip0.3em \Pi $ is consistent iff $ A\ne \varnothing $ . A proposition $ A $ is consistent with a proposition $ B $ iff $ A\hskip0.3em \cap \hskip0.3em B\ne \varnothing $ . $ A $ entails $ B $ iff $ A\hskip0.3em \subseteq \hskip0.3em B $ . The negation ¬ $ {A} $ of a proposition is given by its complement $ \Pi \hskip0.3em \backslash \hskip0.3em A $ , the conjunction $ A\hskip0.3em \wedge \hskip0.3em B $ of two propositions by their intersection $ A\hskip0.3em \cap \hskip0.3em B $ , and the disjunction $ A\hskip0.3em \vee \hskip0.3em B $ by their union $ A\hskip0.3em \cup \hskip0.3em B $ . The probability distribution $ {P}_{\Pi} $ is defined for all and only subsets of $ \Pi $ .

We are now in a position to define justified belief in terms of credence:

The conjunct “you expect $ {B}_{\theta } $ to remain more likely than not” means you consider no relevant proposition that would lower your credence $ {P}_{\Pi}\left({B}_{\theta}\right) $ to $ 1/2 $ or below. You consider a proposition $ B $ to be relevant to $ {B}_{\theta } $ iff your credence in it is non-zero and it is consistent with $ {B}_{\theta } $ . In symbols, you consider a proposition $ B\hskip0.3em \subseteq \hskip0.3em \Pi $ to be relevant to $ {B}_{\theta}\hskip0.3em \subseteq \hskip0.3em \Pi $ iff $ {P}_{\Pi}(B)>0 $ and $ {B}_{\theta}\hskip0.3em \cap \hskip0.3em B\ne \varnothing $ . In sum, you expect $ {B}_{\theta } $ to remain more likely than not iff your conditional credence $ {P}_{\Pi}\left({B}_{\theta }|B\right)>1/2 $ for any proposition $ B\hskip0.3em \subseteq \hskip0.3em \Pi $ you consider relevant. Equivalently, you expect $ {B}_{\theta } $ to remain more likely than not iff you consider no relevant proposition $ B\hskip0.3em \subseteq \hskip0.3em \Pi $ which would lower your credence $ {P}_{\Pi}({B}_{\theta }|B) $ to $ 1/2 $ or below.Footnote ⁷

My notion of justified belief is synchronic: you are justified in believing a proposition at time t iff you have a high credence in it at t and you expect it to remain more likely than not at t. In what follows, the reference to a point in time is left implicit.

Let us revisit the prisoners example. Your total evidence is that 99 out of 100 prisoners killed a prison guard. Each of the 100 prisoners may be innocent and you have no reason to believe that any one is more or less likely to be guilty than any of the others. All the 99:1 statistic says is that there are 100 equiprobable possibilities that each prisoner is innocent and all the others are guilty. So your total evidence assigns to only 100 possibilities a definite positive probability value, and so partitions the underlying set of possibilities into exactly 100. Let $ {\pi}_i $ be the possibility where prisoner i ( $ 1\le i\le 100 $ ) is innocent and all the other prisoners are guilty. Your total evidence makes you consider the partition $ \Pi =\left\{{\pi}_1,\dots, {\pi}_{100}\right\} $ of mutually exclusive and jointly exhaustive possibilities. Moreover, you assign to each possibility the same definite credence: $ {P}_{\Pi}\left(\left\{{\pi}_1\right\}\right)=\dots ={P}_{\Pi}\left(\left\{{\pi}_{100}\right\}\right)=1/100 $ .

Let us name the prisoner standing trial “prisoner 1.” Are you justified to believe that prisoner 1 is guilty? Well, your credence in the guilt of prisoner 1 is high: $ {P}_{\Pi}\left(\Pi \backslash \left\{{\pi}_1\right\}\right)=99/100 $ . However, you do not expect any non-empty $ {B}_{\theta}\subseteq \Pi \backslash \left\{{\pi}_1\right\} $ to remain more likely than not:

$$ {P}_{\Pi}\left({B}_{\theta }|\left\{{\pi}_1,b\right\}\right)=\frac{P\left(\left\{b\right\}\right)}{P\left(\left\{{\pi}_1,b\right\}\right)}=1/2\hskip0.62em \mathrm{for}\hskip0.5em \mathrm{any}\hskip0.52em b\in {B}_{\theta }. $$

For illustration, let $ {B}_{\theta }=\Pi \backslash \left\{{\pi}_1\right\} $ . You assign the proposition that prisoner 1 or prisoner 2 is innocent a positive credence and the proposition is consistent with the proposition that prisoner 1 is guilty. Hence, you consider the proposition relevant. And your credence that prisoner 1 is guilty given that prisoner 1 or prisoner 2 is innocent does not strictly exceed $ 1/2 $ :

$$ {P}_{\Pi}\left(\Pi \backslash \left\{{\pi}_1\right\}|\left\{{\pi}_1,{\pi}_2\right\}\right)=\frac{P\left(\left\{{\pi}_2\right\}\right)}{P\left(\left\{{\pi}_1,{\pi}_2\right\}\right)}=1/2. $$

This means you do not expect $ \Pi \backslash \left\{{\pi}_1\right\} $ to remain more likely than not. A similar argument applies to each prisoner. Hence, you are not justified in believing of any prisoner that he is guilty—even though your respective credence is very high. My notion of justified belief solves this paradigmatic example of statistical evidence.

My notion can also deliver justified belief where appropriate. To see this, consider a modification of the prisoners example. There is no video recording, but the prison director walks by. She then testifies about prisoner 1: “I saw him killing the guard!” Let’s suppose you think that the director is very reliable but not perfectly so: she raises your credence that prisoner 1 is guilty to $ 99/100 $ .

Your total evidence is the prison director’s eyewitness testimony “I saw prisoner 1 killing the guard.” This testimony is only about prisoner 1; it does not say anything about another prisoner. It does not answer at all the question of how likely it is that prisoner 2 is guilty. As you deem the prison director not perfectly reliable, her testimony makes you assign a definite positive credence to exactly two possibilities: the testimony is true and so prisoner 1 killed the guard, or else the testimony is false and the prisoner is innocent. Indeed, all the testimony bears on is the question whether or not prisoner 1 is guilty of having killed the guard. So your total evidence makes you consider the partition $ \Pi '=\left\{\pi {'}_1,\pi {'}_2\right\} $ , where $ \pi {'}_1 $ is the possibility that prisoner 1 is innocent and $ \pi {'}_2 $ the possibility that prisoner 1 is guilty.

Are you justified in believing that prisoner 1 is guilty based on the eyewitness evidence? Well, your credence in the guilt of prisoner 1 is high: $ {P}_{\Pi '}\left(\Pi '\backslash \left\{\pi {'}_1\right\}\right)=99/100 $ . And you expect it to remain high. There are only three possibilities. $ \left\{\pi {'}_1\right\} $ is inconsistent with $ \Pi '\backslash \left\{\pi {'}_1\right\} $ and so irrelevant; $ {P}_{\Pi '}\Big(\left\{\pi {'}_2\right\}|\left(\left\{\pi {'}_2\right\}\right)=1 $ ; and $ {P}_{\Pi '}\left(\left\{\pi {'}_2\right\}|\Pi '\right)=99/100 $ . By choosing $ {B}_{\theta }=\Pi '\backslash \left\{\pi {'}_1\right\}, $ you are justified to believe that prisoner 1 is guilty.

We have seen that the different pieces of evidence in the prisoners example and the director example, respectively, determine the same credence in the guilt of prisoner 1. And yet my threshold view says that there is justified belief in his guilt in the former but not in the latter example. The reason is that the different pieces of evidence give rise to different partitions of the underlying possibilities. The 99:1 statistic induces a uniform credence function expressing a symmetry between the prisoners: each prisoner is just as likely as any other to be innocent. I suggest this symmetry is why it feels so random to convict one of the prisoners based on statistical evidence alone: looking at the probability values, it could likewise have been any other prisoner.Footnote ⁸

The director’s eyewitness testimony, by contrast, biases the fact-finder’s credences towards prisoner 1 being guilty. It induces an uneven credence function expressing that the eyewitness evidence supports the guilt possibility much more than the innocence possibility. The uneven credence over the two-cell partition—prisoner 1 killed the guard, or else he did not—gives a rather strong indication of what to believe about prisoner 1. The eyewitness evidence induces no air of randomness: the two possibilities are far from being equally likely. From the vantage point of my notion of justified belief, the distinction between the 99:1 statistic and the eyewitness testimony hints at a general notion of statistical evidence.

IV. Statistical Evidence

I have said that statistical evidence supports a high credence but no justified belief.Footnote ⁹ This common characterization is directly derived from the prisoners example, where the 99:1 statistic supports a high credence but no justified belief. Indeed, the term “statistical evidence” is usually characterized only by pointing to examples.Footnote ¹⁰ Enoch and Spectre make this explicit:

I believe lacking a definition of statistical evidence is a problem. As long as we don’t know what pieces of evidence are statistical, we simply cannot answer the question of whether or not statistical evidence may give rise to justified belief—on my notion or another. We need a general definition of “statistical evidence” in order to know whether my notion of justified belief solved the problem posed by it.

My notion of justified belief suggests the following definition of statistical evidence. A piece of evidence is purely statistical iff it would assign a uniform probability distribution over the partition it induces if it were the only piece of evidence received. So defined, a piece of purely statistical evidence alone may support a high credence but never gives rise to justified belief. The problem posed by statistical evidence for the simple threshold view is solved by my notion of justified belief on the suggested definition. Furthermore, we may say that your total evidence is purely statistical iff it assigns uniform credences over the partition it induces. Non-statistical total evidence assigns non-uniform credences over the induced partition, and so may give rise to justified belief. The degree to which a piece of evidence (or the total evidence) is statistical may be measured by how much it deviates from the uniform distribution. The exact measure is left for future work.

In the literature, statistical evidence is usually contrasted with “individual evidence.”Footnote ¹² A paradigm example of individual evidence is eyewitness testimony. Indeed, the prison director’s testimony is non-statistical (to a high degree) and induces justified belief. Based on the testimony alone, you expect the prisoner’s guilt to remain more likely than not. My statistical/non-statistical distinction (or continuum) accounts for the intuitive dichotomy between “statistical” and “individual” evidence in our two examples.

Many statistics are not statistical evidence on my definition. This sounds paradoxical but it is not. All statistics which, if they were the sole piece of evidence, would assign a non-uniform probability distribution over the induced partition are non-statistical (to some degree). And any sufficiently uneven distribution can give rise to justified belief depending on the induced partition and the specific probability values. An uneven distribution based on a statistic may well give rise to justified belief.

Base rates, by contrast, are statistical evidence. A base rate is a relative frequency—a proportion of individuals in a population who have a certain feature. A base rate without any further evidence induces a uniform distribution over a fixed number of possibilities. My definition, therefore, can explain why “a base rate unaccompanied by other evidence” is purely statistical evidence.Footnote ¹³ But note that my definition counts any piece of evidence inducing a uniform distribution “purely statistical”—not only base rates.

In my framework, all evidence is probabilistic. Each piece of evidence—be it statistical or not—leads to some credence distribution. Pieces of statistical evidence may still help give rise to justified belief when combined with other pieces of evidence. So “the” law should have no aversion to statistics in general, let alone probabilistic evidence.Footnote ¹⁴ The point is merely that a piece of purely statistical evidence alone cannot give rise to justified belief. And neither does purely statistical total evidence.

V. Rational Belief States

One might wonder, “why am I not justified in believing that prisoner 1 is guilty in the prisoners example? After all, my credence in his guilt is very high. Why is this not sufficient for justified belief of guilt?” This objection sides with the intuition expressed in the simple threshold view. My answer is that you have no justified belief in his guilt because your belief state would be irrational. Let me explain.

We have assumed that your beliefs are rational: consistent and deductively closed. To be more precise, we say your state of belief is consistent iff the strongest proposition $ {B}_{\theta } $ you believe—the conjunction of all the propositions you believe—is non-empty. We say your state of full belief is deductively closed iff (1) you believe the proposition $ B $ if you believe $ A $ and $ A $ entails $ B $ , and (2) you believe $ A\hskip0.3em \wedge \hskip0.3em B $ if you believe $ A $ and you believe B. Hence, you believe a proposition $ A $ iff your strongest belief $ {B}_{\theta } $ is consistent and entails $ {A} $ .

We show now that your beliefs in the prisoners example cannot be rational if you adhere to the simple threshold view. If so, you are justified in believing that one prisoner is innocent because your credence $ {P}_{\Pi}\left(\Pi \right)=1 $ is maximal. At the same time, you are justified in believing of each prisoner that he is guilty because your credence $ {P}_{\Pi}\left(\Pi \backslash \left\{{\pi}_i\right\}\right)=99/100 $ is high enough ( $ 1\le i\le 100 $ ). If your justified beliefs were closed under conjunction, you would also be justified in believing that all prisoners are guilty:

$$ \left(\Pi \backslash {\pi}_1\right)\hskip0.3em \cap \hskip0.3em \left(\Pi \backslash {\pi}_2\right)\hskip0.3em \cap \dots \cap \hskip0.3em \left(\Pi \backslash {\pi}_{100}\right)=\varnothing . $$

But you should never be justified in believing the contradiction that all prisoners are guilty and one is innocent. Indeed, the “simple threshold you” is not justified in believing that all prisoners are guilty because your credence $ P\left(\varnothing \right) $ in that proposition is minimal. Your “justified” beliefs are rather not closed under conjunction and so they are not deductively closed. The simple threshold view cannot satisfy the two standard rationality norms on justified belief.

Consequently, a simple threshold believer with non-maximal threshold is not guaranteed to have a rational state of belief: there is no strongest but consistent belief that entails all her other beliefs. This is a cost. To see that, imagine you are found guilty but the fact-finder’s “justified” beliefs are inconsistent. Surely you have a reason to contest the verdict. For one, the fact-finder believes everything, in particular that you are innocent. Moreover, imagine there are only three elements to be proven in court for a finding of guilt. The fact-finder is well justified in believing each element $ A $ , $ B $ , and $ C $ in isolation. However, the fact-finder’s beliefs are not closed under conjunction. So it could be that they are not justified in believing all elements $ A\wedge B\wedge C $ . This is indeed a possibility on the simple threshold view. Should the defendant then be found guilty? If so, the simple threshold view of legal proof is false. If not, what else is required for a justified finding of guilt besides the justified beliefs in each element? And how should the verdict of not guilty be explained considering the justified beliefs in $ A $ , $ B $ , and $ C $ ? To give up deductive closure seems to lead to more questions than answers.

Unlike the simple threshold view, my threshold view guarantees that you have a rational belief state. There is always a consistent and deductively closed belief state represented by the strongest non-empty proposition $ {B}_{\theta } $ you are justified in believing. Your credence $ {P}_{\Pi}\left({B}_{\theta}\right) $ is typically non-maximal. In the prisoners example, however, you are not justified in believing of a particular prisoner that he is guilty even though your credence in his guilt is very high. For then you would be justified in believing of each prisoner that he is guilty due to the symmetry of the example. But then, as your justified beliefs are deductively closed, you would believe that all prisoners are guilty and that one is innocent. Fortunately, you are rational enough not to believe such a contradiction. All you are justified in believing is that one prisoner is innocent and the other 99 are guilty—you just don’t know which one. Your rational belief state is $ {B}_{\theta }=\Pi $ , where $ \Pi $ is both the partition induced by your total evidence and the strongest proposition of which you are certain. Assuming your total evidence is non-empty, you are at the very least justified in believing your total evidence—and more generally any proposition to which you assign maximal credence.

We can assume that a notion of justified belief that guarantees rationality is better than one that doesn’t. This suggests that the rationality of our belief states explains the intuition that belief is not justified in the prisoners example—the unease to say yes that is felt by many when asked whether they believe the prisoner is guilty. There is no such unease in the director’s example. And no wonder. In this example, the consistency, deductive closure, and evidential support of your beliefs are not in any tension: you have a high enough credence in his guilt and expect it to remain more likely than not.

VI. Thresholds and Legal Proof

My notion of justified belief entails a threshold view. Your state of justified belief is represented by the “strongest” non-empty proposition $ {B}_{\theta } $ you are justified in believing. Hence, you are justified in believing a proposition $ A\subseteq \Pi $ iff $ {B}_{\theta}\subseteq A $ . $ {B}_{\theta}\subseteq \Pi $ is then the smallest set of possibilities of which you have a high enough credence and you expect it to remain more likely than not. As $ {B}_{\theta } $ is non-empty, it is consistent with the partition Π induced by your total evidence such that $ {P}_{\Pi}\left(\Pi \right)=1 $ . As your $ {P}_{\Pi}({B}_{\theta }|\Pi ) $ must exceed $ 1/2 $ , so does $ {P}_{\Pi}\left({B}_{\theta}\right) $ . And any superset of $ {B}_{\theta } $ has a credence that is at least as high. This means you are justified in believing a proposition $ A\subseteq \Pi $ iff $ {P}_{\Pi}(A)\ge {P}_{\Pi}\left({B}_{\theta}\right) $ .Footnote ¹⁵ Your (variable) threshold for justified belief is your (current) credence in the strongest proposition you are justified in believing.

I propose that a legal standard of proof should be tantamount to justified belief of guilt relative to a certain credence threshold. Murder is a matter of criminal law. A criminal conviction requires the prosecution to prove the defendant’s guilt “beyond a reasonable doubt.” This means the admissible evidence presented in court must be enough to “remove any reasonable doubt in the mind of the fact-finder that the accused is guilty of the crime with which they are charged.” What the phrase is supposed to mean, however, is less clear. It can be explained as follows: proof beyond a reasonable doubt should be tantamount to justified belief in a criminal trial. A defendant’s guilt should be proven beyond a reasonable doubt iff your justified beliefs $ {B}_{\theta } $ entail his guilt. You have a reasonable doubt if you consider a proposition to be relevant which would lower your credence in $ {B}_{\theta } $ to $ 1/2 $ or below. If you consider a set of possibilities consistent with your beliefs that would make the defendant’s innocence more likely than not, you have a reasonable doubt. If so, your belief in guilt is not justified.Footnote ¹⁶

Let us turn to the long-anticipated question: what credence is high enough for justified belief? The answer is that your credence should exceed a threshold that depends on your value assessments or “stakes.” You are fair in that you consider how valuable the possible outcomes of your verdict are. The argument rests on orthodox decision theory, where utility is a proxy for value. The theory says we should maximize expected utility. In this paradigm, a defendant should be found guilty iff finding him guilty has greater expected utility than finding him innocent.Footnote ¹⁷ The legal decision problem can be summed up as follows:

Your available options are finding him guilty or innocent. Let’s denote your credence that the prisoner is guilty by P(G), and so your credence that he is innocent by 1 − P(G). Each option—together with the prisoner’s actual guilt or innocence—determines an outcome: a true finding of guilt (TG), a false finding of guilt (FG), a false finding of innocence (FI), and a true finding of innocence (TI). Your utility function U assigns each outcome a value. According to the principle of maximizing expected utility, you should find the prisoner guilty iff

$$ U(TG)\cdotp P(G)+U(FG)\cdotp \left(1-P(G)\right)\ge U(FI)\cdotp P(G)+U(TI)\cdotp \left(1-P(G)\right). $$

The inequality is equivalent to

$$ P(G)\ge \frac{U(TI)-U(FG)\;}{U(TG)-U(FG)+U(TI)-U(FI)}=\theta . $$

Standard decision theory recommends finding guilty just in case your credence of guilt meets the threshold obtained from the utility values you assign to the outcomes. We borrow the threshold $ \theta $ to choose among the sets $ {B}_{\theta } $ you expect to remain more likely than not: pick the least probable $ {B}_{\theta } $ such that $ {P}_{\Pi}\left({B}_{\theta}\right)\ge \theta . $

Plausibly, you neither disvalue a true finding of guilt nor a true finding of innocence. Let’s say the cost of true findings is zero: $ U(TG)=U(TI)=0 $ . Under this assumption, we obtain

$$ P(G)\ge \frac{-U(FG)}{-U(FG)-U(FI)}. $$

Furthermore, let’s say you disvalue falsely finding the prisoner guilty much more than falsely finding him innocent. Blackstone thought “the law holds that it is better that ten guilty persons escape than that one innocent suffer.”Footnote ¹⁸ Assuming you deem a decision-theoretic version of the Blackstone ratio to be true of the prisoners example, you think a false finding of guilt is ten times worse than a false finding of innocence:

$$ U(FG)=-10\hskip4.00pt \mathrm{a}\mathrm{n}\mathrm{d}\hskip2pt U(FI)=-1. $$

Your credence threshold for justified belief is then $ 10/11 $ , or approximately 0.91. You deem a credence above this threshold to be high enough for justified belief in the prisoner’s example. This is how we use orthodox decision theory to obtain a threshold for your “high enough” credence.

To be clear, we propose a justified belief account of legal proof, not one that maximizes expected utility. And we do not aim to reconcile the two approaches. Indeed, you think finding prisoner 1 guilty has greater expected utility than finding him innocent in the prisoners example. Yet you are not justified in believing that prisoner 1 is guilty—at least on my notion of justified belief. Rational truth-seekers and expected utility maximizers may come apart.Footnote ¹⁹ The maximizers achieve a much higher accuracy than our truth-seeker in the prisoners example: 99 correct verdicts and only one incorrect one. But they cannot justify their verdicts in terms of full belief without further ado. Perhaps they could do so by adopting the simple threshold view. But there is some tension. Their value assessments in certain cases may be as follows: finding guilty has greatest expected utility but the credence threshold for finding guilty, and so for justified belief in guilt, is below $ 1/2 $ . As I explained above, such a simple credence threshold leads to “justified” beliefs in contradictions. Our rational truth-seeker does not maximize expected utility and is not prepared to sacrifice the rationality of belief for a gain in accuracy.

I leave a thorough comparison of my justified belief account of legal proof to one of maximizing expected utility for future work. There, I will discuss an avenue for reconciling the two approaches to a large extent, namely by investigating what value assessments are appropriate in what legal cases. For now, I merely use decision theory to borrow a credence threshold for justified belief. And I can do this without further problem: if the threshold imported from decision theory happens to be exactly $ 1/2 $ or below, your threshold is just above $ 1/2 $ —you must still expect your justified beliefs $ {B}_{\theta } $ to remain more likely than not.

My account of legal proof is unifying. Evidential standards other than beyond a reasonable doubt just require a less-high credence of guilt. The reason seems to be this: the difference between disvaluing a false finding of guilt and a false finding of innocence diminishes in these cases, compared with the “high stakes” cases like murder.

The evidential standard of civil law is known as “preponderance of the evidence” and is typically interpreted thus: a plaintiff’s claim counts as proven in court just in case the claim is established to be more likely than not. In a civil trial, you should have no preference for either a finding for the plaintiff or a finding for the defendant. This means in our framework that a false finding of liability and a false finding of innocence should be equally costly. Together with the above assumption that the cost of true findings is zero, the borrowed threshold for finding liable should then be exactly $ 1/2 $ . A defendant’s liability should be established by preponderance of the evidence iff the fact-finder is justified in believing that the defendant is liable. And this is the case whenever $ {B}_{\theta } $ entails the defendant’s liability and they expect $ {B}_{\theta } $ to remain more likely than not. This solves the “blue bus” case—a case of civil law that is structurally similar to the prisoners exampleFootnote ²⁰—and other structurally similar cases such as the “paradox of the gatecrasher.”Footnote ²¹ In sum, my account of legal proof solves the “proof paradoxes.”Footnote ²²