Abstract
Dynamic epistemic-doxastic logics describe the new knowledge or new beliefs indexBelief of agents after some informational event has happened. Technically, this requires an update rule that turns a doxastic-epistemic modelM(recording the current information state of the agents) and a dynamic ‘event model’ E (modeling the relevant, possibly complex event taking place right now) into a new model M×E, with in particular new epistemic accessibility and new doxastic plausibility relations. Concrete rules have been proposed for this purpose, and they work well in examples, and generate complete logics. But can such rules be justified in a deeper sense by appealing to underlying considerations? This technical Note proposes a way of thinking about this, by making a connection between belief revision and social choice theory, broadly construed. Our analysis mixes the updates proposed by Baltag & Smets Baltag and Smets (2008) in dynamic epistemic logic with the framework of Andr’eka, Ryan Ȧ Schobbens Andr&eka et al (2002) for merging of binary relations. Reasonable social choice-like postulates turn out to leave very few candidates for update rules. In particular, we show how ‘Priority Update’ for new plausibility relations can be viewed as a process of social choice between the various signals that make up an agent over time. We also discuss how our style of analysis might be extended.
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Notes
- 1.
An alternative mechanism are the dynamic logic programs of van Benthem and Liu (2007) .
- 2.
Franz Dietrich has pointed out with concrete suggestions that interesting connections might be made with more standard social choice literature but this will have to wait till another occasion.
- 3.
Indices may have multiple occurrences in the graph, but we will ignore this subtlety here.
- 4.
Priority graphs have natural operations of sequential composition (put one graph above another) and parallel composition (take a disjoint union of graphs). These yield an elegant calculus: (a) disjoint union leads to intersection of relations, (b) sequential composition to lexicographic order.
- 5.
In general, we may only need to order a subset of this full product space, since some relevant constraint may rule out pairs: as we have seen already with the DEL event preconditions.
- 6.
We can also rephrase things over one set of “pair objects”. First lift component relations to pairs: \((a, c) R_1 (b, d)\) iff \(a R_1 c, (a, c) R_2 (b, d)\) iff \(c R_2 d\), then merge in single domain style. As the referee points out, in social choice , this would be a rather ego-centric ordering of bundles of objects.
- 7.
By a simple computation, we also get the equivalence \((s, e) < (t, f)\) iff \((s < t \land \ e \leq f) \lor e < f\).
- 8.
For instance, if singletons are reflexive, agents will be indifferent between x and x itself, and by the Abstentions principle below, we only need to look at the other pair relations.
- 9.
In fact, the other option of giving priority to the first argument: say, the initial model M, is an interesting conservative anti-Jeffreyan variant where little learning takes place.
- 10.
Both are instances of the so-called “But” operator of ARS, i.e., a “Leader/Follower” pattern.
- 11.
Christian List has suggested that the results in this chapter are close to a characterization of “lexicographic dictatorships” by Luce and Raiffa (1957): cf. (D’Aspremont (1985). These links are yet to be explored, again looking at social choice postulates for their belief revision content. Another result that List has suggested as an alternative belief revision mechanism is May’s Theorem capturing the essence of democratic majority voting: cf. Goodin and List (2006).
- 12.
Franz Dietrich notes that our condition of Permutation Invariance is very strong, and that the literature on Arrow-style theorems suggests better results using only weaker versions of it.
- 13.
As for more technical issues, more might be said about relative power of different update rules in achieving new relational patterns on models. Compare Priority Update versus Flat Product Update. Which rule is more general in its dynamic effects, if we allow re-encoding of arguments? Priority Update can never make an established strict preference for x over y “indifferent” again, while I think Democracy can mimic any effect of Priority by suitably re-encoded sequences of events.
- 14.
Again this follows up on a question from Franz Dietrich.
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Acknowledgments
I thank Christian List for his useful feedback suggesting deeper forays into social choice theory – and especially, Franz Dietrich for many informative critical comments, only part of which I have been able to process in this short note.
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van Benthem, J. (2011). Belief Update as Social Choice. In: Girard, P., Roy, O., Marion, M. (eds) Dynamic Formal Epistemology. Synthese Library, vol 351. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0074-1_8
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