Abstract
In a logic with a dimension that represents social networks, for example friendship, it is natural to add hierarchies. We can then talk about friends being better than others, and isolate best friends. However, hierarchies are not rigid: majors can become lieutenant, friendship may be strengthened or compromised, and experts can loose or gain credibility. A proper analysis of the dynamics of hierarchies is thus essential to the logic of social networks. Hierarchies of agents are structurally very similar to plausibility orders of possible worlds central to logics for belief dynamics. I use this formal analogy to show how standard policies of belief revision can be applied in social networks, thus providing systematic mechanisms of promotion and demotion in social networks.
I would like to thank Shaun White, Marcus Triplett and the anonymous referee for comments that improved the paper greatly.
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Notes
- 1.
I use friendship as a basic social relation for simplicity. I thus only assume F to be symmetric. Other social relations could be used, but friendship is all I need for the interpretations of promotion and demotion I have in mind.
- 2.
Preorders are reflexive and transitive relations. Total preorders make any two friends comparable. Friends may be equally ranked, as you should expect.
- 3.
Because it is defined in terms of H, H  <  is redundant in models. But it is not redundant in the logic, as it is well-known that strict subrelations are not modally definable. For uniformity, I thus keep H  <  in models.
- 4.
Here and throughout the paper, I omit transitive and reflexive links whenever it improves readability in pictures.
- 5.
I choose this notation for the definition of the semantics over the more common \(M,w,a\models \varphi\) for uniformity and easier integration of PDL in the next sections. In the more common notation, instead of writing \((w,a) \in [\![\langle \pi \rangle \varphi ]\!]^{M}\), we would write \(M,w,a\models \langle \pi \rangle \varphi\).
- 6.
Maybe not very intuitive, but that’s how it works.
- 7.
- 8.
We only accept transformations that produce hierarchical models. Here’s a technical problem for the inclined reader: how do you characterise acceptable transformations for different logics? That is, if I give you a class of models, how can you isolate transformations that will produce models within the same class?
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Girard, P. (2015). Logic of Promotion and Demotion. In: Herzig, A., Lorini, E. (eds) The Cognitive Foundations of Group Attitudes and Social Interaction. Studies in the Philosophy of Sociality, vol 5. Springer, Cham. https://doi.org/10.1007/978-3-319-21732-1_5
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