Abstract
A group is in a state of pluralistic ignorance (PI) if, roughly speaking, every member of the group thinks that his or her belief or desire is different from the beliefs or desires of the other members of the group. PI has been invoked to explain many otherwise puzzling phenomena in social psychology. The main purpose of this article is to shed light on the nature of PI states – their structure, internal consistency and opacity – using the formal apparatus of Dynamic Doxastic Logic, and also to study the sense in which such states are “fragile”, i.e. to identify plausible conditions under which a PI state cascades into a state of shared belief as the result of announcement.
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Notes
The term ‘norm’ should here be understood in a descriptive sense as signifying a general tendency of behavioral conformity.
According to a large scientific literature, this feature is also found in a number of equilibria and dynamics (e.g. informational cascades) that are grounded on a state of pluralistic ignorance (see [5, 6], and [11]), although some cases of PI reported in the litterature display a more robust behaviour.
Here “relatively few” means that in a group composed of n agents the process terminates in n rounds of communication between agents.
By convention, non-labelled points are \(\neg N\)-states.
Simple safety of \(\phi \) at w already guarantees such robustness for atomic proposition (as N) but is not in general preserved by update. For example the “moorean” \(p \wedge \neg K_{i}p\) may very well be safe at some state w but it is never preserved after a \(!p\)-update.
Formal approaches to judgment aggregation (JA) and preference aggregation (PA) express individual opinions and preferences via an ordering among states. Aggregation policies can be represented as functions from the set of the individual orderings of the agents, having as value some other ordering among states, e.g. in the simple case where agent i is the dictator the output will simply consist in agent’s i ordering. Most literature in PA develops the theory of aggregation of total orders (or even linear ones), not of preorders such as the ones we are dealing with here, but there are exceptions (see [21]). The logical properties of different operators of preference aggregation have been largely investigated in [1] and, for what specifically concerns doxastic and proairetic logics based on plausibility frames, the reader may also refer to [8].
An interesting issue, to be left for future work, is to develop this conceptual affinity into a technical one, integrating our framework with JA theories and studying how different policies of belief aggregation influence PI-scenarios. For example, the presence of a fashion leader or trendsetter, i.e. a person whose preference deserves bigger consideration than those of other agents, presumably has a big impact for creation and dissolution of PI. Here, for the sake of simplicity we treat all agents as epistemic peers.
It is easy to verify that in the updated model \(B_{(1,2)}P_{\frac {1}{2}G(1,2)}N\) and \(B_{(2,1)}P_{\frac {1}{2}G(2,1)}N\).
Several forms of belief revision are available in DDL, essentially as operations of order change among states (see [4] and [3] for examples). In our specific case two policies might be explored: (a) simple order upgrade where \(\neg B_{j}\phi \)-states become minimal in i’s information cell, and (b) addition of \(\neg B_{j}\phi \)-states in i’s information cell.
Of course, nothing in this semantics prevents a group member from realizing at a later point that the group was at an earlier point in a state of PI.
This semantics corresponds to a normal modal system KD45 for B, i.e. validates both \(B\phi \rightarrow BB\phi \) and \(\neg B\phi \rightarrow B\neg B\phi \) which are sufficient to derive a contradiction from Eq. 5 (see [3]). It is worth commenting, in this connection, on an alternative form of a PI scenario in which every member believes \(\phi \) and also that the others members, rather than failing to believe \(\phi \), believe \(\neg \phi \). From the assumption that a given member believes this to be so, we may easily derive a proposition expressing that this member believes all the other members to have contradictory beliefs.
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We would like to especially thank Davide Grossi for many highly useful comments and suggestions on earlier versions of this paper. We are also indebted to the referees for several valuable remarks that helped improve the final version. We also wish to thank Rasmus Rendsvig, Vincent Hendricks, Frank Zenker, Jens Ulrik Hansen, Bengt Hansson and the participants to the Lund-Copenhagen Workshop on Social Epistemology as well as audiences in Bologna (EEN meeting 2012), Lund, Paris and Bochum for many helpful observations.
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Proietti, C., Olsson, E.J. A DDL Approach to Pluralistic Ignorance and Collective Belief. J Philos Logic 43, 499–515 (2014). https://doi.org/10.1007/s10992-013-9277-3
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DOI: https://doi.org/10.1007/s10992-013-9277-3