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Actions, Belief Update, and DDL

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Krister Segerberg on Logic of Actions

Part of the book series: Outstanding Contributions to Logic ((OCTR,volume 1))

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Abstract

Two prominent topics in Krister Segerberg’s works are, on the one hand, actions, and on the other hand, belief change. Both topics are connected in multiple ways; one of these connections is via KGM belief update, since, as we argue, belief update is a specific case of feedback-free action progression. We discuss the links between update and action, and, starting from Segerberg’s works, discuss further other possible interpretations of belief update, its differences with AGM belief revision, and why it is interesting to develop further KGM-based Dynamic Doxastic Logic.

A significant part of this article is a revised version of [23]

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Notes

  1. 1.

    (U5), (U6) and (U7) are much more controversial than the other ones (see [18]); they characterize the specific class of updates based on a similarity-based minimization process (which is known to lead to several counterintuitive results).

  2. 2.

    Our assumption that the language is finite allows us to consider revision operators as acting on propositional formulas as in [22] (instead of belief sets).

  3. 3.

    Note that this scenario is also a case for belief extrapolation [8], which is a particular form of time-stamped revision.

  4. 4.

    This formulation appears in [21], which may be one of the explanations for such a long-lasting ambiguity.

  5. 5.

    Note that without time stamps (and in particular within the framework of belief update), we cannot distinguish between “\(B\) has become false” (as in ”I see Bob go out of the office”) and “the world has evolved in such a way that \(B\) is now false” (as in “I now see Bob out of his office”). Anyway, for Example 3.2, the expected outcome is the same in both cases (provided that \(A\) and \(B\) are expected to persist with respect to the granularity of time considered).

  6. 6.

    The only case where belief update could be compatible with interpreting \(\alpha \) as an observation would therefore be the case where not the faintest correlation exists between the state of the world at different time points; in this case, we would have \(\varphi \diamond \alpha \equiv \alpha \) whenever \(\alpha \) is consistent—a totally degenerate and uninteresting case.

  7. 7.

    More sophisticated models may involve graded uncertainty such as probabilities, delayed effects etc.

  8. 8.

    Unless the state of the world after the action is performed is totally disconnected from the state of the world before the action is performed, which only happens if \(R_{A}(s) = S\) for all \(s\). In this case, a feedback never allows for learning anything about the past state of the world. Clearly, this case is a very degenerated one.

  9. 9.

    The distinction between actions and events is mostly irrelevant to our discussion. Actions are usually thought of as agent-trigerred, whereas events don’t, or don’t necessarily (see for instance [31]). Who triggers what has no impact on our discussion: an action performed consciously and intentionally by an agent, or a nature-trigerred event, or an action performed by another agent, have the same effects on the agent’s belief state provided that, in all cases, the agent is perfectly aware of the action or the event taking place.

  10. 10.

    However, this point is somewhat debatable: update would work as well if we don’t assume that the agent’s initial beliefs is correct—of course, in this case the final beliefs may be wrong as well.

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Acknowledgments

In my conference paper [23], I wrote that I would never have thought of writing that chapter without these years of discussion with Andreas Herzig about the very meaning of belief update. This is still true now, with a few more years in the count.

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  1. LAMSADE-CNRS, Paris, France

    Jérôme Lang

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Correspondence to Jérôme Lang .

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  1. The John Paul II Catholic University of, Lublin, Poland

    Robert Trypuz

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Lang, J. (2014). Actions, Belief Update, and DDL. In: Trypuz, R. (eds) Krister Segerberg on Logic of Actions. Outstanding Contributions to Logic, vol 1. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7046-1_11

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