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JuDAS: a theory of rational belief revision

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Abstract

Although the AGM theory established a paradigm for the theory of belief revision, which is generally regarded as a kind of standard in the field, it is also frequently criticized as inadequate because it neglects justificational structures. Other theories of belief revision are similarly remiss in this regard. So far, little has been done to address this shortcoming. This paper aims to fill this gap. Following a critical analysis of the AGM theory, a justification operator is introduced as a formal means to incorporate justificational structures into a belief revision theory. An AGM style belief revision theory is proposed that is based on such a justification operator. The theory is presented in an axiomatic form. Representation theorems link the axioms with constructive belief change mechanisms. The proposed theory addresses the problems of the AGM theory and similar approaches that neglect justificational structures.

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Notes

  1. I would like to thank two anonymous reviewers for their valuable comments on an earlier drafts of this paper. Cf. Alchourrón et al. (1985) and Gärdenfors (1988).

  2. A proof for this well-known result can, for instance, be found in Gärdenfors (1988, App. A).

  3. A⊥ denotes the set of all sentences and is called the absurd belief set.

  4. These two sets of postulates reproduce how AGM characterize belief change functions, e.g. in Gärdenfors (1988, Chap. 3). Note, however, that this characterization can be simplified in various ways. In the case of expansion, the postulates Closure, Success, Inclusion, and a modified minimality requirement actually suffice to uniquely determine the AGM expansion function Cn(A ∪ {x}). It can then be shown that this function also satisfies the remaining two AGM postulates Redundancy and Monotony. In the case of the revision postulate Consistency, it is actually sufficient to require the left-to-right direction, given the postulates Closure and Success.

  5. 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.

  6. See Gärdenfors (1988, p. 67f).

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

  8. See Baroni et al. (2013, p. 25).

  9. 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. Pagnucco (1996) for his belief change procedures that are based on abductive reasoning. Cf. Doyle (1979, 1992) for his RMS software. Cf. also Alchourrón and Makinson (1985) on their proposal of so-called safe contractions which is akin to Doyle’s RMS, and Rott and Hansson (2014) for a recent overview thereof.

  10. Although this terminology is standard practice (cf. e.g. Hansson 1999b, p. 235), it should be noted that the term “rejection” is used with two different meanings in this context. On the one hand, the term denotes one of the three epistemic attitudes a subject can adopt with respect to a given sentence (cf. Sect. 2). On the other hand, it denotes one of the two decisions a subject can take with respect to new information it has been offered. To avoid any misunderstanding, let us elucidate how the two combine: if one decides to incorporate a new piece of information x, one will ultimately accept x. If one decides to reject a new piece of information x, one may either end up indeterminate with respect to x (namely if ¬x is also not accepted), or adopt the epistemic attitude of rejecting x (namely if ¬x is accepted). Thus, typically non-prioritized belief change theories that allow the epistemic subject to decide whether or not it takes a new piece of information x on board, are compatible with the subject ultimately adopting any of the three epistemic attitudes with respect to x. This is exemplified, e.g. by Hansson’s (1997) semi-revisions.

  11. In Haas (2018), I elaborate on this point and distinguish between two interpretations of the AGM theory: (1) The scope of AGM expansions and revisions is supposed to cover all cases in which a new piece of information is being offered to an epistemic subject. (2) The scope of AGM expansions and revisions is only supposed to cover those cases in which a new piece of information is being offered to an epistemic subject and it is actually incorporated. Regardless of which of these two interpretations one considers to be exegetically correct, I argue that it is desirable in both cases to formulate a criterion based on which one can determine when new information should be incorporated and when it should be rejected.

  12. In Haas (2018), I refer to this postulate as Weak Monotony. After an anonymous reviewer pointed out to me that this term already had been coined by Rott (2001, p. 113) for a different postulate, I decided to change its name to Conditional Monotony.

  13. See Hansson (1999b, Sect. 2.1).

  14. The only genuine cases of contractions are so-called contractions for the sake of argument, where one pretends to abandon a belief in order to enter a discussion without begging the question. In Haas (2005, Chap. 6), I argue that the AGM theory is well-suited to model this special case of hypothetical belief change. In philosophy, belief change theories are used to model the doxastic dynamics of (idealized) persons. These theories can also, however, be used to model the dynamics of legal codes in law and databases in artificial intelligence. In these domains, contractions might well be in order.

  15. In Haas (2018, Sect. 6), I discuss the AGM postulate Recovery, which is arguably the most controversial element of their contraction procedure. Similar to the analysis in this paper, I argue that Recovery ought to be replaced by a weaker postulate that takes justificational structures into account. Makinson (1997a) and Hansson (1999a) were the first to discuss the role of justificational structures in the context of Recovery. Despite their differing views, they agree that Recovery is implausible if justificational structures are taken into account. In Makinson’s own words: “[R]ecovery is […] an inappropriate condition for the operation of contraction when the theory is seen as comprising not only statements but also a relation or other structural element indicating lines of justification, grounding, or reasons for belief.” Hansson adds to this: “Actual human beliefs always have such a justificatory structure; at least I have not been able to find a case in which they do not. It is difficult if not impossible to find examples about which we can have intuitions, and in which the belief set is not associated with a justificatory structure […]”. In Haas (2018), I also discuss Hansson’s (1999b) postulates Relevance and Core-retainment as possible alternatives to Recovery, and find them wanting because they also fail to adequately take justificational structures into account. Cf. also Wassermann (1999, Sect. 5.2.3) for revision operations that build on these two postulates.

  16. Note that the characteristics of postulate (Js1) differ from those of (Js2)–(Js6). While the latter postulates can be described as rationality requirements, (Js1) only states the syntactic form of the justification operator. Although the syntactic form required by (Js1) is one that many explications of the notion “justification” exhibit, I discuss alternatives to it in Haas (2005, p. 70f).

  17. Note that here A ∈  A is a set of sentences that may or may not be deductively closed.

  18. The introduction of a set of justified sentences Js(A) relative to an acceptance system A has similarities to the introduction of a set of credible sentences by credibility-limited revisions discussed by Hansson et al. (2001) and the introduction of a set of retractable sentences by so-called shielded contractions, which were discussed by Fermé and Hansson (2001).

  19. In another attempt to propose an improved belief revision operation, Hansson (1992) introduced a so-called conclusion operator Cn′, which is an extension of the standard consequence operator Cn. Since the operator Cn′ and the justification operator Js discussed in this essay have some common properties, 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.

  20. Here, it is irrelevant whether the theories of justification represented by Jsmin and Jsmax have actually been advocated by some authors. Nevertheless, it is worth mentioning in passing that the theory of justification represented by Jsmin corresponds to a skeptic who only exempts logically valid sentences from his doubts. Jsmax, on the other hand, can be associated with the rudimentary type of coherence theory that has been proposed, e.g. by Neurath (1932/1933).

  21. Various other formal justification operators that satisfy (Js1)–(Js6) can be found in Haas (2005). Cf. especially App. A.

  22. Note that the right to left inclusion Js(A) ⊆ A holds trivially because of (Js2).

  23. The straightforward proof is left to the reader.

  24. The proof of Lemma 5 is rather technical and can be found in Haas (2005, App. A).

  25. Note that AGM expansions fulfill Idempotence trivially because it is entailed by the unconditional Success postulate and Redundancy.

  26. As pointed out in Fn. 4, this parallels the situation with AGM’s axiomatic characterization.

  27. To reduce the number brackets and to thus improve the readability of formulas, it is advisable to introduce the composition “∘” of two operators O1 and O2 in the usual way: O1∘O2(A) ≔ O1[O2(A)].

  28. Here, “[x]” denotes the equivalence class of x, i.e., [x] ≔ {y ∈  L | Cn(y) = Cn(x)}.

  29. See Hansson (1999b, p. 238).

  30. 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).

  31. Cf. Pagnucco (1996, p. 105 + 166).

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

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    Gordian Haas

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Haas, G. JuDAS: a theory of rational belief revision. Synthese 197, 5027–5050 (2020). https://doi.org/10.1007/s11229-018-01958-0

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