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A Panorama of Iterated Revision

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David Makinson on Classical Methods for Non-Classical Problems

Part of the book series: Outstanding Contributions to Logic ((OCTR,volume 3))

Abstract

In 1985, David Makinson, together with Carlos Alchourron and Peter Gärdenfors published an article, the now renowned “AGM paper”, that gave rise to an entire new area of research: Belief Revision. The AGM paper set the stage for studying belief revision and provided the first fundamental results in the area. There was however one aspect of belief revision that was not addressed in the AGM paper: iterated belief revision. Since 1985, there have been numerous attempts to tackle this problem. In this chapter, we shall review some of the most influential approaches to the problem of iterated belief revision, and discuss their strengths and shortcomings.

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Notes

  1. 1.

    There is some overlap between this article and an earlier survey of ours on belief revision Peppas (2008).

  2. 2.

    We note that this assumption is made only to simplify the exposition; many of the approaches discussed herein can work, at least technically, in a more general setting.

  3. 3.

    Although these postulates where first proposed by Gärdenfors alone, they were extensively studied in collaboration with Alchourron and Makinson (1985); thus their name.

  4. 4.

    Although \((K*7)\) and \((K*8)\) seem to relate different belief sets, as will become apparent from the constructive models later on, this is not really the case.

  5. 5.

    Recall that \(\text {I}\!{{\text {M}}_L}\) is the set of all consistent complete theories of \(L\), and for a theory \(K\) of \(L\), \([K]\) is the set of all consistent complete theories that contain \(K\).

  6. 6.

    We note that the Limit Assumption is similar to smoothness as defined by Kraus et al. (1990), also known as stoppering in Makinson (1994).

  7. 7.

    For a set of possible worlds \(V\), \(min(V,\le )\) denotes the set of minimal worlds in \(V\) with respect to \(\le \); i.e. \(min(V,\le )\) = \(\{ r\in V:\) for all \(r' \in V\), if \(r' \le r\) then \(r\le r'\}\).

  8. 8.

    Notice that the original system of spheres \(S\) is centered on \([K]\), not on \([K*\varphi ]\), and therefore cannot be used to direct further revisions.

  9. 9.

    In this sense an ordinal conditional function \(\kappa \) is quite similar to a system of spheres \(S\): both are formal devices for ranking possible worlds in terms of plausibility. However \(\kappa \) not only tells us which of any two worlds is more plausible; it also tells us by how much is one world more plausible than the other.

  10. 10.

    The left subtraction of two ordinals \(\alpha , \beta \) such that \(\alpha \ge \beta \), is defined as the unique ordinal \(\gamma \) such that \(\alpha \) = \(\beta + \gamma \).

  11. 11.

    That is, given an OCF \(\kappa \) and any \(d>0\), the function \(*\) defined as \(K*\varphi \) = \(\bigcap \{ r\in \text {I}\!{{\text {M}}_L}:\) \(\kappa *\langle \varphi , d\rangle (r) = 0 \}\) satisfies the AGM postulates \((K*1)\)–\((K*8)\).

  12. 12.

    This is the case where the new information \(\varphi \) contradicts the original belief set (since \(\kappa (\varphi ) > 0\), the agent originally believes \(\lnot \varphi )\).

  13. 13.

    There is a well known connection between a system of spheres \(S\) and an epistemic entrenchment \(\le \). In particular, the latter can easily be constructed from the former (while preserving the induced revision function) as follows: \(\varphi \le \psi \) iff \(c(\lnot \varphi ) \subseteq c(\lnot \psi )\), for all contingent \(\varphi ,\psi \in L\).

  14. 14.

    Loosely speaking, the bands of a system of spheres are the sets of worlds between successive spheres.

  15. 15.

    Recall that for any sentence \(\psi \), \(c(\psi )\) denotes the smallest sphere in \(S\) intersecting \([\psi ]\).

  16. 16.

    It should be noted that Darwiche and Pearl use different notation, and as already mentioned, they leave open the representation of a belief state (it is not necessarily represented as a system of spheres).

  17. 17.

    Like with (S*1)–(S*8), the original formulation of (DP1)–(DP4) is slightly different. Herein we have rephrased the Darwiche and Pearl postulates in AGM notation.

  18. 18.

    Although it should be noted that Darwiche and Pearl argue that this shift is not necessitated by technical reasons alone; conceptual considerations also point the same way.

  19. 19.

    It should be noted that (Ind) was also studied independently by Booth et al. (2005), Booth and Meyer (2006).

  20. 20.

    Distance between possible worlds does not have to be expressed in terms of real numbers; this is an assumption made herein for simplicity.

  21. 21.

    For any propositional variable \(x\), by \(\overline{x}\) we denote the negation of \(x\).

  22. 22.

    The distance between a world \(r\) and a set of worlds \(V\) can be defined as the smallest distance between \(r\) and a world in \(V\). Hence, according to Fig. 4, the closest world to \(\{ r_2, r_3 \}\) is \(r_1\) followed by \(r_0\).

  23. 23.

    This is not the name used in Booth and Meyer (2011). In fact the overall exposition herein is slightly different from Booth and Meyer (2011), but the essence remains the same.

  24. 24.

    As already noted, a system of spheres is just another way of representing a total preorder on possible worlds.

  25. 25.

    The BM approach has also been characterized axiomatically in Booth and Meyer (2011).

  26. 26.

    That is, all \(\lnot \varphi \)-worlds that are initially strictly more plausible than \(r_{\lnot \psi }\), are placed at the same level as \(r_{\lnot \psi }\).

  27. 27.

    It should be noted that the selection function employed in choice revision, does not depend on the initial belief set \(K\).

  28. 28.

    See Katsuno and Mendelzon (1991) for a formal treatment of belief update and its difference from belief revision.

  29. 29.

    Perhaps the best way to explain this is that there is no “right” input as such; different types of input are appropriate for different kinds of iterated revision scenarios.

  30. 30.

    Depending on the model, \(\mathcal {U}\), \(\mathcal {U'}\) can be systems of spheres, ordinal conditional functions, preorders on possible worlds, etc.

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Acknowledgments

I am grateful to the anonymous reviewers for their valuable comments, and to the editor of this book, Sven Ove Hansson, for his excellent work in coordinating our joint efforts. Since this book is devoted to David Makinson’s work, I would like to take this opportunity to express my gratitude to David for co-founding the area of Belief Revision that has constantly fueled my intellectual curiosity for over 20 years. More importantly though, I would like to thank David for setting such an inspiring example of a true scholar of the very best in academic traditions.

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Authors and Affiliations

  1. Department of Business Administration, University of Patras, Patra, Greece

    Pavlos Peppas

  2. Faculty of Engineering and IT, University of Technology, Sydney, Australia

    Pavlos Peppas

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  1. Pavlos Peppas
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Correspondence to Pavlos Peppas .

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  1. Royal Institute of Technology (KTH) Div. Philosophy, Stockholm, Sweden

    Sven Ove Hansson

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Peppas, P. (2014). A Panorama of Iterated Revision. In: Hansson, S. (eds) David Makinson on Classical Methods for Non-Classical Problems. Outstanding Contributions to Logic, vol 3. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7759-0_5

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