Believability Relations for Select-Direct Sentential Revision

A set of sentential revision operations can be generated in a select-direct way within a new framework for belief change named descriptor revision firstly introduced in Hansson [8]. In this paper, we adopt another constructive approach to these operations, based on a relation ⪯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\preceq}$$\end{document} on sentences named believability relation. Intuitively, φ⪯ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varphi \preceq \psi}$$\end{document} means that the subject is at least as prone to believe or accept φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varphi}$$\end{document} as to believe or accept ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\psi}$$\end{document}. We demonstrate that so called H-believability relations and basic believability relations, the second of which the is axiomatically characterized with a set of weak postulates, are faithful alternative models for two typical select-direct sentential revision operations. Then we investigate additional postulates on believability relations that correlate with properties of the generated revision operations. Finally, we show that traditional AGM revision operations can be reconstructed from a strengthened variant of the basic believability relation and there is a close connection between this relation and the standard epistemic entrenchment relation.


Introduction
In Hansson [8] it is argued that in the classical literatures (such as Alchourrón et al. [1] and Grove [6]) on logic of belief change which mainly focus the operations of contraction and revision, we can find a standard methodology which can be summarized as "select-and-intersect": Select the most plausible sets that satisfy the success condition (for example, removing a specified sentence from original beliefs in the case of contraction, or adding a specified sentence to original beliefs in the case of revision or expansion), and then take their intersection as outcome.Hansson [7,8] further argued that this method has at least three major disadvantages.Firstly, the property of being an optimal potential outcome is not generally preserved under intersection.This point can be illustrated by an example in Sandqvist [15].
Secondly, the fact that a success condition holds for all elements of a family of sets does not always imply that their intersection also satisfies this condition.For an instance, the success condition of "making up one's mind" about ϕ, i.e. adding ϕ or adding ¬ϕ to original beliefs, is not generally preserved to be satisfied after intersection.Thirdly, the adequacy of the options available for selection and intersection is contestable.In the traditional framework, the selection is made among remainders (Alchourrón et al. [1]) or possible worlds (Grove [6]).It is not difficult to show that neither of them are plausible belief sets (Alchourrón & Makinson [2], Hansson [7]).
In order to avoid these disadvantages, a new approach to belief change named "descriptor revision" was introduced in a series of papers by Hansson [7][8][9][10].In contrast to "select-and-intersect" among the implausible options, the methodology of this approach can be summarized as "select-direct" among the plausible options: It is assumed that there is a set of belief sets (not necessarily being remainders or possible worlds) as potential outcomes of belief change, and the mechanism of change is a direct choice among these potential outcomes.On the other hand, this is a very general framework since success conditions for various types of belief changes are described in a general way with the help of a metalinguistic belief operator B. For example, the success condition of contraction by ϕ is ¬Bϕ, that of revision by ϕ is Bϕ, that of making up one's mind about ϕ is Bϕ ∨ B¬ϕ.A descriptor is a set of such formulas with B that encode the relevant success condition and descriptor revision on belief set K is performed with a unified operator • which applies to all descriptors (Hansson [8]).
This new construction for belief change cannot only be used to investigate some interesting belief change patterns which cannot be represented in the "select-and-intersect" way, such as making up one's mind (Zhang & Hansson [16]), but also throw new light on the traditional operations.For example, sentential revision can be reconstructed in a select-direct way from descriptor revision as K ϕ = K • Bϕ.Another interesting consequence of this new proposal is that an alternative construction with a relation * of epistemic proximity defined on descriptors was proved being an alternative modelling for the descriptor revision (Hansson [9]).Let Ψ and Φ be any two descriptors.Informally, Ψ * Ξ means that "the subject is at least disposed to perform a change in the belief system resulting in assent to Ψ as one resulting in assent to Ξ" (Hansson [9]).Just as a sentential revision can be derived from a descriptor revision •, it seems a relation on sentences can also be derived from a proximity relation * .This kind of relation is named as believability relation (Hansson [9]) since the intended interpretation of ϕ ψ is as same as that of Bϕ * Bψ. 1 Intuitively, ϕ ψ means that the subject is at least as prone to believing or accepting ϕ as incorporating ψ into her beliefs.
As it was shown that descriptor revision can be reconstructed from the relations of epistemic proximity (Hansson [9]), the question naturally arises whether the believability relation derived from the relations of epistemic proximity can be used as an alternative modelling for the sentential revision derived from descriptor revision.The main purpose of the present contribution is to show that some revision operations generated in this way indeed can be reconstructed by believability relations satisfying some suitable conditions too.It should be noted that the idea of using relations on sentences reflecting degree of acceptance to construct revision operations also appeared in models in Cantwell [3] and Rott [4].However, in both of them the relations reflecting degree of acceptance were not proposed in their own right, but determined by or generalized from the epistemic entrenchment relation (Gärdenfors & Makinson [5]).So these models are essentially different from the approach investigated here in both respects of the way of representing the degree of acceptance and the way of constructing the revision operations.
The structure of this paper will be organized in this way: in the next section, we will introduce some formal preliminaries, in particular, we will differentiate two different kinds of select-direct sentential revision derived from descriptor revision, i.e. dependent revision and independent revision.In Section 3, we will specify the believability relations that can generate the dependent revision, which are derived from so called standard epistemic proximity relations.Section 4 will be devoted to the case of the independent revision, which is proved to be obtainable through a set of believability relations axiomatically characterized with four weak postulates.In Section 5, more potential properties on believability relations and their impact on the properties of the relevant revision operations will be studied.As it has been shown that traditional AGM revision operations (Alchourrón et al. [1]) can be reconstructed in the select-direct way (Hansson [10]), we will investigate in Section 6 whether they can also be reconstructed using believability relations as well as the possible connection between the believability relations and the epistemic entrenchment relations.Section 7 concludes. 1In this sense, the believability relation is the same as a restricted variant of epistemic proximity relation on descriptors of the most simple form (such as Bϕ, Bψ, etc.).The reason why we define it on sentences but not on descriptors is mainly for simplicity in expression and convenience in comparison with the epistemic entrenchment relation (Gärdenfors & Makinson [5]), which is also defined on sentences.

Preliminaries
The object language L is defined inductively by a set of propositional variables and the truth functional operations ¬, ∧, ∨ and → in the usual way.Sentences in L will be denoted by lower-case Greek letters in the second half of the alphabet and sets of such sentences by upper-case Roman letters.Cn is a consequence operation for L satisfying supraclassicality (if ϕ can be derived from A by classical truth-functional logic, then ϕ ∈ Cn(A)), compactness (if ϕ ∈ Cn(A), then there exists some finite B ⊆ A such that ϕ ∈ Cn(B)) and deduction property (ϕ ∈ Cn(A ∪ {ψ}) if and only if ψ → ϕ ∈ Cn(A)).X ϕ is an alternative notation for ϕ ∈ Cn(X).{ϕ} ψ will be simply written as ϕ ψ. ϕ ≡ ψ means ϕ ψ and ψ ϕ.
The beliefs of an agent are represented by a belief set, which is a set X ⊆ L such that X = Cn(X).K is fixed to denote the set of the original beliefs of the agent.We assume that K is consistent unless stated otherwise.
An atomic belief descriptor is a sentence Bϕ with ϕ ∈ L. The symbol B is not part of the object language L by which the agent's beliefs are expressed.A molecular belief descriptor is a truth-functional combination of atomic descriptors.A composite belief descriptor (in short: descriptor; denoted by upper-case Greek letters) is a set2 of molecular descriptors.Bϕ is satisfied by a belief set X if and only if ϕ ∈ X. Conditions of satisfaction for molecular descriptors are defined inductively, hence (letting α and β stand for molecular descriptors) X satisfies ¬α if and only if it does not satisfy α, it satisfies α ∨ β if and only if it satisfies either α or β, etc.It satisfies a composite descriptor Φ if and only if it satisfies all its elements.X Φ denotes that X satisfies Φ and Φ Ξ that all belief sets satisfying Φ also satisfy Ξ (Hansson [8]).
We use to denote the descriptor B ∧ ¬B .It is easy to see that there is no belief set satisfying .
Descriptor revision on belief set K is performed with a unified operator • such that K • Φ is an operation with Φ as its success condition.Several constructions for concrete descriptor revision operations can be found in Hansson [8], of which the relational model defined as follows is of special importance.
Definition 2.1 (Hansson [8], modified 3 ).Let S be any set of descriptors, (X, ) S is a relational select-direct model (in short: relational model) with respect to K if and only if it satisfies:4 (X1) X is a set of belief sets.
( 1) K X for every X ∈ X.
( 2) For any Φ ∈ S, if {X ∈ X | X Φ} (we denote it as X Φ ) is not empty, then it has a unique -minimal element denoted by X Φ < .A descriptor revision • on K is based on (or determined by) some relational model (X, ) S with respect to K if and only if for any Φ, to The intuitive meaning of X is an outcome set which represents all the potential outcomes under several belief change patterns.(with the strict part <) is an ordering used to select the best among candidates satisfying a certain success condition.Condition ( 2) guarantees that this kind of directselection is achievable for any success condition which is satisfiable in X.To some extent, descriptor revision is in a more abstract level than the AGM revision.Consider the Definition 2.1, for any descriptor revision •, it assumes that there exists an outcome set which contains all the potential outcomes of the operation •, but it says little about what these outcomes should be like.In contrast, in the AGM framework, the belief change is supposed to perform in a way of preserving consistence and reserving information as much as possible.Therefore, the intersection in the AGM framework as a useful tool to construct the intended outcome becomes dispensable in the context of descriptor revision.
Sentential revision operation on K can be reconstructed from descriptor revision in the following way: 2.An operator on K is a independent select-direct sentential revision (in short: independent revision) if and only if it is determined by a sentential descriptor revision.
It is easy to see that the dependent revision operations are special cases of independent ones.Compared with the independent, properties of dependent revision are more subtle since they are sensitive to those of other change patterns.How to axiomatically characterize the dependent revision is still an open problem.But the axiomatization of the independent revision can be obtained as an immediate corollary of the Theorem 1 in Zhang & Hansson [16].In the next two sections, we will show that both of these operations can be reconstructed by the believability relations satisfying certain suitable conditions.

Believability Relations for Dependent Revision
As we have mentioned in Section 1, a relation * (with the strict part ≺ * and the symmetric part * ) on descriptors called epistemic proximity relation was introduced in Hansson [9].Ψ * Ξ intuitively means that Ψ is at least as epistemically proximal as Ξ.It is assumed that a standard epistemic proximity relation should satisfy the following five postulates: Proposition 3.1 (Hansson [9]).Let • be any descriptor revision operation on K.Then, the following two propositions are equivalent.
1. • is a global descriptor revision operation on K.

2.
• is derived from some standard epistemic proximity relation with respect to K8 in the following way: Hansson further proposed that a set of believability relations can be derived from the standard epistemic proximity relation in the following way: * to − ϕ ψ if and only if Bϕ * Bψ.
However, some information contained in the original epistemic proximity relation will be lost after this kind of restriction.Consider two relations of epistemic proximity through the way of * to − .As a result, the dependent revision derived from the global descriptor revision is not able to be reconstructed by these believability (see the Observation 5 in Hansson [9]).However, as we will show in what follows, this limitation can be resolved by a simple modification on the derivation method * to − .Consider the believability relations derived from the standard epistemic proximity relation in this way: * to ϕ ψ if and only if Bψ≺ * and Bϕ * Bψ.
Although * to is a little different from * to − proposed in Hansson [9], still let us call relations obtained through this way H-believability relations.It is easy to see that in this way if * ϕ then ϕ will not be in the domain of the derived .This modification is crucial for the main result of this section as follows.
Note that Ref( ) denote the domain of .
Theorem 3.2.Let be any sentential revision operation on K.Then, the following two propositions are equivalent.
1. is a dependent revision operation on K.
2. is derived from some H-believability relation in the following way: Theorem 3.2 shows that H-believability relations are faithful alternative models for dependent revision operations, though the axiomatization of this kind of believability relations has not been settled.In the next section, we will study the believability relations for constructing independent revision operations.A positive result is that an axiomatic characterization of these relations can be obtained.

Believability Relations for Independent Revision
In this section, we will prove that believability relations which exactly generate all the independent revision operations on K through the way of to can be axiomatically characterized by following postulates: weak transitivity: Let ϕ ψ and ψ ϕ, then (i) ϕ ξ if and only if ψ ξ, and (ii) ξ ϕ if and only if ξ ψ.It is arguable that these postulates represent a minimum set of conditions a believability relation should satisfy.Weak transitivity is just a very weak version of transitivity which is assumed for almost all orderings.The rationale for weak coupling is that if the agent will consequently add ψ and ξ to her beliefs in case of accepting ϕ, then she also add ψ ∧ ξ to her beliefs in this case.This is reasonable if we assume that the beliefs of the agent are represented by a belief set.The justification of relative counter-dominance is that if ϕ logically entails ψ, and K must be revised to incorporate either ϕ or ψ, then it will be a smaller change to accept ϕ rather than to accept ψ, because then ψ must be added too, if we assume that the beliefs of the agent is closed under consequence operation.Relative minimality is justifiable since it needs to do nothing to add ϕ to K if it is already in K.So we call relations characterized by these postulates basic believability relations with respect to K. It is easy to see that the H-believability relations satisfy these four conditions.
In what follows, we prove the equivalence between the sentential relational models and the basic believability relations in terms of their expressive power for constructing revision operations.The work can be divided into the following three lemmas.
Note that we use [ϕ] to denote the set {ψ | ϕ ϕ ∧ ψ} for simplicity Lemma 4.1.Let (X, ) be any sentential relational model and a relation on L retrieved from (X, ) in the following way: Then, is a basic believability relation.
Moreover, (X, ) with X = Ref( ) is a sentential relational model for revision and can be retrieved from (X, ) through to .
We say that a sentential revision is based on (or determined by) some sentential (global) relational model if it is based on the sentential (global) descriptor revision determined by this model, and is based on (or determined by) some believability relation if it is generated from this relation in the way of to .Now we can state the third lemma as follows.
Lemma 4.3.Let (X, ) be any sentential relational model and a believability relation retrieved from (X, ) through to .Then, a revision operation is based on (X, ) if and only if it is based on .Lemmas 4.1 and 4.2 jointly show that all basic believability relations can be derived from the sentential relational models.Moreover, Lemma 4.3 says that the sentential revision based on certain sentential relational model is the same as the revision determined by the believability relation that the model gives rise to.Hence, the set of the sentential revision operations generated from basic believability relation coincide with the independent revision operations.Moreover, as we have mentioned in the last paragraph of the Section 2, the independent revision can be axiomatically characterized with several postulates on the operators.We present these results together in the following representation theorem.
Theorem 4.4.Let be any operation on K.Then, the following three propositions are equivalent: 1. is an independent revision.

is generated from some basic believability relation through
to .
3. satisfies the following conditions: Although the postulates in Theorem 4.4 are so weak that they actually characterize a broad set of operations, the maxi-choice revision proposed in Alchourrón & Makinson [2] is not covered by them since postulate "reciprocity" cannot be derived from the first six AGM postulates (Alchourrón & Makinson [2]).However, there exist other constructions for descriptor revision (and hence for the derived select-direct sentential revision), in one of which the direct choice among the outcome set is specified by a selecting function instead of the ordering in Definition 1.The resulting revision operation from this setting is even weaker and hence maxi-choice revision is a special case of it.(For more details on the construction for descriptor revision using selecting function, see Hansson [8], p. 958) Besides, Lemma 4.2 tells us that to is an injection from basic believability relations to the sentential relational model.But it is easy to find an example in which two different sentential relational models generate the same believability relation through to .So it is not a bijection.However, with some restriction, one-to-one correspondences between these two kinds of constructions are obtainable.We will present some this kind of results in the next section.

More Properties on Believability Relations
The properties of the basic believability relations introduced and studied in the previous section does not necessarily cover all plausible properties of the believability relations.In this section, we will impose some additional properties on the basic believability relations and investigate their consequences for properties of the equivalent sentential relational models and derived revision operations.

Transitivity
Given that ϕ ψ is explained as meaning that it is not more difficult for the subject to become to believe ϕ than ψ, the postulates characterizing the basic believability relations appear to be a bit too weak.It seems that even in a very general setting a suitable believability relation at least needs to satisfy: transitivity: If ϕ ψ and ψ ξ, then ϕ ξ.
We call the basic believability relations satisfying this additional condition transitive believability relations.
The main goal of this subsection is just to argue for the plausibility and generality of the transitive believability relation through a formal result demonstrating that there is no distinction between the transitive and general basic believability relations from the angle of expressive power.
As a preparation, we prove that to (or to ) is a bijection between the transitive believability relations and a special subset of sentential relational model which is defined as follows.
Definition 5.1.(X, ) is a canonical sentential model with respect to K if and only if it is a sentential relational model additionally satisfying: ( 3) is reflexive, i.e.X X for all X ∈ X, and transitive.
It is easy to see that there exist sentential relational models which are not canonical sentential models.However, the following lemma shows that these two kinds of models are equivalent in expressive power.That is the reason why we call models defined in this way canonical.Lemma 5.2.Let (X, ) be any sentential relational model.Then, a canonical sentential model (Y, ) can be constructed as follows: (with the strict part < ) is the transitive closure of ∩(X × X).
Moreover, A revision operator is determined by (X, ) if and only if it is based on (Y, ).Now we demonstrate that a bijection between the transitive believability relations and the canonical relational models can be obtained by to and to .
Lemma From these two lemmas and the Lemma 4.3 in previous section, the result we claimed previously follows immediately.

Theorem 5.4. A revision operator is based on some basic believability relation if and only if it is determined by some transitive believability relation.
Hence, we can focus on the transitive believability relation or the subsets of them in the following part of this paper without loss of generality.And we will see that correspondence between the transitive believability relations and the canonical sentential models offers us a useful tool to investigate properties of the revision operations based on the believability relations.

Exhaustiveness, Maximality, Coupling and Completeness
Another natural strengthening of basic believability relations is to require their domain to contain all sentences from L, i.e. every should satisfy: Given that is a transitive believability relation, it is obvious that satisfies exhaustiveness if and only if it satisfies: counter-dominance: If ϕ ψ, then ψ ϕ. minimality: ϕ ∈ K if and only if ϕ ψ for all ψ.
Moreover, that a believability relation satisfies exhaustiveness means that every ϕ from L is possible to be accepted by the agent.It is plausible to suppose that in this situation it is strictly more difficult for a rational agent to accept or believe ⊥ than to believe any non-falsum.In other words, should satisfy: Weak coupling has been the only untouched postulate of those the characterizing basic believability relations as yet.All the same, there is a natural strengthening of it as follows: coupling: If ϕ ψ and ϕ ξ, then ϕ ψ ∧ ξ.
We call believability relations characterized by transitivity, coupling, counter-dominance, maximality and minimality strengthened believability relations.Let K = Cn({ }) and let the relation on L be defined as ϕ ψ if and only if ψ ϕ.It is easy to see that is a strengthened believability relation with respect to K but does not satisfy the condition: We call strengthened believability relation additionally satisfying completeness quasi-linear believability relation, since it is easy to see that a linear order can be obtained on equivalence classes generated by the symmetric part of a quasi-linear believability relation.Moreover, the corresponding sentential relational models obtained by to from these two types of believability relations are called strengthened sentential models and linear strengthened sentential models respectively.The following theorem explains why they are named in this way.
in addition satisfies coupling if and only if (X, ) additionally satisfies

in addition satisfies completeness if and only if (X, ) additionally satisfies
( 5): is complete, i.e.X Y or Y X for all X, Y ∈ X.
Theorem 5.5 summarises the impact of strengthening the basic believability relations on the properties of the correspondent relational model.One the other hand, the correspondent impact on the properties of the generated sentential revision operations is demonstrated in the following two representation theorems.
Theorem 5.6.Let be any sentential revision on K.Then, is determined by a transitive believability relation satisfying exhaustiveness and maximality if and only if it satisfies closure, confirmation, reciprocity and the following two postulates: Theorem 5.7.Let be any sentential revision operation on K, then the following propositions are equivalent: 1. can be constructed from some strengthened believability relation through to .
2. can be constructed from some quasi-linear believability relation through to .

(strong reciprocity)
These two representation theorems show that sentential revision operations generated from the strengthened versions of believability relations can also be axiomatically characterized by certain plausible postulates, of which the strong reciprocity is closely relative to a non-monotonic reasoning rule named as "loop" introduced in Klaus et al. [12].For more discussion on this, see Makinson & Gärdenfors [13].
Moreover, although the quasi-linear believability relations are a proper subset of strengthened believability relations, Theorem 5.7 shows that they have the same expressive power.It was proved in Hansson [8] that if (X, ) is a global relational model, then is linear.So as a consequence of Theorem 5.7, if a revision operation is a dependent revision based on some global relational model satisfying ( 2), then it is also an independent revision derived from some sentential relational model (X, ) with is linear.But it is still unknown whether the opposite implication also holds.If so, with the results obtained in this paper, the axiomatic characterizations of dependent revision operations and H-believability relations can be obtained immediately.

Believability Relations for AGM Revision
The classic AGM revision operation was introduced in Alchourrón, et al. [1] as follows: Definition 6.1 (Alchourrón et al. [1]).For any belief set K, an operation on K is called AGM revision operation if and only if it satisfies the following eight AGM postulates: Note that we assume that K is consistent.It is easy to see that confirmation can be derived from inclusion and vacuity under this assumption.And it has been proved in Makinson & Gärdenfors [13] that strong reciprocity holds for all operations which satisfy those eight AGM postulates.So, by Theorem 5.7, all AGM revision operations on K can be generated from strengthened believability relation with respect to K.
A closely related result was given in Hansson [10] from the point of view of model construction.In that paper a set of conditions on relational models was specified and it was shown that a revision operation is based on some linear strengthened sentential model satisfying those conditions if and only if it satisfies the eight AGM postulates.
Can we similarly find the conditions characterizing the believability relations that exactly give rise to all the AGM revision operations?There is a positive answer.
And their "iteration" versions: ( ( Given satisfies transitivity and counter-dominance, it is easy to see that the ( 1) and ( 2) imply the ( 1) and ( 2).The following theorem shows that if we further strengthen the believability relations with ( 1) and ( 2), we can construct exactly the AGM belief revision operations from these strengthened relations.Theorem 6.2. 1.If a revision operation is based on some strengthened believability relation satisfying ( 1) and ( 2), then it satisfies the first six AGM postulates.

A revision operation is determined by some strengthened believability relation satisfying ( 1) and ( 2) if and only if it satisfies all eight AGM postulates.
The fact that AGM revision can be reconstructed in the select-direct way to some extent indicates that it is hard to draw a clear line between "select-and-intersect" and "select-direct" revision operations.It is possibly not suitable to regard the "select-direct" approach as an opposite to the "select-and-intersect" one, instead they are more like two perspectives at different levels.

Epistemic Entrenchment Relation
In what follows, we discuss the connection between the strengthened believability relation and the so called standard epistemic entrenchment relation which was firstly introduced in Gärdenfors & Makinson [5] in the following way: The intuitive meaning of maximal outcome believability relations is that the possible outcomes (including K) of revision based on these relations with inputs of consistent formulas are all maximal consistent sets.In other words, in these cases, believing or accepting ¬ϕ is equivalent to giving up ϕ.So a bridge can be built from maximal outcome believability relations to standard entrenchment relations in the following way: to ϕ ψ if and only if ¬ϕ ¬ψ.
In fact, the following theorem shows that from the maximal outcome believability relations we can exactly derive through to all the standard epistemic entrenchment relations satisfying the following property: Theorem 6.5.Let K be any maximal consistent belief set.

Let be a maximal outcome believability relation with respect to K and obtained by
to from , then is a standard epistemic entrenchment relation with respect to K satisfying ( ).

Let be a standard epistemic entrenchment relation with respect to K satisfying ( ) and obtained from by means of the following definition:
to ϕ ψ if and only if ¬ϕ ¬ψ.

Then, is a maximal outcome believability relation with respect to K and can be reconstructed from by
to .
It is interesting to make a further comparison between the strengthened believability relations and the standard epistemic entrenchment relations.A direct translation between these two orderings is expected as both of them can exactly generate the AGM revision. 10Moreover, weakened version of believability relation (such as the basic believability relations) and its connections with the basic entrenchment proposed in Rott [14] are also worth being studied.We left all these as future work.

Conclusion
The main purpose of this paper is to show that select-direct revision operations generated by descriptor revisions can be reconstructed from a type of relation on sentences called believability relations.More specifically, so called H-believability relations and basic believability relations were proved to be faithful alternative models for the dependent select-direct revision and the independent select-direct revision respectively.Particularly, an axiomatic characterization of the basic believability relation was obtained.In Section 5, we investigated more potential properties of the believability relations except those characterizing the basic believability relations and obtained a strengthened variant of believability relations.This type of relations is of special interest since independent revision operations constructed from relational models of the most well-ordered form, i.e. in (X, ) is a linear ordering, can be exactly derived from them, and all traditional AGM revision operation can be exactly generated from certain subset of them.Moreover, we showed that there is a close connection between the standard epistemic entrenchment relation and the strengthened believability relation.All these facts together confirm the importance and plausibility of the believability relation as a construction for revision operations.
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Appendix: Proofs
Proof (for Theorem 3.2).Let * be any proximity relation and a Hbelievability relation derived from it through * to .Furthermore, let • be descriptor revision derived from * through * to • and the sentential revision from through to .By Proposition 3.2, • is a global descriptor revision.So according to the definition of dependent revision operation, we only need to prove that K ϕ = K • Bϕ for every ϕ.There are only two cases.
Moreover, it follows from Bϕ≺ * that B(ϕ ∧ ψ)≺ * since * satisfies transitivity and counter-dominance.So, due to * to , ϕ ϕ ∧ ψ if and only if In order to prove Lemmas 4.1 and 4.2, we first prove the following two observations on sentential relational models and relations, which are also useful in some other proofs.

Let
due to a result in the first part of this observation.It follows that ϕ ϕ∧ψ and ψ ψ∧ϕ.So ϕ∧ψ ξ due to weak transitivity.Moreover, we can conclude ψ ϕ ∧ ψ from ψ ψ ∧ ϕ due to relative counter-dominance and weak transitivity.Thus, ξ ψ due to weak transitivity.(ii): The proof is similar to that for (i).
Observation 7.2.Let (X, ) be any sentential relational model and ψ ϕ, then the following three propositions are equivalent: Proof (for Lemma 4.1).We only need to check that satisfies the four postulates for the basic believability relation.Weak transitivity: Let ψ ϕ and ϕ ψ, then X ψ < X ϕ < due to to .So X ϕ < = X ψ < due to observation 7.2.Hence, ξ ϕ if and only if X ξ < X ϕ < due to to , and if and only if

and if and only if ξ ψ due to
to .And by a similar argument, we can also prove that ϕ ξ if and only if ψ ξ.Thus, weak transitivity holds for .
It follows that X ψ < exists and X ψ < X ϕ < due to observation 7.2.Thus, ψ ϕ due to to .Relative minimality: ϕ ∈ K if and only if X ϕ < = K due to K = X < and observation 7.2, and if and only if X ϕ < X ψ < for all ψ such that X ψ < exists, and if and only if ϕ ∈ Ref( ) and ϕ ψ for all ψ ∈ Ref( ) due to to .
Proof (for Lemma 4.2).Part 1: The relative minimality of guarantees that X is not empty and it follows from observation 7.1 that the map to is well-defined.Hence, there exists (X, ) which can be constructed from by to .Part 2: Now we check that (X, ) meets the requirements for the sentential relational model.[ψ] due to to , and if and only if X ϕ < exists, X ψ < exists and X ϕ < X ψ < as we have proved [ϕ] = X ϕ < when [ϕ] ∈ X, and if and only if ϕ ψ since is constructed through to from (X, ).Thus, = .
Proof (for Lemma 4.3).We only need to prove (i) X ϕ < exists if and only if ϕ ∈ Ref( ), and (ii) Y by the definition of .Suppose for contradiction that there exists some Y = X ϕ < such that Y ∈ Y ϕ and Y < X ϕ < .It follows from the definition of that there exists X = X ϕ < such that X ∈ Y ϕ ⊆ X ϕ and Y X < X ϕ < .This shows that X ϕ < is not the unique -element in X ϕ .Thus, Y ϕ < exists and Y ϕ < = X ϕ < .( * 1): Suppose for contradiction that there exists some X ∈ Y such that there is no ϕ satisfying X = Y ϕ < .This means that for any ϕ ∈ X, there exists some Y ∈ Y ⊆ X such that ϕ ∈ Y and Y < X. Due to the definition of , it follows that for any ϕ ∈ X, there exists some Y ∈ X such that ϕ ∈ Y and Y < X.It follows from the definition of Y that X / ∈ Y. ( 3): It follows immediately from the definition of that it is transitive.Moreover, ( * 1) yields that is reflexive.Thus, is a pre-order on Y.Then, we prove that a revision operator is based on (X, ) if and only if it is based on (Y, ).Note that we have shown that when X ϕ < exists, Y ϕ < exists as well and [⊥] = Cn({⊥}) since ϕ ⊥ due to the relative counterdominance of .If there exists ψ such that Cn({⊥}) X ψ < , then ⊥ ψ and hence ξ ψ for all ξ due to the relative counter-dominance and transitivity of .Then ψ ≡ ⊥ by maximality.Thus, for any ϕ ≡ ⊥, X ϕ < < Cn({⊥}).From right to left: Suppose there exists some ϕ such that ψ ϕ for all ψ and ϕ ≡ ⊥.It follows that ⊥ ϕ and hence Cn({⊥}) = [⊥] [ϕ] = X ϕ < .By Observation 7.2, [⊥] = [ϕ] and hence ϕ ≡ ⊥ which contradicts the hypothesis.Moreover, Cn({⊥}) ∈ X yields ⊥ ∈ Ref( ).It follows that ϕ ⊥ for every ϕ due to relative counter-dominance of .Thus, satisfies exhaustiveness and maximality.2. From left to right: Let X ϕ < X ψ < and X ψ < X ϕ < , then ϕ ψ by to .So ϕ ϕ ∧ ψ and ψ ψ ∧ ϕ due to ϕ ϕ, ψ ψ and coupling of .Hence, X ϕ∧ψ < X ϕ < and X ϕ∧ψ < X ψ < by to .It follows from Observation 7.2 that X ϕ < = X ϕ∧ψ < = X ψ < .Thus, is anti-symmetric.From right to left: Let L. Zhang ϕ ψ and ϕ ξ, then X ϕ < = X ψ < = X ξ < due to to and the antisymmetry of .So ψ ∈ X ξ < and hence X ϕ < = X ψ < = X ψ∧ξ < due to observation 7.2.Thus, by to , ϕ ψ ∧ ξ. 3. ϕ ψ or ψ φ holds if and only if X ϕ < X ψ < or X ψ < X ϕ < due to to .Note that (X, ) is a canonical sentential model, i.e. for any X ∈ X, X = X ξ < for some ξ.Thus, satisfies completeness if and only if does so.
Proof (for Theorem 5.6).It has been proved in Zhang & Hansson [16] that revision operations based on sentential relational models satisfying ( 2) can be characterized by the first five postulates listed in the proposition.So by Lemma 5.2, revision operations based on canonical sentential models satisfying that condition can also be represented by those postulates.Moreover, Lemmas 4.3 and 5.3 and Theorem 5.5 jointly imply that a revision operation is based on this kind of canonical sentential model if and only if it is based on a transitive believability relation satisfying exhaustiveness and maximality.Thus, this theorem holds.
Proof (for Theorem 5.7).It has been proved in Zhang & Hansson [16] that a revision operation is based on some strengthened sentential model if and only if it satisfies closure, confirmation, success, consistency and strong reciprocity.Next we will prove that a revision operation is based on a strengthened sentential model if and only if it is based on a linear strengthened sentential model.Let (X, ) be any strengthened sentential model.Let Y = X \ Cn({⊥}), as we have shown in Theorem 5.5, ∩(Y × Y) is a partial order.Given the axiom of choice, there is a linear order * on X which extends the partial order .11Let = * ∪{(X, Cn(⊥)) | X ∈ X}.Now we prove that (X, ) is a linear strengthened sentential model.It is easy to see that (X, ) satisfies (X1), (X2), ( 2), ( 1) and ( 3) − ( 5).For ( 1) and ( 2), we only need to show that for every ϕ, if X ϕ = ∅, then X ϕ < = X ϕ < .Let X ϕ = ∅, then X ϕ < exists and X ϕ < X for every X ∈ X ϕ since (X, ) satisfies ( 2) and extends .And for any Y ∈ X ϕ , if Y X ϕ < , then Y = X ϕ < since is anti-symmetric.So X ϕ < exists and X ϕ < = X ϕ < .This result also means that a revision operation is based on (X, ) if and only if it is determined by (X, ).Thus, by Lemmas 4.3, 5.3 and Theorem 5.5, the propositions 1 and 2 in this theorem are equivalent.
Proof (for Theorem 6.2).Conditions ( 1), ( 2), ( 1) and ( 2) are just "translations" of inclusion, vacuity, superexpansion and subexpansion through following rules: For the from right to left part of the proposition 2, let be a AGM revision operation, then can be determined by some strengthened believability relation as we have mentioned.So and also satisfy those translation rules.Thus, satisfies ( 1) and ( 2).

Lemma 4 . 2 .
Let be any basic believability relation.Then, a binary relation on the power set of L can be constructed from in the following way:

Definition 2 .2 (Hansson [8]). Let
• be some descriptor revision on K.A sentential revision on K is based on (or determined by) • if and only ifK ϕ = K • Bϕ for all ϕ ∈ L.Let (X, ) S be any relational model, we name it global relational model if S contains all descriptors, and sentential relational model if S = {Bϕ | ϕ ∈ L}.Accordingly, • will be called global descriptor revision if it is based on some global relational model, and sentential descriptor revision if it is determined by some sentential relational model.With these two kinds of descriptor revision operations, at least two types of sentential revision operations can be constructed.6 , she will not accept ψ as her new belief in any case.The situation is different if she is in the belief status represented by * 2 .However, this difference cannot be expressed by the believability relations derived from * 1 and * * 1 and * 2 that respectively satisfies Bϕ * 1 * 1 Bψ and Bϕ * 2 Bψ * 2 .If the agent is in the belief status represented by * 1 5.3.1.Let be any transitive believability relation and let (X, ) [16]f (forTheorem 4.4).The equivalence between propositions 1 and 2 follows from Lemmas 4.1, 4.2 and 4.3.For the equivalence between propositions 1 and 3 (i.e.axiomatization of independent revision), see Zhang & Hansson[16].Let Y ϕ = ∅, then X ϕ = ∅ since Y ⊆ X and hence X ϕ By Lemma 4.2, (X, ) is a sentential relational model.Now we check that it is also a canonical model.( * 1): X ∈ X if and only if there exists some ϕ such that ϕ ∈ Ref( ) and Since satisfies exhaustiveness we have ⊥ ∈ Ref( ), and hence [⊥] ∈ X.Moreover, by the relative counter-dominance of , ⊥ ⊥ ∧ ϕ for every ϕ.Hence, [⊥] = Cn({⊥}) ∈ X.
Proof (for Lemma 5.3).1.Part 1:Proof (for Theorem 5.5). 1.From left to right: 1. ψ ∈ K ϕ if and only if ϕ ϕ ∧ ψ 2. ϕ ∈ K if and only if ϕ 3. ψ / ∈ K ϕ if and only if ϕ ≺ ϕ ∧ ψ 4. ϕ / ∈ K if and only if ≺ ϕ Let be a strengthened believability relation and determined by this relation.Then it is obvious that and satisfy the above four rules.It immediately follows that the proposition 1 and the from left to right part of the proposition 2 in this theorem hold.