In Lotteries, Knowledge, and Rational Belief. Essays on the Lottery Paradox. Cambridge University Press. pp. 223-54 (2021)

Authors
Franz Dietrich
Centre National de la Recherche Scientifique
Christian List
Ludwig Maximilians Universität, München
Abstract
Agents are often assumed to have degrees of belief (“credences”) and also binary beliefs (“beliefs simpliciter”). How are these related to each other? A much-discussed answer asserts that it is rational to believe a proposition if and only if one has a high enough degree of belief in it. But this answer runs into the “lottery paradox”: the set of believed propositions may violate the key rationality conditions of consistency and deductive closure. In earlier work, we showed that this problem generalizes: there exists no local function from degrees of belief to binary beliefs that satisfies some minimal conditions of rationality and non-triviality. “Locality” means that the binary belief in each proposition depends only on the degree of belief in that proposition, not on the degrees of belief in others. One might think that the impossibility can be avoided by dropping the assumption that binary beliefs are a function of degrees of belief. We prove that, even if we drop the “functionality” restriction, there still exists no local relation between degrees of belief and binary beliefs that satisfies some minimal conditions. Thus functionality is not the source of the impossibility; its source is the condition of locality. If there is any non-trivial relation between degrees of belief and binary beliefs at all, it must be a “holistic” one. We explore several concrete forms this “holistic” relation could take.
Keywords Degrees of belief  Binary beliefs  Credences  Lottery paradox  Deductive closure  Holism  Credence-belief relation  Qualitative and quantitative belief
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References found in this work BETA

The Stability Theory of Belief.Hannes Leitgeb - 2014 - Philosophical Review 123 (2):131-171.
The Paradox of the Preface.David C. Makinson - 1965 - Analysis 25 (6):205.
Arrow's Theorem in Judgment Aggregation.Franz Dietrich & Christian List - 2007 - Social Choice and Welfare 29 (1):19-33.
Generalizing the Lottery Paradox.Igor Douven & Timothy Williamson - 2006 - British Journal for the Philosophy of Science 57 (4):755-779.
Propositional Reasoning That Tracks Probabilistic Reasoning.Hanti Lin & Kevin Kelly - 2012 - Journal of Philosophical Logic 41 (6):957-981.

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Perception and Probability.Alex Byrne - 2021 - Philosophy and Phenomenological Research:1-21.

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