Lottery Paradox

Formal theories of belief-credence interaction that satisfy the standard logical requirements on belief, such as conjunctive closure, face the problem of partition-sensitivity. According to these theories, a rational agent can believe X relative to one partitioning of possibilities, but the same belief may not be rational relative to some other partitioning, even when the agent’s evidence remains the same. Focusing on Leitgeb’s stability theory (Leitgeb, The stability of belief: How rational belief coheres with probability, Oxford University Press, 2017), which exemplifies (...) this problem, this paper aims to go beyond the simple partition-sensitivity of such theories. To address this and related problems, I connect the issues surrounding belief-credence interaction with Bayesian network theory and the notion of epistemic basing. The view that I defend and explicate is that rational belief corresponds to stably high probability within a network of interconnected propositions, where the network is structured by the relation of direct explanatory priority. As I argue, this view provides solutions to important problems regarding partition-sensitivity, specifically the problems involving irrelevant propositions and large partitions. In the final section, I extend the proposed theory and articulate the network-based notion of robustness of belief, which differentiates lottery-type beliefs from more ordinary beliefs. (shrink)

This paper employs epistemic decision theory to explore rational bridge principles between probabilistic beliefs and deductively cogent beliefs. I re-examine Hempel and Levi’s epistemic decision theories and generalize them by introducing a novel rationality norm for belief binarization. This norm posits that an agent ought to have binary beliefs that maximize expected utility in light of their credences. Our findings reveal that the proposed norm implies certain geometrical principles, namely convexity norms. Building upon this framework, I critically evaluate the Humean (...) thesis in Leitgeb’s stability theory of belief and Lin-Kelly’s tracking theory. We establish the impossibility results, demonstrating that those theories violate the proposed norms and consequently fail to do the job of expected utility maximization. In contrast, we discover alternative approaches that align with all of the proposed norms, such as generating beliefs that minimize a Bregman divergence from credences. Our epistemic decision theory for belief binarization can be compared to Dorst’s accuracy argument for the Lockean thesis. We conclude that deductively cogent expected accuracy maximizers are neither Lockean nor Humean. (shrink)

There is a wide consensus among epistemologists that we fail to know lottery propositions (that is, highly likely propositions solely supported by statistical evidence), but there is no agreed-upon explanation of why we fail to know them. Yet, paradoxes surrounding lotteries continue pressing on the need to identify the correct explanation. This dissertation evaluates various explanations of why we do not know lottery propositions, which normally take one of two forms. The first argues that lottery beliefs are true by mere (...) epistemic luck (roughly, luck, chance, or accident in possessing a true belief in a proposition p, which prevents us from knowing p), and provides an account of epistemic luck that entails that lottery beliefs are epistemically luckily true. Here I evaluate and reject an explanation that relies on the largely dominant account of epistemically lucky beliefs as “unsafe true beliefs”. The second argues that lottery beliefs are unjustified (where being justified in believing p is necessary for knowing p), and provides an account of justification with a condition that lottery beliefs or propositions do not satisfy. I divide conditions on justification into “probabilistic” and “non-probabilistic” and argue, contrary to Douven and Williamson (2016), that justification can have a probabilistic condition that lottery beliefs fail to satisfy, thus restoring such accounts as a theoretical possibility to solve the “lottery paradox”. I also argue that among the dominant candidates of non-probabilistic conditions, Smith’s (2016) normic support condition on justification is the only suitable one to explain why we do not know lottery propositions. Alternatively, if lottery propositions are justified, there is nonetheless a solution to the lottery paradox compatible with their possessing such epistemic status, which consists in rejecting the “aggregativity of justification”. I argue that this principle is incorrect, and that preserving it obstructs solving what I identify as the most basic form of the lottery paradox. Finally, I address the legal correlates of lottery propositions –i.e., litigated claims supported by bare statistical evidence. I divide explanations of why such claims do not meet a given standard of proof into two types: externalist and internalist. I present independent problems for two externalist explanations –Pritchard’s (2018, 2022) modal explanation and Blome-Tillmann’s (2017) knowledge-first explanation, and argue that they face such problems in virtue of being externalist. This finding motivates the need to evaluate externalist accounts qua externalist to determine if an externalist account is a viable solution to the “proof paradox”. (shrink)

This paper investigates the question of how subjective probability should relate to binary belief. We propose new distance minimization methods, and develop epistemic decision-theoretic accounts. Both approaches can be shown to get “close” to the truth: the first one by getting “close” to a given probability, and the second by getting expectedly “close” to the truth. More specifically, we study distance minimization with a refined notion of Bregman divergence and expected utility maximization with strict proper scores. Our main results reveal (...) that the two ways to get “close” to the truth can coincide. (shrink)

A necessary criterion of Duncan Pritchard’s Anti-luck Virtue Epistemology is his safety condition. A believer cannot know p unless her belief is safe. Her belief is safe only if p could not have easily been false. But “easily” is not to be understood probabilistically. The chance that p is false might be extremely low and yet p remains unsafe. This is what happens, Pritchard argues, in lottery examples and explains why knowledge is not a function of the probabilistic strength of (...) one’s evidence. This paper argues that, contra Pritchard, modality holds no epistemic advantage over this type of “probabilistic evidentialism” that he criticizes. I begin with a review of Pritchard’s argument supporting modality over probability; second, I explain the problems with this argument, and third, I offer an alternative explanation of the lottery example. At the completion of the paper, modality and probability are on equal epistemic footing. (shrink)