Abstract
We present a logical setting that incorporates a belief-revision mechanism within Dynamic-Epistemic logic. As the “static” basis for belief revision, we use epistemic plausibility models, together with a modal language based on two epistemic operators: a “knowledge” modality K (the standard S5, fully introspective, notion), and a “safe belief” modality □ (“weak”, non-negatively-introspective, notion, capturing a version of Lehrer’s “indefeasible knowledge”). To deal with “dynamic” belief revision, we introduce action plausibility models, representing various types of “doxastic events”. Action models “act” on state models via a modified update product operation: the “Action-Priority” Update. This is the natural dynamic generalization of AGM revision, giving priority to the incoming information (i.e., to “actions”) over prior beliefs. We completely axiomatize this logic, and show how our update mechanism can “simulate”, in a uniform manner, many different belief-revision policies.
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Notes
- 1.
Or “doxastic events”, in the terminology of van Benthem (2007).
- 2.
To verify that a higher-level belief about another belief is “true” we need to check the content of that higher-level belief (i.e., the existence of the second, lower-level belief) against the “real world”. So the real world has to include the agent’s beliefs.
- 3.
But our approach can be easily modified to incorporate ontic changes, along the lines of van Benthem et al. (2006b).
- 4.
One could argue that our plausibility pre-order relation is equivalent to a quantitative notion (of ordinal degrees of plausibility, such as in Spohn (1988)), but unlike in Aucher (2003) and van Ditmarsch (2005) the way belief update is defined in our account does not make any use of the ordinal “arithmetic” of these degrees.
- 5.
Note that, as in untyped lambda-calculus, the input-term encoding the operation (i.e., our “action model”) and the “static” input-term to be operated upon (i.e., the “state model”) are essentially of the same type: epistemic plausibility models for the same language (and for the same set of agents).
- 6.
E.g., our update cannot deal with “forgetful” agents, since “perfect recall” is in-built. But finding out what exactly are the “natural limits” of our approach is for now an open problem.
- 7.
Imposing syntactic-dependent conditions in the very definition of a class of structures makes the definition meaningful only for one language; or else, the meaning of what, say, a plausibility model is won’t be robust: it will change whenever one wants to extend the logic, by adding a few more operators. This is very undesirable, since then one cannot compare the expressivity of different logics on the same class of models.
- 8.
In the Economics literature, connectedness is called “completeness”, see e.g., Board (2002).
- 9.
I.e., there exists no infinite descending chain s0 > s 1 > ⋯ .
- 10.
This interpretation is the one virtually adopted by all the proponents of the defeasibility theory, from Lehrer to Stalnaker.
- 11.
This of course assumes agents to be “rational” in a sense that excludes “fundamentalist” or “dogmatic” beliefs, i.e., beliefs in unknown propositions but refusing any revision, even when contradicted by facts. But this “rationality” assumption is already built in our plausibility models, which satisfy an epistemically friendly version of the standard AGM postulates of rational belief revision. See Baltag and Smets (2006a) for details.
- 12.
This identity corresponds to the definition of “necessity” in Stalnaker (1968) in terms of doxastic conditionals.
- 13.
The logic in Baltag and Smets (2006a) has these operators, but for simplicity we decided to leave them aside in this presentation.
- 14.
See e.g., Blackburn et al. (2001) for the general theory of modal correspondence and canonicity.
- 15.
This choice can be seen as a generalization of the so-called “maximal-Spohn” revision.
- 16.
Van Benthem calls this an “event model”.
- 17.
We stress this is a minor restriction, and it is very easy to extend this setting to “ontic” actions. The only reason we stick with this restriction is that it simplifies the definitions, and that it is general enough to apply to all the actions we are interested here, and in particular to all communication actions.
- 18.
As in Baltag and Moss (2004), we will be able to represent non-deterministic actions as sums (unions) of deterministic ones.
- 19.
Of course, at a later moment, the above-mentioned belief about action (now belonging to the past) might be itself revised. But this is another, future update.
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Acknowledgements
Sonja Smets’ contribution to this research was made possible by the post-doctoral fellowship awarded to her by the Flemish Fund for Scientific Research. We thank Johan van Benthem for his insights and help, and for the illuminating discussions we had with him on the topic of this paper. His pioneering work on dynamic belief revision acted as the “trigger” for our own. We also thank Larry Moss, Hans van Ditmarsch, Jan van Eijck and Hans Rott for their most valuable feedback. Finally, we thank the editors and the anonymous referees of the LOFT7-proceedings for their useful suggestions and comments.
During the republication of this paper in 2015, the research of Sonja Smets was funded by the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant agreement no.283963.
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Baltag, A., Smets, S. (2016). A Qualitative Theory of Dynamic Interactive Belief Revision. In: Arló-Costa, H., Hendricks, V., van Benthem, J. (eds) Readings in Formal Epistemology. Springer Graduate Texts in Philosophy, vol 1. Springer, Cham. https://doi.org/10.1007/978-3-319-20451-2_39
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