Filtered Belief Revision: Syntax and Semantics

In an earlier paper [Rational choice and AGM belief revision, Artificial Intelligence, 2009] a correspondence was established between the set-theoretic structures of revealed-preference theory (developed in economics) and the syntactic belief revision functions of the AGM theory (developed in philosophy and computer science). In this paper we extend the re-interpretation of those structures in terms of one-shot belief revision by relating them to the trichotomous attitude towards information studied in Garapa (Rev Symb Logic, 1–21, 2020) where information may be either (1) fully accepted or (2) rejected or (3) taken seriously but not fully accepted. We begin by introducing the syntactic notion of filtered belief revision and providing a characterization of it in terms of a mixture of both AGM revision and contraction. We then establish a correspondence between the proposed notion of filtered belief revision and the above-mentioned set-theoretic structures, interpreted as semantic partial belief revision structures. We also provide an interpretation of the trichotomous attitude towards information in terms of the degree of implausibility of the information.


Introduction
The dominant paradigm in belief revision is the so-called AGM theory [Alchourrón et al. (1985) and Gärdenfors (1988)], which is a syntactic theory that represents the belief state of an agent as a pair (K , * ), where K is a consistent and deductively closed set of formulas in a propositional language (interpreted as the agent's initial beliefs) and I am grateful to two anonymous reviewers for helpful and constructive comments. A first version of this paper was presented at the 17th TARK conference (July 2019) and an extended abstract published in Bonanno (2019).
B Giacomo Bonanno gfbonanno@ucdavis.edu 1 University of California, Davis, USA * : → 2 (where denotes the set of formulas and 2 the set of subsets of ) is a function that associates with every formula φ ∈ (interpreted as new information) a set K * φ ⊆ , representing the agent's revised beliefs in response to information φ. If the function * satisfies a set of six properties, known as the basic AGM postulates, then it is called a basic AGM belief revision function, while if it satisfies two additional properties (the so-called supplementary postulates) then it is called a supplemented AGM belief revision function. One of the six basic postulates is the success axiom, according to which every item of information φ is incorporated into the revised beliefs: φ ∈ K * φ. In the past twenty years a large literature has emerged centered on relaxing the success axiom: the so-called non-prioritized belief revision approach [for surveys of this literature see Hansson (1999b), Fermé andHansson (2011, Section 6.2, 2018, Chapter 8)]. This literature acknowledges the fact that there may be some items of information that an agent is not disposed to accept, for a number of reasons: 1. the agent may doubt the reliability of the source of the information, 2. the information may be in conflict with some highly entrenched beliefs of the agent, 3. the agent might judge the information to be too far-fetched or implausible. 1 In Case 1 it is not the intrinsic content of the information that leads the agent to reject it, but the lack of trust in its source; thus the same information may be accepted it if originates from source A but rejected if it originates from source B. For a recent investigation of the role of trust in belief revision see Booth and Hunter (2018).
In Case 2 it is the conflict of the information with some core beliefs of the agent that leads to the information being rejected: such core beliefs are given a privileged status by the agent and are essentially immune to revision. This possibility was first studied in Makinson (1997) and Fermé and Hansson (2001). 2 In Case 3, the agent's belief state is described not only by the pair (K , * ), but also by a partition of the set of sentences into two sets: the set C of credible sentences and the set R of rejected sentences. The belief revision function * : → 2 is then such that K * φ = K if φ ∈ R (if the information is rejected then beliefs remain unchanged), while the success postulate applies to credible information: φ ∈ K * φ if φ ∈ C . This line of inquiry is pursued in Hansson et al. (2001). 3 This approach is taken a step further in Garapa (2020) [and, independently, Bonanno (2019)] with the introduction of a third, and intermediate, possibility, namely formulas that are credible but with a lower level of credibility: When revising by a sentence that is considered to be at the second level of credibility, that sentence is not incorporated but all the beliefs that are inconsistent with it are removed. The intuition underlying this behavior is that, the belief is not credible enough to be incorporated in the agent's belief set, but creates in the agent some doubt making him/her remove all the beliefs that are inconsistent with it. (Garapa (2020), p. 2.) Garapa (2020) calls this trichotomous stance towards information "two level credibility-limited revision". To illustrate these different attitudes towards a particular item of information, consider the case of three husbands, A, B and C, each of whom believes that his wife is, and has always been, faithful to him. Each husband is approached by a trusted friend who tells him "your wife is having an affair with another man".
• Husband A accepts the information, comes to believe that his wife is unfaithful and files for divorce. • Husband B is unshakable in his belief that his wife is faithful and discards the information as not credible. • Husband C's reaction is somewhere in the middle: he abandons his belief in his wife's faithfulness, but does not go all the way to believing that she is unfaithful; in other words, he becomes open-minded about the possibility of her being unfaithful (and, perhaps, hires a private investigator to resolve the uncertainty).
Thus we partition the set of sentences into three sets: the set C of credible sentences [these correspond to the "high credibility" sentences in Garapa (2020)], the set R of rejected sentences and the set A of sentences that we call "allowable" [these correspond to the "low credibility" sentences in Garapa (2020)]. 4 For the latter we have that {φ, ¬φ} ∩ (K * φ) = ∅, that is, when informed that φ the agent believes neither φ nor ¬φ (in other words, she is open-minded towards both φ and ¬φ). While the approach in Garapa (2020) is entirely syntactic (see Sect. 2 for details), our focus is on the semantics (Sect. 3). However, in Sect. 2 we start with the AGM-style syntactic approach and put forward the notion of filtered belief revision and provide a characterization of it in terms of basic AGM belief revision, as follows (Proposition 1). Let • : → 2 be a filtered belief-revision function (Definition 2, Sect. 2); then for some basic AGM belief-revision function * : → 2 , ∀φ ∈ :  [Harper (1976)] the contraction by ¬φ coincides with taking the intersection of the original beliefs with the revision by φ). 5 Note that this implies that the underlying contraction function satisfies the recovery axiom: see Sect. 5 for a discussion of this point).
It is worth stressing that the focus of this paper is on one-shot belief revision (see Sect. 5, for a discussion of this point).
In Sect. 3 we turn to the main focus of this paper by proposing a semantics for filtered belief revision that extends the semantics for AGM belief revision put forward in Bonanno (2009), based on the structures of revealed-preference theory in economics. 6 Revealed-preference theory considers choice structures , E, f consisting of a nonempty set (whose elements are interpreted as possible alternatives to choose from), a collection E of subsets of (interpreted as possible menus, or choice sets) and a function f : E → 2 (2 denotes the set of subsets of ), representing choices made by the agent, conditional on each menu. Given this interpretation, the following restriction on the function f is a natural requirement (the alternatives chosen from menu E should be elements of E): ∀E ∈ E, The objective of revealed-preference theory is to characterize choice structures that can be "rationalized" by a total pre-order on , interpreted as a preference relation, 7 in the sense that, for every E ∈ E, f (E) is the set of most preferred alternatives in E: f (E) = {ω ∈ E : ω ω , ∀ω ∈ E}. In Bonanno (2009) a re-interpretation of choice structures in terms of semantic partial belief revision functions was put forward. A model of , E, f (where is now thought of as a set of states) is obtained by adding a valuation V that assigns to every atomic formula p the set of states at which p is true. Truth of an arbitrary formula at a state is then defined as usual. Given a model , E, f , V , the initial beliefs of the agent are taken to be the set of formulas φ such that f ( ) ⊆ ||φ||, where ||φ|| denotes the truth set of φ; thus f ( ) is interpreted as the set of states that are initially considered possible. The events (sets of states) in E ⊆ 2 are interpreted as possible items of information. If φ is a formula such that ||φ|| ∈ E, the revised belief upon learning that φ is defined as the set of formulas ψ such that f (||φ||) ⊆ ||ψ||. Thus the event f (||φ||) is interpreted as the set of states that are considered possible after learning that φ is the case. In light of this interpretation, condition (2) above corresponds to the success postulate of AGM theory. A model of a structure , E, f thus gives rise to a partial syntactic belief revision function whose domain is typically a proper subset of the set of formulas. The objective of Bonanno (2009) was to find necessary and sufficient conditions on the structure that guarantee the existence of an AGM belief-revision function that extends the partial belief revision function obtained from an arbitrary model of it.
In this paper we continue the above analysis by removing restriction (2). First of all, we allow for some events-in the set of potential items of information E-to be treated as not credible, so that Secondly, for information E ∈ E which is credible we postulate the success property (2): Finally, we also consider a third type of information, which is taken seriously but not given the same status as credible information, and call it allowable; we capture this possibility by means of the following condition, which says that allowable information is not ruled out by the revised beliefs: We model credibility, allowability and rejection by partitioning the set E of possible items of information into three sets: the set E C of credible items, the set E A of allowable items and the set E R of rejected items. Thus we consider partial belief revision structures (PBRS for short) , As explained above, we use valuations to link syntax and semantics and obtain, from every model of a PBRS, a syntactic partial belief revision function. We then define a PBRS to be basic-AGM consistent if, for every model of it, the associated partial belief revision function can be extended to a full-domain belief-revision function • : → 2 such that, for some basic AGM belief-revision function * : → 2 , equation (1) is satisfied. Proposition 3 in Sect. 3 provides necessary and sufficient conditions for a PBRS to be basic-AGM consistent. In Sect. 4 we provide an interpretation of credible, allowable and rejected information in terms of the degree of implausibility of the information and then extend the analysis of the previous section to supplemented AGM consistency. Section 5 contains a discussion of several aspects of the proposed framework and of related literature.

The syntactic approach
Let be the set of formulas of a propositional language based on a countable set At of atomic formulas. 8 We write φ to denote that the formula φ is a tautology, that is, a theorem of Propositional Logic. Given a subset K ⊆ , its PL-deductive closure [K ] P L (where 'PL' stands for Propositional Logic) is defined as follows: ψ ∈ [K ] P L if and only if there exist φ 1 , ..., φ n ∈ K (with n ≥ 0) such that (φ 1 Let K ⊆ be a consistent and deductively closed set of formulas representing the agent's initial beliefs. A (syntactic) belief revision function is a function * : → 2 that associates with every formula φ ∈ (thought of as new information) a set K * φ ⊆ (thought of as the revised beliefs upon learning that φ). A belief revision function * : → 2 is called a basic AGM function if it satisfies the first six of the following properties and it is called a supplemented AGM function if it satisfies all of them. The following properties are known as the AGM postulates: ∀φ, ψ ∈ , AGM1 requires the revised belief set to be deductively closed. AGM2 postulates that the information be believed. AGM3 says that beliefs should be revised minimally, in the sense that no new formula should be added unless it can be deduced from the information received and the initial beliefs. 10 AGM4 says that if the information received is compatible with the initial beliefs, then any formula that can be deduced from the information and the initial beliefs should be part of the revised beliefs. AGM5 requires the revised beliefs to be consistent, unless the information φ is a contradiction (that is, unless ¬φ).
AGM6 requires that if φ is propositionally equivalent to ψ then the result of revising by φ be identical to the result of revising by ψ.
AGM1-AGM6 are called the basic AGM postulates, while AGM7 and AGM8 are called the supplementary AGM postulates. AGM7 and AGM8 are a generalization of AGM3 and AGM4 that concern composite belief revisions of the form K * (φ ∧ ψ): The idea is that, if K is to be changed minimally so as to include two sentences φ and ψ, such a change should be possible by first revising K with respect to φ and then expanding K * φ by ψ-provided that ψ does not contradict the beliefs in K * φ (Gärdenfors & Rott, 1995, p. 54).
For an extensive discussion of the rationale behind the AGM postulates see Gärdenfors (1988), Gärdenfors and Rott (1995).
We now extend the notion of belief revision by allowing the agent to discriminate among different items of information.
Definition 1 Let be the set of formulas of a propositional language. A credibility partition is a partition { C , A , R } of such that 1. C is the set of credible formulas and is such that

2.
A is the (possibly empty) set of allowable formulas. We assume that if φ ∈ A then φ is consistent and that A is closed under logical equivalence. 3. R is the set of rejected formulas, which contains (at least) all the contradictions.
The properties that we have postulated for C and A are called "element consistency" and "closure under logical equivalence" in Hansson et al. (2001). There are other, natural, properties to consider, but for the time being we restrict attention to a minimal set of properties that are sufficient for the representation result of Proposition 1. We will address the issue of what are additional "natural" properties for the sets C , A and R in Sect. 5.
Following Hansson et al. (2001), given a belief set K , we use the symbol • to denote a general belief revision function based on K (that is, a function • : → 2 that associates with every φ ∈ a new belief set K • φ) and the symbol * to denote a (basic or supplemented) AGM belief-revision function based on K . Definition 2 Let K be a consistent and deductively closed set of formulas (representing the initial beliefs) and { C , A , R } a credibility partition of (Definition 1). A belief-revision function • : → 2 is called a filtered belief revision function (based on K and { C , A , R }) if it satisfies the following properties: ∀φ, ψ ∈ , By (F1), if information φ is rejected (φ ∈ R ), then the original beliefs are preserved. The next two properties deal with the case where φ / ∈ R . (F2) says that if, initially, the agent did not believe ¬φ, then (a) if φ is credible then the new beliefs are given by the expansion of K by φ, while (b) if φ is allowable then the agent does not change her beliefs.
(F3) says that if, initially, the agent believed ¬φ, then (a) if φ is credible, then the agent switches from believing ¬φ to believing φ, (b) if φ is allowable, then the agent suspends her belief in ¬φ in a conservative way, that is, she contracts her belief set by ¬φ in such a way as to retain as much as possible of her original beliefs and if she were to re-introduce ¬φ into her revised beliefs and close under logical consequence then she would go back to her initial beliefs. As noted in the introduction, this means that contraction satisfies the so-called recovery postulate. See Sect. 5 for a discussion of this assumption.
By (F4) belief revision satisfies extensionality: if φ is logically equivalent to ψ then revision by φ coincides with revision by ψ.
The following proposition provides a characterization of filtered belief revision in terms of basic AGM belief revision. The proof is given in Appendix A. We take the belief state of the agent to be represented by the triple (K , { C , A , R } , •): initial beliefs, credibility partition and revision function. 11 Garapa (2020) shows that the credibility partition can be derived from the pair Garapa (2020) proves the following equivalence. Proposition 2 Garapa (2020, Theorem 3.14, p.10) Let K be a consistent and deductively closed set of formulas and • : → 2 a belief revision function based on K . Then the following are equivalent:

Proposition 1 Let K be a consistent and deductively closed set of formulas,
(A) • satisfies the following properties:

(B) there exists a basic AGM belief revision function * :
→ 2 and a credibility partition While in the AGM approach the domain of a belief revision function is the entire set of formulas (that is, every formula is viewed as a potential item of information), in the following sections we will consider the possibility that the domain of a belief revision function is a subset of the set of formulas . Let K be a consistent and deductively closed set of formulas, representing the agent's initial beliefs, and let ⊆ be a set of formulas representing possible items of information. Let • : → 2 be a function that associates with every formula ψ ∈ a set K • ψ ⊆ . If = then • is called a partial belief revision function, while if = then it is called a full-domain belief revision function. If • is a partial belief revision function with domain and • is a full-domain belief revision function, we say that • is an extension of • if, for all ψ ∈ , K • ψ = K • ψ. The rationale for considering partial belief revision functions is discussed in Sect. 5.
We now turn to the main focus of this paper, namely the semantic structures outlined in the introduction.

Semantics: belief revision structures
Next we turn to the notion of a model, or interpretation, of a PBRS.
Fix a propositional language based on a countable set At of atomic formulas and let be the set of formulas. A valuation is a function V : At → 2 that associates with every atomic formula p ∈ At the set of states at which p is true. Truth of an arbitrary formula at a state is defined recursively as follows (ω | φ means that formula φ is true at state ω): 12 We do not assume that is finite.
Given a valuation V , define: 14 Since f ( ) is interpreted as the set of states that the individual initially considers possible, (5) is the initial belief set. It is straightforward to show that K is consistent (since, by 3(a) of Definition 3, f ( ) = ∅) and deductively closed.
(6) is the set of formulas that are potential items of information. (7) is the partial belief revision function encoding the agent's disposition to revise her beliefs in response to items of information in (for E ∈ E, f (E) is interpreted as the set of states that the individual considers possible after receiving information represented by event E).
Equivalently, by Proposition 1, a PBRS is basic-AGM consistent if, for every model of it, there exists a filtered belief revision function (Definition 2) that extends the partial belief revision function generated by the model.
The following proposition gives necessary and sufficient conditions for a PBRS to be basic-AGM consistent. The proof is given in Appendix B.
f be a partial belief revision structure. Then the following are equivalent: (A) C is basic-AGM consistent (Definition 5).

Plausibility and supplemented AGM consistency
In this section we investigate what additional properties a PBRS needs to satisfy in order to obtain a correspondence result analogous to Proposition 3 but involving supplemented, rather than basic, AGM belief revision (that is, belief revision functions that satisfy the six basic AGM postulates as well as the two supplementary ones). 15 To obtain a characterization in terms of supplemented AGM belief revision we need to add more structure. In particular, we are interested in determining when a PBRS can be "rationalized" by a plausibility order. 16 In parallel to the partition of the set of sentences into three sets ( C , the set of credible sentences, A , the set of allowable sentences, and R , the set of rejected sentences), we postulate a "plausibility-based" partition of the set of states into three sets: the set C of credible states, which-in a precise sense explained below -are the states with low implausibility, the set A of allowable states, which are the states with intermediate implausibility, and the set R of rejected states, which are the states with high implausibility. Note that, throughout this section, we shall assume that the set of states is finite. 17 Definition 6 A plausibility order is a total pre-order on , that is, a binary relation ⊆ × which is complete or total (∀ω, ω ∈ either ω ω or ω ω or both) and transitive (∀ω, ω , ω ∈ if ω ω and ω ω then ω ω ). The interpretation of ω ω is that state ω is at least as plausible as state ω . We denote by the strict component of : ω ω if ω ω and ω ω (thus ω ω means that ω is more plausible than ω ) and by ∼ the equivalence component of : ω ∼ ω if ω ω and ω ω (thus ω ∼ ω means that ω is just as plausible as ω ). For every F ⊆ we denote by best F the set of most plausible elements of F, that is, Given a plausibility order we can partition into ranked equivalence classes as follows. 18 15 In Theorem 3.16 (p. 11) Garapa (2020) provides a syntactic characterization that is analogous to Proposition 2 but in terms of supplemented, rather than basic, AGM belief revision. 16 The notion of a pre-order on the set of possible worlds as a semantics for the AGM theory of belief revision first appeared in Katsuno and Mendelzon (1991). 17 Alternatively, we could allow to be infinite and restrict attention to well-founded plausibility orders. 18 Plausibility orders are equivalent to a special case of the ordinal ranking functions introduced by Spohn (1988), namely ranking functions r : → N (where N denotes the set of non-negative integers) with no gaps, that is, if r (ω) = k for some k > 0 then there exists an ω such that r (ω ) = k − 1. Starting from -Let 0 be the set of most plausible states in : -Let 1 be the set of most plausible states in \ 0 : If ω ∈ k we say that k is the degree of implausibility of state ω (thus the most plausible states are those with degree of implausibility 0, the next most plausible states are those with degree of implausibility 1, etc.). Clearly, for every k ≥ 0, if ω, ω ∈ k then ω ∼ ω . Letn be the largest degree of implausibility, that is,n is such that n = ∅ andn j=0 j = .
Given two integers m and n such that 0 ≤ m ≤ n ≤n, let C be the set of states with degree of implausibility at most m, A be the set of states with degree of implausibility greater than m but at most n and R the set of the remaining states: 19 Definition 7 When a partition { C , A , R } of is obtained from a plausibility order of as explained above, we call it a plausibility-based partition; clearly, the following holds: if ω ∈ C and ω ∈ A ∪ R then ω ω , and if ω ∈ A and ω ∈ R then ω ω .
That is, credible states (those in C ) are more plausible than allowable or rejected states (those in A ∪ R ) and allowable states (those in A ) are more plausible than rejected states (those in R ). From now on we will focus on basic-AGM-consistent PBRS, which-in virtue of Definitions 3 and 5 and Proposition 3-can be redefined as follows.
such a ranking function one would define j = {ω ∈ : r (ω) = j}. Furthermore, when is taken to be the set of maximally consistent sets of sentences, plausibility orders are also equivalent to the systems of spheres introduced by Grove (1988). 19 If n = m then A = ∅ and if n =n then R = ∅.

Definition 8 A basic-AGM-consistent partial belief revision structure
Thus, • by Point 1, if information E has a credible content (E ∩ C = ∅), then the agent revises her beliefs based exclusively on the credible content of the information does not consist entirely of rejected states either (E ∩ A = ∅), then the agent revises her beliefs based exclusively on the "allowable" content of the information We now turn to the issue of what properties of a plausibility-based BPBRS are necessary and sufficient for supplemented AGM consistency. The following definition mirrors Definition 5.

Definition 10 A plausibility-based BPBRS
• be the corresponding partial belief revision function, there exist 1. a full-domain belief revision function • : → 2 that extends • (that is, for every φ ∈ , K • φ = K • φ) and 2. two supplemented AGM belief revision functions * C : → 2 and * A : → 2 such that, for every φ ∈ , If (12) is satisfied, we say that the plausibility order rationalizes the BPBRS.
Note that, by 2(c) of Definition 6, best ⊆ C and thus best = best C ; furthermore, Properties 1 and 2 of Definition 9 are consistent with (12): for example, Proposition 5 in Appendix C provides necessary and sufficient conditions for a plausibility-based BPBRS to be rationalizable. Since that result is rather technical and independent of the characterization of supplemented AGM consistency (Proposition 4 below), we have relegated it to an appendix.
The following proposition, which mirrors Proposition 3 of the previous section, extends Propositions 7 and 8 in Bonanno (2009) to the current framework. The proof is given in Appendix D.

Proposition 4 Let
As explained in Bonanno (2009), the assumption that is finite is needed to ensure that best E = ∅, for every ∅ = E ⊆ . If one strengthens the definition of plausibility order by requiring that, ∀E ⊆ , if E = ∅ then best E = ∅, that is, if one requires the plausibility order to be well founded, then the assumption of finiteness of can be dropped.

Discussion
We investigated a notion of belief revision that, besides acceptance of information, allows for two additional possibilities: (1) that an item of information be discarded as not credible and thus not allowed to affect one's beliefs and (2) that an item of information be treated as a serious possibility without assigning full credibility to it. We started with a syntactic version of this notion, which we called "filtered belief revision", and characterized it in terms of basic AGM belief revision. We then introduced the notion of partial belief revision structure, which provides a simple set-theoretic semantics for belief revision, and provided a characterization of filtered belief revision in terms of properties of these structures. Finally, we considered the notion of rationalizability of a choice structure in terms of a plausibility order and established a correspondence between rationalizability and AGM consistency in terms of the full set of AGM postulates (that is, the six basic postulates together with the supplementary ones). We also provided an interpretation of credibility, allowability and rejection of information in terms of the degree of implausibility of the information.
In this Section we discuss related literature and several aspects of the proposed approach. 1. Alternative theories of non-prioritized belief revision. Several theories of "non-prioritized" belief revision have been proposed in which the AGM success postulate is relaxed. 22 Fermé and Hansson (1999) put forward a syntactic model of "selective revision" in which it is possible for the agent to accept only part of the information. In a similar vein, Booth and Hunter (2018) develop a semantic model where only part of the information originating from a given source is accepted, namely that part relative to which the source is qualified or competent. Fermé and Rott (2004) [see also Rott (2012)] introduce the notion of "revision by comparison" based on the idea that the degree of entrenchment of the information φ ought to be compared with the degree of entrenchment of a reference sentence ψ which is currently believed; the revision is performed according to the principle "see to it that the entrenchment of φ is at least as firm as the entrenchment of ψ" (Fermé and Rott (2004), p. 8). 23 Our contribution is along the lines of Hansson et al. (2001) where the set of sentences is partitioned into two sets, the set of credible sentences and the set of rejected sentences: if information is credible then it is incorporated in the new beliefs while if it is rejected then the initial beliefs are unaffected. Garapa (2020) [and, independently, Bonanno (2019)] goes a step further by separating credible sentences into two categories: the high-credibility sentences, for which the success property holds, and the low-credibility sentences which induce a contraction of the original belief set by the negation of the information (instead of 'high credibility' and 'low credibility' we used the terms 'credibility' and 'allowability', respectively). While the analysis in Garapa (2020) is exclusively focused on the AGM-style syntactic approach, our focus has been on set-theoretic structures that can be viewed as the semantic counterpart of partial belief revision functions.

Why a semantics based on partial belief revision functions?
The semantic structures we considered in Sections 3 and 4 are such that not every conceivable proposition (or event) constitutes a possible item of information, so thattypically-any given interpretation (or valuation) gives rise to a partial syntactic belief revision function, that is, one whose domain is not the entire set of sentences . What is the rationale for the proposed semantics? It can be viewed as complementing the traditional syntactic approach by "inverting" it, in the following sense. The syntactic approach starts with an initial belief set and determines, for every input sentence φ, a new belief set representing the revised beliefs in response to φ. On the other hand, the partial belief revision structures considered in Sections 3 and 4 can be thought of as providing a (possibly small) collection of "revision scenarios", each of which consists of a piece of new information and its corresponding revision result (the set of pairs (E, f (E)), for every E in some set E ⊆ 2 ) and the question is "does there exist an AGM belief-revision function that rationalizes those scenarios?" In other words, is the given collection of revision scenarios consistent with a rational belief-revision policy? For example, an agent might introspectively consider how she would react to a number of hypothetical items of information and ask herself "would it be rational for me to revise my beliefs in this way?" Alternatively, one could probe an agent with some hypothetical informational inputs to see how she would react to those inputs and ask the question "are the agent's answers consistent with a rational disposition to revise her beliefs?" The characterization results of Propositions 3 and 4 provide an answer to these questions. 24 3. Properties of the credibility partition. In Sect. 2 (Definition 1) we gave a list of properties for a credibility partition { C , A , R }. The intention there was to identify a minimal set of properties that would be sufficient for our results. It should be noted that the two main representation results provided in Garapa (2020) (Theorems 3.14, p. 10 and 3.16, p. 11) also make use of the same two properties that we considered for C and A . However, there are other, natural, properties that have been discussed in the literature concerning the set of credible/allowable sentences ; Fermé et al. (2003); Garapa et al. (2018); Garapa (2020)] and we now turn to a discussion of this issue. Let us start with the set of credible sentences C . In Definition 1 we required this set to contain only consistent sentences and to be closed under logical equivalence  call the first requirement "element consistency" and the second requirement "closure under logical equivalence"]. Another natural property to consider is closure under logical consequence, which Hansson et al. (2001) call "single sentence closure": if φ ∈ C and φ → ψ then ψ ∈ C ; that is, if φ is credible then any sentence that is logically implied by φ is also credible. Hansson et al. (2001), Fermé et al. (2003) and Garapa et al. (2018) consider three more properties: "disjunctive completeness": if φ ∨ ψ ∈ C then either φ ∈ C or ψ ∈ C , "negation completeness": either φ ∈ C or ¬φ ∈ C , and "expansive credibility": if φ / ∈ K then ¬φ ∈ C . In order to gain some insight as to whether these four additional properties (closure under logical consequence, disjunctive completeness, negation completeness and expansive credibility) represent natural requirements and-at the same time-confirm the appropriateness of the two assumed properties (element consistency and closure under logical equivalence), let us take a semantic point of view. Consider an arbitrary finite set of states , an arbitrary plausibility partition { C , A , R } of (Definition 7) and an arbitrary valuation V : → 2 . Label a sentence φ With this interpretation and focusing first on the set C of credible sentences, it is clear that the properties assumed in Definition 1 are satisfied: (1) element consistency (if φ is a contradiction then ||φ|| = ∅ and thus ||φ|| ∩ C = ∅, so that φ is not credible), (2) closure under logical equivalence (if φ ↔ ψ then ||φ|| = ||ψ|| and thus ||φ||∩ C = ∅ if and only if ||ψ||∩ C = ∅). Now let us go through the additional properties: (3) closure under logical consequence is satisfied (if φ → ψ then ||φ|| ⊆ ||ψ|| and thus ||φ|| ∩ C = ∅ implies ||ψ|| ∩ C = ∅), (4) disjunctive completeness is satisfied (since ||φ ∨ ψ|| = ||φ|| ∪ ||ψ||, if ||φ ∨ ψ|| ∩ C = ∅ then either ||φ|| ∩ C = ∅ or ||ψ|| ∩ C = ∅); (5) negation completeness is also satisfied (if ||φ|| ∩ C = ∅ then C ⊆ ||¬φ|| and thus, since C = ∅, C ∩ ||¬φ|| = C = ∅); on the other hand, (6) expansive credibility does not hold: it is possible that, for a credible sentence φ, φ / ∈ K and yet ||¬φ|| ∩ C = ∅, so that ¬φ is not credible. Thus natural properties to impose on C -besides the two given in Definition 1-are: closure under logical consequence (or single sentence closure: if φ ∈ C then [φ] P L ⊆ C ), disjunctive completeness (if φ ∨ ψ ∈ C then either φ ∈ C or ψ ∈ C ) and negation completeness (either φ ∈ C or ¬φ ∈ C ). Turning now to the set A of allowable sentences, we postulated (Definition 1) that, like C , also A contains only consistent sentences and is closed under logical equivalence. These two properties pass the semantic test suggested above. On the other hand, closure under logical consequence does not apply to A . To see this, let φ ∈ A and note that φ → ( p ∨ ¬p); yet, since ( p ∨ ¬p) is a tautology, ( p ∨ ¬p) is credible, rather than allowable. A property that does satisfy the semantic test is the following: if φ ∈ A then ¬φ ∈ C (indeed, if φ is allowable then ||φ||∩ C = ∅ and thus, as shown above, C ∩||¬φ|| = C = ∅). Another property that passes the semantic test is: if φ ∈ A and ψ ∈ A then φ ∨ ψ ∈ A (indeed, if φ and ψ are allowable then ||φ|| ∩ C = ||ψ|| ∩ C = ∅, ||φ|| ∩ A = ∅ and ||ψ|| ∩ A = ∅, so that (||φ|| ∪ ||ψ||) ∩ C = ∅ and (||φ|| ∪ ||ψ||) ∩ A = ∅).
The properties discussed so far pertain to the sets C and A . One can also relate properties of the belief operator to properties of the sets C and A : Garapa (2020, Proposition 3.12, p. 9) provides a thorough analysis of 14 such possible links. Finally, we turn to the set R of rejected sentences. For this set natural properties are: (1) closure under logical equivalence (if φ ↔ ψ then ||φ|| = ||ψ|| and thus ||φ|| ⊆ R if and only if ||ψ|| ⊆ R ), (2) if φ ∈ R then ¬φ ∈ C (indeed, if φ is rejected then ||φ|| ⊆ R and thus, since R ∩ C = ∅, ||φ|| ∩ C = ∅ and therefore, as shown above, ||¬φ|| ∩ C = ∅), (3) if φ ∈ R and ψ ∈ R then φ ∨ ψ ∈ R . 4. Categorical matching. We identified the belief state of an agent with a triple -K ⊂ is a consistent and deductively closed set of sentences, representing the initial beliefs, -{ C , A , R } is a partition of the set of sentences into credible, allowable and rejected sentences, representing the disposition of the agent whether to accept any given sentence as information, -• : → 2 is a belief revision function transforming the initial belief set K into a new belief set K • φ in response to a sentence φ (thought of as information).
However, a belief-revision theory should satisfy what Gärdenfors and Rott (1995, p. 37) called the principle of categorical matching, that is, the principle that "the representation of a belief state after a belief change has taken place should be of the same format as the representation of the belief state before the change" [see also Rott (1999)]. This is particularly important in the context of iterated belief revision [see Darwiche and Pearl (1997)]. In our context, if S is the set of belief states of the form K , { C , A , R } , • then a belief-revision function ought to be defined as a function f : × S → S. The function • : → 2 gives only one of the three components of f (φ, K , { C , A , R } , • ), namely the new belief set K • φ, but-in general-also the function • and the credibility partition { C , A , R } could change as a result of receiving informational input φ. Since this paper is not concerned with iterated belief revision, that is, its focus is on "one-shot" revision, our analysis (which follows the traditional approach) is without loss of generality. We leave the extension of the analysis to the context of iterated belief revision as a topic for future research. 5. The recovery postulate. We proposed a notion of belief change, called "filtered belief revision", which was shown to be equivalent to a combination of classical AGM belief revision and contraction, the latter defined via the Harper Identity. Contraction by ¬φ occurs when the agent is informed of a sentence φ which she deems not credible but allowable. The agent's reaction is then to suspend judgment about φ, in particular to abandon her belief in ¬φ if the latter was in her initial belief set. For such a belief change we postulated the recovery property, according to which the agent would return to her initial belief set if she were to then accept ¬φ. The recovery postulate appears to be a natural way of capturing a "minimal" way of suspending belief in ¬φ, but has been subject to scrutiny (see Makinson, 1987;Fuhrmann, 1991;Hansson, 1991Hansson, , 1996Hansson, , 1999aLevi 1991;Lindström & Rabinowicz, 1991;Niederée, 1991). In Makinson's terminology Makinson (1987), contraction operations that do not satisfy the recovery postulates are called withdrawals. Alternative types of withdrawal operators have been studied in the literature: contraction without recovery (Fermé (1998)), semi-contraction (Fermé and Rodriguez (1998)), severe withdrawal (Rott and Pagnucco (1999)), systematic withdrawal (Meyer et al. (2002)), mild contraction (Levi (2004)). A potential topic for future research is whether one can relax the recovery postulate and still obtain a simple characterization, in terms of AGM belief revision, of the appropriately modified notion of filtered belief revision.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

A Proof of Proposition 1 (A) implies (B).
Given a belief set K , a credibility partition { C , A , R } and a filtered belief revision function • : → 2 , define the function * : → 2 as follows: First we show that the function * so defined is a basic AGM belief revision function. Fix an arbitrary φ ∈ .
2. Suppose now that φ is consistent, so that, by (13), if ¬φ / ∈ K then, by Definition 2, • The 'only if' part of (AGM5) is satisfied because -if ¬φ ∈ K then, by Definition 2, K • φ is consistent and thus not equal to , and thus, since K is consistent and does not imply ¬φ, K • φ = .
Since, by (F1) of Definition 2, K • φ = K when φ ∈ R , we only need to show that (a) Fix an arbitrary φ ∈ C .
-Consider next the case where E ∈ E A , so that p ∈ A . By (19), since q ∈ K • p, q ∈ K • p = K ∩(K * p). From q ∈ K it follows that f ( ) ⊆ ||q|| = f (E). It remains to prove that the converse is also true, namely that f (E) ⊆ f ( ).
• AGM1 is satisfied by construction.
In cases 4 and 5 the property holds trivially since ¬φ ∈ K . • AGM5 is satisfied by construction.
Next define the following full-domain belief revision function • : → 2 : where K * φ is given by (33). Then, by definition of basic-AGM consistent PBRS (Definition 5), it only remains to prove that • is an extension of • (given by (32)), that is, that, for every φ ∈ , χ ∈ K • φ if and only if χ ∈ K • φ. Fix an arbitrary φ ∈ , that is, a formula φ such that ||φ|| ∈ E.

C Necessary and sufficient conditions for rationalizability of a BPBRS
In this appendix we provide necessary and sufficient conditions for a plausibility-based BPBRS to be rationalizable. 25

Proposition 5 A plausibility-based BPBRS
The proof of Proposition 5 makes repeated use of Proposition 6 below due to Hansson (1968, Theorem 7, p. 455). We begin with a definition.  25 Conditions A and B in Proposition 5 below are a generalization of what is known in the revealed preference literature as the Strong Axiom of Revealed Preference (SARP), which is a necessary, but not sufficient, condition for rationalizability by a total pre-order (see Hansson, 1968). Let , E, f be a simple choice structure (Definition 12 below) and let E 1 , . . . , E n , E n+1 be a sequence in E with E n+1 = E 1 . Then SARP is the following condition: if E k ∩ f (E k+1 ) = ∅, ∀k ∈ {1, . . . , n}, then there exists a j ∈ {1, . . . , n} such that For the case where n = 2, SARP says the following (interpreting f (E) as the set of alternatives chosen from E): if from a set of alternatives E 1 , x is chosen when y and z are available, and if from some other set of alternatives E 2 , y is chosen while z is available, then there can be no set of alternatives containing x and z where z is chosen and x is not (SARP extends this condition to chains of any finite length).

Proof of Proposition 5
First we show that if rationalizes the plausibility-based BPBRS { C , A , R } , {E C , E A , E R } , f then (A) of Proposition 5 is satisfied. Construct the following simple choice structure W , F, h : By Definition 9, C = ∅. By 2 of Definition 9, if E ∈ E C then E ∩ C ∈ E C and by 2 of Definition 8, ∅ / ∈ F and C ∈ F. By 3(c) of Definition 8, h(F) = ∅, ∀F ∈ F. By hypothesis, since the BPBRS is rationalized by the total pre-order ⊆ × , if Hence we have indeed defined a simple choice structure. Let C be the restriction of to C (that is, C = ∩ ( C × C )). By construction, since rationalizes the given PBRS, we have that Now fix an arbitrary sequence .., F n , F n+1 be the corresponding sequence in F, that is, for every k = 1, ..., n, F k = E k ∩ C (thus F n+1 = F 1 ). Then, for every k = 1, ..., n, F k ∩ h(F k+1 ) = ∅. Thus, by (36) and Proposition 6, , ∀k = 1, ..., n; that is, (A) of Proposition 5 holds. Next we show that if rationalizes the plausibility-based BPBRS then Part (B) of Proposition 5 is satisfied. If A = ∅ there is nothing to prove. Assume, therefore, that A = ∅ (so that, by 3 of Definition 9, A ∈ E). Construct the following simple choice structure W , G, g : By 3(d) of Definition 8, for every G ∈ G, g(G) = ∅; furthermore, by hypothesis, since the BPBRS is rationalized by the total pre-order Hence we have indeed defined a simple choice structure. Let A be the restriction of to A (that is, A = ∩ ( A × A )). By construction, since rationalizes the given PBRS, we have that Now fix an arbitrary sequence E 1 , ..., E n , E n+1 in E A with E n+1 = E 1 (thus, by 3(a) of Definition 9 E k ∩ C = ∅, ∀k = 1, ..., n) such that (E k ∩ A )∩ f (E k+1 ∩ A ) = ∅, ∀k = 1, ..., n. Let G 1 , ..., G n , G n+1 be the corresponding sequence in G, that is, for every k = 1, ..., n, G k = E k ∩ A (thus G n+1 = G 1 ). Then, for every k = 1, ..., n, G k ∩ g(G k+1 ) = ∅. Thus, by (38) and Proposition 6, .., n, and let F 1 , ..., F n , F n+1 be the corresponding sequence in F (that is, F k = E k ∩ C , for all k = 1, ..., n). By the Property Hence, since the sequence was chosen arbitrarily, it follows from Proposition 6 that there exists a total pre-order C on W × W = C × C such that Two cases are possible. Case 1: A = ∅. In this case, define ⊆ × as follows: = C ∪ (ω, ω ) : ω ∈ C and ω ∈ R ∪ (ω, ω ) : ω, ω ∈ R . (40) Then is a plausibility order (Definition 6) and satisfies the properties given in (10). Fix an arbitrary E ∈ E. If E ∩ C = ∅ then, by 2(c) of Definition 9, f (E) = f (E ∩ C ) de f = h(E ∩ C ) and by (39) h(E ∩ C ) = best C (E ∩ C ). By (40), if ω ∈ E ∩ C and ω ∈ E \ C then ω ω so that best E = best (E ∩ C ) = best C (E ∩ C ); thus f (E) = best E. If E ∩ C = ∅ then E ⊆ R and, by 3(b) Definition 3, = h( C ) and by (39) h( C ) = best C ( C ); by (40), best = best C C , so that f ( ) = best . Case 2: A = ∅. In this case let W , G, g be the choice structure defined in (37). Fix an arbitrary sequence E 1 , ..., E n , E n+1 in E with E n+1 = E 1 such that E k ∩ C = ∅, ∀k = 1, ..., n, and (E k ∩ A )∩g (E k+1 ∩ A ) = ∅, ∀k = 1, ..., n and let G 1 , ..., G n , G n+1 be the corresponding sequence in G (that is, G k = E k ∩ A , for all k = 1, ..., n). By Property (B) of Proposition 5, (E k ∩ A ) ∩ g (E k+1 ∩ A ) = g (E k ∩ A ) ∩ (E k+1 ∩ A ) , ∀k = 1, ..., n, that is, G k ∩ g (G k+1 ) = g (G k ) ∩ G k+1 , ∀k = 1, ..., n. Hence, since the sequence was chosen arbitrarily, it follows from Proposition 6 that there exists a total pre-order A on W × W = A × A such that ∀G ∈ G, g(G) = best A G de f = {ω ∈ G : ω A ω , ∀ω ∈ G}.
Then is a plausibility order (Definition 6) and satisfies the properties given in (10). Fix an arbitrary E ∈ E. If E ∩ C = ∅ or E ⊆ R , then f (E) = best E by the argument developed in Case 1. If E ∩ C = ∅ and E ∩ A = ∅, then, by 3(d) of Definition 9, f (E) = f (E ∩ A ) de f = g(E ∩ A ) and by (41) g(E ∩ A ) = best A (E ∩ A ). By (42), if ω ∈ E ∩ A and ω ∈ R then ω ω so that best E = best (E ∩ A ) = best A (E ∩ A ); thus f (E) = best E.
Case 2: A = ∅. In this case extract from C the simple choice structure , G, g where G = {E ∩ A : E ∈ E A } ∪ { } and g is the restriction of f to G. By 3(d) of Definition 9, G de f = {φ ∈ : ||φ|| ∈ G} ⊆ . By construction, since g( ) = f ( ), {φ ∈ : g( ) ⊆ ||φ||} = K . Then, by (43), the supplemented AGM function * A is an extension of • G and thus the simple structure , G, g is AGM consistent in the sense of Definition 3 in Bonanno (2009) so that, by Proposition 8 in Bonanno (2009), there exists a total preorder A on such that, for every G ∈ G, Let A = A ∩ ( A × A ) and define ⊆ × as in (42). Then the argument to show that, ∀E ∈ E, f (E) = best E is a repetition of the argument used in Case 2 of the proof of Proposition 5.

(B) implies (A).
Let C (with a finite set of states ) be a plausibility-based BPBRS which is rationalized by a plausibility order on (Definition 6), in the sense that, for every E ∈ E, We want to show that C is supplemented AGM consistent (Definition 10). Let C , F, h be the simple choice structure described above (F = {E ∩ C : E ∈ E C } and h is the restriction of f to F). Then, by (47), C , F, h is rationalized by the total pre-order C de f = ∩ ( C × C ), so that, by Proposition 7 in Bonanno (2009), there exists a supplemented AGM function * C that extends the function • F defined above (K • F φ = {χ ∈ : h(||φ||) ⊆ ||χ ||}). Similarly, let , G, g be the other simple choice structure described above (G = {E ∩ A : E ∈ E A } ∪ { } and g is the restriction of f to G). Then, by (47), , G, g is rationalized by the total pre-order A de f = ∩ ( A × A ), so that, by Proposition 7 in Bonanno (2009), there exists a supplemented AGM function * A that extends the function • G defined above (that is, K • G φ = {χ ∈ : g(||φ||) ⊆ ||χ ||}). Now define • : → 2 by We need to show that • is an extension of • . This is a consequence of the following facts: