Relations of epistemic proximity for belief change

Artificial Intelligence 217 (2014) 76–91

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Artificial Intelligence www.elsevier.com/locate/artint

Relations of epistemic proximity for belief change Sven Ove Hansson Royal Institute of Technology, Sweden

a r t i c l e

i n f o

Article history: Received 19 September 2013 Received in revised form 4 August 2014 Accepted 5 August 2014 Available online 8 August 2014 Keywords: Descriptor revision Epistemic proximity Epistemic entrenchment Believability

a b s t r a c t Relations of epistemic proximity are closely related to relations of epistemic entrenchment, but contrary to the latter they do not refer to sentences but to belief patterns that are expressed with a metalinguistic belief predicate B. Hence ¬B p  B¬q means that disbelief in p is more close at hand (obtainable with less far-reaching changes in belief) than belief in not-q. The logic of epistemic proximity is investigated, and it is used to construct a uniform operation of belief change that has the standard operations as special cases, specified with success conditions such as B p for revision and {¬B p 1 , . . . , ¬B pn } for multiple contraction. Standard entrenchment relations are obtained by defining p to be less entrenched than q if and only if ¬B p  ¬Bq. © 2014 Published by Elsevier B.V.

1. Introduction The literature on belief revision has a strong focus on two types of belief change, namely the removal of a sentence from the agent’s set of beliefs (contraction) and the addition of a sentence to that set (revision or expansion).1 Some other types of operations have also been introduced, such as the simultaneous removal of several sentences (multiple contraction) and the simultaneous addition of several sentences (multiple revision) (Hansson [5]). The various types of belief change operations are distinguished by their success conditions. In revision, the success condition is that a given sentence p should be believed, in contraction that a given sentence p should not be believed, in multiple contraction that none of a given set of sentences should be believed, etc. In Hansson [6] a unified approach to success conditions for belief change was introduced. It is based on a metalinguistic belief operator B that takes elements of the object language as arguments. For any sentence p in the object language, B p signifies that p is believed. A belief descriptor (henceforth: descriptor) is a set of molecular combinations of such B-sentences, for instance {¬B p , ¬B¬ p } means that neither p nor its negation is believed and {B p ∨ Bq} that either p or q is believed. With this notation, a wide variety of belief change operations can be subsumed under a unified operator ◦. Then K ◦ {B p } corresponds to revision by the sentence p, K ◦ {¬B p } to contraction of the sentence p, K ◦ {B p ∨ B¬ p } to making up one’s mind about p, etc. The standard approach to belief revision relies heavily on set-theoretical intersection. The approach can be described as a two-step select-and-intersect method. Operations of change take place in two steps. In the first step several belief sets (logically closed sets) are selected, all of which satisfy the success criterion for the operation. Then the intersection of those sets is formed, and is taken to be the outcome of the operation. (In the original AGM model for contraction, the sets selected in the first step are maximal subsets of the original belief set not containing the sentence to be removed. In the so-called sphere model of revision, the sets selected in the first step are possible worlds that contain the sentence to be added.)

1 In revision, elements of the original belief set are removed if that is necessary to retain consistency. In expansion, all elements of the original belief set are retained.

http://dx.doi.org/10.1016/j.artint.2014.08.001 0004-3702/© 2014 Published by Elsevier B.V.

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The select-and-intersect method has the obvious advantage of providing an elegant solution to ties. When we hesitate between two or more potential outcomes, then we use their intersection, i.e. the part that they have in common, as the output. However, this method also has two major disadvantages. First, as explained by Sandqvist [9], the property of being an optimal potential outcome is not necessarily preserved under intersection. Secondly, and in a sense more fundamentally, the fact that a success condition holds for all members of a collection of sets does not always imply that their intersection also satisfies this condition. True, the success conditions of ordinary contraction and revision ({¬B p } respectively {B p }) are intersection-resistant in this sense, but there are other success conditions of interest in artificial intelligence that can be lost in intersection. This applies for instance to the two success conditions {B p ∨ Bq} and {B p ∨ B¬ p }. The latter is of particular interest for artificial intelligence since it corresponds to the practically important belief change operation of making up one’s mind. Consider two consistent belief sets X 1 and X 2 such that p ∈ X 1 and ¬ p ∈ X 2 . Although both X 1 and X 2 satisfy the said success condition, X 1 ∩ X 2 does not. In order to account for belief change operations that are not intersection-resistant we will have to give up the selectand-intersect method. It can be replaced by a select-direct method in which a selection function operates directly on the set of potential outcomes, and selects exactly one of them to be the outcome. Then intersection-resistance becomes irrelevant, and at the same time the problem pointed out by Sandqvist is solved. But obviously, for this to work the selection function will have to be mono-selective, i.e. always deliver exactly one belief set as outcome when presented with a non-empty set of belief sets as input. The investigation of the select-direct method, as applied to operations with descriptors as inputs, was begun in Hansson [6], where the following operation was introduced: Definition 1. (See Hansson [6].) An operation ◦ on a belief set K is a relational descriptor revision if and only if there is a set X of belief sets with K ∈ X (its outcome set),2 and a relation  on X, such that (i) K  X for all X ∈ X, and (ii) for all descriptors Ψ : K ◦ Ψ is the unique -minimal element of X that satisfies Ψ , unless Ψ is unsatisfiable within X, in which case K ◦ Ψ = K . It was shown in Hansson [6] that the conditions of this definition can only be satisfied if  is a total ordering3 that satisfies the somewhat technical property descriptor-wellfoundedness, according to which it holds for all descriptors Ψ that if some element in the domain of  satisfies Ψ , then there is some -minimal element that does so. Furthermore, it was shown that the operations introduced in Definition 1 are axiomatically characterized by the following properties: (K  Ψ means that K satisfies Ψ and Ψ  Ξ that all belief sets satisfying Ψ also satisfy Ξ . The corresponding equivalence relation is written .) If K If K If If

Ψ  Ψ  then K ◦ Ψ = K ◦ Ψ  (extensionality) ◦ Ψ = Cn( K ◦ Ψ ) (closure) K  Ψ then K ◦ Ψ = K (confirmation) ◦ Ψ  Ψ or K ◦ Ψ = K (relative success) K ◦ Ξ  Ψ for some Ξ , then K ◦ Ψ  Ψ (regularity) K ◦ Ψ  Ξ then K ◦ Ψ = K ◦ (Ψ ∪ Ξ ) (cumulativity)

One of the most important results in the belief revision literature is the equivalence between on the one hand the original AGM framework (Alchourrón et al. [1]) that employs orderings on sets of sentences and on the other hand an approach employing relations on sentences, called relations of epistemic entrenchment (Gärdenfors [3], Gärdenfors and Makinson [4]). Intuitively, p < q (“p is less entrenched than q”) means that in a contraction, if we can choose between giving up p and giving up q, then we will opt for the former. If the entrenchment relation satisfies certain conditions (listed in Section 5), then an operator of contraction can be constructed from it that will satisfy the full set of AGM postulates for contraction. Conversely, every operator that satisfies these postulates can be constructed from an entrenchment relation.4 Obviously, in the context of descriptor revision we can talk of entrenchment as a relation not between two sentences p and q to be removed but between two descriptors {¬B p } and {¬Bq}. Instead of saying that p is less entrenched than q we can then say that the agent is more inclined to have a belief state satisfying {¬B p } than one that satisfies {¬Bq}. This reformulation gives rise to the question whether that relation can be meaningfully extended to all descriptors, not only those of the form {¬B p }. It is the purpose of the present contribution to show that this can indeed be done. The generalized relation on descriptors will be called a relation of epistemic proximity. After some formal preliminaries have been provided in Section 2, relations of epistemic proximity will be formally introduced in Section 3. For any two descriptors Ψ and Ξ we will say that Ψ is at least as epistemically proximal as Ξ

2 The name was chosen since as can be seen from Definition 4, for every X ∈ X there is some descriptor Ψ such that X = K ◦ Ψ , i.e. X is the outcome of revision by some descriptor. 3 A total ordering is a binary relation that satisfies transitivity, completeness ( X  Y or Y  X ) and anti-symmetry (if X  Y and Y  X , then X = Y ). 4 For further developments of epistemic entrenchment see Rott [7]. On safe contraction, another way to use relations on sentences in belief change, see Alchourrón and Makinson [2] and Rott and Hansson [8].

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(written: Ψ Ξ ) if and only if the subject is at least as disposed to perform a change in the belief system resulting in assent to Ψ as one resulting in assent to Ξ . It will be shown that operators based on such relations are essentially interchangeable with operators constructed as in Definition 1. Section 4 is devoted to the special case in which epistemic proximity is applied only to descriptors of the form {B p }. It can then be replaced by a relation on sentences to be called the relation of believability. In Section 5 we will apply epistemic proximity to descriptors of the form {¬B p }. It can then be replaced by a relation on sentences that has all the standard properties of epistemic entrenchment. Some general conclusions are drawn in Section 6. All formal proofs are deferred to Appendix A. 2. Formal preliminaries Sentences in the object language L will be denoted by lower-case letters (p, q . . . ) and sets of such sentences by upper-case letters ( A , B . . .). The usual truth-functional operations are denoted ¬, &, ∨, →, and ↔. is a tautology and ⊥ a contradiction. The consequence operator Cn for L expresses a supraclassical and compact logic satisfying the deduction property (q ∈ Cn( A ∪ { p }) if and only if p → q ∈ Cn( A )). X  p is an alternative notation for p ∈ Cn( X ). A belief set is a set X of sentences such that X = Cn( X ). We use K to denote a consistent belief set. An atomic belief descriptor is a sentence B p with p ∈ L. The symbol B is not part of the object language L in 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 Ψ, Ξ . . .) is a set of molecular descriptors. Descriptors may be infinite. ⍊ (descriptor falsum) denotes {B p , ¬B p } for an arbitrary p. Definition 2. The descriptor disjunction  is defined by the relationship

Ψ  Ξ = {α ∨ β | α ∈ Ψ, β ∈ Ξ } Lemma 1. K  Ψ  Ξ holds if and only if either K  Ψ or K  Ξ holds. A belief set K satisfies a composite descriptor Ψ if and only if it satisfies all its elements. A descriptor is satisfiable within a set of belief sets if and only if it is satisfied by at least one of its elements. The symbols  and  denote relations of satisfaction, as indicated in Section 1. Furthermore: Definition 3. A descriptor Ψ is maxispecified (maximally specified) if and only if there is exactly one belief set Y in ℘ (L) such that Y  Ψ . It is then a maxispecified descriptor for Y . There are many (equivalent) maxispecified descriptors for each belief set. For instance, if X = Cn({q}) then both {Bq} ∪ {¬B p | p ∈ / X } and {B p | p ∈ X } ∪ {¬B p | p ∈ / X } are maxispecified descriptors for X . For convenience, one of the maxispecified descriptors for a belief set X will be denoted as follows: Definition 4. (See Hansson [6].) For any belief set X , we denote by Π X the maxispecified descriptor for it such that:

Π X = {B p | p ∈ X } ∪ {¬B p | p ∈ / X} 3. Epistemic proximity The relation of epistemic proximity was introduced informally in Section 1. Its strict part is denoted  and its symmetric part . We will assume that it has the following five properties: Definition 5. A relation on descriptors is a relation of epistemic proximity if and only if it satisfies: Transitivity: If Ψ Ξ and Ξ Σ , then Ψ Σ Counter-dominance: If Ψ  Ξ then Ξ Ψ Coupling: If Ψ  Ξ then Ψ  Ψ ∪ Ξ Amplification: Either Ψ ∪ {B p } Ψ or Ψ ∪ {¬B p } Ψ Absurdity avoidance: Ψ  ⍊ for some Ψ Contrary to the standard definition of epistemic entrenchment (Gärdenfors [3]; Gärdenfors and Makinson [4]), Definition 5 does not mention the belief set towhich the relation is associated. However, it can be retrieved as the belief set specified by the maxispecified descriptor {Ψ | Ψ  B }.5   {B }, i.e. either  {Ψ | Ψ  B } is maxispecified,  note that for all p ∈ L, due to amplification either {B , B p }  {B } or {B , ¬B p } {Ψ | Ψ   B } or {B , ¬B p } ∈ {Ψ | Ψ  B }. It follows that for all p ∈ L, either B p ∈ {Ψ | Ψ  B } or ¬B p ∈ {Ψ | Ψ  B }. It remains to show that {Ψ | Ψ  B } specifies a belief set. If not, then there is some p with B  B p and B  ¬B p. Due to coupling and transitivity, B  ⍊. Due to counter-dominance, B Ψ for all Ψ , and transitivity yields ⍊ Ψ for all Ψ , contrary to absurdity avoidance. 5

To see that

{B , B p } ∈

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Fig. 1. The transformation functions for descriptor revision.

The following observation reports two important properties of that follow from the ones already given. Observation 1. Let be a relation on descriptors. (1) If satisfies transitivity and counter-dominance, then it satisfies: If Ψ  Ψ  and Ξ  Ξ  , then Ψ Ξ if and only if Ψ  Ξ  (intersubstitutivity). (2) If satisfies transitivity, counter-dominance, coupling, and amplification, then it satisfies: Ψ Ξ or Ξ Ψ (completeness). A useful class of maxispecified descriptors can be defined with the use of epistemic proximity: Definition 6. Let be a relation on descriptors and let Ψ be an element of its domain. For each p ∈ L, let β( p ) = B p if  such that Ψ  Ψ ∪ {B p }, and otherwise let β( p ) = ¬B p. If Ψ  ⍊ then the maximal amplification of Ψ is the descriptor Ψ  = {β( p ) | p ∈ L}. If Ψ  ⍊ then Ψ  = {B p | p ∈ L} ∪ {¬B p | p ∈ L}. Ψ

 be Observation 2. Let be a relation on descriptors that satisfies transitivity, counter-dominance, coupling, and amplification. Let Ψ defined as in Definition 6. Then:   Ψ , and (1) Ψ   Ψ ∪ Ξ if and only if Ψ   Ξ. (2) Ψ Furthermore: (3) For any maxispecified descriptors Ψ and Ψ  : Ψ  Ψ  if and only if Ψ  Ψ  . We are now going to show that relations of epistemic proximity with these properties are interchangeable with the belief set orderings introduced in Definition 1. There is a one-to-one relationship between the two types of orderings, and also a one-to-one relationship between each of them and operators (◦) of descriptor revision. These relationships can be expressed with a set of transformation functions (Fig. 1). The terminology is based on the abbreviations b for relations on belief sets and d for relations on descriptors. Thus t db takes us from a relation (of epistemic proximity) on descriptors to a relation on belief sets, and t ◦d from an operator of change to a relation on descriptors. Definition 7. Let  be a total and descriptor-wellfounded ordering on belief sets. Let X be its domain and let K be the -minimal element of X. Then: t b◦ () is the operation ◦ on K such that (i) if Ψ is satisfiable within X, then K ◦ Ψ is the unique -minimal element of X that satisfies Ψ , and (ii) for all other Ψ , K ◦ Ψ = K . t bd () is the relation on descriptors such that Ξ Ψ if and only if either (i) there is some X with X  Ξ such that X  Y for all Y with Y  Ψ , or (ii) Ψ is not satisfiable within the domain of . Let be a relation on descriptors with the strict part  and the symmetric part , and let K = { p | B p  B }. Then:

  B p } for some Φ with Φ  ⍊, such that X  Y iff there is some t db ( ) is the relation  on sets constructible as { p | Φ descriptor Ψ with Y  Ψ such that Ξ Ψ for all Ξ with X  Ξ . t d◦ ( ) is the descriptor operation ◦ on K such that q ∈ K ◦ Ψ if and only if either (i) Ψ ∪ {Bq}  Ψ  ⍊ or (ii) q ∈ K and Ψ  ⍊.

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Let ◦ be a descriptor operation. Then: t ◦b (◦) is the relation  on belief sets obtainable as K ◦ Ψ for some Ψ , such that X  Y if and only if there are Ξ and Ψ such that X = K ◦ Ξ , K ◦ Ξ  Ξ , Y = K ◦ Ψ , K ◦ Ψ  Ψ , and K ◦ Ξ = K ◦ (Ξ  Ψ ). t ◦d (◦) is the relation on descriptors such that Ξ Ψ if and only if either (i) K ◦ Ξ  Ξ , K ◦ Ψ  Ψ , and K ◦ Ξ = K ◦ (Ξ  Ψ ), or (ii) K ◦ Ψ  Ψ . The interchangeability referred to above is established in the following two theorems: Theorem 1. Let  be a total and descriptor-wellfounded relation on a set of belief sets. Then: (1) t bd () is a relation on descriptors that satisfies transitivity, counter-dominance, coupling, amplification, and absurdity avoidance, (2) t db (t bd ()) = , (3) t d◦ (t bd ()) = t b◦ (), (4) t ◦b (t b◦ ()) = , and (5) t ◦d (t b◦ ()) = t bd (). Theorem 2. Let be a relation on descriptors that satisfies transitivity, counter-dominance, coupling, amplification, and absurdity avoidance. Then: (1) t db ( ) is a complete, transitive, antisymmetric and descriptor-wellfounded relation on a set of belief sets, (2) t bd (t db ( )) = , (3) t b◦ (t db ( )) = t d◦ ( ), (4) t ◦d (t d◦ ( )) = , and (5) t ◦b (t d◦ ( )) = t db ( ). In combination, Theorems 1 and 2 show that the relationship between relational descriptor revision (following t b◦ ) and descriptor revision based on epistemic proximity (following t d◦ ) is essentially the same as that in AGM between transitively relational partial meet contraction and contraction based on epistemic entrenchment: these are different constructions that yield exactly the same classes of operations of change. 4. Believability relations for sentential (semi)revision By “belief revision” is usually meant an operation (denoted ∗) in which a single sentence is incorporated into the belief set, and consistency is upheld if possible (Alchourrón et al. [1]). To avoid ambiguity, the term “sentential revision” will be used here for this type of operation. By a “semi-revision” is meant a similar (sentential) relation that does not satisfy the success postulate, p ∈ K ∗ p for all p. The present framework can easily be restricted to these sentential operations. For that purpose we can replace ◦ by an operation with sentences as inputs, by a relation on sentences, and  by a relation whose domain only contains those belief sets that are the outcome of some sentential revision: Definition 8. Let K be a belief set, ◦ a descriptor revision on K , X its outcome set,  (with the strict part <) a total ordering on X with K as its minimal element, and a relation of epistemic proximity. Then: (1) The sentential operation ∗ such that K ∗ p = K ◦ {B p } for all p is the sentential semirevision on K that is derived from ◦. It is a sentential revision if and only if the domain of  contains Cn({⊥}).







(2) The total belief set ordering  such that Z  W if and only if Z  W , Z  {Y ∈ X | Y < Z } and W  {Y ∈ X | Y < W } is the additive restriction of . Furthermore,  is additively restricted if and only if it is its own additive restriction.



(3) The relation on sentences, such that p q iff B p Bq is the believability relation that is based on .

The additive restriction  of  is the restriction of  to those elements of its domain that can be obtained as the





outcome of revising K by some descriptor of the form B p. The strict part of  is denoted <. The strict part of is



denoted  and its symmetrical part . See Hansson [6] for an axiomatic characterization of the form of sentential revision introduced in part 1 of Definition 8. It is straight-forwardly verified that the derived semirevision ∗ satisfies the success postulate ( p ∈ K ∗ p ) if and only if it is a revision according to part 1 of the definition. Not surprisingly, each of these three restrictions involves a loss of information: Observation 3. (1) Let ◦ be a descriptor revision on the belief set K and ∗ its derived semirevision operator. It does not hold in general that ◦ is derivable from ∗.



(2) Let  be a total belief set ordering and  its additive restriction. It does not hold in general that  is derivable from .

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Fig. 2. Derivability diagram for sentential (semi)revision.

(3) Let be a proximity relation and the believability relation that is based on it. It does not hold in general that is derivable

from . The following three observations explore the relationships among the three restrictions introduced in Definition 8. Observation 4. Let K be a belief set,  a total and descriptor-wellfounded belief set ordering with K as its minimal element, and

◦ = t b◦ (). Furthermore, let  be the additive restriction of  and ∗ the sentential semirevision that is derived from ◦. Then:





(1) ∗ is derivable from  as follows: For all p, (i) if p is satisfiable within the domain of , then K ∗ p is the unique -minimal

p-containing element of the domain of , and (ii) otherwise, K ∗ p = K .

(2)  is not derivable from ∗. Observation 5. Let K be a belief set,  a total and descriptor-wellfounded belief set ordering with K as its minimal element, ◦ = t b◦ ()

and = t bd (). Furthermore, let ∗ be the sentential semirevision that is derived from ◦ and the believability relation that is based on . Then:

(1) ∗ is not derivable from .

(2) is not derivable from ∗.

Observation 6. Let  be a total and descriptor-wellfounded belief set ordering and let = t bd (). Furthermore, let  be the additive

restriction of  and the believability relation that is based on . Then:







(1) is derivable from  as follows: p q if and only if either (i) p and q are both satisfiable within the domain of , and the first

p-containing belief set does not come after the first q-containing one, or (ii) q is unsatisfiable within the domain of .



(2) It does not hold in general that  is derivable from . These results are summarized in Fig. 2. As can be seen there, the interdefinabilities illustrated in Fig. 1 are not retained

after the restriction. For a given belief set ordering , let  be its additive restriction, ∗ the sentential revision it gives rise





to and the believability relation that it gives rise to. Then we can derive ∗ and from , but in neither case is derivation

in the other direction possible. Furthermore, neither ∗ nor can be derived from the other. Part 1 of Observation 5 and part 2 of Observation 6 may both seem counterintuitive, but they can both easily be verified by comparing the two total belief set orderings  and  that are exhaustively characterized by Cn({ p })  Cn({ p&q})  Cn({⊥}) respectively Cn({ p })  Cn({ p&q}). They give rise to the same believability relation but differ in that K ∗¬q = Cn({⊥}) but K ∗ ¬q = K . 5. Revocation, contraction, and entrenchment Just as we restricted our attention to descriptors of the form B p we can restrict it to descriptors of the form ¬B p. Operations with a success condition of that form have been called operations of revocation (Hansson [6]). Similarly, operations with success conditions of the form {¬B p 1 , . . . , ¬B pn } can be called operations of multiple revocation. These are more

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general categories than operations of (multiple) contraction, since the latter are required to satisfy the inclusion postulate, K  p ⊆ K (for multiple contraction K  A ⊆ K ). The following definition introduces the corresponding restrictions on belief set orderings and orderings of epistemic proximity: Definition 9. Let K be a belief set, ◦ a descriptor revision on K , X its outcome set,  (with the strict part <) a total ordering of X with K as its minimal element, and a relation of epistemic proximity. Then: (1) The sentential operation such that K  p = K ◦ ¬B  p for all p is the revocation on K that is derived from ◦. It is a contraction if and only if both { K  p | p ∈ L} ⊆ K and { K  p | p ∈ L} = Cn(∅). −



−



(2) The total belief set ordering  such that Z  W if and only if Z  W , {Y ∈ X | Y < Z }  Z and {Y ∈ X | Y < W }  W is the subtractive restriction of . Furthermore,  is subtractively restricted if and only if it is its own subtractive −

−

restriction. The strict part of  is denoted <. (3) The relation  on sentences such that p  q if and only if ¬B p ¬Bq is the entrenchment relation derived from . ˙. The strict part of  is denoted  and its symmetrical part  Concerning part 1 of the definition it should be noted that the conditions for a revocation to be a contraction according to the above definition coincide with what is required to make it satisfy the postulates inclusion (K  p ⊆ K ) and success −

(if  p then K  p  p).6 Concerning part 2, the subtractive restriction  of  is the restriction of  to those elements of its domain that can be obtained as the outcome of revising K by some descriptor of the form ¬B p. In part 3 it is important to note that the entrenchment relation is defined not only for contraction but also for the more general operation of revocation. As the following observation shows, the entrenchment relation  obtained in this way turns out to satisfy all the standard properties of epistemic entrenchment, even if the operation is a revocation and not a contraction7 : Observation 7. Let be a relation on descriptors and let  be the entrenchment relation derived from it. Then: (1) If satisfies transitivity then so does . (2) If satisfies counter-dominance, then  satisfies dominance (if p  q then p  q). (3) If satisfies transitivity, counter-dominance, coupling, and amplification, then  satisfies conjunctiveness (either p  p&q or q  p&q). / {r | ⊥  r } if and (4) If satisfies transitivity, counter-dominance, coupling, and amplification, then  satisfies minimality (p ∈ only if p  q for all q). (5) If satisfies transitivity, counter-dominance, coupling, and amplification, then  satisfies maximality (if q  p for all q, then  p) if and only if satisfies: If  p then ¬B p  ⍊.

/ K if and only if p  q for all q) in The definition of minimality in Observation 7 differs from the standard definition (p ∈ not mentioning the belief set K . However, K is derivable from  as {r | ⊥  r }, and therefore it is not necessary to mention K in the axiomatization.8 Just as for revision, each of the three restrictions involves loss of information: Observation 8. (1) Let ◦ be a descriptor revision on a belief set K and  its derived revocation operator. It does not hold in general that ◦ is derivable from  (not even if  is a contraction). −

−

(2) Let  be a total belief set ordering and  its subtractive restriction. It does not hold in general that  is derivable from  (not even if the revocation obtainable from  is a contraction).

6 The operations obtainable through Definition 9 satisfy the supplementary AGM postulates, but there are AGM operators satisfying these postulates that are not obtainable through the construction of Definition 9. To see this it is sufficient to note that the operators obtainable through Definition 9 satisfy the postulate:

K  ( p&q) = K  p

or

K  ( p&q) = K  q.

AGM contractions only satisfy the weaker postulate:

K  ( p&q) = K  p

or

K  ( p&q) = K  q

or

K  ( p&q) = K  p ∩ K  q.

It should also be noted that operations constructed according to Definition 9 do not in general satisfy the recovery postulate, which shows that they are not in general AGM operations. 7 Note that for maximality to hold, ¬B p  ⍊ must hold for all non-tautological p. 8 In the present framework, we assume that  is derivable from a total belief set ordering  as stated in Definition 9. Then K is the -minimal set, and

/

˙ ⊥. Thus p ∈ K if and only if p  ˙ ⊥, and due to dominance p∈ / K holds if and only if p is -minimal. Since ⊥ ∈ / K , this means that p ∈ / K if and only if p  this means that p ∈ K if and only if ⊥  p. Thus K = { p | ⊥  p }. For this to hold, the revocation  does not have to be a contraction. The same definition of K is obtainable in the AGM model. According to Gärdenfors’ definition of entrenchment (Gärdenfors [3], Gärdenfors and Makinson [4]), p ∈ K  q if and only if either q  p ∨ q or  p. Assuming that K = K  ⊥ it follows that p ∈ K if and only if either ⊥  p or  p. Unless  is the identity relation, this is equivalent with ⊥  p.

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Fig. 3. Derivability diagram for sentential revocation.

(3) Let be a proximity relation and  its corresponding entrenchment relation. It does not hold in general that is derivable from  (not even if is derivable from a total belief set ordering that gives rise to a contraction). The following three observations explore the relationships among the three restrictions introduced in Definition 9. Observation 9. Let K be a belief set,  a total and descriptor-wellfounded belief set ordering with K as its minimal element, and −

◦ = t b◦ (). Furthermore, let  be the subtractive restriction of  and  the operation of revocation that is derived from ◦. Then: −

−

−

(1)  is derivable from  as follows: (i) If the domain of  contains some element not containing p, then K  p is the -minimal element not containing p, and (ii) otherwise, K  p = K . −

−

−

(2)  is derivable from  as follows:  is the relation on the outcome set of  such that X  Y if and only if there are p and q such that X = K  p, Y = K  q, K  p  p, K  q  q, and K  p = K  ( p&q). Observation 10. Let K be a belief set,  a total and descriptor-wellfounded belief set ordering with K as its minimal element, ◦ = t b◦ () and  = t bd (). Furthermore, let  be the revocation operator derived from ◦ and  the entrenchment relation derived from . Then: (1)  is derivable from  as follows: p  q if and only if K  p = K  ( p&q). (2)  is not derivable from , not even under the assumption that  is a contraction. −

Observation 11. Let  be a total and descriptor-wellfounded belief set ordering and = t bd (). Furthermore, let  be the subtractive restriction of  and  the entrenchment relation derived from . Then: −

−

(1)  is derivable from  as follows: p  q if and only if in the domain of , no set not containing q precedes any set not containing p.9 −

(2)  is not derivable from .

−

These results are summarized in Fig. 3. We can conclude that the interdefinability between  and ◦ is retained between −

 and , and from either of these we can derive . However, neither  nor  can be derived from . All contraction (and revocation) operators that are obtainable as restrictions of an operator of relational descriptor revision have an entrenchment relation associated with them in the manner shown in Definition 9. But as noted above, this is a different category of operations than those that have an entrenchment relation associated to them in the usual AGM way. Therefore it should not come as a surprise that Gärdenfors’ definition of contraction in terms of entrenchment cannot be used in the present framework:

9

−

Note that this holds if all sets in the domain of  contain q.

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Observation 12. Let  be a relation on sentences that satisfies transitivity, dominance, conjunctiveness, minimality, and maximality. It does not hold in general that there is a total belief set ordering  such that  is the entrenchment ordering derived from  and that the operation  of revocation derived form  satisfies:

q∈K p

if and only if either p  p ∨ q or  q.

It follows from Observation 11, part 2, that a given entrenchment relation may be obtainable from several different belief set orderings. The following theorem and observation provide an exact characterization of these belief set orderings. Theorem 3. Let  satisfy transitivity, dominance, and conjunctiveness. Then  is the entrenchment relation derivable from a total belief set ordering  if and only if  is constructible by assigning to each equivalence class in  exactly one logically closed set E p such that

   {s | p  s} ⊆ E p ⊆ s  (∀t )(s ∨ t  /˙ p )

for some p in that equivalence class, and letting E p  E q iff p  q.10 Observation 13. Let  satisfy transitivity, dominance, and conjunctiveness. Then the sets {s | p  s} and {s | (∀t )(s ∨ t  /˙ p )} are both logically closed. The lower limit identified in Theorem 3 can be further characterized as follows: Definition 10. A total belief set ordering  is shrinking if and only if it holds for all X and Y in its domain that if X < Y then Y ⊂ X . Theorem 4. Let K be a belief set and  a subtractively restricted total belief set ordering for K that generates the contraction  and the entrenchment relation . Then the following four conditions are equivalent: (1)  is shrinking, (2)  satisfies: K  p ⊆ K  q or K  q ⊆ K  p, (3)  is derivable from  as follows: K  p = {s | p  s}, and (4)  is derivable from  as follows:  is the operation on the sets obtainable as {s | p  s} for some p ∈ L, such that X  Y iff Y ⊆ X. Corollary to Theorems 3 and 4. An ordering  on sentences is the entrenchment relation derivable from some total belief set ordering if and only if it is the entrenchment relation derivable from some shrinking total belief set ordering. In summary, we have seen in this section that the restriction of a relation of epistemic proximity to descriptors that represent revocations (that need not be contractions) gives rise to a relation satisfying all the standard properties of epistemic entrenchment. Different belief set orderings may give rise to one and the same relation of epistemic entrenchment, and we have characterized those orderings. 6. Conclusions The purpose of this investigation was to introduce a relation on descriptors that has a similar role in descriptor revision as epistemic entrenchment has in AGM contraction. This was achieved in Section 3, whose main result was an interdefinability between on the one hand descriptor revisions based on a linear ordering of belief sets and on the other hand descriptor revisions based on a relation of epistemic proximity on descriptors. This result is analogous to the interdefinability in AGM between transitively relational partial meet contraction and contraction based on epistemic entrenchment. In Section 4, we restricted our attention to descriptors corresponding to sentential revision, and derived a binary relation (believability) on sentences, based on that restriction. In Section 5, we performed a similar restriction to descriptors corresponding to revocation or contraction, and derived a relation on sentences for that case. In the latter case the derived relation satisfies all the standard properties of epistemic entrenchment. Based on these results, epistemic proximity can be seen as a generalization of epistemic entrenchment in the sense that all relations of epistemic proximity on descriptors give rise to relations of epistemic entrenchment on sentences. However, the construction of contraction operators is not the same as in the AGM framework. In the present framework, a contraction operator cannot in general be reconstructed from its associated entrenchment relation and, in particular, Gärdenfors’ recipe for deriving contraction from entrenchment is not applicable. Nevertheless, the appearance of entrenchment relations with the standard properties in a framework quite different from the AGM framework in which these relations were originally developed confirms the importance and generality of these relations. 10

/

/

˙ s} instead of {s | (∀t )(s ∨ t  ˙ p )} as is upper As can be seen from the proof of this theorem, it could alternatively have been formulated with {s | p 

/

˙ p )} is logically closed, and therefore it is an unambiguous upper bound with which E p limit. The present formulation is preferred because {s | (∀t )(s ∨ t 

/

/

˙ p )} if and only if it is a subset of {s | p  ˙ s}. can coincide. Note that a logically closed set is a subset of {s | (∀t )(s ∨ t 

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Acknowledgements I would like to thank Zhang Li and the three journal referees for unusually helpful comments on an earlier version of this paper. Appendix A. Proofs Proof of Lemma 1. We need to show that for all belief sets X : (1) If X  Ψ or X  Ξ , then X  Ψ  Ξ , and (2) If X  Ψ  Ξ , then either X  Ψ or X  Ξ . (1) follows directly. For (2), let X  Ψ  Ξ . It is sufficient to show that if X  Ψ then X  Ξ . Let X  Ψ . Then there is some

α  ∈ Ψ such that X  α  , and consequently X  ¬α  . It follows from X  Ψ  Ξ that X  {α  ∨ β | β ∈ Ξ }. Combining this with X  ¬α  we obtain X  {β | β ∈ Ξ }, i.e. X  Ξ , as desired. 2 Lemma 2. Let  be a complete, transitive, antisymmetric, and descriptor-wellfounded relation on a set X of belief sets. For each descriptor Ψ that is satisfiable within X, let M Ψ be the -minimal Ψ -satisfying element of X. Then t bd () is the relation such that Ξ Ψ if and only if either M Ξ  M Ψ or Ψ is unsatisfiable within X. Proof of Lemma 2. Leaving out the case when Ψ is unsatisfiable within X we have:

Ξ Ψ

iff there is some X with X  Ξ such that X  Y for all Y with Y  Ψ (t bd ) iff M Ξ  Y for all Y with Y  Ψ ( is transitive and M Ξ  X for all X with X  Ξ ) iff M Ξ  M Ψ ( is transitive and M Ψ  Y for all Y with Y  Ψ ) 2 Lemma 3. Let be a relation on sentences that satisfies transitivity, counter-dominance, coupling, and amplification. Let t db ( ) = . Then X  Y if and only if Π X ΠY . Proof of Lemma 3. Let t b◦ () = ◦. We first note that if X is an element of the domain of , then there is some Ψ such , and then due to t b◦ , that X = K ◦ Ψ and K ◦ Ψ  Ψ . This follows since due to t db , X is completely characterized by some Ξ  and K ◦ Ξ Ξ . X = K ◦Ξ Next, let X and Y be elements of the domain of . As we have just seen, there are Ξ and Ψ such that X = K ◦ Ξ , K ◦ Ξ  Ξ , Y = K ◦ Ψ , and K ◦ Ψ  Ψ . We then have: XY iff K ◦ Ξ  K ◦ Ψ iff there is some Φ with K ◦ Ψ  Φ such that Σ Φ for all Σ with K ◦ Ξ  Σ (t db ) iff there is some Φ with Π K ◦Ψ  Φ such that Σ Φ for all Σ with Π K ◦Ξ  Σ iff there is some Φ with Π K ◦Ψ  Φ such that Π K ◦Ξ Φ (counter-dominance and the transitivity of ) iff Π K ◦Ξ Π K ◦Ψ (counter-dominance and the transitivity of ) iff Π X ΠY 2

Ψ. Lemma 4. Ψ   Ψ holds trivially. When Ψ  ⍊, note that Ψ ∪ {β( p )}  ⍊ holds for all p due to Proof of Lemma 4. When Ψ  ⍊, Ψ Ψ ∪ {β( p )}  Ψ and transitivity. Due to the coupling postulate and the compactness of the logic there is some belief set X  there can only be one such set. such that for all sentences p, X  Ψ ∪ {β( p )}, and due to the exhaustive construction of Ψ  , and it satisfies Ψ . 2 It is the set specified by Ψ Lemma 5. Let satisfy transitivity, counter-dominance, coupling, and amplification. Then: (1) either Ψ (Ψ  Ξ ) or Ξ (Ψ  Ξ ) (disjunctiveness), and (2) (Ψ  Ξ )  Ψ if and only if Ψ Ξ .

 Proof of Lemma 5. Part 1: Ψ Ξ Ψ Ξ  Either Ψ  Ξ  Ψ or Ψ Ξ Ξ

(Lemma 4) (Lemma 1)

  Ξ or Ξ Ψ Ξ (counter-dominance) Either Ψ Ψ Either Ψ (Ψ  Ξ ) or Ξ (Ψ  Ξ ) (transitivity and Observation 2, part 1) Part 2: For one direction, let (Ψ  Ξ )  Ψ . Using counter-dominance and Lemma 1, (Ψ  Ξ ) Ξ , and with transitivity we can derive Ψ Ξ . For the other direction, let Ψ Ξ . Suppose for contradiction that (Ψ  Ξ ) ≄ Ψ . Then due to counter-dominance and Lemma 1, (Ψ  Ξ )  Ψ . Part 1 yields Ξ (Ψ  Ξ ), and transitivity yields Ξ  Ψ , contrary to the assumption. We can conclude that (Ψ  Ξ )  Ψ . 2

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  B p } is the Lemma 6. Let satisfy transitivity, counter-dominance, coupling, and amplification. Let t db ( ) = . Then { p | Ψ -minimal Ψ -satisfying element of the domain of .   B p }  Ψ and: Proof of Lemma 6. Suppose not. Then due to t db there is some descriptor Ξ such that { p | Ξ   Bp} < { p | Ψ   B p }. (1) { p | Ξ Due to the definition of t db and the completeness of (Observation 1), (1) is equivalent with:

  B p }  Σ there is some Φ with { p | Ψ   B p }  Φ and Σ  Φ . (2) For all Σ such that { p | Ξ Due to counter-dominance and transitivity this is equivalent with:

Ψ  (3) Ξ   Ψ . It follows from counter-dominance that Ψ Ξ   B p }  Ψ we also have Ξ  and from Observation 2, Since { p | Ξ   Ψ . Thus Ψ Ξ Ψ   Ψ . This contradicts transitivity and we can conclude from this contradiction that part 1, that Ψ   B p } is indeed the -minimal Ψ -satisfying element of the domain of . 2 {p | Ψ Lemma 7. (See Gärdenfors and Makinson [4].) Let  be a relation on sentences that satisfies transitivity, dominance, and conjunctiveness. Then it satisfies completeness. Proof of Lemma 7. Due to conjunctiveness, either p  p&q or q  p&q. In the former case we use dominance to obtain p&q  q and transitivity to obtain p  q. In the latter case q  p is obtained in the same way. 2 Proof of Observation 1. Part 1: Counter-dominance yields Ψ  Ψ  and Ξ  Ξ  . The rest follows from transitivity. Part 2: Due to Lemma 5, part 1, either Ψ (Ψ  Ξ ) or Ξ (Ψ  Ξ ). In the former case we use counter-dominance and Lemma 1 to obtain (Ψ  Ξ ) Ξ and then transitivity to obtain Ψ Ξ . In the latter case Ξ Ψ is obtained in the same way. 2

  Ψ . Since Ψ  is maxispecified it follows from Ψ   Ψ that Ψ  ∪ Ψ  Ψ , Proof of Observation 2. Part 1: Due to Lemma 4, Ψ ∪Ψ  Ψ  . Now suppose that Ψ  ∪ Ψ ≄ Ψ . Due to the axiom of choice and the construction thus due to counter-dominance, Ψ  ∪ Ψ and Ψ  Δ and Ψ ≄ Δ ∪ {β( p )}. in Definition 6 there is then some Δ and some p such that Ψ ⊆ Δ ⊂ Δ ∪ {β( p )} ⊆ Ψ But due to counter-dominance, amplification and the construction in Definition 6, Ψ ∪ {β( p )}  Ψ , thus by transitivity  ∪ Ψ  Ψ , thus by and coupling, Δ ∪ (Ψ ∪ {β( p )})  Ψ , or equivalently Δ ∪ {β( p )}  Ψ . This contradiction shows that Ψ ∪Ψ Ψ  we have Ψ   Ψ , as desired. transitivity and Ψ Part 2, left to right: Ψ ∪Ξ Ψ   Ψ Ψ ∪Ξ  = Ψ Ψ ∪Ξ Ξ Ψ

(part 1 and transitivity)  and Ψ ∪ Ξ are both maxispecified) (since Ψ

  Ξ . Since Ψ   Ψ due to Lemma 4, we then have Ψ   Ψ ∪ Ξ . Counter-dominance yields Part 2, right-to-left: Let Ψ 

 . Due to Ψ ∪ Ξ  Ψ and counter-dominance we also have Ψ (Ψ ∪ Ξ ). Part 1 and transitivity yield Ψ (Ψ ∪ Ξ ) Ψ   (Ψ ∪ Ξ ). (Ψ ∪ Ξ ). We can conclude that Ψ Part 3: For one direction, let Ψ  Ψ  . Then Ψ  Ψ  follows from counter-dominance. For the other direction, let Ψ  Ψ  and let X be the unique belief set with X  Ψ and X  the unique belief set with X   Ψ  . Then Ψ  Π X and Ψ   Π X  . Suppose for contradiction that X = X  . Without loss of generality we can then assume that there is some p ∈ X \ X  . We then have Π X  B p and Π X   ¬B p, thus Π X  Π X  , thus Ψ  Ψ  . Contradiction. 2 Proof of Theorem 1. Part 1 follows directly, using Lemma 2.  Part 2: Let t bd () = , t db ( ) =  , and t b◦ () = ◦. Due to the definition of t db , the domain of  is the set of belief sets

 such that Ψ is satisfiable within the domain of , i.e.  has the same domain as . Furthermore: specified by some Ψ  X Y (Lemma 3, whose conditions are satisfied due to part 1) iff Π X ΠY (Lemma 2) iff K ◦ Π X  K ◦ ΠY iff X  Y Part 3: Let t b◦ () = ◦. Let t bd () = and t d◦ ( ) = ◦ . Let ◦i be the part of ◦ that is based on clause (i) of the definition of t b◦ and ◦i the part of ◦ that is based on clause (i) of the definition of t d◦ . Since  is descriptor-wellfounded, it follows from the definition of t b◦ that the domain of ◦i is the set of descriptors that are satisfiable within the domain of . It

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follows from t d◦ that the domain of ◦i is the set of descriptors Ψ satisfying Ψ  ⍊, and due to t bd these are the descriptors that are satisfiable within the domain of . Thus ◦i and ◦i have the same domain. Therefore, in order to prove ◦ = ◦ it is sufficient to show that ◦i = ◦i . We have: q ∈ K ◦i Ψ

Ψ ∪ {Bq}  Ψ  ⍊ (t d◦ ) Ψ ∪ {Bq}  Ψ (domain of ◦i ) Ψ ∪ {Bq} Ψ and Ψ Ψ ∪ {Bq} K ◦i (Ψ ∪ {Bq})  K ◦i Ψ and K ◦i Ψ  K ◦i (Ψ ∪ {Bq}) (Lemma 2) K ◦i (Ψ ∪ {Bq}) = K ◦i Ψ (antisymmetry of ) q ∈ K ◦i Ψ (t b◦ )  b◦ ◦ b b◦ ◦ b  Part 4: Let t () = ◦ and t (◦) =  . Due to the definitions of t and t , both  and  have the domain { K ◦ Ψ | K ◦ Ψ  Ψ }. Within that domain: K ◦ Ψ  K ◦ Ξ iff K ◦ Ψ = K ◦ (Ψ  Ξ ) (t ◦b ) (t b◦ and Lemma 1) iff K ◦ Ψ  K ◦ Ξ Part 5: Let t b◦ () = ◦, and t ◦d (◦) = , and t bd () =  . Then: Ξ Ψ iff either K ◦ Ξ  Ξ , K ◦ Ψ  Ψ , and K ◦ Ξ = K ◦ (Ξ  Ψ ) or K ◦ Ψ  Ψ (t ◦d ) iff either Ξ and Ψ are both satisfiable within X and the -minimal Ξ  Ψ -satisfying element satisfies Ξ , or Ψ is unsatis(t b◦ ) fiable within X (t bd ) 2 iff Ξ  Ψ

iff iff iff iff iff iff

Proof of Theorem 2. Part 1. Let t db ( ) = . Transitivity: Directly from the transitivity of , using Lemma 3. Completeness: With Lemma 3 from Observation 1, part 2. Antisymmetry: XY X

Π X  ΠY Π X  ΠY X=Y

(Lemma 3) (Observation 2, part 3) (Definition 4)

Descriptor-wellfoundedness: From Lemma 6. Part 2: Let t db ( ) = , t bd () =  , and t b◦ () = ◦. Ψ  Ξ iff either K ◦ Ψ  K ◦ Ξ or Ξ is unsatisfiable within the domain of  (Lemma 2, whose conditions are satisfied due to part 1) (Lemma 311 ) iff Π K ◦Ψ Π K ◦Ξ or Ξ  ⍊   (Lemma 6) iff Ψ Ξ or Ξ  ⍊ (Observation 2, part 1, and the transitivity of ) iff Ψ Ξ or Ξ  ⍊ (Ψ ⍊ from counter-dominance, transitivity) iff Ψ Ξ Part 3: Let t db ( ) = , t b◦ () = ◦ , and t d◦ ( ) = ◦. Let X be the domain of . Since B holds in all belief sets, it holds  in the minimal element of the domain of , and it follows from t db that this element is { p | B

 B p }, which according to Definition 6 is identical to { p | B p  B }. q ∈ K ◦ Ψ   Bq and Ψ is satisfiable within X or q ∈ K and Ψ is unsatisfiable within X iff either Ψ (Lemma 6)   Ψ ∪ {Bq} and Ψ  ⍊ or q ∈ K and Ψ  ⍊ (Observation 2, part 2) iff either Ψ (Observation 2, part 1, transitivity of ) iff either Ψ  Ψ ∪ {Bq} and Ψ  ⍊ or q ∈ K and Ψ  ⍊ (t d◦ ) iff q ∈ K ◦ Ψ d◦ ◦ d  Part 4: Let t ( ) = ◦ and t (◦) = . Then:

Ψ  Ξ

iff either K ◦ Ψ  Ψ , K ◦ Ξ  Ξ , and K ◦ (Ψ  Ξ ) = K ◦ Ψ or K ◦ Ξ  Ξ iff either K ◦ Ψ  Ψ , K ◦ Ξ  Ξ , and for all q: (Ψ  Ξ )  (Ψ  Ξ ) ∪ {Bq}  ⍊ iff Ψ  Ψ ∪ {Bq}  ⍊ or Ξ  ⍊

(t ◦d ) (t d◦ )

   ⍊ or Ξ  ⍊ either K ◦ Ψ  Ψ , K ◦ Ξ  Ξ , and Ψ (Definition 6) Ξ Ψ either K ◦ Ψ  Ψ , K ◦ Ξ  Ξ , and Ψ  Ξ  Ψ  ⍊ or Ξ  ⍊ (Observation 2) Ψ Ξ  ⍊ or Ξ  ⍊ (Lemma 5, part 2) Ψ Ξ (Ξ ⍊ and Ψ ⍊ from counter-dominance, transitivity of )  Part 5: Let t d◦ ( ) = ◦, t ◦b (◦) = , and t db ( ) =  . First note that due to t ◦b the domain of  is the set of belief sets X d◦ such that X = K ◦ Ξ for some Ξ , and due to t and t db this is equal to the domain of  .

iff iff iff iff

11 To see that Ξ is unsatisfiable within the domain of  if and only if Ξ  ⍊, let Ξ be unsatisfiable within the domain of . Then due to t db it is not the case that Ξ  ⍊, and since ⍊  Ξ counter-dominance yields Ξ  ⍊. The other direction follows directly.

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For any K ◦ Ξ and K ◦ Ψ in that domain we have: K ◦Ξ  K ◦Ψ iff K ◦ Ξ = K ◦ (Ξ  Ψ ) (t ◦b ) (t d◦ ) iff Ξ  (Ξ  Ψ )  ⍊ (Lemma 5, part 2) iff Ξ Ψ  ⍊  Ψ ⍊ (Observation 2, part 1) iff Ξ   Bq}  Φ such that Σ Φ for all Σ with {q | Ξ   Bq}  Σ . iff there is some Φ with {q | Ψ  whenever {q | Ψ   Bq}  Φ , (Due to counter-dominance Φ Ψ  whenever {q | Ξ   Bq}  Σ .) and Σ Ξ   Bq}  {q | Ψ   Bq} (t db ) iff {q | Ξ (t d◦ , Lemma 6) 2 iff K ◦ Ξ  K ◦ Ψ Proof of Observation 3. Part 1: In a language with only the two atoms p and q, let  and  be exhaustively described by:





























Cn(∅) < Cn { p&q} < Cn { p } < Cn { p&¬q} < Cn {¬ p&q} < Cn {¬ p&¬q} < Cn {⊥} respectively:

















Cn(∅) < Cn { p&q} < Cn { p&¬q} < Cn {¬ p&q} < Cn {¬ p&¬q} < Cn {⊥}

Let ◦ be the descriptor revision generated by  and ◦ that generated by  . Then K ◦ {B p , ¬Bq} = Cn({ p }) but K ◦ {B p , ¬Bq} = Cn({ p&¬q}). However,  and  generate the same sentential revision. Part 2: We can use the same example as in part 1, and note that  is the additive restriction of both  and itself. Part 3: Let  and  be as in part 1. They generate different proximity relations on descriptors, namely and  , such that {B p , ¬Bq}  {B p , B¬q} but {B p , B¬q}  {B p , ¬Bq}. However,  and  generate the same believability relation. 2 Proof of Observation 4. Part 1: Directly from the definitions. Part 2: We need to construct two additively restricted total belief set orderings that generate the same sentential revision. Let K 0 = K 0 = Cn({s ∨ t }) K 1 = Cn({s}) and K 1 = Cn({t }) K 2 = Cn({t }) and K 2 = Cn({s}) K n = K n for all n > 2 Let  and  (with the strict parts < and < ) be the total belief set orderings that are completely characterized by the series K 0 < K 1 < K 2 . . . and K 0 < K 1 < K 2 . . . . Let ∗ and ∗ be the revisions based on the two orderings. They have the same outcome set, to be denoted X. We need to show that for all p ∈ L and all X ∈ X, K ∗ p = X if and only if K ∗ p = X . This is straightforwardly shown for all X ∈ X except possibly for Cn({s}) and Cn({t }). The proofs in these two cases are symmetrical. One of them is: / K0, p ∈ / K 1 , and p ∈ K 2 K ∗ p = Cn({t }) iff p ∈ iff s ∨ t  p, s  p, and t  p iff s ∨ t  p and t  p / K 0 and p ∈ K 1 iff p ∈ iff K ∗ p = Cn({t }) 2 Proof of Observation 5. Part 1: In a language with only the two atoms p and q, consider the total belief set orderings exhaustively characterized as follows:





















Cn { p }  Cn { p&q}  Cn {⊥} Cn { p }  Cn { p&q}

They give rise to the same believability relation but different sentential semirevisions (e.g. K ∗¬q = Cn({⊥}) but K ∗ ¬q = K ).

Part 2: In the example presented in the proof of Observation 4, part 2, the believability relation generated from



 and the corresponding believability relation  generated from  are not the same, as can be seen from st and



 t  s. However, as shown in the proof of Observation 4, part 2, and  are associated with the same sentential revision ∗. 2 Proof of Observation 6. Part 1:

p q iff B p Bq iff either M B p  M Bq or Bq is unsatisfiable within the domain of . Part 2: Use the same example as in the proof of Observation 5, part 1.

(Definition 8) (Lemma 2; M defined as in that lemma)

2

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Proof of Observation 7. Part 1: Directly from Definition 9. Part 2: p q B p  Bq ¬Bq  ¬B p ¬B p ¬Bq (counter-dominance) (Definition 9) p q Part 3: ¬B p (¬B p ∨ ¬Bq) or ¬Bq (¬B p ∨ ¬Bq) (Lemma 5) ¬B p ¬B( p&q) or ¬Bq ¬B( p&q) (intersubstitutivity) (Definition 9) p  ( p&q) or q  ( p&q) Part 4: / {r | ⊥  r } p∈ iff ⊥ /p (Parts 1–3, Lemma 7) iff p  ⊥ (⊥  q due to part 2; transitivity from part 1) iff p  q for all q Part 5: If q  p for all q, then  p iff: If ¬Bq ¬B p for all q, then  p (Observation 1) iff: If  p then ¬B p  ¬Bq for some q (¬B  ⍊, ¬Bq ⍊ from counter-dominance, transitivity) 2 iff: If  p then ¬B p  ⍊ Proof of Observation 8. Part 1: In a language with only the two atoms p and q, let  and  be completely characterized by:

























Cn { p&q} < Cn { p ∨ q} < Cn { p } < Cn {q → p }





Cn { p&q} < Cn { p ∨ q} < Cn {q → p }

The descriptor revisions ◦ and ◦ that  respectively  give rise to differ since K ◦ {B(q → p ), ¬Bq} = Cn({ p }) and K ◦ {B(q → p ), ¬Bq} = Cn({q → p }). However, they generate the same revocation, which is a contraction. Part 2: In the example introduced in part 1,  is the subtractive restriction of both  and itself. Part 3: In a language with only the two atoms p and q, let  and  be completely characterized by:





















Cn { p&q} < Cn { p } < Cn {q} < Cn(∅)





Cn { p&q} < Cn { p } < Cn { p ∨ q} < Cn(∅)

 and  give rise to different proximity relations and  , as can be seen from {¬B( p ∨ q)} {¬B p , ¬Bq} and {¬B p , ¬Bq}  {¬B( p ∨ q)}. To prove that they give rise to the same relation of entrenchment, let  and  be the entrenchment relations based on  respectively  . We need to show that for all sentences z, Cn({q}) is the -minimal belief set in the domain of  not containing z if and only if Cn({ p ∨ q}) is the  -minimal belief set in the domain of  not containing z. This is done as follows: / Cn({q}) z ∈ Cn({ p&q}) and z ∈ Cn({ p }) and z ∈ iff z ∈ Cn({ p }) \ Cn({q}) iff z ∈ Cn({ p }) \ (Cn({ p }) ∩ Cn({q})) iff z ∈ Cn({ p }) \ Cn({ p ∨ q}) z ∈ Cn({ p&q}) and z ∈ Cn({ p }) and z ∈ / Cn({ p ∨ q})

2

Proof of Observation 9. Part 1: Due to Definition 9, K  p = K ◦ ¬B p. According to t b◦ , if ¬B p is satisfiable within the domain of , then K  p is the -minimal set in the domain of  not containing p. According to Definition 9, this is also −

the -minimal set not containing p. The case when ¬B p is not satisfiable within the domain of  follows directly. −

Part 2: It follows from Definition 9 that the domain of  consists of the sets K ◦ ¬B p such that K ◦ ¬B p  ¬B p, i.e. equivalently the sets K  p such that K  p  p. Furthermore: −

K  p  K  q iff K ◦ ¬B p  K ◦ ¬Bq iff K ◦ ¬B p = K ◦ (¬B p ∨ ¬Bq) iff K ◦ ¬B p = K ◦ ¬B( p&q) iff K  p = K  ( p&q) Proof of Observation 10. Part 1: p  q iff ¬B p ¬Bq iff K ◦ ¬B p  K ◦ ¬Bq or ¬Bq is unsatisfiable within the domain of 

(Definition 9) (t ◦b , part 4 of Theorem 1) (extensionality) (Definition 9) 2

(Definition 9) (Lemma 2)

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S.O. Hansson / Artificial Intelligence 217 (2014) 76–91 −

iff K  p  K  q or q is included in all elements of the domain of  (Definition 9) iff K  p = K  ( p&q) or q is included in all elements of the domain of  iff K  p = K  ( p&q) (part 2 of Observation 9) Part 2: In the example presented in the proof of Observation 8, part 3,  and  give rise to different revocations (that are both contractions), as can be seen from revoking p (contracting by p). But as shown in the proof of Observation 8, part 3, they give rise to the same entrenchment relation . 2 Proof of Observation 11. Part 1: p  q iff ¬B p ¬Bq iff M ¬B p  M ¬Bq or ¬Bq is unsatisfiable within the domain of  −

(Definition 9) (Lemma 2; M defined as in the lemma)

−

iff M ¬B p  M ¬Bq or ¬Bq is unsatisfiable within the domain of 

(Definition 9) −

−

Part 2: In the example given in the proof of Observation 8, part 3, we have  = and  different. However, they correspond to the same entrenchment relation . 2



= . Thus

−

−

 and  are

Proof of Observation 12. In a language with only the two atoms p and q, let  be the entrenchment ordering that is derivable from the total belief set ordering 1 that is completely characterized by Cn({ p , q}) <1 Cn({ p ∨ q}) <1 Cn(∅). Let 2 be another belief set ordering that gives rise to . Let X be the second element of its domain (i.e. Cn({ p , q}) <2 X and there is no Y with Cn({ p , q}) <2 Y <2 X ). Then Cn({ p , q}) \ X = Cn({ p , q}) \ Cn({ p ∨ q}), thus p ∈ / X, q ∈ / X , and p ∨ q ∈ X . Thus q ∈ / K  p, in spite of p  p ∨ q. 2 Proof of Theorem 3. For all sentences p that are included in some element of the domain of , let X p be the -minimal element of the domain of  that does not contain p, if there is such an element. We are first going to show that if we let E p = {s | p  s}, then  is the entrenchment relation derivable from . Since  is then shrinking (cf. Definition 10), {s | p  s} = X p and we have: X p  X q or q is not represented in the domain of  iff {s | p  s}  {s | q  s} or q is not represented in the domain of  iff E p  E q or q is not represented in the domain of  iff p  q. (The logical closure of {s | p  s} is shown in Observation 13.) Next we are going to identify the belief sets E p that can replace {s | p  s} without changing the derived entrenchment ordering. Since the construction using {s | p  s} yields a shrinking belief set ordering, the -minimal belief set not containing p is {s | p  s}, and the intersection of all preceding belief sets is equal to the immediately preceding one, i.e. {s | p  s}. Therefore our criterion for E p is that {s | p  s} \ E p = {s | p  s} \ {s | p  s}. We have: {s | p  s} \ E p = {s | p  s} \ {s | p  s} iff E p ∩ ({s | p  s} \ {s | p  s}) = ∅ and {s | p  s} ⊆ E p (since {s | p  s} ⊆ {s | p  s}) ˙ s} = ∅ and {s | p  s} ⊆ E p iff E p ∩ {s | p  iff {s | p  s} ⊆ E p ⊆ {s | p  /˙ s} iff {s | p  s} ⊆ E p ⊆ {s | (∀t )(s ∨ t  /˙ p )} (since E p is logically closed) 2 Proof of Observation 13. For the first set, let {s | p  s}  r. We are going to show that r ∈ {s | p  s}. Due to compactness there is a finite subset {s1 , . . . , sn } of {s | p  s} such that {s1 , . . . , sn }  r, equivalently s1 & . . . &sn  r. It follows by repeated use of conjunctiveness that there is some sk ∈ {s1 , . . . , sn } with sk  (s1 & . . . &sn ), and dominance yields (s1 & . . . &sn )  r. Thus p  sk  (s1 & . . . &sn )  r, thus by transitivity p  r, i.e. r ∈ {s | p  s}. For the second set, let {s | (∀t )(s ∨ t  /˙ p )}  u and suppose that u ∈/ {s | (∀t )(s ∨ t  /˙ p )}. Then there is some v with ˙ p. Due to {s | (∀t )(s ∨ t  (u ∨ v )  /˙ p )}  u ∨ v and compactness there is a finite set {s1 , . . . , sn } ⊆ {s | (∀t )(s ∨ t  /˙ p )} such that {s1 , . . . , sn }  u ∨ v. It follows truth-functionally that  u ∨ v ≡ (s1 ∨ u ∨ v )& . . . &(sn ∨ u ∨ v ), and dominance ˙ (s1 ∨ u ∨ v )& . . . &(sn ∨ u ∨ v ). Due to conjunctiveness there is some sk such that (sk ∨ u ∨ v )  (s1 ∨ u ∨ yields u ∨ v  ˙ (s1 ∨ u ∨ v )& . . . &(sn ∨ u ∨ v ). We already have (u ∨ v )  ˙ p and v )& . . . &(sn ∨ u ∨ v ). Dominance yields (sk ∨ u ∨ v )  ˙ (s1 ∨ u ∨ v )& . . . &(sn ∨ u ∨ v ). Transitivity yields (sk ∨ u ∨ v )  ˙ p, contrary to sk ∈ {s | (∀t )(s ∨ t  u∨v /˙ p )}. We can conclude from this contradiction that u ∈ {s | (∀t )(s ∨ t  /˙ p )}. 2 Proof of Theorem 4. The proofs from (1) to (2), from (2) to (1) and from (1) to (3) are straightforward. From (3) to (2): We have K  p = {s | p  s} and K  q = {s | q  s}. Due to Observation 7 and Lemma 7,  is transitive and complete. Thus either p  q or q  p. Due to transitivity, {s | q  s} ⊆ {s | p  s} in the former case and {s | p  s} ⊆ {s | q  s} in the latter. From (1) to (4): Due to Observation 9, since  is subtractively restricted, the domain of  is the outcome set of , which according to (3) is the set of sets {s | p  s}. Since  is shrinking and subtractively restricted, X  Y if and only if Y ⊆ X . From (4) to (2): Due to the transitivity and completeness of , it follows from (4) that either K  p ⊆ K  q or K  q ⊆ K  p. 2

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