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Contraction, Revision, Expansion: Representing Belief Change Operations

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Krister Segerberg on Logic of Actions

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

Abstract

The underlying idealizations in Krister Segerberg’s Dynamic Doxastic Logic (DDL) are investigated in comparison with other belief revision models. It is argued that the doxastic voluntarism of the proposed interpretation is problematic but can be discarded. The treatment of conditional operators in DDL is discussed, and it is proposed that the use of conditional operators not satisfying the Ramsey test should be further investigated.

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Notes

  1. 1.

    Or more precisely: the sentences I was committed to believe to be true (Cf. [18]).

  2. 2.

    If belief change is interpreted as referring to other belief-holders than individual persons, then the sentential format may be less problematic. One example of this is database management. The contents of databases are more readily representable by sentences than human beliefs or memories. Another example is changes in collectively created and maintained stocks of information or knowledge, such as the corpus of scientific beliefs. Collective processes are usually based on sentential representations since these are needed for interindividual communication.

  3. 3.

    Or set of sentences [9].

  4. 4.

    These are the most elementary demands of conservatism. In addition the following, somewhat less precise, demands are common: (iii) In incorporation, no sentences are added unless this is needed to incorporate the input. (ii) In contraction, no sentences are removed unless this is needed to remove the input.

  5. 5.

    An alternative approach takes as primitive an operation that both removes some sentence(s) and adds some other sentence(s). It is then possible to develop a model of belief change on the basis of one single primitive operation instead of two as in Levi’s model [13].

  6. 6.

    For arbitrary \(K_1\) and \(K_2\), this recipe only works if the language is finite. In a framework with an infinite language, two operations on sentences are not sufficient to take us from a belief set \(K_1\) to any other belief set \(K_2\). If there is a countably infinite number of logical atoms, then the number of belief sets expressible in the language is uncountable. (This can be shown with a standard diagonalization argument.) On the other hand, there is only a countable number of sequences \(\div p + q\) of a contraction by one sentence (\(p\)) followed by an expansion by another (\(q\)). This problem can be solved by introducing multiple contraction and expansion.

  7. 7.

    This refers to the modelling of human beliefs. Pure contraction of databases is unproblematic.

  8. 8.

    Heinrich Wansing is another prominent proponent of this view. He has proposed that developing the semantics of belief ascriptions from the viewpoint of doxastic voluntarism can be a way to avoid closure of belief under logical consequence. In his view, a variant of seeing-to-it-that (stit) logic of agency can be used to represent voluntary acquisition and abandonment of belief. [41] In later work he has further specified this as dstit-theory, where dstit stands for “deliberately sees to it that” [42]. He proposes the introduction of a belief formation operator to be read “\(\alpha \) sees to it that \(\alpha \) believes that \(p\)” (p. 212).

  9. 9.

    Issac Levi [19] avoids most of the common difficulties with Ramsey test conditionals by accepting (4) but not (5).

  10. 10.

    However, it is trivially unproblematic in an approach where it holds for all potential belief sets \(K_1\) and \(K_2\) that \(K_1 \nsubseteq K_2\). In such a framework there cannot be any pure contraction, or any other operation that takes us from a belief set to one of its proper subsets. Furthermore, there cannot be any pure expansion, or any other operation that takes us from a belief set to one of its proper supersets. Cf. [11].

  11. 11.

    The notation in the quoted formulas has been slightly modified.

  12. 12.

    This conflation is perhaps stimulated by the standard theory of probabilities, in which two notions of degree of belief are merged: the current strength of a belief and the propensity to retain it when it is challenged are represented by the same number. However, this is a limitation that should not necessarily be transferred to other frameworks.

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

    Sven Ove Hansson

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Correspondence to Sven Ove Hansson .

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

    Robert Trypuz

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Hansson, S.O. (2014). Contraction, Revision, Expansion: Representing Belief Change Operations. 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_7

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