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Four Ways in Which Theories of Belief Revision Could Benefit from Theories of Epistemic Justification

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Abstract

Belief revision theories aim to model the dynamics of epistemic states. Besides beliefs, epistemic states comprise most importantly justificational structures. Typically, belief revision theories, however, model the dynamics of beliefs while neglecting justificational structures over and above logical relations. Despite some awareness that this approach is problematic, how devastating the consequences of this neglect are has not yet been fully grasped. In this paper, I argue that taking justificational structures into account could solve four well-known problems of belief revision.

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Notes

  1. I would like to thank Sven Ove Hansson and two anonymous reviewers for their valuable comments on earlier drafts of this paper.

  2. Cf. Alchourrón et al. (1985).

  3. Because logically closed sets of sentences comprise infinitely many sentences, there has always been some quarrel as to whether they can adequately represent the epistemic state of subjects with finite cognitive capacities. Bear in mind, however, that belief revision theories are meant to be normative, i.e., aiming at describing, how ideal rational subjects (with ideal cognitive capacities) revise their beliefs. Thus, I do not want to criticize AGM on these ground, but would rather side with Levi (1977), who urged to interpret a logically closed set of sentences as representing the beliefs that the subject is committed to believe, rather than those that s/he actually believes in. Furthermore, Hansson (2008) proposed a contraction procedure, that ensures “finite representability”, i.e., a set of sentences A representing an epistemic state is always identical to the deductive closure of some finite set of sentences A′.

  4. Although both belief bases and deductively closed belief sets are unable to express non-logical forms of justification, belief bases can express certain aspects of justification which the latter cannot. For instance, if p is removed from the belief base {p, p → q} then q is no longer justified. But if p is removed from the belief base {p, q}, which expresses the same belief set as the former belief base, then q is still justified.

  5. See Gärdenfors (1988, p. 67f).

  6. See Spohn (2012, p. 118f).

  7. Cf. Spohn (1988) for a concise presentation of his ranking theory (which he then called Ordinal Conditional Functions) and Spohn (2012) for a truly comprehensive exposition thereof. Cf. Haas (2015, Chap. 7) for a concise presentation of my JuDAS theory. A more comprehensive presentation of it in German can be found in Haas (2005). Cf. Doyle (1979, 1992) for his RMS software.

  8. Cf., e.g., Gärdenfors (1990) and Doyle (1992). In this paper, I do not seek to take a stand on the question which type of justification theory is best suited to complement a belief revision theory. It is worth mentioning, however, that Hansson and Olsson (1999) have provided some evidence that “[r]epresenting a belief state as a logically closed set of sentences […] is […] difficult to reconcile with a coherentist approach.“ Thus, it might seem as though the AGM theory cannot be combined with such a theory of justification. Hansson and Olsson’s argument is not decisive, however, as it is based on a number of assumptions. Although they claim that these “[…] theses [are] commonly associated with coherentism […]”, they do not, for instance, apply to Lehrer’s (2000) theory, which is arguably one of the most prominent coherentist approaches. Bender (1989) has classified coherence theories into those that take coherence to be a property and those that take it to be a relation making it quite clear that Lehrer’s theory is of the latter type. Hansson and Olsson, on the other hand, assume a theory that takes coherence to be a property, i.e. applying to belief states (cf. p. 245). Similarly, Hansson and Olsson’s so-called Supraclassicality requirement (If A ⊢ p, then A supports p) is not fulfilled by Lehrer’s theory, either. If the epistemic subject does not accept a sentence p, then p is not justified relative to its acceptance system, because the sentence “You do not accept p” is a competitor to p that cannot be beaten or neutralized. Consequently, even if the antecedent of Supraclassicality holds, the consequent may not.

  9. I am grateful to an anonymous reviewer for suggesting to include del Val (1997) in my discussion of justificational structure in AGM. del Val argues that the latter distinction collapses because “the coherence and foundational theories of belief revision are mathematically equivalent” (Sect. 5). If his argument were sound, it would provide further evidence that the resources of standard belief revision theories are insufficient to capture the notion of justification. However, another reviewer pointed out that del Val’s argument seems to only take a limiting case of belief base operations into account. Since it would go beyond the scope of this essay to analyze del Val’s argument in detail, it remains open whether it can actually lend further support to this claim.

  10. The Success postulate for expansions is sometimes also referred to as Weak Success. Cf., e.g., Fermé and Hansson (1999) and Hansson (1997a).

  11. See Cross and Thomason (1992, p. 251).

  12. See Levi (1991, p. 108).

  13. A collection of these approaches can be found in the special issue Hansson (1997b), while Falappa et al. (2012) is a recent addition to the debate on non-prioritized belief change.

  14. Cases of knowledge by testimony, such as the Martians example, can also be modeled in a different way, which I have discussed in Haas (2015, p. 79). My belief in p could also be derived and justified schematically as follows: P1 “If S asserts that p, then it is reasonable to believe that p.”, P2 “S asserts that p.”, therefore C “It is reasonable to believe that p”. Premise P1, in turn, has to be inferred from a more general premise such as P0 “S is a reliable source of information (on the subject matter of p)”. In the expert version of the Martians example, which we discussed above, I believe P0, while in the dreamer version, I do not. Although I will incorporate P2 in both versions—thus complying with AGM’s Success postulate—I will end up believing p in the expert version but not in the dreamer version. Note, however, that a general statement such as P0 can only be derived inductively from beliefs describing S’s track record as an informant. Thus, P0 could only enter an AGM belief set if we combine their theory with a theory of justification that allows for derivations beyond truth-functional consequences. To sum up, in order to correctly model the Martians example, we need a theory of justification, irrespective of whether we take the input to be p or P2. In the next section, we will also discuss an example violating Success for expansions, which does not involve knowledge by testimony.

  15. One aim of this paper is to propose modifications of some of the AGM postulates. Although these new postulates will constrain belief change operators, this paper does not seek to fully characterize them. Thus, it should be understood that the operator + and * may well be distinct despite sharing the same criterion of incorporation according to the Conditional Success properties. In that regard, the modified properties are just like AGM’s original Success postulates.

  16. See Hansson (1999, p. 238).

  17. The same holds true of relationally screened revisions that have been proposed by Makinson (1997b) as a more refined version of his procedure as well as of entrenchment-based non-prioritized revisions proposed by Hansson et al. (2001).

  18. Cf. Haas (2016).

  19. I have adapted this example for the present context from BonJour (1985).

  20. Unlike Success and Monotony, Preservation is not a postulate but a theorem of the AGM theory, which is an immediate consequence of the expansion postulate Inclusion (A ⊆ A + p) and the revision postulate Vacuity (If ¬p ∉ A, then A + p ⊆ A * p).

  21. Note that the lack of any requirement to give up beliefs which can no longer be justified is exactly what belief bases have proposed to remedy. As pointed out in Sect. 1, however, this proposal leaves much to be desired as it only takes logical relations into account and neglects non-logical structures of justification.

  22. Cf. also Gärdenfors (1988, Chap. 7).

  23. Cf. Katsuno and Mendelzon (1992, Lemma 3.3).

  24. This example is discussed by (Hansson 1999, p. 71ff).

  25. This is a slightly modified version of an example by Hansson (1991).

  26. I am grateful to an anonymous reviewer for suggesting to discuss Hansson’s postulates because they “are closer to the idea of a justificational view”. Although Hansson (1999, p. 74) briefly discusses Relevance and Core-retainment as possible weakenings of Recovery and finds them inadequate as such, it should be noted that he pointed out to me that this was not his main reason for proposing these postulates to axiomatically characterize belief base operations. It should furthermore be mentioned that the reviewer’s suggestion “follows the content of the postulates” and that s/he “never had in mind that the original intention of Sven Ove Hansson was to substitute Recovery on belief sets […]”. Rott and Pagnucco (1999) provide an excellent survey of withdrawal functions.

  27. See Makinson (1997a, p. 478).

  28. A proof can, for instance, be found in Haas (2005, Sect. 2.10).

  29. This proposal is similar to Fuhrmann’s so-called Filtering Condition: If q is abandoned in A − p, then A − p should not contain any beliefs the epistemic subject held just because it believed in q. Cf. Fuhrmann (1991, p. 184). In the Cleopatra example, this applies to the conditional p → q. I believed the material conditional just because I believed in q. According to the Filtering Condition, p → q should thus be given up in A − p because q is given up.

  30. Cf. Haas (2005, Sect. 5.1).

  31. See Hansson (1999, p. 68).

  32. The introduction of a set of justified sentences Js(A) relative to a belief set A has similarities to the introduction of a set of credible sentences by credibility-limited revisions, which were discussed in Sect. 2. The same holds true for the introduction of a set of retractable sentences by so-called shielded contractions, which were discussed by Fermé and Hansson (2001).

  33. This undoubtedly is a technical use of the term “justification operator”, as some operators that we intuitively would not consider to be justification operators might meet the postulates (Js1)–(Js6). In another attempt to propose an improved belief revision operation, Hansson (1992a) introduced a so-called conclusion operator Cn′, which is an extension of the usual consequence operator Cn. Since the operator Cn′ and the justification operator Js discussed in this essay have some properties in common, it would be interesting to study the relationship between belief revision constructions based on Cn′ and Js, respectively. However, this would go beyond the scope of this essay.

  34. A proof for Lemma 6.1 is given in Haas (2005, p. 76).

  35. A proof for Lemma 6.3 is given in Haas (2005, p. 77).

  36. In a personal conversation, Keith Lehrer agreed that the postulates (Js1)–(Js6) are met by his theory. In Haas (2005, Sect. 8.4), I presented a formal argument to the same effect. Various other formal justification operators satisfying (Js1)–(Js6) can be found in App. A of the same publication.

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    Gordian Haas

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Haas, G. Four Ways in Which Theories of Belief Revision Could Benefit from Theories of Epistemic Justification. Erkenn 85, 295–316 (2020). https://doi.org/10.1007/s10670-018-0028-2

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