I propose a relevance-based independence axiom on how to aggregate individual yes/no judgments on given propositions into collective judgments: the collective judgment on a proposition depends only on people’s judgments on propositions which are relevant to that proposition. This axiom contrasts with the classical independence axiom: the collective judgment on a proposition depends only on people’s judgments on the same proposition. I generalize the premise-based rule and the sequential-priority rule to an arbitrary priority order of the propositions, instead of a (...) dichotomous premise/conclusion order resp. a linear priority order. I prove four impossibility theorems on relevance-based aggregation. One theorem simultaneously generalizes Arrow’s Theorem (in its general and indifference-free versions) and the well-known Arrow-like theorem in judgment aggregation. (shrink)

This paper addresses how multiple individual credences on logically related issues should be aggregated into collective binary beliefs. We call this binarizing belief aggregation. It is vulnerable to dilemmas such as the discursive dilemma or the lottery paradox: proposition-wise independent aggregation can generate inconsistent or not deductively closed collective judgments. Addressing this challenge using the familiar axiomatic approach, we introduce general conditions on a binarizing belief aggregation rule, including rationality conditions on individual inputs and collective outputs, and determine which rules (...) (if any) satisfy different combinations of these conditions. Furthermore, we analyze similarities and differences between our proofs and other related proofs in the literature and conclude that the problem of binarizing belief aggregation is a free-standing aggregation problem not reducible to judgment aggregation or probabilistic opinion pooling. (shrink)

The article proceeds upon the assumption that the beliefs and degrees of belief of rational agents satisfy a number of constraints, including: consistency and deductive closure for belief sets, conformity to the axioms of probability for degrees of belief, and 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. (shrink)

What is the relationship between degrees of belief and binary beliefs? Can the latter be expressed as a function of the former—a so-called “belief-binarization rule”—without running into difficulties such as the lottery paradox? We show that this problem can be usefully analyzed from the perspective of judgment-aggregation theory. Although some formal similarities between belief binarization and judgment aggregation have been noted before, the connection between the two problems has not yet been studied in full generality. In this paper, we seek (...) to fill this gap. The paper is organized around a baseline impossibility theorem, which we use to map out the space of possible solutions to the belief-binarization problem. Our theorem shows that, except in limiting cases, there exists no belief-binarization rule satisfying four initially plausible desiderata. Surprisingly, this result is a direct corollary of the judgment-aggregation variant of Arrow’s classic impossibility theorem in social choice theory. (shrink)

Ordinary common knowledge is formally expressed by strong probabilistic common belief. How strong exactly? The question can be answered by drawing from the similar equivalence, recently explored, between plain and probabilistic individual beliefs. I argue that such a move entails that common knowledge displays a double fragility: as a description of a collective state and as a phenomenon, because it can respectively disappear as group size increases, or more worryingly as the epistemic context changes. I argue that despite this latter (...) fragility, the effects of common knowledge on action are robust. Unfortunately, this in turn leads to a third fragility, that of the concept of common knowledge, which threatens to collapse on probabilistic common belief. This also reveals a disanalogy between the individual and the collective cases. I finally pinpoint the subtle difference entailed by the two concepts, expressed in terms of the attitude towards contrary evidence or of the agents’ awareness. As a result, common knowledge can be defended as a concept, which refers to a fragile yet distinct collective attitude. (shrink)

Group beliefs in social networks are often construed as arising from individual beliefs through processes of update and aggregation. In this paper, we explore an alternative ‘arational’ perspective. More specifically, we focus on group attitudes as neutral tendencies toward alignment of opinions driven by influence patterns among agents modeled in a Markov dynamics. In addition, we investigate logical patterns in the resulting potential group beliefs or, in more neutral arational terminology: collective opinion structures.