The joint aggregation of beliefs and degrees of belief
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The joint aggregation of beliefs and degrees of belief Paul D. Thorn1 Received: 8 March 2018 / Accepted: 26 September 2018 © Springer Nature B.V. 2018
Abstract The article proceeds upon the assumption that the beliefs and degrees of belief of rational agents satisfy a number of constraints, including: (1) consistency and deductive closure for belief sets, (2) conformity to the axioms of probability for degrees of belief, and (3) the Lockean Thesis concerning the relationship between belief and degree of belief. Assuming that the beliefs and degrees of belief of both individuals and collectives satisfy the preceding three constraints, I discuss what further constraints may be imposed on the aggregation of beliefs and degrees of belief. Some possibility and impossibility results are presented. The possibility results suggest that the three proposed rationality constraints are compatible with reasonable aggregation procedures for belief and degree of belief. Keywords Belief aggregation · Opinion pooling · Discursive dilemma · Full and partial belief · The Lockean thesis
1 Introduction There are two sorts of doxastic states that have received considerable attention from epistemologists. The first sort of doxastic state is belief. Belief is an all or nothing affair: For every proposition, ϕ, one either believes it or one does not. If one does not believe ϕ, one may either disbelieve ϕ (which is equivalent to believing not ϕ) or remain uncommitted (neither believing nor disbelieving). A second sort of doxastic state is degree of belief. The latter sort of state reflects the fact that beliefs may be held with varied degrees of conviction. Degrees of belief also correspond to personal (or subjective) probabilities, which are of considerable importance to decision theory. In order to cement the link between degree of belief and rational action, it is typically assumed that rational degrees of belief satisfy the axioms of probability.
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Paul D. Thorn [email protected] Heinrich Heine Universitaet Duesseldorf, Universitaetsstr. 1, 40225 Duesseldorf, Germany
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Synthese
The two sorts of doxastic state give rise to two distinct aggregation problems. The first is the problem of aggregating beliefs: Given a group of agents, the problem is to pool individual beliefs regarding a domain of propositions, in order to form a collective belief set.1 The second problem concerns the aggregation of degrees of belief: Given a group of agents, the problem is to pool their individual degrees of belief regarding a domain of propositions, in order to form collective degrees of belief.2 The determination of what principles are appropriate in forming a collective doxastic state may vary by context, and depend on the relative importance of respecting procedural concerns (e.g., for the ‘fairness’ with which individual doxastic states have a bearing on the collective) versus veritistic concerns (e.g., for the tendency of the aggregation procedure to yield an accurate collective doxastic state). As a rough guide to forming intuitions about
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