Proof Paradoxes: An Analysis of Thomson's Argument and Redmayne’s Objection

Hezareh, Mohammad Reza; Hezareh, Mohammad Reza

doi:10.30465/ps.2026.52906.1800

Proof Paradoxes: An Analysis of Thomson's Argument and Redmayne’s Objection

Document Type : Research Paper

Authors

Mohammad Reza Hezareh ¹

Mohammad Reza Hezareh ²

¹ MA in Philosophy of Science, Amirkabir University of Technology, Tehran, Iran.

² Assistant Professor of Science and Technology Studies Group, Department of Management, Science and Technology, Amirkabir University of Technology, Tehran, Iran

https://doi.org/10.30465/ps.2026.52906.1800

Abstract

The proof paradoxes refer to situations where the foundational claim supporting a verdict relies solely on naked statistical evidence. These examples are constructed so that the decision-maker’s degree of belief, derived solely from this statistical data, exceeds the probabilistic thresholds set by relevant standards of proof, thus leading to a verdict. However, despite the high probability, many people intuitively feel that this alone is insufficient to justify a conviction. A closer examination of other types of evidence typically considered in court reveals that such evidence is often just as susceptible to error as naked statistical evidence (Di Bello, 2019).
This raises the question: what explains this intuitive preference for other forms of evidence? Two main approaches address this issue. The first is the “revisionary” approach. According to this perspective, because the probability of guilt based on other evidence—such as eyewitness testimony or DNA matches—is generally comparable to, or even less than, the probabilistic values derived from statistical evidence in these paradoxical cases, our intuition that such evidence is more convincing is unwarranted. Therefore, this intuition warrants reconsideration.
The second approach is the “non-revisionary” stance. This perspective seeks to justify why convictions based solely on naked statistical evidence seem unjustified and aims to clarify the features that distinguish valid evidentiary proof from naked statistical evidence. Many philosophers and legal scholars have endeavored to elucidate this distinction. For example, Williams (Williams) argues that valid evidence must be dependent on and tied to the defendant, whereas naked statistical evidence does not satisfy this requirement. Similarly, Moss and Duff rely on the notion of knowledge to differentiate between statistical evidence and eyewitness testimony (Moss, 2018; Duff, 2007). One of the latest attempts to articulate this distinction is Smith’s (Smith, 2018) “normic support” approach, which has been further refined by Di Bello (Di Bello, 2020). Smith argues that the relevant difference is that an explanation would be expected if a conviction based on eyewitness testimony turned out to be mistaken, while no such explanation would be expected if a conviction based on naked statistical evidence were mistaken. In contrast, Di Bello contends that Smith's proposal would lead to overgeneralization and sees the problem with naked statistical evidence as the lack of access to an undercutting defeater in the cross-examination process.
This article proceeds by analyzing the views of Thomson and the criticisms raised by Redmayne. Thomson employs the blue bus example—one of the paradigmatic illustrations of proof paradoxes—to demonstrate his criterion for distinguishing valid evidence. In this scenario, an individual is involved in a collision with a bus. In the city where the incident occurred, only two bus companies operate: Blue and Red. The Blue company owns 70% of the buses, and the Red company owns 30%. The buses are colored accordingly. No other evidence is available to determine which company’s bus was involved in the collision. Based on ownership data, the probability that the bus involved belonged to the Blue company is 70%, exceeding the standard of preponderance of evidence (Hamer, 2004). This standard is commonly used in civil cases. Thomson argues that the absence of a causal relationship between ownership rates and the liability of the company indicates an intuitive inconsistency in convicting the Blue company based solely on naked statistical evidence. According to Thomson, valid proof must be causally connected to the defendant’s liability—that is, there must be a causal link between the evidence and the facts of the case (Thomson, 1986).
Redmayne, however, criticizes Thomson’s conception of causality as ambiguous, arguing that his criteria do not clearly differentiate naked statistical evidence from other types of evidence. Specifically, Redmayne (2008) contends that, when considering Thomson’s definition of causality, it remains possible to establish a causal link between ownership rates and the incident, thereby undermining Thomson’s criterion.
The primary aim of this research is to demonstrate that Thomson’s proposed criterion can be refined to address Redmayne’s objections. To do so, we first clarify Thomson’s argument and Redmayne’s critiques. A detailed analysis reveals that Redmayne’s critique relies on two main assumptions: first, that it is justified to infer the rate of accidents from ownership rates; and second, that a causal relationship exists between ownership rates and the probability of liability.
To challenge the first assumption, we employ Pollock’s (Pollock, 1990) nomic probability framework, arguing that an objective probability value is epistemically justified only if it satisfies the criteria of nomic probability. This demonstrates that inferring accident rates solely from ownership data in proof paradox examples cannot constitute a justified probabilistic inference. Regarding the second assumption, Redmayne’s critique—that Thomson’s definition of causality lacks sufficient precision—is well-founded. This paper adopts Woodward’s (Woodward, 2005) causal model, particularly its reproducibility criterion, to establish a clear and meaningful distinction between naked statistical evidence and epistemologically valid evidence. Consequently, the absence of a causal link—according to Woodward’s model—explains why conviction based on other valid evidentiary grounds is intuitively superior to one based solely on naked statistical evidence.

Keywords

Pollock's Nomic probability

legal standards of proof

Proof Paradoxes

Woodward's causal model

Thomson

Redmayne

Statistical evidence

Subjects

Philosophy of science

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