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Hezareh M, Etemadoleslami Bakhtiari S. Di Bello on Smith's Solution to the Proof Paradoxes. jpt 2025; 5 (1) :81-95
URL: http://jpt.modares.ac.ir/article-34-79377-en.html
URL: http://jpt.modares.ac.ir/article-34-79377-en.html
1- Department of Science and Technology Studies, Faculty of Management, Science and Technology, Amirkabir University of Technology, Tehran, Iran
Keywords: Proof Paradoxes, Normic Support, Naked Statistical Evidence, Undercutting Defeater, Base Rate Fallacy, Smith, Di Bello, Pollock
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Introduction
In a general classification within the legal system, cases are divided into two categories: civil and criminal. Civil cases are adjudicated based on the standard of "balance of probabilities." In contrast, in criminal cases, the required standard for issuing a verdict is "proof beyond a reasonable doubt" [Hamer, 2004: 74]. According to the probabilistic interpretation of these standards of proof, the decision threshold is set at 50% for civil cases. In contrast, the criminal justice system protects defendants against wrongful convictions and upholds the presumption of innocence; therefore, the threshold for criminal cases must be significantly higher than that established for civil cases. Typically, 90 or 95% thresholds are necessary to decide the criminal standard [Hamer, 2004: 74].
After this introduction, we will discuss proof paradoxes. Proof paradoxes arise when the hypothesis of a defendant's guilt or conviction is supported by naked statistical evidence. Despite the high probability associated with these hypotheses, exceeding the thresholds established for decision-making, they frequently appear intuitively insufficient to justify a verdict. Such counterintuitive situations can occur in both civil and criminal cases. The prison scenario [Nesson, 1979: 1192] in criminal cases and the blue bus company case [Di Bello, 2020: 1] in civil cases are two examples of this subject that will be examined in this article.
Although the probability that the defendant is the perpetrator may exceed the threshold established by the relevant standard of proof, this evidence does not seem sufficient for a conviction. Consider a scenario where we have eyewitness testimony instead of relying on naked statistical data. Can we assert that the probability of an error in the eyewitness's testimony is lower than the probability determined in the statistical scenario? It appears that eyewitnesses can be equally, if not more, fallible. Despite this fallibility in eyewitness accounts and other forms of evidence, many jurists and philosophers argue that such evidence possesses an intuitive superiority over naked statistical data and is, therefore, justifiable in proving a crime [Wells, 1992: 52-793]
What accounts for this intuitive superiority? There are two primary approaches to addressing this question. The first is the "Revisionary Approach" [Di Bello, 2019: 1046]. According to this approach, based on other probative evidence, the probability of the defendant's guilt is essentially the same as, or even lower than, that found in examples of proof paradoxes. Consequently, our intuition regarding the superiority of other evidence is unwarranted and requires revision [Schoeman, 1987; Roth, 2010].
The second perspective discussed is the "Non-Revisionary" approach, which is more widely endorsed among scholars addressing this issue and will be the focus of this article. The central argument of this approach is that the intuitive superiority of certain types of evidence is justified. The primary objective of the non-revisionary view is to differentiate between valid probative evidence and naked statistical evidence.
Many philosophers and jurists have attempted to clarify this distinction. For instance, Williams argues that probative evidence must be directly tied to the defendant, whereas naked statistical evidence does not meet this criterion [Williams, 1979: 297-312, 340]. Similarly, Thomson posits that there must be a causal relationship between the evidence and the desired proposition [Thomson, 1986: 199-219]. Furthermore, Moss and Duff propose a specific version of knowledge relevant to judicial decisions that can differentiate between naked statistical evidence and other compelling evidence for conviction [Moss, 2018; Duff, 2007; Redmayne, 2008].
Smith's solution and Di Bello's amendment
The efforts of Di Bello and Smith represent some of the most significant contributions to this field. Smith argues that the key distinguishing feature between naked statistical evidence and other forms of justified probative evidence is the normic support criteria [Smith, 2018]. According to Smith, normic support reflects our expectations regarding what is likely to happen. For instance, when we place a letter in a mailbox, we expect it to be delivered to the recipient within a few days. If the letter does not reach the recipient as anticipated, there must be an explanation for this unexpected outcome. One possible explanation could be that the vehicle transporting the mail was stolen en route. Smith asserts that naked statistical evidence does not satisfy this criterion [Gardiner, 2018].
Di Bello identified a flaw in this aspect of Smith's theory, specifically the assertion that the mere possibility of a coincidental match in evidence renders that evidence incapable of satisfying normic support. Di Bello demonstrated that coincidental matches can also occur in other forms of evidence, such as eyewitness testimony, DNA matches, and fingerprints [Zabell, 2005]. As a result, according to Smith's argument, other types of evidence also fail to meet the criteria for normic support, leading to a challenge of overgeneralization.
To address this issue, Di Bello disregarded the possibility of a coincidental match as an obstacle to fulfilling the normic support and employed it as a justified explanation in the process of satisfying normic support. Subsequently, utilizing Pollock's classification of defeaters in a defeasible argument, Di Bello demonstrated that the defendant can respond to the evidence presented against him in two ways during cross-examination. First, the defendant can use additional information to undercut the connection between the evidence and the conviction. Second, the defendant can present independent evidence that rebuts the hypothesis supporting the conviction and supports an alternative hypothesis incompatible with the defendant's conviction. Di Bello asserts that when naked statistical evidence serves as the probative evidence, only rebutting defeaters are available, while undercutting defeaters are not. Access to undercutting defeaters restricts the defendant's ability to mount a defense. Di Bello contends that this limitation represents an epistemological weakness of naked statistical evidence compared to other forms of probative evidence, contributing to the intuitive preference for the latter.
Critical Examination of Di Bello's Proposal
According to Di Bello’s argument, only rebutting defeaters are available when naked statistical evidence serves as the basis for conviction. For instance, in the Blue Bus scenario, one might consider the ownership ratio of Blue Buses as evidence while citing the Blue Company’s lower accident rate over the past year as a counterpoint. However, Pollock argues that when two pieces of statistical evidence rebut, the evidence that provides additional information relative to the other will have a rebutting and undercutting effect [Pollock, 2001]. Therefore, Di Bello's assertion that undercutting defeaters are unavailable in such cases appears flawed. This is because rebutting defeaters in statistical evidence will inherently provide an undercutting effect against the opposing evidence.
We then utilized the base rate fallacy to highlight the flaws present in Di Bello’s argument [Bar-Hillel, 1980]. The base rate fallacy quantitatively represents Pollock's argument. If base rate information is not incorporated into the diagnostic information calculation, we risk falling into the base rate fallacy. The effect of base rate information is the quantitative representation of the undercutting defeater that Pollock deemed essential for reaching an accurate conclusion.
Conclusion
Smith established normic support as a criterion for distinguishing between naked statistical evidence and other valid forms of probative evidence. Di Bello criticized Smith's definition of normic support, arguing that it leads to the challenge of overgeneralization. Consequently, while modifying Smith's criteria, he contended that the absence of an undercutting defeater in naked statistical evidence sets it apart from other valid probative evidence. By examining Pollock's classification of defeaters, we argued that undercutting defeaters still applies to naked statistical evidence. Specifically, in the Blue Bus scenario, we demonstrated the impact of undercutting defeaters, drawing on Pollock's arguments, which quantitatively illustrate the base rate fallacy. Therefore, Di Bello's argument for distinguishing between naked statistical evidence and other probative evidence appears questionable.
In a general classification within the legal system, cases are divided into two categories: civil and criminal. Civil cases are adjudicated based on the standard of "balance of probabilities." In contrast, in criminal cases, the required standard for issuing a verdict is "proof beyond a reasonable doubt" [Hamer, 2004: 74]. According to the probabilistic interpretation of these standards of proof, the decision threshold is set at 50% for civil cases. In contrast, the criminal justice system protects defendants against wrongful convictions and upholds the presumption of innocence; therefore, the threshold for criminal cases must be significantly higher than that established for civil cases. Typically, 90 or 95% thresholds are necessary to decide the criminal standard [Hamer, 2004: 74].
After this introduction, we will discuss proof paradoxes. Proof paradoxes arise when the hypothesis of a defendant's guilt or conviction is supported by naked statistical evidence. Despite the high probability associated with these hypotheses, exceeding the thresholds established for decision-making, they frequently appear intuitively insufficient to justify a verdict. Such counterintuitive situations can occur in both civil and criminal cases. The prison scenario [Nesson, 1979: 1192] in criminal cases and the blue bus company case [Di Bello, 2020: 1] in civil cases are two examples of this subject that will be examined in this article.
Although the probability that the defendant is the perpetrator may exceed the threshold established by the relevant standard of proof, this evidence does not seem sufficient for a conviction. Consider a scenario where we have eyewitness testimony instead of relying on naked statistical data. Can we assert that the probability of an error in the eyewitness's testimony is lower than the probability determined in the statistical scenario? It appears that eyewitnesses can be equally, if not more, fallible. Despite this fallibility in eyewitness accounts and other forms of evidence, many jurists and philosophers argue that such evidence possesses an intuitive superiority over naked statistical data and is, therefore, justifiable in proving a crime [Wells, 1992: 52-793]
What accounts for this intuitive superiority? There are two primary approaches to addressing this question. The first is the "Revisionary Approach" [Di Bello, 2019: 1046]. According to this approach, based on other probative evidence, the probability of the defendant's guilt is essentially the same as, or even lower than, that found in examples of proof paradoxes. Consequently, our intuition regarding the superiority of other evidence is unwarranted and requires revision [Schoeman, 1987; Roth, 2010].
The second perspective discussed is the "Non-Revisionary" approach, which is more widely endorsed among scholars addressing this issue and will be the focus of this article. The central argument of this approach is that the intuitive superiority of certain types of evidence is justified. The primary objective of the non-revisionary view is to differentiate between valid probative evidence and naked statistical evidence.
Many philosophers and jurists have attempted to clarify this distinction. For instance, Williams argues that probative evidence must be directly tied to the defendant, whereas naked statistical evidence does not meet this criterion [Williams, 1979: 297-312, 340]. Similarly, Thomson posits that there must be a causal relationship between the evidence and the desired proposition [Thomson, 1986: 199-219]. Furthermore, Moss and Duff propose a specific version of knowledge relevant to judicial decisions that can differentiate between naked statistical evidence and other compelling evidence for conviction [Moss, 2018; Duff, 2007; Redmayne, 2008].
Smith's solution and Di Bello's amendment
The efforts of Di Bello and Smith represent some of the most significant contributions to this field. Smith argues that the key distinguishing feature between naked statistical evidence and other forms of justified probative evidence is the normic support criteria [Smith, 2018]. According to Smith, normic support reflects our expectations regarding what is likely to happen. For instance, when we place a letter in a mailbox, we expect it to be delivered to the recipient within a few days. If the letter does not reach the recipient as anticipated, there must be an explanation for this unexpected outcome. One possible explanation could be that the vehicle transporting the mail was stolen en route. Smith asserts that naked statistical evidence does not satisfy this criterion [Gardiner, 2018].
Di Bello identified a flaw in this aspect of Smith's theory, specifically the assertion that the mere possibility of a coincidental match in evidence renders that evidence incapable of satisfying normic support. Di Bello demonstrated that coincidental matches can also occur in other forms of evidence, such as eyewitness testimony, DNA matches, and fingerprints [Zabell, 2005]. As a result, according to Smith's argument, other types of evidence also fail to meet the criteria for normic support, leading to a challenge of overgeneralization.
To address this issue, Di Bello disregarded the possibility of a coincidental match as an obstacle to fulfilling the normic support and employed it as a justified explanation in the process of satisfying normic support. Subsequently, utilizing Pollock's classification of defeaters in a defeasible argument, Di Bello demonstrated that the defendant can respond to the evidence presented against him in two ways during cross-examination. First, the defendant can use additional information to undercut the connection between the evidence and the conviction. Second, the defendant can present independent evidence that rebuts the hypothesis supporting the conviction and supports an alternative hypothesis incompatible with the defendant's conviction. Di Bello asserts that when naked statistical evidence serves as the probative evidence, only rebutting defeaters are available, while undercutting defeaters are not. Access to undercutting defeaters restricts the defendant's ability to mount a defense. Di Bello contends that this limitation represents an epistemological weakness of naked statistical evidence compared to other forms of probative evidence, contributing to the intuitive preference for the latter.
Critical Examination of Di Bello's Proposal
According to Di Bello’s argument, only rebutting defeaters are available when naked statistical evidence serves as the basis for conviction. For instance, in the Blue Bus scenario, one might consider the ownership ratio of Blue Buses as evidence while citing the Blue Company’s lower accident rate over the past year as a counterpoint. However, Pollock argues that when two pieces of statistical evidence rebut, the evidence that provides additional information relative to the other will have a rebutting and undercutting effect [Pollock, 2001]. Therefore, Di Bello's assertion that undercutting defeaters are unavailable in such cases appears flawed. This is because rebutting defeaters in statistical evidence will inherently provide an undercutting effect against the opposing evidence.
We then utilized the base rate fallacy to highlight the flaws present in Di Bello’s argument [Bar-Hillel, 1980]. The base rate fallacy quantitatively represents Pollock's argument. If base rate information is not incorporated into the diagnostic information calculation, we risk falling into the base rate fallacy. The effect of base rate information is the quantitative representation of the undercutting defeater that Pollock deemed essential for reaching an accurate conclusion.
Conclusion
Smith established normic support as a criterion for distinguishing between naked statistical evidence and other valid forms of probative evidence. Di Bello criticized Smith's definition of normic support, arguing that it leads to the challenge of overgeneralization. Consequently, while modifying Smith's criteria, he contended that the absence of an undercutting defeater in naked statistical evidence sets it apart from other valid probative evidence. By examining Pollock's classification of defeaters, we argued that undercutting defeaters still applies to naked statistical evidence. Specifically, in the Blue Bus scenario, we demonstrated the impact of undercutting defeaters, drawing on Pollock's arguments, which quantitatively illustrate the base rate fallacy. Therefore, Di Bello's argument for distinguishing between naked statistical evidence and other probative evidence appears questionable.
Article Type: Original Research |
Subject:
Philosophy of Science (Analytical)
Received: 2025/01/22 | Accepted: 2025/02/27 | Published: 2025/03/19
Received: 2025/01/22 | Accepted: 2025/02/27 | Published: 2025/03/19
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