Severe Withdrawal (and Recovery)

Hans RottDepartment of PhilosophyUniversity of AmsterdamNieuwe Doelenstraat 151012 CP AmsterdamThe Netherlands

Tel: +31-20-525 4510Fax: +31-20-525 4503Email:rott@philo.uva.nlMaurice PagnuccoKnowledge Systems GroupDepartment of Artificial IntelligenceSchool of Computer Science and EngineeringUniversity of New South WalesNSW, 2052, Australia

Tel: +61-2-9385 1441Fax: +61-2-9385 1083Email:morri@cse.unsw.edu.auDecember 23, 1997

Abstract. The problem of how to remove information from an agent’s stock of beliefs is ofparamount concern in the belief change literature. An inquiring agent may remove beliefs for avariety of reasons: a belief may be called into doubt or the agent may simply wish to entertainother possiblities. In the prominent AGM framework [1, 8] for belief change, upon which thework here is based, one of the three central operations,contraction, addresses this concern (theother two deal with the incorporation of new information). Makinson [23] has generalised thiswork by introducing the notion of awithdrawaloperation.

Underlying the account proffered by AGM is the idea ofrational belief change. A beliefchange operation should be guided by certain principles or integrity constraints in order tocharacterise change by a rational agent. One of the most noted principles within the contextof AGM is the Principle of Informational Economy. However, adoption of this principle inits purest form has been rejected by AGM leading to a more relaxed interpretation. In thispaper, we argue that this weakening of the Principle of Informational Economy suggests thatit is only one of a number of principles which should be taken into account. Furthermore, thisweakening points toward a Principle of Indifference. This motivates the introduction of a newbelief removal operation that we callsevere withdrawal. We provide rationality postulates forsevere withdrawal and explore its relationship with AGM contraction. Moreover, we furnishpossible worlds and epistemic entrenchment semantics for severe withdrawals.

Key words: AGM, belief change, belief contraction, epistemic entrenchment, severe with-drawal, systems of spheres.

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1. Introduction

An inquiring agent must, among other things, deal with the problem ofbeliefchange(or belief revision) — how to modify its current epistemic state (stockof beliefs) in light of new information. One of the more popular accountsof belief change in recent times has been that introduced by Alchourron,Gardenfors and Makinson [1] (henceforth referred to as theAGM frame-work). The AGM framework for belief change distinguishes three types oftransformations on epistemic states:belief contraction(K .��) — removalof belief � from epistemic stateK without addition of any further beliefs;belief expansion(K + �) — addition of belief� and its consequences with-out removal of any existing beliefs; and,belief revision(K � �) — additionof belief� and its consequences with possible removal of existing beliefs inorder to maintain consistency.

In this paper we are predominantly concerned with the process of beliefremoval — “contraction” and “withdrawal”.1 It is our aim here to introducea new, principled, belief removal operation. Although distinct from AGMcontraction, the two are related through their emergent revision behaviour.Moreover, when attention is restricted to all functions related in this way andsatisfying certain intuitive principles, we find that thesetwo proposals lie atopposite ends of the spectrum with respect to their degree ofbelief removalmeasured in terms of set-theoretic inclusion. The alternative proposal intro-duced here is contrasted with AGM contraction which can be seen as a pointof reference in helping to understand the vagaries of this new approach. Inthe AGM vein, rationality postulates are provided for our proposal and twoconstructions of central importance to the AGM framework — systems ofspheres and epistemic entrenchment — are adapted and used tofurther pro-mote this comparison.

There are a number of reasons why an inquiring agent would be inter-ested in removing beliefs from its current epistemic state.If the agent findsitself in an inconsistent state — believing contradictory information — thenit can give up certain beliefs in an attempt to regain consistency.2 On theother hand, an agent may want to suspend belief in a particular propositionbecause it no longer has any confidence in that proposition orsimply becauseit would like to consider other possibilities. In either case, the overriding con-cern is that the agent no longer include the proposition in question amongits beliefs. Moreover, if one subscribes to Levi’sCommensurability Thesis[18, p. 65] which states that any reasonable transition between two epistemicstates can be achieved through a sequence of expansions and contractions,then the importance of contraction is clearly evident.3

Principally, we are concerned with characterising that belief change under-gone by those agents which act in accord with certain principles or “integrityconstraints” commonly referred to asrationality criteria (see also Gardenforsand Rott [11, p. 38]). Arguably the most well known of these criteria (espe-

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cially in the context of the AGM framework) is thePrinciple of InformationalEconomy[8] which we shall present here in slightly more general guise as thePrinciple of Economy:4� The Principle of Economy:

Keep loss to a minimum.

This principle has, in fact, become largely synonymous withthe AGM frame-work. A special instance of this constraint, where loss is measured in terms ofset-theoretic inclusion (of epistemic states), is known asthePrinciple of Con-servatism[16]. It can be considered the starting point for the AGM accountof contraction; the motivating concern underlying the AGM notion of “maxi-choice contraction” and the pathway to that of “partial meetcontraction” [1].5An important point to note is that such a comparison, on the basis of set-theoretic inclusion, presupposes the association of a positive value of utilitywith every single item of belief. The Principle of (Informational) Economy(and consequently that of Conservatism) is a restricted case of thePrincipleof Minimal Change[16] which states that addition as well as loss should bekept to a minimum.

It is our contention here that the Principle of Economy has been severelycompromised in the AGM framework. In its purest form, as the Principle ofConservatism, it has been shown to lead to undesirable consequences whenapplied to logically closed belief sets [2].6 As a result its imposition is effect-ed to a much less stringent degree. We claim that this principle is not, in fact,an overriding criterion but, instead, must be applied in combination with oth-er, equally important, principles in order to obtain an intuitively satisfactoryaccount of belief change. Moreover, these principles are ina state of tensionwith respect to each other (i.e., they have conflicting concerns). In this paperwe advocate, in particular,Principles of Indifference and Preference. Briefly,taken together they state that an object held in equal or higher regard thananother should be treated equally or more favourably than the latter. In fact,we argue that AGM contraction does embrace these principlesto a limitedextent. However, this partial adoption does not appear to beclearly motivatedor even justified. Therefore, we propose a stricter adherence to the Princi-ples of Indifference and Preference. As a result of this change in view, wepropose a new form of contraction differing from that put forward by AGM.Moreover, the rationality criteria proposed are equally applicable to the twoconstructive modellings that we investigate here —systems of spheresandepistemic entrenchment orderings. Interestingly enough, this new contractionoperation does not affect the AGM beliefrevisionoperation.

Let us return our focus of attention to the AGM development ofbeliefcontraction. Applying the Principle of Economy in the form of the Principleof Conservatism, it was at first suggested that the contraction of a belief setK by a sentence� could be achieved by selecting some maximal subset ofK

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that does not imply�. As mentioned above, this proposal was immediatelyabandoned as it has undesirable consequences. Instead, a selection function was applied to the set of all such maximal non-implying subsets,K?�7, inorder to select a set of the “best” elements which are then intersected to obtainapartial meet contraction function(K .�� = T (K?�)). It should be notedthat the selection function is defined for all sentences� but withK heldfixed (i.e., for some belief setK, may take as an argumentK?� for any� 2L). It is clear that this development leads away from conservatism. Certainobjects are under scrutiny. A mechanism is used to discriminate among themalthough it may not be possible to distinguish some apart andso these areall retained and processed together. This leads us to formulate Indifference asfollows:� The Principle of Indifference

Objects held in equal regard should be treated equally.

The situation in which all maximal non-implying subsets areheld in equalregard is an AGM full meet contraction8 which stands at the opposite end ofthe spectrum to AGM maxichoice contraction.

In settling on partial meet we realize that the Principle of Economy and thePrinciple of Indifference are in a state of tension with respect to one anoth-er; Economy advocates the selection of a single element fromK?� whileIndiffence recommends to give up more than necessary if the selection mech-anism does not single out a unique “best” solution. Both of these principlesfigure in the rationale behind the choice of “best” elements implicitly adopt-ed in partial meet contraction. We shall extend this strategy by accepting thefollowing, intuitively appealing, principle:� The Principle of Strict Preference

Objects held in higher regard should be afforded a more favourabletreatment.

Taken together with the Principle of Indifference, this principle can be seenas advancing the following rather general principle:� The Principle of Weak Preference

If one object is held in equal or higher regard than another, the formershould be treated no worse than the latter.

Such a principle can already be seen to be at work in AGM partial meet con-traction with “importance” being judged through the selection function .However, consideration is restricted to the elements ofK?� for a particular

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sentence� 2 L. Our aim here is to emphasise an alternative to AGM con-traction which we believe adheres more faithfully to the Principles of Indif-ference and Preference. However, adopting the AGM format for belief changeprocesses (i.e., epistemic inputs as sentences from a suitable object languageand epistemic states as sets of sentences that are deductively closed undersome consequence relation) allows us to effect a straightforward comparisonof the two proposals.

In the following section we outline some technical preliminaries. In sec-tions 3 and 4 we present an intuitive overview of two important constructivemodellings for AGM belief contraction. We describe how theyfail to live upto the requirements demanded by the Principles of Indifference and Prefer-ence and outline an approach that resolutely favours these principles over thePrinciple of Economy. A common method of presenting AGM contractionoperations is through rationality postulates which we survey in section 5 andcontrast, in section 6, with rationality postulates for thebelief removal opera-tion advocated in the present paper. The relationship between AGM contrac-tion operations and our account of belief removal is more directly addressedin section 7. In sections 8 through 11 we return to the constructive modellingsdiscussed in sections 3 and 4, investigating the technical aspects of their appli-cation in our belief removal operation. This leads us to an investigation of therelationship between the two constructive modellings adopted here — sys-tems of spheres and epistemic entrenchment — in section 12. We concludewith a discussion of the insights stemming from our approach, its relationshipto other work in the literature (section 13) and, in section 14, a summary ofthe contributions made here.

2. Technical Preliminaries

Throughout this paper we assume a fixed propositional languageLwith count-ably many propositional symbols. We assume thatL avails of the standardlogical connectives, namely:; ^; _; !; and$, together with the proposi-tional constants> (truth) and? (falsum). The underlying logic will be iden-tified with its consequence operator Cn: 2L ! 2L which is assumed tosatisfy the following properties.� � Cn(�) (Inclusion)

If � � �, thenCn(�) � Cn(�) (Monotonicity)Cn(�) = Cn(Cn(�)) (Iteration)If � can be derived from� by classical

truth-functional logic, then� 2 Cn(�) (Supraclassicality) 2 Cn(� [ f�g) if and only if (�! ) 2 Cn(�) (Deduction)If � 2 Cn(�), then� 2 Cn(�0) for some finite

subset�0 � � (Compactness)

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We often write� ` � to mean� 2 Cn(�) and` � for ; ` �.We refer to any set of sentencesK in L as abelief setor theory if K is

closed underCn (i.e.,K = Cn(K)). One special belief set is the absurdbelief setK? containing all sentences inL. A belief setK is consistentinL if and only if it does not contain sentences� and:� for any� 2 L, i.e.,if K does not equalK?. A belief set iscompletein L if either � 2 K or:� 2 K for every� 2 L. The set of all belief sets is denotedK. We adoptthe convention of denoting sentences by lower case Greek letters�; ; : : :and sets of sentences by upper case Roman lettersH; K; : : :.

3. Sphere-Based Withdrawal

An interesting way of viewing the process of belief change isin terms of pos-sible worlds. A construction in this vein, specifically focussed on the AGMframework, has been proposed by Grove [12] who adapted Lewis’ [21] possi-ble worlds modelling for counterfactual conditionals. This approach possess-es a highly intuitive appeal through the pictorial representation by systems ofspheres.9 In this section we concentrate on motivating our approach throughthis intuition, deferring the main technical details to section 8.

Grove [12] characterises the current beliefs of an agent by the collectionof those possible worlds that are consistent with the agent’s beliefs. But thisis not the entire representation of an epistemic state. The remaining worlds —those inconsistent with the agent’s current beliefs — are grouped around thiscore collection in decreasing order of plausibility. This results in a systemof spheres centred on the set of worlds consistent with the agent’s beliefs.Change in belief involves the determination of those worldscharacterisingthe agent’s new beliefs and is guided by the preference ordering over worlds.

More specifically, we denote the possible worlds consistentwith a set ofsentencesK by [K] and the set of all possible worlds byW. We also adopt theshorthand[�] for [f�g]. A sphereis simply a set of possible worldsX � W.A system of spheres centred on[K] is a set of nested spheres (in the senseof set inclusion) in which the smallest or innermost sphere is [K] and theoutermost sphere isW. This is a generalisation of Lewis [21] whose systemsof spheres are centred on a single worldw 2 W (the actual world) if weallow ourselves to neglect the fact that Lewis does not require W to be anelement of every system of spheres. Essentially, a system ofspheres centredon [K] orders those worlds inconsistent with the agent’s epistemic stateK.Intuitively, the agent believes the actual world to be one oftheK-worlds butdoes not have sufficient information to establish which one.However, theagent may be mistaken, in which case it believes that the actual world is mostlikely to be one of those in the next greater sphere and so on. As such, asystem of spheres can be considered an ordering ofplausibility over worlds;

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[K]

ML

[ φ]¬

c ( )

f ( )

S

S

φ ¬

φ ¬

Figure 1. Sphere semantics for AGM belief contraction showing[K .��] shaded.

the more plausible worlds lying further towards the centre of the system ofspheres.

This ordering provides us with a powerful tool for investigating the pro-cess of belief change. In this paper we concern ourselves with the operation ofbelief contraction. That is, the situation in which an agentwishes to suspendone of its beliefs. In this scenario, information is being removed, openingup more possibilities. In other words, the agent’s candidate worlds increase;more worlds being added to[K]. In order to suspend belief in a sentence�the agent must have some candidate worlds in which� is false and there-fore, considering the principal case in which� is initially believed (� 2 K),it must at least introduce some:�-worlds into [K]. The AGM approach tothis problem is motivated to a large extent by the Principle of InformationalEconcomy. Accordingly, simply the closest:�-worlds — those in the small-est sphere containing:�-worlds — are added to[K]. This situation is illus-trated in Figure 1. If byfS(�) we denote the�-worlds closest toK and weintroduce a functionth that returns the belief set corresponding to a set ofworlds, then we have the following method for defining an AGM contractionfunction .� from a system of spheres:

(Def .� from S) K .�� = th([K] [ fS(:�))It will also be convenient to refer to the smallest sphere intersecting� whichwe denote bycS(�). ThenfS(�) is given bycS(�) \ [�].

Now, if one were to apply the Principle of Informational Economy inits unadulterated form (i.e., Conservatism), then the aim of contraction —removal of a belief� from epistemic stateK — would be achieved throughthe addition ofa single:�-world to [K] rather than a number of:�-worldsas depicted in Figure 1. This form of contraction corresponds tomaxichoicecontractionin the AGM literature [1], i.e., the idea of taking belief contrac-tion of K by � to be some maximal subset ofK that fails to imply�.10However, this proposal has been shown to possess a number of drawbacks.

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Foremost among these is the fact that any revision function defined from sucha contraction function via the Levi Identity would always lead to a completetheory, i.e.,Cn(K .��[f:�g) is maximallyconsistent. This indicates that toolittle information is being removed. As a result, the Principle of InformationalEconomy is imposed on a much weaker level as indicated above.Instead ofincluding only one:�-world in contraction, AGM incorporate a number of:�-worlds — those held to be most plausible — into the agent’s epistemicstate. However, none of the rationality postulates mentioned thus far specifyprecisely how to deal with worlds that are equally preferredby the agent. Asa remedy, we suggest the employment of the Principle of Indifference. As wehave seen, AGM have gone part of the way to adopting such a principle. How-ever, they limit their embracement of such a strategy to the area covered by:�-worlds only. Presumably, this is due to a desire to remain asfaithful to thePrinciple of Informational Economy as possible, despite its recognised short-comings. Yet the Principle of Informational Economy has been compromisedand its relevance called into question. We propose to place still less emphasison its application and subordinate it to the Principle of Indifference.

Another principle, relating to the preference structure supplied by thesphere modelling, that we suggest to respect is the Principle of Strict Prefer-ence. According to this principle, worlds considered more plausible should begiven more favourable treatment. When contracting its belief set with respectto �, the agent must at least include some (one, at any rate):�-world intoits epistemic state. But the aforementioned principles, asapplied to possibleworlds and systems of spheres, advocate that any�-worlds just as plausi-ble as the innermost:�-worlds should be included also. Thus, together theysanction the following specialisation of the Principle of Weak Preference:

If one world is considered at least as plausible as another, then the formershould be admitted in the agent’s epistemic state if the latter is.

The Principle of Informational Economy, in a weak form, can be viewed aslimiting the extent of change to that sphere containing the closest:�-worldsand not beyond. The Principle of Weak Preference determineswhich worldsinside this limited region should be included in the new epistemic state. With-out any further restrictions it suggests that all worlds inside this region shouldform part of the contracted epistemic state. In a way, even AGM appeal to thisprinciple. There, however, the principle is only applied relative to:�-worlds,not all worlds inW. However, no principle authorising a restricted impositionof this principle is established. The new situation is illustrated in Figure 2.The agent has determined a preference over worlds and does not prefer the(closest):�-worlds over the (closer)�-worlds just because it is giving upbelief in�. Its preferences are established prior to the change and we assumethat there is no reason to alter them in light of the new information (epistemicinput). It is for this reason that the Principle of Conservatism (the Principle

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ML

[ φ]¬

[K]

c ( ) S φ ¬

Figure 2. Sphere semantics for severe withdrawal showing[K ..��] shaded.

of Informational Economy in its pure form) must give way. We shall refer tothis type of belief removal assevere withdrawal.

Denoting bycS(�) the closest sphere toK containing�-worlds, as men-tioned above, we obtain a severe withdrawal function as follows:

(Def..� from S) K ..�� = th(cS(:�))

It is the study of this class of functions to which we devote ourselves here.When considered in a common setting, this form of belief removal has beenindependently advocated by Levi [20] who refers to such functions asmildcontractions. Levi argues for mild contractions in terms of an informationtheoretic argument. We shall return to a consideration of Levi’s arguments inthe discussion (in Section 13).11

4. Entrenchment-Based Withdrawals

Lewis [21, Section 2.5] was perhaps the first to realise that atotal order-ing over possible worlds could be rephrased as a total ordering over the sen-tences of a language. Grove [12] provides such an ordering based on systemsof spheres centred on[K]. Gardenfors and Makinson [9] also introduce anordering over sentences known as anepistemic entrenchment.12 Intuitively,an epistemic entrenchment relation� is an ordering over the agent’s beliefswhich reflects the plausibilities or degrees of retractability from a given beliefstateK. The relation� � can be read as “it is at least as hard to discard than it is to discard�.” Epistemic entrenchment relations are thought of assatisfying a number of structural constraints which we needpresent only inSection 11.

Now let a relation� of epistemic entrenchment be given, and let< be itsasymmetric part. Then the contraction based on< as suggested by Gardenforsand Makinson [9, (C-) condition)] is as follows.

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(Def .� from�) K .�� = �K \ f : � < � _ g if � 2 K and 6` �K otherwise

Although Gardenfors and Makinson [9, p. 89] offer a motivation for this def-inition, it is rather hard to understand. Besides, as Gardenfors and Makinsonpoint out, their argument in support of (Def.� from �) depends on the con-troversial postulate of recovery which we will discuss below.

In contraction, a basic idea seems to be that less epistemically entrenchedsentences are to be given up in favour of more entrenched sentences [8,pp. 17–18, 75, 87]. Such an interpretation is vaguely reminiscent of an impo-sition of the Principle of Preference. In a related fashion,a more straightfor-ward way of using� was aired by Rott [29] (also compare Gardenfors andRott [11, p. 73]):

(Def ..� from�) K ..�� = �K \ f : � < g if � 2 K and 6` �K otherwise

We shall see that condition (Def..� from�) can in fact be used in both direc-tions. On the reverse reading,� is epistemically less entrenched then , exact-ly when the successful removal of� from epistemic stateK results in theretention of .

As in the case of Grovean sphere-based contractions (alias AGM contrac-tions), the pure idea of minimising the amount of information lost is compro-mised in GM entrenchment-based contractions determined by(Def .� from�). In general, the result of an entrenchment contraction is not a maximalsubset of the theoryK that does not entail�.13 If � is in K, then everysuch maximal non-implying subset includes either� _ or � _ : ; if it didnot, then it would not be maximal by disjunctive reasoning (which followsfrom our assumptions forCn). However, it need not be the case that either� < � _ (� _ ) or � < � _ (� _ : ). The reason for this is that there maybe ties in the plausibility of beliefs — just as there were ties in theplausi-bility of models. In the technical framework used for relations of epistemicentrenchment, the situation that neither� _ nor � _ : is in the contrac-tion of K with respect to� arises just in case� _ is as equally plausibleor entrenched as� _ : . So Gardenfors and Makinson are ready to let the“Principle of Indifference” override the Principle of Minimal Change at leastas far as disjunctions of� and some other sentence are concerned. However, itis generally not the case that if remains untouched inK .�� and� is equal-ly or more entrenched than , then� remains untouched inK .�� as well.Thus the Principles of Indifference and Preference are violated. We shall ful-ly install these principles for the entrenchment-based removal of beliefs byendorsing (Def

..� from �) in this paper. According to this definition, onlypreferences matter, the content of the beliefs remains totally disregarded.

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5. The AGM Postulates for Contraction

In the preceding two sections we have discussed, from an intuitive standpoint,two constructive modellings for AGM belief change. These modellings aremore often characterised byrationality postulateswhich specify axiomaticconstraints that should be satisfied by any contraction operator of that partic-ular type. As with the constructive modellings, they are guided by the ratio-nality criteria outlined in the introduction. The following postulates are thosefor AGM contraction over a belief setK.

( .�1) K .�� = Cn(K .��)( .�2) K .�� � K( .�3) If � =2 K, thenK � K .��( .�4) If 6` �, then� =2 K .��( .�5) K � Cn((K .��) [ f�g)( .�6) If Cn(�) = Cn( ), thenK .�� = K .� ( .�7) K .�� \K .� � K .�(� ^ )( .�8) If � =2 K .�(� ^ ), thenK .�(� ^ ) � K .��

The reader familiar with the AGM postulates for contractionwill noticethat postulate (.�3) is given in a slightly weaker form than usual [8, p. 61]. Theusual consequent,K .�� = K, is easily recovered with the help of (.�2). Italso follows from (.�1), ( .�2) and (.�5) thatK .�� = K for every� 2 Cn(;).This condition is sometimes referred to asFailure (c.f. [14, p. 109]).

The most controversial of the AGM postulates for contraction is ( .�5)which is commonly referred to asRecovery. In the presence of postulates( .�1) and (.�2) it implies thatK = Cn((K .��) [ f�g) if � is in K. Thatis, removing a sentence� and then restoring it leads to the original belief setwhenever� is in that belief set to begin with. Interestingly, recoveryhas nocounterpart among the postulates for AGM revision14 which may be definedfrom contraction via the Levi Identity.15 We shall not enter into the polemicsurrounding the recovery property but, instead, refer the interested reader tothe relevant literature [13, 18, 22, 23, 25].

We find it important to also consider the following weaker versions of( .�7) and (.�8). We note, given postulates (.�1) — ( .�6), that (.�7) implies( .�7c) and (.�8) implies (.�8c) (see [30, Lemma 1]).

( .�7c) If 2 K .�(� ^ ) , thenK .�� � K .�(� ^ )( .�8c) If 2 K .�(� ^ ) , thenK .�(� ^ ) � K .��

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The derivation of (.�7c) from (

.�7) uses (.�5), while the derivation of

(.�8c) from (

.�8) uses (.�4) and Failure. Postulates (

.�7) and (.�8) are, respec-

tively, the contraction counterparts of the rules Or and Rational Monotonyused in nonmonotonic reasoning. On the other hand, postulates (.�7c) and( .�8c) are the contraction counterparts of the rules Cut and Cumulative Monotonyrespectively.16 In nonmonotonic reasoning, Cut and Cumulative Monotonyare considered to be much more fundamental than Or and Rational Monotony.Postulates (.�7c) and (.�8c) are indeed exceedingly plausible in the contextof belief revision as well. Taken together, they state that if is still presentafter the removal of� ^ , then that removal just boils down to the removalof �.

6. Postulates for Severe Withdrawals

As we shall soon see, the following postulates characterisethe new beliefremoval operation advocated in sections 3 and 4. The most obvious differencewith AGM contractions is marked by the absence of the Recovery postulate.17( ..�1) K ..�� = Cn(K ..��)( ..�2) K ..�� � K( ..�3) If � =2 K or ` �, thenK � K ..��( ..�4) If 6` �, then� =2 K ..��( ..�6) If Cn(�) = Cn( ), thenK ..�� = K ..� ( ..�7a) If 6` �, thenK ..�� � K ..�(� ^ )( ..�8) If � =2 K ..�(� ^ ), thenK ..�(� ^ ) � K ..��

Postulates (..�1), (..�2) (..�4), (..�6) and (..�8) are simply those for AGMcontraction overK. Postulate (..�3) contains an additional antecedent in orderto take care of the limiting case of Failure (which was previously handledwith the aid of Recovery). We shall call the collection (..�1), (..�2), (..�3), (..�4)and (..�6) thebasicpostulates. Postulate (..�7) has been replaced by the muchstrongerantitony condition( ..�7a). It states that anything that is given up inorder to remove a strong sentence (the conjunction of� and ) should also begiven up when removing a weaker sentence (�) from the belief set, providedthe latter is not logically true. Intuitively, this makes quite a bit of sense. Ingiving up� ^ at least one of� or must be abandoned. If� is given up inK ..�(�^ ), we can simply achieveK ..�� by abandoning the same beliefs. If is given up instead, we may have to give up more. If we are serious aboutadhering to the Principles of Preference and Indifference we should at least

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give up as much because and the beliefs that have been given up thus farwere apparently held in lower regard.

Clearly, the postulate of recovery does not follow from the present col-lection of postulates. Makinson [23] refers to belief removal operations sat-isfying postulates (.�1) – ( .�4) and (.�6), but not necessarily Recovery, aswithdrawal functions. In this sense, any withdrawal function would be con-sidered weaker than an AGM contraction function. On the other hand, wehave decisively strengthened (.�7) through its replacement by (..�7a). In thisrespect (and by the introduction of the Failure condition in( ..�3)), the result-ing withdrawal function is stronger than an AGM contractionfunction. Forreasons that will become clear later, we call the operationscharacterised bythe above set of postulatessevere withdrawal functions.

Notice also, in the context of the basic postulates (..�1) – (..�4) and (..�6),that (

..�7a) implies (..�7c) and (

..�8) implies (..�8c). Recovery is not required

for these derivations.An alternative axiomatisation of severe withdrawal is given by Pagnucco

[27]. It consists of the AGM postulates (..�1) – (..�4) and (..�6) together withthe following two postulates:

( ..�9) If � =2 K ..� , thenK ..� � K ..��( ..�10) If 6` � and� 2 K ..� , thenK ..�� � K ..� It is shown that postulates (..�7) and (..�8) follow from these postulates. In fact,postulate (..�10) is redundant as we show below. We shall soon see (Lemma 3)that these postulates do not hold in general for AGM contraction.

Postulate (..�9) states that, if� is given up in removing fromK, then any-thing given up in removing� fromK should also be given up when removing fromK. Casting our thoughts back to the principles outlined at theoutset,the antecedent tells us that� is held in no higher regard than (possibly low-er) and therefore no more (perhaps less) need be given up in order to remove when compared to removing�. That is, at least as much work needs tobe done in removing as is required to remove� from K. Postulate (..�10)states that, if a non-tautological sentence� is retained when removing fromK, then whatever is given up to remove should also be given up to remove� from K. When� is held in higher regard than , more work may need tobe done in giving up� than is required to give up .

The following lemma shows that these two proposed axiomatisations areequivalent.

LEMMA 1. Let the basic postulates (..�1) – (..�4) and (..�6) be given. Then(i) ( ..�7a) and (..�8) taken together are equivalent with (..�9);(ii) ( ..�7a) and (..�8) imply (..�10).

The second part shows that postulate (..�10) is indeed redundant and we can

omit it from further consideration although it is of course aproperty of severe

14

withdrawals. Let us briefly look at some further properties following fromour postulates in order to gain a clearer insight into the nature of severe with-drawal.

LEMMA 2. Let..� be a severe withdrawal function overK. Then

(i) EitherK ..�� � K ..� or K ..� � K ..��.(ii) Either K ..�(� ^ ) = K ..�� or K ..�(� ^ ) = K ..� .(iii) If K ..�� ^ � K ..� , then =2 K ..�� or ` � or ` .(iv) If 6` � and 6` , then either� 62 K ..� or 62 K ..��.

The first part of the lemma tells us that severe withdrawals are nested onewithin the other. This attests to the strength of the introduced postulates. Thesecond part states that withdrawal by a conjunction is equivalent to with-drawal by of one its conjuncts (give up the least preferred).This factoringcondition, calledDecomposition[1, p. 525], characterises maxichoice con-traction within the class of AGM partial meet contraction functions [1, Obser-vation 6.3(a)]. In the current context however, we are concerned with with-drawal functions and thus recovery is lacking. The third part of the lemma isthe condition calledConverse Conjunctive Inclusionin Ferme and Rodriguez[7, p. 4]. Our proof shows that this condition is redundant inthe axiomatisa-tion of these authors (which includes (..�9)). The last property is referred toasExpulsiveness[15, Observation 2.52].18 It says that for any two arbitrarynon-theorems� and , in the removal of one of them the other will also beremoved. Expulsiveness is an undesirable property since wedo not necessari-ly want sentences that intuitively have nothing to do with one another to affecteach other in belief contractions. This is the bitter pill wehave to swallow ifwe want to adhere to the Principles of Indifference and Preference.

Before we progress it will be useful to adopt some uniform terminologyin order to better classify the belief removal operations wehave come acrossthus far. This will also serve to give a clearer picture of howsevere withdrawalfits into the overall scheme of such functions.

DEFINITION 1. Any function .� satisfying (.�1) – ( .�4), ( .�6) is referred toas awithdrawal function. Moreover, any function.� satisfying (.�1) – ( .�4),( .�6) and8>><>>: ( .�7c); ( .�8c)( .�7); ( .�8c)( .�7); ( .�8)( ..�3); ( ..�7a) and ( .�8)9>>=>>; is called a

8>><>>: cumulative withdrawalpreferential withdrawal

rational (or AGM) withdrawalsevere withdrawal

9>>=>>;function.Any withdrawal function satisfying the Recovery postulate(

.�5) is called acontractionfunction.

It should be clear from the foregoing discussion that the class of with-drawals (without recovery) form a linear hierarchy. The labels ‘cumulative’,

15

‘preferential’ and ‘rational’ are borrowed from the corresponding notions inthe theory of nonmonotonic reasoning [24, 30]. Severe withdrawal is the mostrestricted class of those present and then, in order of increasing generality, wehave rational (or, AGM) withdrawals, preferential withdrawals, cumulativewithdrawals and (unrestricted) withdrawals. Adding recovery to a withdraw-al function leads to the corresponding contraction function. What, then, isthe nature of the even more restricted class ofsevere contraction functions?Significantly, no such contraction function exists (on painof triviality).

LEMMA 3. There is no contraction function over a non-trivial belief setKthat satisfies postulates (.�1) – ( .�8) and (..�9).

Here we call a belief setK trivial if it does not contain a non-tautologicalsentence� that does not already axiomatiseK, i.e., for whichK 6= Cn(�).

We are thus faced with genuine alternatives. It is clear that( ..�7a) is notsatisfied by AGM contractions. Unable to obtain a hybrid of AGM contractionand severe withdrawal however, we devote the remainder of this paper toexplaining the relationship between the two and providing two precise AGM-like constructive modellings for severe withdrawal functions.

7. Relating AGM Contraction and Severe Withdrawals

Thus far we have motivated our investigation of belief removal primarilythrough the intuition behind two constructive modellings dealt with in Sec-tions 3 and 4 and the way in which they satisfy certain principles of rational-ity. However, we can study the correspondence between AGM contractionsand severe withdrawals without reference to systems of spheres or entrench-ment relations and, in fact, without reference to any constructive modellingat all. Severe withdrawals are far more “skeptical” than AGMcontractions inthat they lead to theories that are smaller in terms of set-theoretic inclusion.This is but one of the interesting relationships between thetwo.

Makinson [23] observes that withdrawal functions can be partitioned intorevision equivalent classes. Two withdrawal functions,.� and ..� say, arerevi-sion equivalentif the corresponding revision functions, defined from themvia the Levi Identity, are equivalent, i.e., ifK .�� = Cn((K .�:�) [ f�g) =Cn((K ..�:�) [ f�g) = K..�� for all � in L. Moreover, he noted [23, Obser-vation p. 389] that in each revision equivalent class[ .�], the maximal element(in terms of set-theoretic inclusion) was an AGM (partial meet) contractionfunction.

The problem addressed in this section is to find the correspondence betweenrevision equivalent AGM contraction functions and severe withdrawal func-tions. The idea at the back of our minds is that the relationship betweenmatching functions should be exactly as that in the constructions by means

16

of systems of spheres or entrenchment relations. When talking aboutthecor-respondence we presuppose that there is a unique AGM contraction functionand a unique severe withdrawal function in each class[ .�] of revision equiva-lent withdrawal functions. The following lemma shows that this is indeed thecase.

LEMMA 4. Let .� and .�0 be two withdrawal functions that are revisionequivalent. Then.� and .�0 are identical whenever either of the followingtwo clauses holds:

(i).� and

.�0 satisfy (.�1), (

.�2) and Recovery (.�5);

(ii) .� and .�0 are severe withdrawal functions.

Building on earlier results of Gardenfors, Makinson [23, p. 389] gives a some-what roundabout proof of the fact that there is only one element in [ .�] whichsatisfies (.�1) – ( .�6). Part(i) of the above lemma shows that Recovery almostalone guarantees an identity in this case. On the other hand,lacking Recovery,the proof for severe withdrawals in part(ii) makes essential use of postulates(

.�7a) and (.�8c).

The constructive modellings considered in Sections 3 and 4 have indicatedthat, for a severe withdrawal function..� and its revision equivalent AGMcontraction function.�, the belief setK ..�� will contain no more beliefs thanK .��.19 Letting .� be an AGM contraction function, the corresponding severewithdrawal function..� can be defined as follows.

(Def ..� from .�) K ..�� = � f : 2 K .�(� ^ )g if 6` �K otherwise

Intuitively, in giving up�, (Def ..� from .�) tells us to retain those beliefs that would be retained when given a choice to remove either� or (orboth). If is considered more important than� when there is a possibility ofdeciding between them, then this consideration should alsobe kept in mindwhen deciding what to remove in the severe withdrawal ofK by �.

An alternative idea is expressed by the following definition.

(Def0 ..� from .�) K ..�� = � TfK .�(� ^ ) : 2 Lg if 6` �K otherwise

According to (Def0 ..� from.�), in giving up� we should retain those beliefs

that are always retained when given a choice between giving up � or anotherbelief. That is, we retain those beliefs that are always retained when there isthe possibility of removing either� or another sentence (or both).

It turns out that these two approaches are, in fact, equivalent.

LEMMA 5. If .� satisfies (.�1), ( .�2), ( .�5), ( .�6), ( .�7) and (.�8), then(Def..� from

.�) and(Def0 ..� from.�) are equivalent.

17

It remains, however, to show that these definitions are in fact adequate. More-over, we ought to show that revision equivalent severe withdrawals are (set-theoretically) smaller than AGM contractions. The relevant result is as fol-lows. result.

OBSERVATION 6. If .� is an AGM contraction function, then..� as obtainedby (Def ..� from .�) is a severe withdrawal function revision equivalent to.�,andK ..�� � K .�� for all � 2 L.

To indicate that severe withdrawals are in fact very severe compared toother withdrawals in regard to the volume of beliefs removed, we note thefollowing result.

OBSERVATION 7.Let.� be an AGM contraction function. Then the severe

withdrawal function ..� defined from.� by definition (Def..� from .�) is thesmallest withdrawal function satisfying postulate (.�8c) which is revisionequivalent to.�.

Smallness is measured here in terms of set-theoretic inclusion. Thus, severewithdrawal removes more beliefs than a large class of (revision equivalent)withdrawals which encompasses cumulative, preferential and rational with-drawals (as well as their contraction counterparts of course). This is an inter-esting and significant class of belief removal functions because they sat-isfy the contraction counterpart (.�8c) of Cumulative Monotony which isan important and widely accepted property in the study of (nonmonoton-ic)consequence relations.

However, severe withdrawals are not the smallest withdrawal functions.This distinction belongs to a more iron-fisted or procrustean withdrawal func-tion which may be defined as follows.

(Def ...� from .�) K ...�� = � Cn(�) \K .�� if � 2 K and 6` �K otherwise

This definition would work equally well with..� substituted for.�. It deter-mines an excessive type of belief removal. First we show thatit is, in fact, thesmallest revision equivalent withdrawal function.

OBSERVATION 8.Let .� be an AGM contraction function. Then the with-drawal function...� defined from.� by definition (Def...� from .�) is the smallestwithdrawal function which is revision equivalent to.�.

From (Def ...� from .�) it is easy to see that the following property holds:If� 2 K and 6` �, thenK ...�� � Cn(�). Such a withdrawal, uniformly applied,is drastic indeed and it is also counterintuitive. Why should we retain onlyconsequences of the very belief we want to retract?

Returning to our discussion of the relationship between severe withdrawaland AGM contraction, going back in the other direction (i.e., from

..� to.�) is

18

quite simple. Let..� be a severe withdrawal function. Then the corresponding

AGM contraction function.� is defined by

(Def.� from

..�) K .�� = �K \Cn(K ..�� [ f:�g) if 6` �K otherwise

This method consists of consecutively applying the Levi andHarper identi-ties. It has been advocated as a trick of enforcing the Recovery postulate byMakinson [23, pp. 389, 391]. Its adequacy is demonstrated bythe followingresult.

OBSERVATION 9. If ..� is a severe withdrawal function, then.� as obtainedby (Def .� from ..�) is an AGM contraction function revision equivalent to.�,andK ..�� � K .�� for all � 2 L.

The appropriateness of the definitions in this section is further indicated bythe following result demonstrating that.� and ..� induce isomorphic structures[5] via the definitions above. The first part states that successive applicationsof (Def

..� from.�) and (Def

.� from..�), in that order, result in the same AGM

contraction function. The second part states that the corresponding result,mutatis mutandis, holds for severe withdrawal functions.

OBSERVATION 10.(i) If we start with an AGM contraction function.�, turnit into a severe withdrawal function..� by (Def ..� from .�) and turn the latterinto an AGM contraction function

.�0 by (Def.� from

..�), then we end up with.�0 = .�.(ii) If we start with a severe withdrawal function..�, turn it into an AGMcontraction function.� by (Def .� from ..�) and turn the latter into a severewithdrawal function..�0 by (Def ..� from .�), then we end up with..�0 = ..�.

This result implies that (Def..� from

.�) and (Def.� from

..�) induce a one-one correspondence between (revision equivalent) AGM contraction func-tions and severe withdrawal functions.

In this section we have taken a closer look at the interrelationship betweenAGM contraction and severe withdrawal. One very important point to noticeis that, although we are contrasting different belief removal behaviour, thereis no effect on the respective revision operations obtainedvia the Levi Iden-tity. Since different revision behaviour is not, in general, linked with identi-cal belief removal operations, there is a greater degree of freedom in beliefremoval than in belief revision functions. It is our aim hereto indicate thatthere are types of belief removal behaviour, differing fromAGM contractionyet revision equivalent to it, that can be motivated by rational means. In fact,our main aim is to promote severe withdrawal as a highly principled memberof this community. We have also witnessed another member — viz. the “iron-fisted” withdrawal — which is the smallest withdrawal function in a class ofrevision equivalent withdrawals. Makinson [23, p. 389] points out that AGM

19

contraction is the largest withdrawal function in this class. Other possibilitiesto be found in the literature include Levi’s [18] saturatable contractions (usingundamped informational value) and the partial meet varietystudied by Hans-son and Olsson [14], Levi [18, 20] contraction using damped informationalvalue of type 1, Levi [19] contraction using damped informational value oftype 2 (i.e.,mild contractionsor, using our terminology,severe withdraw-al), Cantwell’s [4] fallback-based contraction, Meyeret al.’s systematic with-drawal [26], Lindstrom and Rabinowicz’s [22] interpolation operator, Fermeand Rodriguez’s [6] semi-contraction operator and Nayak’s[p.c.] withdraw-al. Appendix B briefly contrasts these various approaches interms of systemsof spheres.

Having investigated the relationship between severe withdrawal and AGMcontraction functions, we now return to the system of spheres construction forbelief removal functions.

8. Retrieving Systems of Spheres from Rational Withdrawals

In this section we elaborate upon the ideas presented in Section 3 in a moretechnical manner. Grove [12] views maximally consistent sets of sentences(consistent complete theories) as “possible worlds”. An ordering is then imposedover the set of all such possible worldsML. The set of all possible worldsconsistent with a set of sentencesK (not necessarily closed underCn) isdenoted[K] and may be determined as[K] = fm 2 ML : K � mg. Weuse[�] as a shorthand for[f�g]. We also define a functionth : 2ML ! Kmapping sets of possible worlds to belief sets by puttingth(X) = TX foranyX �ML.

Now recall that a system of spheres is a nested collection of sets of worldsin which [K] is the smallest sphere andML is the largest. Formally, we havethe following definition due to Adam Grove.

DEFINITION 2. [12] Let S be any collection of subsets ofML. We callS a system of spheres, centred onX � ML, if it satisfies the followingconditions:

(S1) S is totally ordered by�; that is, ifU; V 2 S, thenU � V or V � U(S2) X is the�-minimum ofS(S3) ML is the�-maximum ofS(S4) If � 2 L and 6` :�, then there is a smallest sphere inS intersecting[�]

(i.e., there is a sphereU 2 S such thatU \ [�] 6= ;, andV \ [�] 6= ;impliesU � V for all V 2 S)

20

That spheres are nested is specified by condition (S1). Condition (S4) guar-antees that there is a smallest or innermost sphere intersecting [�] for any� 2 L. This corresponds to Lewis’ [21, p. 19] limit assumption. Wedenotethis spherecS(�) (cf. Section 3). More formally we have a functioncS : L !2ML defined as follows:

(Def cS) cS(�) = 8>><>>: the sphereU 2 S such thatU \ [�] 6= ; andV \ [�] 6= ;impliesU � V for all V 2 S whenever 6` :�[K] otherwise

This allows us to formally define a functionfS : L ! 2ML returning the�-worlds closest to[K] (cf. Section 3). With each system of spheresS centredon [K] we can associate a functionfS(�) = [�] \ cS(�). Note that, in thecase where �, by (DefcS), we automatically have via (Def.� from S) and(Def ..� from S) thatK .�� = K ..�� = K.

Our main interest in this section is the method used to construct the sys-tem of spheres centred on[K] corresponding to an AGM contraction or severewithdrawal function. Before turning to severe withdrawal functions, we firstadapt Lewis’ [21, pp. 59 and 133–134] and Grove’s [12, p. 162]methods ofconstructing systems of spheres from counterfactuals and revisions, respec-tively, to the context of AGM contraction functions. The idea is to specify amethod by which each sphereX� (the minimal sphere intersecting[:�]) canbe determined. A system of spheresS is then obtained by accumulating allsetsX� so determined and the setML of all worlds just in case it is not iden-tified with one of theX�’s. More specifically,S = fX� : � 2 Lg [ fMLgwheneverK 6= L andS = fX� : � 2 Lg [ fMLg [ f;g otherwise.

A set of possible worldsX � ML is in the Lewis-Grovean system ofspheresS (i.e., is a sphere inS) derived from .� if and only ifX � fm : there is a� such thatK .�� � mg and

for all ; if X \ [: ] 6= ; then[K .� ] � X:This condition, which we shall refer to as thefirst construction of Lewis andGrove, can be rephrased by the following equation:X is in S if and only ifX = S f[K .� ] : X 6� [ ]g20.

However, this is not what is actually used in the completeness proofs ofLewis and Grove. The spheresX� they need for their proofs (Lewis [21, p.59], Grove [12, p. 162]) have the following form, here again transferred fromthe context of counterfactuals and revisions to the contextof contractions. AsetX� of worlds is inS obtained from.� if and only if

(Def S from.�) X� = � Sf[K .� ] : [ ] � [�]g whenever 6` �[K] otherwise

21

We shall refer to this condition as thesecondconstruction of Lewis andGrove. EveryX� thus constructed is a Lewis-Grovean sphere according tothe first construction and it is actually the�-minimal such sphere intersect-ing [:�].21 However, there is no guarantee that all spheres of the first con-struction can be captured by theX�’s. Using either of the first or the secondLewis-Grove construction results in a system of spheres where each spherecan be represented as the union of model sets of a certain collection of theo-ries.

The situation changes if we consider severe withdrawal instead of AGMcontraction. If the second construction is applied to severe withdrawals, thenwe shall see that each sphere consists of the model set of exactly one theory.22

In order to construct a system of spheresS centred on[K] from a severewithdrawal function ..� overK, we essentially identify a sphere inS withthe collection[K ..��] for some� 2 L. Any worlds not accounted for in thismanner (i.e., “irrelevant worlds” — see below) are thrown into the outermostsphereML (by the construction ofS noted above). More precisely, we have:

(Def S from ..�) X� = [K ..��]Note that we do not need a special case for6` � because in this scenario,due to the Failure property captured by postulates (..�2) and (..�3), we haveK ..�� = K soX� = [K]. We now show that for severe withdrawal functions..�, this definition coincides with the second Lewis-Grove condition (Def Sfrom .�).

LEMMA 11. If..� is a severe withdrawal function, then the two conditions

(Def S from .�) and(Def S from ..�) are equivalent.

This result also highlights the special nature of severe withdrawal functions.Due to their properties, we obtain a much simplified way to construct systemsof spheres.

We now briefly investigate several transformations that maybe applied to asystem of spheres without affecting the AGM contraction or severe withdraw-al generated from it. They give rise to systems of spheres that are equivalentin the sense of the following definition.

DEFINITION 3. LetS andS 0 be two systems of spheres, let.� and .�0 bethe contraction functions based onS andS 0 and ..� and ..�0 be the severewithdrawal functions based onS andS 0, respectively. ThenS andS 0 arecalledequivalentif and only if for every sentence� it holds thatK .�� =K .�0� andK ..�� = K ..�0�.

Consider, now, the following operations on systems of spheres.

DEFINITION 4. LetS be a system of spheres centred on[K]. Then

22St (the trimmingof S) is obtained by removing fromS all the spheresS suchthatS is never the smallest sphere intersecting[�], i.e., allS such thatS 6= cS(�) for all � 2 L.Su (theclosure under unionsof S) is obtained by adding toS all the unionsU of classes of spheres inS, i.e., allU such thatU = SS 0 for somesubsetS 0 of S.Scl (the topological closureof S) is obtained by replacing all spheresS inS by the sets of worlds (models) that satisfy the theory ofS, i.e., byreplacing allS in S by [th(S)].

In this last case we can define an operatorcl on sets of worlds (models)ascl(S) = [th(S)]. This operation is clearly a closure operator, i.e.,(i) S �cl(S), (ii) cl(cl(S)) � cl(S) and(iii) S � S0 impliescl(S) � cl(S0). More-over,S andcl(S) have the same theory,th(S) = th(cl(S)). Notice that forevery� 2 L, th(cS(�)) = th(cScl(�)) andth(fS(�)) = th(fScl(�)). Clear-ly, all the operations onS result in systems of spheres, andSt � S � Su,but Scl is in general not comparable to any of the other systems of spheres.Nevertheless, we have the following result relating systems of spheres andtransformations applied to them.

LEMMA 12. S, St, Su andScl are all equivalent.

The final result in this section lends further weight to the suitability of thepairing of AGM contractions and severe withdrawals that we suggested inSection 7. It shows that any two functions related by the appropriate defini-tions generate equivalent systems of spheres.

OBSERVATION 13.Let .� and ..� be corresponding AGM contraction andsevere withdrawal functions either via (Def..� from .�) or via (Def .� from..�). Then .� and ..� lead to equivalent systems of spheres, via (DefS from .�)and (DefS from ..�). More precisely, the system of spheres obtained from..�is the topological closure of that obtained from.�.

For a contraction or withdrawal function.�, call a worldm irrelevant(with respect to .�) if there is no� 2 L such thatm containsK .�� (i.e.,m 62 [K .��]). Lewis-Grove spheres relegate irrelevant worlds to the out-ermost sphere, thereby sacrificing the�-elementarity of their spheres. Insevere withdrawal we, as it were, add the irrelevant worlds to the Lewis-Grove spheres in such a way that they “make no difference” forwithdrawalsbut make sure that the resulting spheres are�-elementary.23

9. Representation Theorems for Sphere-Based Withdrawals

The following analogues of Grove’s results concerning belief contraction(which we state without proof) show that the construction interms of systems

23

of spheres outlined in Sections 3 and 8 is in fact an appropriate rendering ofthe AGM rationality postulates for belief contraction overK.24 The first partof the observation states that the method of adding the innermost:�-worldsto [K] as described by (Def.� fromS) does indeed produce an AGM contrac-tion function. The second part shows that for any AGM contraction functionand belief setK, one can construct a system of spheresS centred on[K]using (DefS from .�) for which the addition of the innermost:�-worlds to[K] corresponds to the contraction ofK by �.

OBSERVATION 14. [12, Theorems 1 and 2](i) If S satisfies (S1) – (S4),then the function.� obtained fromS by (Def .� from S) is an AGM contrac-tion function.(ii) If .� is an AGM contraction function, then.� can be represented as asphere-based contraction, where the sphere systemS on which .� is based isobtained by(Def S from .�) andS satisfies (S1) – (S4).

This result shows the mutual adequacy of definitions (Def.� from S) and

(Def S from.�) introduced here. The corresponding representation theorem

can now be established for severe withdrawal overK.

OBSERVATION 15.(i) If S satisfies (S1) – (S4), then the function..� obtainedfromS by (Def ..� from S) is a severe withdrawal function.(ii) If ..� is a severe withdrawal function, then..� can be represented as asphere-based withdrawal, where the sphere systemS on which ..� is basedis obtained by(Def S from .�) (or equivalently, by(Def S from ..�)) andSsatisfies (S1) – (S4).

The first part shows that the method of taking the smallest sphere intersect-ing [:�], expounded by (Def..� from S) is an accurate rendering of a severewithdrawal function. The second part states that the methodfor construct-ing systems of spheres via (DefS from ..�) or, equivalently (DefS from .�)by Lemma 11, does give a system of spheres for which the smallest sphereintersecting[:�] corresponds toK ..��.

This concludes our direct treatment of severe withdrawal interms of sys-tems of spheres. We shall return to systems of spheres in a slightly differentcontext later.

10. Retrieving Epistemic Entrenchment Relations from RationalWithdrawals

While systems of spheres encode an ordering on worlds (consistent and com-plete sets of sentences), epistemic entrenchment orders sentences. In this sec-tion we concentrate on the methods used to generate an epistemic entrench-ment (relative toK) from a given AGM contraction or severe withdrawalfunction (overK).

24

The fundamental idea of how to retrieve an entrenchment relation frombelief change behaviour is this. A sentence� is epistemically less entrenchedin a belief stateK than a sentence if and only if an agent in belief stateKwho is forced to give up either� or will give up � and hold on to . Thisidea can be set in motion when we realise that to give up either� or canvery well be rephrased as the task of giving up� ^ . So let a contractionfunction .� (of any kind) be given.

(Def< from .�) � < iff 2 K .�(� ^ ) and � =2 K .�(� ^ )The second clause is necessary since the agent may just refuse to withdraw�^ . Rott [11, 30, 31] argues that it is indeed best to work with strict relations< of epistemic entrenchment provided one is interested in having the flexi-

bility to sensibly weaken the postulates involved (in particular, to drop therequirement that everything is comparable in terms of entrenchment) and infinding one-to-one correspondences between postulates forentrenchment andpostulates relating to contraction behaviour or to rational choices. We shallnot, however, pursue this project further here but keep to the original, moresimple, if less flexible, account of Gardenfors and Makinson. Thus we shallwork with non-strict relations� which may be thought of as the conversecomplements of the above-mentioned strict relations and weuse the postu-late (..�4) to restrict refusal of contraction to logically true sentences. Thefollowing is the original definition of Gardenfors and Makinson [9, p. 89].

(Def� from .�) � � iff � =2 K .�(� ^ ) or ` � ^ As with systems of spheres, if..� is a severe withdrawal function, then

the process of retrieving entrenchments from contractionscan be simplifiedconsiderably.

(Def� from ..�) � � iff � =2 K ..� or ` This essentially means that the condition (Def..� from �) (see section 4)can be used in both directions. Except for some limiting cases, is inK ..��if and only if � < . This greatly simplifies the transition between severewithdrawal functions and their associated epistemic entrenchment relations.

We now show that the two conditions above are equivalent as far as severewithdrawals are concerned. Again, as in the case for systemsof spheres, thisis due to the properties induced by the postulates for severewithdrawal.

LEMMA 16. If ..� is a severe withdrawal function, then the two conditions(Def� from .�) and(Def� from ..�) are equivalent.

In order to prove these and subsequent results, we recall thedefinition ofepistemic entrenchment as introduced by Gardenfors and Makinson [8, 9].

25

DEFINITION 5. Let� be an ordering of the sentences ofL. We call� arelation of epistemic entrenchmentwith respect to some belief setK, if itsatisfies the following conditions:(E1) If � � and � � then� � � (Transitivity)

(E2) If � ` then� � (Dominance)

(E3) � � � ^ or � � ^ (Conjunctiveness)

(E4) If K 6= L then:� � for every 2 L iff � =2 K (Minimality)

(E5) If � � for every 2 L, then` � (Maximality)

It follows from (E1) – (E5) that an epistemic entrenchment isa total pre-order over sentences in which tautologies are greatest while non-beliefs aresmallest elements. While an entrenchment ordering is an ordering of beliefsinK, systems of spheres can be seen as ordering worlds outside[K]. We shallreturn to the relationship between entrenchment and systems of spheres in asubsequent section.

The final result in this section lends further weight to our claim that thepairing of AGM contractions and severe withdrawals that we suggested inSection 7 is the right one. The result shows that any two functions related bythe appropriate definitions generate identical relations of epistemic entrench-ment.

OBSERVATION 17.Let .� and ..� be corresponding AGM contraction andsevere withdrawal functions either via (Def..� from .�) or via (Def .� from..�). Then .� and ..� lead to identical entrenchment relations, via (Def� from.�) and (Def� from ..�).

11. Representation Theorems for Entrenchment-Based Withdrawals

In this section we turn to more technical results concerningthe notion of epis-temic entrenchment. In essence, we would like to formally show the appro-priateness of (Def..� from �) and (Def� from ..�) introduced in Sections 4and 10 just as we were able to do for analogous definitions in terms of sys-tems of spheres in Section 9. The following representation theorem is due toGardenfors and Makinson.25OBSERVATION 18. [9, Theorems 4 and 5](i) If � satisfies (E1) – (E5), thenthe function .� obtained from� by (Def .� from�) is an AGM contractionfunction, that is, it satisfies (.�1) – ( .�8).(ii) If .� is an AGM contraction function, then.� can be represented as anentrenchment-based contraction where the relation� on which .� is based isobtained by(Def� from

.�) and� satisfies (E1) – (E5).

26

The first part states that the method of retaining in contracting�when�_ is strictly more entrenched than� gives an AGM entrenchment relation. Thesecond part shows that the appropriate entrenchment relation can be obtainedfrom an AGM contraction function using the recipe given by (Def � from.�).

We can now formulate an entirely parallel representation theorem for severewithdrawals. This result is the epistemic entrenchment analogue of Observa-tion 15 for systems of spheres.

OBSERVATION 19.(i) If � satisfies (E1) – (E5), then the function..� obtainedfrom� by (Def ..� from�) is a severe withdrawal function.(ii) If ..� is a severe withdrawal function, then..� can be represented as anentrenchment-based withdrawal where the relation� on which ..� is basedis obtained by(Def� from .�) (or equivalently, by(Def� from ..�)), and�satisfies (E1) – (E5).

The first part shows that the technique of retaining whenever it is strictlymore entrenched than�, i.e., the technique expounded in (Def

..� from �),gives a severe withdrawal function. The second part states that the method forconstructing entrenchment relations via (Def� from .�), or equivalently via(Def� from ..�), gives an entrenchment relation for which the set of beliefsmore entrenched than� isK ..��.

This result shows that we can use the same sort of entrenchment relation asGardenfors and Makinson but we apply it in a different manner which favoursthe Principles of Preference and Indifference over the Principle of MinimalChange — thereby violating Recovery. We can retain the same definition(Def � from .�) as in the Gardenfors-Makinson framework to reconstructthe underlying entrenchment relation from some observed severe withdraw-al behaviour; in our framework, however, the definition can be simplified to(Def � from ..�). Like Gardenfors and Makinson for the case of AGM con-tractions, we obtain a perfect match between severe withdrawal functions andentrenchment relations.

12. Relating Spheres and Entrenchments

Up till now we have been studying AGM contractions and severewithdrawalsfrom the point of view of both sphere semantics and entrenchment seman-tics; two important constructive modellings in the settingof AGM-style beliefchange. We have found that there is a far-reaching parallel between these twokinds of semantics or constructions for belief change functions. Now we wantto give an explanation of that parallel in terms of a direct bridge between sys-tems of spheres and entrenchment relations, bypassing any particular type ofbelief change function.

27

We begin by considering how to retrieve systems of spheres from epis-temic entrenchment relations. Given some entrenchment relation�, we con-struct the corresponding system of spheresS(�) as follows (that is, we accu-mulate all suchS�’s andML as in Section 8):

(Def S from�) S� = [f : � < g]The set of sentencesf : � < g on the right-hand-side of (DefS from�) isacut in the sense of [29, p. 159]. First we have to check whether we actuallyobtain a system of spheres from this construction.

LEMMA 20. For any entrenchment relation� with respect toK, the systemof spheresS(�) satisfies conditions (S1) – (S4) with respect to[K].

Next we show that the system of spheres obtained from an entrench-ment relation in this way is equivalent with the latter in thesense that itleads to the same AGM contraction and the same severe withdrawal func-tion. More precisely, we show that the AGM contraction function (respec-tively, severe withdrawal function) obtained from a systemof spheresS(�)derived from an entrenchment relation� is the same as the AGM contractionfunction (respectively, severe withdrawal function) obtained directly from theentrenchment relation�.

OBSERVATION 21.For any entrenchment relation�, the AGM contractionsand the severe withdrawals generated from� andS(�) are identical, i.e.,C(S(�)) = C(�) andW(S(�)) =W(�).Here,C(�) refers to the AGM contraction function obtained from the entrench-ment relation� by means of (Def

.� from�). Similarly,C(S(�)) is the AGMcontraction function obtained from the system of spheresS(�) via (Def

.�fromS). Again,W(�) andW(S(�)) refer to the severe withdrawal functionobtained by the relevant definitions in Sections 3 and 4.

Let us now turn our attention to the reverse problem of obtaining an epis-temic entrenchment relation from a system of spheres. Givensome system ofspheresS, we construct the corresponding entrenchment relation�= E(S)as follows.

(Def� from S) � � iff for all S 2 S if S � [�] thenS � [ ]We check whether we actually obtain an entrenchment relation from this

construction.

LEMMA 22. For any system of spheresS with respect to[K], the entrench-ment relationE(S) satisfies conditions (E1) – (E5) with respect toK.

Given the nestedness (S1) and the limit assumption (S4) for systems ofspheres, this condition reduces to the following.26

28

(Def0 � from S) � � iff cS(: ) 6� [�]We first show that this definition fits together with the one forthe converse

direction introduced above. The notation employed in the following observa-tion should be self-explanatory by now.

OBSERVATION 23.Let� be an entrenchment relation andS a system ofspheres. Then

(i) E(S(�)) =�.(ii) S(E(S)) is the topological closure of the trimming ofS, i.e.,(St)cl.

The first part of this result exposes a strong connection between epistemicentrenchment and systems of spheres. The second result, while not quiteas strong, shows that applying (Def0 � from S) followed by (DefS from�) leads to an equivalent (although not necessarily identical — see Lem-ma 12) system of spheres. Together they indicate an isomorphism betweenepistemic entrenchment and a particular subclass (those trimmed and topo-logically closed) of systems of spheres.

We finally show the sphere analogue of Observation 21. That is, that theAGM contraction function (respectively, severe withdrawal function) obtainedfrom an entrenchment relationE(S) derived from a system of spheresS is thesame as that obtained directly fromS itself.

OBSERVATION 24.For any system of spheresS, the AGM contractions andthe severe withdrawals generated fromS andE(S) are identical, i.e.,C(E(S)) =C(S) andW(E(S)) =W(S).Taken together, these results demonstrate the appropriateness of the defini-tions introduced in this section.

13. Discussion

Levi [18] advocates a construction for belief removal basedon saturatablesetsrather than AGM’s maximal consistent subsets ofK not implying �(denotedK?�). He notes that all elementsK 0 of K?� have the propertythat Cn(K 0 [ f:�g) is a consistent complete theory (i.e., obey themaxi-choiceproperty). Yet, there are subsets ofK not in K?� also possessingthis property. These sets Levi refers to assaturatable sets(the collection ofwhich we denoteK??� here). More precisely,K 0 2 K??� if and only if(i) K 0 � K, (ii ) � 62 K 0, and (iii ) Cn(K 0 [ f:�g) is a consistent com-plete theory. Hansson and Olsson [14] place this work in context with theAGM showing that a selection function applied to the set of saturatable setsgenerates a withdrawal function that satisfies postulates (

.�1) – (.�4), (

.�6)and Failure. In other words, this construction can be seen ascapturing thatof a withdrawal function satisfying the Failure property. They extend this

29

work by showing that a selection function defined via a real-valued measure(satisfying a weak monotonicity condition) gives a construction satisfyingthe supplementary postulates (.�7) and (.�8). However, they do not supply a“completeness” result for this extended set of postulates.In light of the workpresented here, severe withdrawal is a further restricted construction that canbe given a complete characterisation. That is, a severe withdrawal representsan axiomatisable subclass of those belief removal operations characterised byHansson and Olsson’s real-valued measure selection function construction.

Let us, however, return to Levi’s arguments on this subject.Levi uses thetermcontractionto denote any function removing, say,� fromK. In Makin-son’s [23] terminology which as adopted in our Definition 1 such functionsare termedwithdrawals; contraction being reserved for those withdrawalssatisfying the additional Recovery postulate (.�5) and characterisable via meetsof maximal non-implying subsets. The class of withdrawals can be obtainedby taking meets of saturatable contractions removing� but not meets of max-imal subsets not implying�. For this reason Levi maintains that one shouldconsider meets of saturatable contractions rather than merely meets of maxi-mal non-implying subsets. While AGM begin with the concept of a maximalnon-implying subset as a way of achieving a minimal (in the sense of setinclusion) change in removing� fromK before settling on (partial) meets ofsuch sets, Levi begins at the “other end.” He embraces saturatable sets sincemeets of these will capture all withdrawal behaviour. Of course, admittingsaturatable sets, and meets of them, violates Recovery (seeFigures 4 and 5 inAppendix B) — a postulate Levi is strongly opposed to.

Now Levi’s major concern in contraction follows the broad aims of thePrinciple of Minimal Change and, more specifically, the Principle of Infor-mational Economy; that is, to minimise the loss of informational value. Assuch, it is important to specify how informational value is measured. Leviconsiders three different measures at various stages during the developmentof his ideas. Initially he consideredundamped (or probability-based) infor-mational value[18, p. 127] where the loss of informational value of the meetof a set of saturatable contractions is calculated using thesum of the losses ofinformational value of the minimal (in the sense of set inclusion) members ofthis set. He rejected this proposal immediately as it leads to a saturated con-traction in every case and therefore satisfies the maxichoice property whichboth Levi and AGM agree is unreasonable. In its place he advocateddampedinformational value (version 1)[18, 20] in which intersections (meets) of sat-uratable contractions incur a loss of informational value equal to the largestloss incurred by a member of the set. However, Levi [20, p. 32]cites anexample where he claims that a class of version 1 contractions which satisfyRecovery are counterintuitive. Moreover, Levi claims lackof uniformity inthat damped informational value (version 1) equals undamped informationalvalue in some cases but the two diverge in others. As a result,this was super-seded bydamped informational value (version 2)[19] where loss of infor-

30

mational value is minimised by taking the meet of those maximal subsetsof K not implying� with minimal undamped informational value and othersaturatable contractions with (undamped) informational value no greater thanthis. In these latter two methods Levi adopts a Rule for Ties where “when twoor more options tie for optimality one should adopt the intersection of all ofthem” [20, p. 27] with the proviso that such a “tie breaking” mechanism beadopted only when the resultant option is optimal. This lastclass of contrac-tion functions are referred to asmild contractionsby Levi [20]. It turns outthat, when placed in a common setting, mild contractions coincide with severewithdrawals. Interestingly enough it has turned out, by ourobservations sur-rounding (Def0 ..� from .�), that Levi could have captured “mild contractions”by considering meets of maximal non-implying subsets — although, to con-tractK by � you would need to consider meets of certain maximal subsets ofK not implying� ^ for all 2 L.

Levi [20] criticises our choice of terminology because it isbased on ameasure of loss in terms of subset inclusion (which we do not deny) and hemaintains that informational value should not be measured in these terms; lossof damped informational value of type 2 is minimised and thusthe contrac-tion (or withdrawal) is mild. We wish to emphasise, however,that while ourterminology is influenced by the fact that severe withdrawals tend to removemore beliefs than other revision equivalent proposals (seeObservation 7), ourarguments in favour of severe withdrawal in this paper are not motivated bythis factor at all (nor, of course, by Informational Economy) but, rather, by theconcerns of principled belief removal behaviour and, most of all, respectingof Indifference and Preference.

In an elegant paper, Kaluzhny and Lehmann [17] give a characterisation ofnonmonotonic inference operationsInf for which Inf (�) can be representedas the set of all monotonic consequences together with some set Ass(�) ofassumptions that are “compatible” with�: Inf (�) = Cn(� [ Ass(�)).27Their intuitive idea is that the assumption operatorAss(�) is antitonic in thesense that for� � � we getAss(�) � Ass(�). The more premises, the lessassumptions are compatible with them.

Given the well-known connections between nonmonotonic reasoning andbelief revision (see for instance [10, 11]), it is easy to recognise that for finite� � L, Kaluzhny and Lehmann’s assumption setAss(�) corresponds to oursevere withdrawalK ..�(:V�), with K = Inf (;) left implicit. Their condi-tion of antitony is the analogue of our condition (..�7a). The constructions forAss(�) they use in their Theorems 2.1 and 2.2, viz.Ass(�) = T fInf (�) :� � �g andAss(�) = T fInf (�) : � � Cn(�)g respectively, are reminis-cent of our definition (Def0 ..� from .�). But there are also important differ-ences. They work on the level of postulates only without considering explicitconstructions of nonmonotonic inference operations or thegeneral principlesthat might motivate them. They work in contexts that do not validate the ruleof Rational Monotony which corresponds to the belief revision postulate (

.�8)

31

alias (..�8). And, perhaps most importantly, nonmonotonic inference relations

correspond torevisionsrather than removals of beliefs. Due to the revisionequivalence of belief contractions and withdrawals, then,the distinction weare most interested in vanishes. To put it differently, Kaluzhny and Lehmanndo not present a study of theirAssoperation in its own right.

14. Conclusions

The AGM account of belief change is guided by principles of rationality.However, contrary to the popular perception given by the literature, the Prin-ciple of Informational Economy cannot be given unrestrained prominenceover other rationality principles. It must be seen as only one of a number offactors to be taken into consideration when deciding which beliefs to discard.In fact, it works in combination with principles such as those of Indiffer-ence and Preference in this regard. Once this is accepted, itcan be seen thatAGM are in fact applying the latter principles only in so far as the:�-worldsare concerned and disregarding the�-worlds. This position seems difficult tomotivate and support. As a result, we propose a new form of belief removaloperation, severe withdrawal, which applies these principles uniformly overall possible worlds. The contentious postulate of recoveryis not satisfied bysevere withdrawal.

In the present work we have attempted a comprehensive treatment of analternative to AGM contraction which takes the Principles of Indifferenceand Preference into account; we call this severe withdrawal. We showed howthese principles point toward a different way of using two important AGMconstructions: systems of spheres and epistemic entrenchment. In these con-structions the objects to which the principles are applied are, in the first case,worlds (or models) and, in the second, sentences of the object language. Bothmethods lead to simple mechanisms for constructing removals of belief.

Interestingly enough, if one prefers to focus on belief revision rather thenbelief removal, then the effects, via the Levi identity, areunnoticeable. Thatis, in any revision equivalent class of withdrawal functions there will be exact-ly one AGM contraction function [23] and one severe withdrawal function.Severe withdrawal functions can be seen as setting a lower bound on inter-esting withdrawal behaviour within each of these revision equivalent classes.We furnished a way of moving backwards and forwards between the cor-responding AGM contraction function and severe withdrawalfunction in agiven class. We also supplied methods for obtaining the desired severe with-drawal behaviour from the constructive modellings of systems of spheres andepistemic entrenchment relations. Furthermore, mechanisms for going backthe other way — extracting the relevant underlying structure (total pre-orderon worlds or one on sentences) — were given. It is interestingto note inregard to this latter point that the definitions for AGM contraction can be

32

used for severe withdrawal to achieve the same effect but, ingeneral, may besimplified. Finally methods were given for mapping directlybetween systemsof spheres and epistemic entrenchment relations that lead to the same AGMcontraction or severe withdrawal function.

One last, and important, moral can be drawn from this exposition withregard to the constructive modellings. Clearly the underlying structure (a sys-tems of spheres or an epistemic entrenchment relation) is important in achiev-ing belief removal (or belief change in general for that matter). However, theway we use this structure is also very crucial. Starting witha fixed struc-ture, different principles give rise to different behaviour. More importantly,this behaviour, through the principles that bring it about,can be motivatedby rational means. Here, the Principles of Indifference andPreference —arguably rational integrity constraints — lead to severe withdrawal.

Acknowledgements

The authors would particularly like to thank Isaac Levi for sharing variousmanuscripts of his work and providing generous comments on the work con-tained here-in. They would also like to thank Allen Courtney, Eduardo Ferme,Norman Foo, Sven Ove Hansson, Rex Kwok, Abhaya Nayak, PavlosPep-pas, Mary-Anne Williams, participants of the Swedish-German Workshopon Belief Revision held in Leipzig during May 1996 and members of theDepartment of Philosophy at Uppsala University for many fruitful discus-sions regarding the content of this paper. Parts of this paper were writtenwhile the first author visited Australia. Support from the Australian ResearchCouncil through the Knowledge Systems Group at the University of Sydney(now at the University of New South Wales), the Department ofManagementat the University of Newcastle and the Deutsche Forschungsgemeinschaft isgratefully acknowledged.

Appendix

A. Proofs

We reproduce the following properties ofth : 2ML ! K, listed by Grove[12], for reference. They will be useful for some of the proofs that follow.

LEMMA 0. Properties ofth [12].

(i) th([K]) = K for all belief sets (i.e., theories)K if the underlying logic iscompact

(ii) th(X) 6= K? if and only ifX is nonempty

33

(iii) For any sentence� 2 L andX �ML, th(X\[�]) = Cn(th(X)[f�g)(iv) For X; X 0 �ML, if X � X 0, thenth(X 0) � th(X)(v) ForK; K 0 2 K, if K � K 0, then[K 0] � [K]LEMMA 1. Let the basic postulates (..�1) – (..�4) and (..�6) be given. Then

(i) ( ..�7a) and (..�8) taken together are equivalent with (..�9);(ii) ( ..�7a) and (..�8) taken together imply (..�10).

Proof. Assume that the basic postulates (..�1) – (..�4) and (..�6) are satis-fied.

(i) ( ..�9) implies (..�7a): Let 6` �. Then by (..�4), � =2 K ..��, so by (..�1),� ^ =2 K ..��. Hence, by (..�9),K ..�� � K ..�(� ^ ).( ..�9) implies (..�8): This is immediate on substituting� ^ for .( ..�7a) and (..�8) imply (..�9): Let � =2 K ..� . With the help of (.�8c),� =2 K ..�(� ^ ). Hence, by (..�8), K ..�(� ^ ) � K ..��. But by (..�7a),K ..� � K ..�(� ^ ) whenever6` and henceK ..� � K ..�� as desired.

If ` �, thenK ..� = K by (..�2) and (

..�3). NowK � K ..�� by (..�2) and

thereforeK ..� � K ..�� trivially.(ii) Let 6` � and� 2 K ..� . If ` , thenK � K ..� by (..�3), soK ..�� �K ..� follows from (..�2). So let be such that6` . From� 2 K ..� and

( ..�7a), we conclude that� 2 K ..�(�^ ). Since6` �^ , ( ..�1) and (..�4) giveus =2 K ..�(� ^ ). Hence, by (..�8), K ..�(� ^ ) � K ..� . On the otherhand, by (..�7a),K ..�� � K ..�(� ^ ). HenceK ..�� � K ..� , as desired.2LEMMA 2. Let ..� be a severe withdrawal function overK. Then

(i) EitherK ..�� � K ..� or K ..� � K ..��.(ii) Either K ..�(� ^ ) = K ..�� or K ..�(� ^ ) = K ..� .(iii) If K ..�� ^ � K ..� , then =2 K ..�� or ` � or ` .(iv) If 6` � and 6` , then either� 62 K ..� or 62 K ..��.

Proof. (i) Consider two cases: (a)� 62 K ..� and (b)� 2 K ..� In theformer case (..�9) givesK ..� � K ..��. In the latter case, if6` �, then (..�10)givesK ..�� � K ..� . Otherwise, � and by (..�3)K � K ..��, and by (..�2)K ..� � K soK ..� � K ..��.

(ii) Using (.�7c) and (.�8c) it is easily seen that if 2 K ..�(� ^ ), thenK ..�(� ^ ) = K ..��, and if� 2 K ..�(� ^ ), thenK ..�(� ^ ) = K ..� .A similar situation holds for� 2 K ..�(� ^ ). Consider then, the case where�; 62 K ..�(� ^ ). By two applications of (..�8),K ..�(� ^ ) � K ..�� andK ..�(� ^ ) � K ..� . Now consider two further subcases: (a) at least one of6` � or 6` holds, and (b) both � and` hold. In the former case, eitherK ..�� � K ..�(� ^ ) or K ..� � K ..�(� ^ ) holds by (

..�7a). It follows

34

that eitherK ..�� = K ..�(� ^ ) or K ..� = K ..�(� ^ ). In the latter case,K ..�� = K ..� = K ..�(� ^ ), by (..�2) and (

..�3).(iii) LetK ..�� ^ � K ..� and 6` � and 6` . Suppose for reductio that 2 K ..��. Then by (..�7a) 2 K ..�(� ^ ). So sinceK ..�� ^ � K ..� ,

we also get 2 K ..� , contradicting (..�4).(iv) Let 6` � and 6` . Suppose for contradiction that both� 2 K ..� and 2 K ..��. Then by (..�10),K ..�� = K ..� , so� 2 K ..�� and 2 K ..� ,

contradicting (..�4). 2LEMMA 3. There is no contraction function over a non-trivial belief setKthat satisfies postulates (.�1) – ( .�8) and (..�9).

Proof. Suppose there is a contraction function.� overK that satisfies all

of (.�1) – (

.�8) and (..�9). Suppose further thatK is non-trivial, i.e., that there

is a� such that� 2 K nCn(;) andK 6� Cn(�). We first show thatK .�� =Cn(;). Suppose, forreductio ad absurdum, that there is a 62 Cn(;) suchthat 2 K .��. Now 2 K by ( .�2). It follows by (..�10), which we showedto follow from ( .�1) – ( .�8) and (..�9) (in Lemma 1(i) and(ii) ), thatK .� �K .��. By ( .�5), ( .�1) and the Deduction Theorem ! � 2 K .� � K .��.However, by (.�4),� 62 K .�� so by (.�1), 62 K .�� contradicting our initialsupposition. ThereforeK .�� = Cn(;). Consequently,K 6� Cn((K .��) [f�g) = Cn(�) violating recovery (.�5). 2LEMMA 4. Let .� and .�0 be two withdrawal functions that are revisionequivalent. Then.� and .�0 are identical whenever either of the following twoclauses holds:

(i) .� and .�0 satisfy (.�1), ( .�2) and Recovery (.�5);(ii)

.� and.�0 are severe withdrawal functions.

Proof. Let .� and .�0 be revision equivalent withdrawal functions.(i) Let .� and .�0 satisfy (.�1), ( .�2) and Recovery (.�5). We need to show

that .� = .�0. Left to right inclusion. Suppose 2 K .��. We need to showthat 2 K .�0� as well. First we show that:� ! 2 K .�0�. From 2K .��, we conclude using monotonicity ofCn and the Levi identity that 2Cn((K .��) [ f:�g) = K � :�. By revision equivalence, we get 2 K �0(:�) = Cn((K .�0�) [ f:�g), so by the Deduction Theorem forCn and( .�1), :� ! 2 K .�0�. On the other hand, we know from (.�2) that 2K .�� � K. So by (.�5) and (.�1), � ! is in K .�0�. From this and thepreviously established fact that:�! is inK .�0�, we conclude with (.�1)that is in fact inK .�0�, as desired.

The right to left inclusion can be proved in the same fashion with.� and

.�0 exchanged.

35

(ii) Let.� and

.�0 be severe withdrawal functions. We need to show that.� = .�0. Left to right inclusion. Suppose 2 K .��. We need to show that 2 K .�0� as well. If� 2 Cn(;), thenK .�� = K = K .�0�, by (..�2) and( ..�3). So let� =2 Cn(;). Then it follows from 2 K .�� that 2 K .�(�^ ),by (..�7a). Thus also 2 Cn(K .�(� ^ ) [ f:(� ^ )g) = K � :(� ^ ),using the monotonicity ofCn and the Levi identity. By revision equivalence,then 2 K �0:(�^ ) = Cn(K .�0(�^ )[f:(�^ )g). By the DeductionTheorem forCn and (..�1), we get that:(�^ )! 2 K .�0(�^ ), whichmeans, by (..�1) again, that 2 K .�0(� ^ ). Using (.�8c) (or alternatively,( ..�4) and (..�8)), we get thatK .�0(� ^ ) � K .�0� and thus 2 K .�0�, asdesired.

The right to left inclusion can be proved in the same fashion with .� and.�0 exchanged. 2LEMMA 5. (Def

..� from.�) and(Def0 .� from

..�) are equivalent.

Proof. It is sufficient to show, for every� 2 L such that6` �, that� 2K .�(� ^ �) iff � 2 TfK .�(� ^ ) : 2 Lg.Right to left is trivial. Let� 2 TfK .�(� ^ ) : 2 Lg. Consequently,� 2 K .�(� ^ ) for all 2 L. Choosing � � we get� 2 K .�(� ^ �) as

desired.From left to right, let� 2 K .�(�^�). We need to show� 2 TfK .�(� ^ ) : 2 Lg. We can do so by showing that� 2 K .�(� ^ ) for arbitrary 2 L. Now�_:�_: 2 K .�(�^�) = K .�((�_:�_: )^ ((�^�)_(� ^ ))) using (.�1) for the former part and (.�6) for the latter. It follows

by (.�8c) and (

.�6) thatK .�(� ^ �) � K((� ^ �) _ (� ^ )). Therefore,� 2 K .�((� ^ �) _ (� ^ )). From our initial supposition and (.�2),� 2 K

giving by (.�5) and (

.�1) that( _ :�)! � 2 K .�( _ :�). Consequently� 2 K .�( _ :�) by ( .�1). It therefore follows by (.�7) and our previousreasoning that� 2 K .�(((� ^ �) _ (� ^ )) ^ ( _ :�)). Hence, by (.�6)� 2 K .�(� ^ ) as desired. 2OBSERVATION 6. If .� is an AGM contraction function, then..� as obtainedby (Def ..� from .�) is a severe withdrawal function revision equivalent to.�,andK ..�� � K .�� for all � 2 L.

Proof. Let .� be an AGM contraction function and..� be obtained from.�via (Def

..� from.�). We first show that

..� is a severe withdrawal function.(By Lemma 5, (Def

..� from.�) and (Def0 ..� from

.�) are equivalent so wecan make use of both definitions to simplify the proof.)

36

(..�1) If 6` �, thenK ..�� = TfK .�(� ^ ) : 2 Lg by (Def0 ..� from

.�).SinceK .�(� ^ ) is a theory for every 2 L by (

.�1), then clearlyK ..�� istoo. Otherwise, � in which caseK ..�� = K and againK ..�� is a theory.

( ..�2) If 6` �, thenK ..�� = TfK .�(� ^ ) : 2 Lg by (Def0 ..� from .�).SinceK .�(�^ ) � K for all 2 L by ( .�2) clearlyK ..�� � K. Otherwise,` � and by (Def0 ..� from .�) K ..�� = K thereforeK ..�� � K trivially.

( ..�3) If ` �,K ..�� = K by (Def ..� from .�) and the desired result followstrivially. Otherwise,6` � and� 62 K. ThenK ..�� = TfK .�(�^ ) : 2 Lgby (Def ..� from .�). Since� 62 K, then� ^ 62 K for all 2 L. ThereforeK � K .�(� ^ ) for all 2 L by ( .�3). HenceK � K ..�� as desired.

( ..�4) Let 6` �. Now � 62 K .�� by ( .�4). It follows by ( .�6) that� 62K .�(� ^ �). Therefore,� 62 K ..�� by (Def ..� from .�).( ..�6) Follows trivially using (.�6).(..�7a) Let 6` �. Suppose� 2 K ..��. Then via (Def

..� from.�) � 2K .�(�^�). It follows by (

.�7) that� 2 K .�((�^�)^ ). (Actually this lastpart follows more directly from condition (

.�P)K .��\Cn(�) � K .�(�^ )— with � � � ^ and � — which is equivalent to (.�7) [1, Observa-tion 3.3 p. 516]). Hence� 2 K .�� ^ by ( .�6) and (Def..� from .�).

( ..�8) Let� 62 K ..�(� ^ ). We need to show thatK ..�(� ^ ) � K ..��. If` �, thenK ..�� = K by (Def ..� from .�) andK ..�(�^ ) � K by (..�2) whichwas shown above to hold. Therefore, it follows directly thatK ..�(� ^ ) �K ..��. Otherwise6` �. Therefore, by (Def..� from .�) K ..�(� ^ ) = f� :� 2 K .�((� ^ ) ^ �)g andK ..�� = f� : � 2 K .�(� ^ �)g. Suppose� 2 K ..�(� ^ ). We need to show that� 2 K ..�� and can do so by showingthat� 2 K .�(�^�). Since� 2 K ..�(�^ ) we have� 2 K .�((�^ )^�) (�)by (Def ..� from .�). It follows that�^ 62 K .�((�^ )^�) by ( .�4) and (.�8)subsequently givesK .�((�^ )^�) � K .�(�^ ) (#). Our initial assumptionthat� 62 K ..�(� ^ ) and (Def

..� from.�) give� 62 K .�((� ^ ) ^ �) which

by (.�6) means� 62 K .�(� ^ ). Using the contrapositive of (#) we get that� 62 K .�((� ^ ) ^ �) and (

.�1) then gives� ^ � 62 K .�((� ^ ) ^ �).Applying ( .�8) again (and an application of (.�6) to the left-hand-side) wesee thatK .�((� ^ ) ^ �) � K .�(� ^ �). It therefore follows from (�) that� 2 K .�(� ^ �) as required.

We now show that.� and ..� are revision equivalent. That is, we show thatK .�� = K..��. NowK .�� = Cn(K .�:� \ f�g) andK..�� = Cn(K ..�:� \f�g) by the Levi identity. We first prove left to right holds. Suppose� 2K .��. Then� ! � 2 K .�:� by the Levi identity, Deduction Theorem and( .�1). Now` [:� ^ (�! �)]$ :� so by (.�6), �! � 2 K .�(:� ^ (�!�)) (�). We consider two cases : (a)6` :�; and, (b)` :�. In the formercase, it follows by (Def..� from .�) and (�) that� ! � 2 K ..�:�. Using theDeduction Theorem,� 2 Cn(K ..�:� [ f�g). Hence� 2 K..�� as required.In the latter caseK .�:� = K = K ..�:� by (

.�1) and (.�5) and (Def

..�from.�) respectively. It follows that�! � 2 K ..�:� and consequently� 2 K..��

37

via applications of the Deduction Theorem and the Levi identity as required.Right to left is similar.

It remains to show thatK ..�� � K .��. This follows straightforwardlyfrom the result by Makinson [23, Observation p. 389] howeverwe includea proof in terms of our own definitions. Suppose� 2 K ..��. If ` �, thenK ..�� = K = K .�� by (Def ..� from .�) for the former part and (.�1) and( .�5) for the latter and the result follows trivially. Otherwise 6` �. Now � 2K .�(� ^ �) by (Def ..� from .�). By ( .�8c)K .�(� ^ �) � K .��. Therefore� 2 K .�� as desired. 2OBSERVATION 7. Let .� be an AGM contraction function. Then the severewithdrawal function..� defined from.� by definition(Def ..� from .�) is thesmallest withdrawal function in terms of set-theoretic inclusion satisfyingpostulate (.�8c) which is revision equivalent to.�.

Proof. Let .� be an AGM contraction function and..� defined from .� via(Def ..� from .�). Let� be any withdrawal function satisfying (.�8c) which isrevision equivalent to.� (and therefore..� also by Observation 6). We need toshow thatK ..�� � K � �.

Suppose 2 K ..��. If ` �, thenK ..�� = K .�� = K (the former by( ..�3) which is satisfied by Observation 6 and the latter by (.�1) and (.�5)).Since� satisfies FailureK � � = K and 2 K � � as desired. Otherwise,6` �. By (Def ..� from .�), 2 K .�(� ^ ) so by the Levi identity:� ^ 2K .�:(� ^ ) = K .�:(� ^ ). By the revision equivalence of.� and� (and..�), :�^ 2 K � :(� ^ ) where� is defined from� via the Levi identity.Using the Levi identity again,:(� ^ ) ! (:� ^ ) 2 K � (� ^ ). Thatis, by (

.�1), 2 K � (� ^ ). But then by (.�8c),K � (� ^ ) � K � �.

Hence 2 K � � as required. 2OBSERVATION 8. Let .� be an AGM contraction function. Then the with-drawal function...� defined from.� by definition(Def ...� from .�) is the smallestwithdrawal function which is revision equivalent to.�.

Proof. Let .� be an AGM contraction function and...� defined from .� via(Def ...� from .�). We first verify that...� is a withdrawal function (i.e., satisfies( .�1) – ( .�4) and (.�5)).

( .�1) If � 2 K and 6` �, we haveK ...�� = Cn(�)\K .�� by (Def ...� from.�) which is obviously closed by (.�1) and the properties ofCn. Otherwise,� 62 K or ` � by (Def

...� from.�). AgainK ...�� is closed.

(.�2) If � 62 K or ` � (Def

...� from.�) givesK ...�� = K in which case

the desired result follows trivially. Otherwise,� 2 K and 6` � soK ...�� =

38

Cn(�) \ K .�� by (Def...� from

.�). Now by (.�2) K .�� � K so clearly

Cn(�) \K .�� � K and thereforeK ...�� � K.( .�3) Let� 62 K. By (Def ...� from .�) K ...�� = K andK � K ...�� follows

trivially.( .�4) Let 6` �. If � 62 K, then by (Def...� from .�)K ...�� = K so� 62 K ...��.

Otherwise,� 2 K andK ...�� = Cn(�)\K .�� by (Def ...� from .�). However,� 62 K .�� by ( .�2) and therefore� 62 Cn(�) \K .�� = K ...��.( .�6) Let` � $ . If � 62 K and` � then clearly 62 K and` . By

(Def ...� from .�) we haveK ...�� = K = K ...� as desired. Otherwise� 2 Kand 6` �. Clearly then 2 K and 6` . NowK ...�� = Cn(�) \ K .�� andK ...� = Cn( ) \K .� . Moreover,Cn(�) = Cn( ) by our supposition atthe outset andK .�� = K .� by ( .�6). HenceK ...�� = Cn(�) \ K .�� =Cn( ) \K .� = K ...� as desired.

(Note: it is easily shown that...� satisfies Failure also.)

Next we show that...� and .� are revision equivalent. Left to right. Suppose� 2 K...��. Then� ! � 2 K ...�:� by the Levi identity, Deduction Theoremand (.�1) (which has been shown above to hold). If:� 62 K or ` :�, thenK ...�:� = K by (Def ...� from .�) andK .�:� = K by ( .�2) and (.�3)/( .�1)and (.�5). Therefore,� ! � 2 K .�:� and� 2 K .�� by the DeductionTheorem and the Levi identity. Otherwise,:� 2 K and` :�. By (Def

...�from

.�), K ...�:� = Cn(:�) \ K .�:� and again it follows that� ! � 2K .�:� whereby we proceed as above.Right to left. Suppose� 2 K .�:�. Then� ! � 2 K .�:� by the Levi

identity, Deduction Theorem and (.�1). If :� 62 K or ` :�, thenK ...�:� =K = K .�:� as above and therefore� ! � 2 K ...�:� whereby the Deduc-tion Theorem and the Levi identity give� 2 K...��. Otherwise,:� 2 K and` :�. Now clearly� ! � 2 Cn(:�). So� ! � 2 Cn(:�) \K .�:� andby (Def ...� from .�) �! � 2 K ...�:�. Hence by the Deduction Theorem andthe Levi identity� 2 K ...�� as desired.

Finally, we show that...� is the smallest withdrawal function revision equiv-alent to .�. Suppose 2 K ...��. We need to show that 2 K � � for anywithdrawal function� revision equivalent to.�. Now 2 K by ( .�2) which...� was shown to satisfy above. If � or � 62 K, thenK .�� = K by ( .�1)and (

.�5)/(.�2) and (

.�3). Since� satisfies Failure and (.�2) and (

.�3) (sinceit is a withdrawal function)K � � = K and 2 K � � as desired. Other-wise, 6` � or � 62 K. By (Def ...� from .�), 2 Cn(�) \ K .��. Therefore, 2 Cn(�) (i.e.,� ` ) and 2 K .��. It follows that:� ^ 2 K .�:� =Cn(K .�� [ f:�g). The revision equivalence of of.� and� (and...�) gives:�^ 2 K � :�. That is,:�^ 2 Cn(K ��[f:�g) and the DeductionTheorem and (.�1) give:�! (:� ^ ) 2 K � �. By ( .�1) again we obtain� _ 2 K � � but since� ` we have 2 K � � as required. 2OBSERVATION 9. If ..� is a severe withdrawal function, then.� as obtainedby (Def

.� from..�) is an AGM contraction function, andK ..�� � K .�� for

39

all � 2 L.

Proof. Let ..� be a severe withdrawal function and.� be obtained from..�via (Def .� from ..�).

We first show that.� is an AGM contraction function.( .�1) In the case that6` �, we haveK .�� = K\Cn(K ..��[f:�g) by (Def

.� from ..�) which is obviously closed by the properties ofCn. Otherwise, �andK .�� = K by (Def .� from ..�). Again,K .�� is closed.

( .�2) If ` �, thenK .�� = K by (Def .� from ..�) and so obviouslyK .�� �K. Otherwise,6` � and (Def .� from ..�) givesK .�� = K \ Cn(K ..�� [f:�g). ClearlyK .�� = K \ Cn(K ..�� [ f:�g) � K soK .�� � K.( .�3) Let � 2 K. If ` �, thenK .�� = K by (Def .� from ..�) andK �K .��. Otherwise, � andK .�� = K \ Cn(K ..�� [ f:�g). Now by (

..�3)we haveK � K ..�� and therefore, by monotonicity ofCn,K � Cn(K ..��[f:�g). HenceK � K \ Cn(K ..�� [ f:�g) and consequentlyK � K .��as desired.

( .�4) Let 6` �. ThenK .�� = K \ Cn(K ..�� [ f:�g) by (Def .� from..�). Suppose� 2 K .��. Then� 2 Cn(K ..�� [ f:�g) by the monotonicityof Cn. By the Deduction Theorem, and (..�1) :� ! � 2 K ..�� or, again by( ..�1),� 2 K ..�� contradicting (..�4). It follows that� 62 K .��.

( .�5) If 6` �, thenK .�� = K \ Cn(K ..�� [ f:�g). By (Def .� from..�). Suppose for reductio ad absurdum that there is a 2 K such that 62Cn(K .�� [ f�g). That is, 62 Cn((K \ Cn(K ..�� [ f:�g)) [ f�g). Bythe Deduction Theorem�! 62 Cn(K \ Cn(K ..�� [ f:�g)). Now either� ! 62 K or � ! 62 Cn(K ..�� [ f:�g). In the former case we havean immediate contradiction since 2 K implies� ! 2 K. In the lattercase we have:�! (�! ) 62 K ..�� by the Deduction Theorem and (

..�1).Equivalently> 62 K ..�� contradicting (

..�1). Otherwise � in which caseK .�� = K andK � Cn(K [ f�g) = Cn(K .�� [ f�g) by monotonicity.( .�6) Let` � $ . Suppose6` �, then 6` . We have by (Def.� from ..�)

thatK .�� = K \ Cn(K ..�� [ f:�g) = K \ Cn(K ..� [ f: g) = K .� .Otherwise,6` � implying 6` andK .�� = K = K .� by (Def .� from ..�).

( .�7) Suppose that6` � and 6` . We need to show thatK .�� \K .� �K .�(� ^ ). By (Def .� from ..�) we have the following:K .�� = K \Cn(K ..� [ f:�g), K .� = K \ Cn(K ..� [ f: g) andK .�(� ^ ) =K \ Cn(K ..� [ f:(� ^ )g). Suppose� 2 K .�� \K .� . Then� 2 K .��and � 2 K .� . So � 2 K by (..�2) and� 2 Cn(K ..�� [ f:�g) and� 2 Cn(K ..� [ f: g) by the monotonicity ofCn. By the Deduction The-orem and (..�1) :� ! � 2 K ..�� and: ! � 2 K ..� . We need to showthat � 2 K .�(� ^ ). We can do so by showing that� 2 K and� 2Cn(K ..�(�^ )[f:(�^ )g) (i.e., by the Deduction Theorem,:(�^ )!� 2 K ..�(� ^ ) or, equivalently,(:� ! �) ^ (: ! �) 2 K ..�(� ^ )). By (

..�7a) and our reasoning above we have that:� ! � 2 K ..�(� ^ )

40

and: ! � 2 K ..�(� ^ ). Therefore, by (..�1), (:�! �) ^ (: ! �) 2K ..�(� ^ ) as desired.

Otherwise, at least one of�, or` holds. If both` � and` hold andtherefore` � ^ , (Def .� from ..�) givesK .�� \K .� = K \K = K =K .�(�^ ) with the result holding trivially. So assume only one of` �, ` holds. Without loss of generality, suppose` � and 6` . NowK .�� = K by(Def .� from ..�). It also follows by (.�2), which we have shown above to hold,thatK .� � K. Therefore,K .��\K .� = K \K .� = K .� � (�^ )(the last of these by (.�7a)).

( .�8) Let� 62 K .�(�^ ). If ` �^ , then (Def .� from ..�) givesK .�(�^ ) = K. It follows by our initial assumption that� 62 K. But this contradictsthe fact that �^ so this case is not possible. Otherwise6` �^ . Moreover,we can assume that6` � for, otherwise,K .�� = K by (Def .� from ..�) andthe result follows directly via (

.�2) which was shown above to hold. Supposenow that� 2 K .�(� ^ ). By (Def

.� from..�) we haveK .�(� ^ ) = K \

Cn(K ..�(�^ )[ f:(�^ )g). This latter fact, together with the DeductionTheorem and (..�1) give:(� ^ ) ! � 2 K ..�(� ^ ) or, in other words(again appealing to (..�1)) (:� ! �) ^ (: ! �) 2 K(� ^ )(#). We alsoknow that either� 62 K or � 62 Cn(K ..�(� ^ ) [ f:(� ^ )g) using theassumption at the outset of this proof. The former does not hold under ourcurrent assumptions so the latter must hold and, via the Deduction Theoremand (..�1), we have:(� ^ ) ! � 62 K ..�(� ^ ) or, in other words� 62K ..�(� ^ ). Now (..�8) and (#) give(:� ! �) ^ (: ! �) 2 K ..��. So,in particular:� ! � 2 K ..�� by (..�1) and the Deduction Theorem gives� 2 Cn(K ..�� [ f:�g). Since� 2 K we have via (Def .� from ..�) that� 2 K .�� as required.

We now show that.� is revision equivalent to..�. That is, we show thatK .�� = K..��. NowK .�� = Cn(K .�:� [ f�g) andK..�� = Cn(K ..�:� [f�g) Left to right. Suppose� 2 K .�� = Cn(K .�:� [ f�g). Now � !� 2 K .�:� by the Deduction Theorem and (.�1) which was shown above to

hold. We consider cases (a)` :�; and, (b)6` :�. In the former case, by (Def..� from .�) K .�:� = K and by (.�1) and (.�5) K ..�:� = K. Clearly then�! � 2 K ..�:� and by the Deduction Theorem� 2 Cn(K ..�:� [ f�g) =K..��. In the latter case, we have� ! � 2 K .�:� by the Levi identity and( .�1). Right to left. Now suppose� 2 K..�� = Cn(K ..�:� [ f�g). By theDeduction Theorem and (..�1) � ! � 2 K ..�:�. We consider two cases (a)` :�; and, (b) :�. In the former case, by (Def.� from ..�)K .�:� = K andby (..�3)K ..�:� = K. Clearly�! � 2 K .�:� and the Deduction Theoremgives� 2 Cn(K .�� [ f�g) = K .��.

Now, in the latter case, by (Def.� from ..�) K .�:� = K \ Cn(K ..�:� [f�g). By the monotonicity ofCn, �! � 2 Cn(K ..�:�[f�g) and�! � 2K. Therefore,� ! � 2 K .�:� by (.�2). Hence� 2 Cn(K .�:� [ f�g) =K .�� by the Deduction Theorem and the Levi identity as desired.

41

It remains to show thatK ..�� � K .�� for all �. This follows straight-forwardly from Makinson’s [23, Observation p. 389] result.However, weinclude a proof in terms of our definitions here. Consider twocases. (a) �;and (b) 6` �. In the former case, by (Def.� from ..�) we haveK .�� = Kand the result follows straightforwardly by (..�2). In the latter case, by (Def.�from ..�) we haveK .�� = K \ (K ..�� [ f:�g). Now suppose� 2 K ..��.Clearly� 2 K by (..�2) and� 2 Cn(K ..�� [ f:�g) by monotonicity ofCn.It therefore follows that� 2 K .��. 2OBSERVATION 10. (i) If we start with an AGM contraction function.�,turn it into a severe withdrawal function..� by (Def ..� from .�) and turn thelatter into an AGM contraction function.�0 by (Def .� from ..�), then we endup with .�0 = .�.

(ii) If we start with a severe withdrawal function..�, turn it into an AGMcontraction function

.� by (Def.� from

..�) and turn the latter into a severewithdrawal function

..�0 by (Def..� from

.�), then we end up with..�0 = ..�.

Proof. Let .�, ..� and .�0 be defined as in the statement above.(i) Left to right. Suppose� 2 K .��. We need to show that� 2 K .�0�.

If ` �, thenK .�� = K ..�� = K andK .�0� = K by ( .�1) and (.�5), (Def..� from .�) and (Def .� from ..�) respectively. Otherwise6` �. By (Def ..�from .�) K ..�� = f : 2 K .�(� ^ )g and by (Def .� from ..�) K .�0� =K \ Cn(K ..�� [ f:�g). Since� 2 K .�� we have� _ � 2 K .�� by ( .�1).Using (.�6) we have� _ � 2 K .�(� _ (� ^ �)) and so� _ � 2 K ..��by (Def ..� from .�). That is, by (..�1) (which is satisfied by Observation 6),:� ! � 2 K ..��. The Deduction Theorem gives� 2 Cn(K ..�� [ f:�g)and by (

..�2) � 2 K. Hence� 2 K .�0� by (Def.�from

..�) as desired.Right to left. Suppose� 2 K .�0�. If ` � we can reason exactly as above.

Otherwise6` �. Now � 2 K and� 2 Cn(K ..�� [ f:�g) by (Def.� from

..�). As a result of applying the Deduction Theorem and (..�1) we have:�!� 2 K ..��. Therefore:�! � 2 K .�(� ^ (:� ! )) = K .�� (the formerpart by (Def ..� from .�) and the latter part by (.�6)). But ( .�5) and (.�1) give� ! � 2 K .��. Putting these together we get by (.�1) that� 2 K .�� asrequired.

(ii) Let ..�, .� and ..�0 be defined as in the statement above. If` � we reasonalong the lines of(i). Otherwise,6` �. Left to right. Suppose� 2 K ..��. By(Def .� from ..�) K .�� = K \ Cn(K ..�� [ f:�g) and by (Def..� from .�)K ..�0� = f� : � 2 K .�(� ^ )g. Now clearly� 2 K by (..�2). So we knowthat� ^� 2 K and 6` � ^ �. We need to show that� 2 K ..�0� which we cando, according to (Def..� from .�), by showing that� 2 K .�(� ^ �). This wecan do, according to (Def

.� from..�) by showing that� 2 K \Cn(K ..�(� ^ ) [ f:(� ^ )g). We have already shown that� 2 K so it remains to show

that� 2 Cn(K ..�(� ^ ) [ f:(� ^ )g) or equivalently, by the Deduction

42

Theorem and (..�1), that:(�^�)! � 2 K ..�(�^ ). That is, by (

..�1) again,� 2 K ..�(� ^ ). Since6` � and� 2 K ..��, this fact follows by (..�7a).

Right to left. Suppose� 2 K ..�0�. Now� 2 K .�(� ^ ) by (Def ..� from.�). By (Def .� from ..�) � 2 K and� 2 Cn(K ..�(� ^ �) [ f:(� ^ �)g).The latter gives:(�^�)! � 2 K ..�(�^�) by the Deduction Theorem and( ..�1). Therefore� 2 K ..�(� ^ �). Hence by (..�4) � 62 K ..�(� ^ �) and by( ..�8) � 2 K ..�� as required. 2LEMMA 11. If ..� is a severe withdrawal function, then the two conditions(Def S from .�) and(Def S from ..�) are equivalent.

Proof. In the case where � both constructions give the sphere[K]. (Forthe right-hand-side use (..�2) and (..�3) and Lemma 0(v)). Therefore, we haveto show that [f[K ..� ] : [ ] � [�]g = [K ..��]whenever6` �.

Left to right. Supposem 2 Sf[K ..� ] : [ ] � [�]g. Thenm 2 [K ..� ]for some[ ] � [�]. We need to show thatm 2 [K ..��]. By (

..�4) � 62 K ..��.Now since[ ] � [�] we have ` � so, by (

..�1), 62 K ..��. Using (..�9) we

haveK ..�� � K ..� . In other words,[K ..� ] � [K ..��] by Lemma 0(v)) asdesired.

Right to left. Supposem 2 [K ..��]. We need to showm 2 [K ..� ] forsome[ ] � [�]. The result follows directly by choosing to be�. 2LEMMA 12. S, St, Su andScl are all equivalent.

Proof. If ` �, then by (DefcS) we havecS(:�) = cSt(:�) = cSu(:�) =cScl(:�) = [K] and by (Def..� from S) K ..�S� = K ..�St� = K ..�Su� =K ..�Scl�. This fact can also be used to showfS(:�) = fSt(:�) = fSu(:�) =fScl(:�) and so, by (Def .� from S), K .�S� = K .�St� = K .�Su� =K .�Scl�.

Therefore we consider the case where6` �. Now by (S4) (and (DefcS))cS(:�) exists. It follows directly by the definition ofSt that cS(:�) =cSt(:�).We now want to show thatcS(:�) = cSu(:�). Suppose to the contrary.

Without loss of generality, since any sphere inS is also inSu by definition,suppose thatcS(:�) � cSu(:�). That is, there is someS 2 Su such thatS \ [:�] 6= ; andS � cSu . By definition ofSu this means that there is someS0 � S andS0 2 S andS0 � cS(:�). But this contradicts the definitionof cS(:�) via (Def cS). HencecS(:�) = cSu(:�). By (Def

..� from S)and using Lemma 0(iv) we getK ..�S� = K ..�St� = K ..�Su�. We can showK .�S� = K .�St� = K .�Su�. in similar fashion.

43

It now remains to considerK .�Scl� andK ..�Scl�. We begin by showingthatth([th(S)]) = th(S) for S �ML (*). Right to left. Suppose� 2 th(S).Now [th(S)] = fm 2 ML : th(S) � mg. Since� 2 the(S), then� 2 mfor all m 2 [th(S)]. Therefore� 2 T[th(S)] and hence� 2 th([th(S)]) asdesired. Left to right. Suppose� 2 th([th(S)]). Further, suppose for reduc-tion that� 62 th(S). Then there is somem 2 ML such thatth(S) � m and:� 2 m by Lindenbaum’s lemma. It follows thatm 2 [th(S)]. Consequent-ly, � 62 T[th(S)] = th([th(S)]) contradicting our initial supposition. Hence� 2 th(S) as desired.

Now th(fScl(:�)) = th([(cS (:�))] \ [:�]) by the definition ofScl andthe definition offS . By Lemma 0(iii) th([(cS(:�))]\[:�]) = Cn(th([th(cS(:�))])[f:�g). Using (*) we have thatth([th(cS(:�))]) = th(cS(:�)). There-fore Cn(th([th(cS (:�))]) [ f:�g) = Cn(th(cS(:�)) [ f:�g) and byLemma 0(iii) againCn(th(cS(:�)) [ f:�g) = th(cS(:�) \ [:�]). Butthis latter part is justth(fS(:�)). Thereforeth(fScl(:�)) = th(fS(:�)).We want to showth([K] [ fS(:�)) = th([K] [ fScl(:�)). Left to right.Now suppose 2 th([K] [ fS(:�)). Then 2 T([K] [ fS(:�)) bydefinition. That is, 2 m for all m2 [K] [ fS(:�) so 2 m for allm 2 [K] and 2 m0 for all m0 2 fS(:�). It follows that 2 th(fS(:�)).Consequently, by the above,th(fScl(:�)). Therefore 2 m for all m 2fScl(:�) and it follows that 2 m for all m 2 [K] [ fScl(:�). As a result 2 th([K] [ fScl(:�)). Right to left is proved similarly. Hence by (Def.�from S) K .�Scl� = K .�S�. Together with the results above we now haveK .�S� = K .�St� = K .�Su� = K .�Scl�.

Now th(cScl(:�)) = th([th(cS))]) and using (*), as above, we haveth([th(cS))]) = th(cS). Thereforeth(cScl(:�)) = th(cS(:�)) and by (Def..� from S) we getK ..�Scl� = K ..�S�. Together with the results above wenow haveK ..�S� = K ..�St� = K ..�Su� = K ..�Scl�. 2OBSERVATION 13. Let .� and ..� be corresponding AGM contraction andsevere withdrawal functions either via (Def..� from .�) or via (Def .� from ..�).Then .� and ..� lead to equivalent systems of spheres, via (DefS from .�) and(DefS from ..�). More precisely, the system of spheres obtained from..� is thetopological closure of that obtained from.�.

Proof. Consider the second Lewis-Grovean construction ofS( .�), givenby

(Def0 S from .�) X� = � Sf[K .�(� ^ )] : 2 Lg whenever 6` �[K] otherwise

and compare it with the construction we get using (DefS from..�) and (Def

..� from.�):

44S(W( .�)): X 0� = [K ..��] = � [TfK .�(� ^ ) : 2 Lg] whenever 6` �[K] otherwise

In order to show thatS(W( .�)) is the topological closure ofS( .�), weprove for each� thatX 0� is the topological closure ofX�, i.e.,X 0� = [th(X�)].Our claim is that[\fK .�(� ^ ) : 2 Lg] = [th([f[K .�(� ^ )] : 2 Lg)]Nowm is in the right-hand side iffm satisfies

T(Sf[K .�(�^ )] : 2 Lg).This means that

(i) m satisfies all� which are satisfied by allm0 that are in[K .�(� ^ )]for some 2 L.

We are done if we can show that this is equivalent tom’s being in the left-hand side which can be reformulated thus:

(ii) m satisfies all� which are contained inK .�(� ^ ) for all 2 L.

To see that(i) and (ii) are equivalent, we finally show that(iii) and (iv) areequivalent:

(iii) � is satisfied by allm0 that are in[K .�(� ^ )] for some 2 L.

(iv) � is contained inK .�(� ^ ) for all 2 L.

That (iv) entails(iii) is trivial for if � is contained in allK .�(� ^ ), thenall m0 that are in some[K .�(� ^ )] satisfy�. To see that(iii) entails(iv),suppose that(iv) is not true, i.e., that there is a such that� =2 K .�(� ^ ).ThenK .�(� ^ ) [ f:�g is consistent, so there is anm00 such thatK .�(� ^ ) [ f:�g � m00, which means that(iii) is not true.

In sum, then, we have shown thatS(W( .�)) = (S( .�))cl.Whereas the construction just considered starts from an AGMcontraction

function .�, we might just as well start from a severe withdrawal function ..�,without changing the result. We know that.� = C( ..�) is an AGM contraction,so by the result just proved, we get(S(C( ..�)))cl = S(W(C( ..�))), but thelatter is, by Observation 10(ii) , identical withS( ..�). 2OBSERVATION 15. (i) If S satisfies (S1) – (S4), then the function..�obtained fromS by (Def ..� from S) is a severe withdrawal function.(ii) If ..� is a severe withdrawal function, then..� can be represented as asphere-based withdrawal, where the sphere systemS on which ..� is basedis obtained by(Def S from

.�) (or equivalently, by(Def S from..�)) andS

satisfies (S1) – (S4).

Proof. (i) Let S satisfy (S1) – (S4) and..� be obtained by (Def

..� fromS). We need to show..� is a severe withdrawal function (i.e., satisfies (

..�1) –(..�4), (

..�6), (..�7a) and (

..�8)).

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(..�1) Directly by (Def

..� fromS) and definition of functionth (see Section3).

( ..�2) By (Def ..� from S) we need to showth(cS(:�)) � K. Now, if ` �,then by (DefcS) we havecS(:�) = [K]. So Lemma 0(i) givesth(cS(:�)) =th([K]) = K and the result holds trivially. Otherwise6` � and by (S2) wehave that[K] is the�-minimum ofS (i.e., [K] � cS(:�)). So by (DefcS)and Lemma 0(iv) the result is established.

( ..�3) Let� 62 K or ` �. In the latter case, using (DefcS), (Def ..� from S)and Lemma 0(i) we have thatK ..�� = K and the result ensues directly. In theformer case and supposing6` � we have that[:�] \ [K] 6= ;. Therefore (S2)and (DefcS) give cS(:�) = [K] and by (Def..� from S) and Lemma 0(i) wehaveK ..�� = K from which the desired result is obtained.

( ..�4) Let 6` �. By definition ofcS(:�), [:�] \ cS(:�) 6= ;. Therefore,using Lemma 0(iv) and (Def

..� from S) we have� 62 th(cS(:�)) = K ..��.(..�6) LetCn(�) = Cn( ). Then[�] = [ ]. If ` �, then` andK ..�� =th([K]) = K ..� . Otherwise6` � and 6` . HowevercS(:�) = cS(: ).

ThereforeK ..�� = th(cS(:�)) = th(cS(: )) = K ..� .( ..�7a) Let 6` �. We need to show thatK ..�� � K ..�(� ^ ). By (Def ..�

from S), we need to show thatth(cS(:�)) � th(cS(:(� ^ ))). That is,by Lemma 0(iv), cS(:(� ^ )) � cS(:�) or, equivalently,cS(:� _ : ) �cS(:�). Since6` �, then6` �^ . Now [:�] � [:�[: ] = [:�][ [: ], soclearlycS(:� _ : ) � cS(:�) as desired.

( ..�8) Let� 62 K ..�(�^ ). By (Def ..� fromS), we haveK ..�� = th(cS(:�)))andK ..�(�^ ) = th(cS(:(�^ ))). Since� 62 K ..�(�^ ) = th(cS(:(�^ ))) = th(cS(:�_: )), thencS(:�_: )\[:�] 6= ;. ThereforecS(:�) �cS(:� _ : ) andth(cS(:� _ : )) � th(cS(:�)) by Lemma 0(iv). ThusK ..�(� ^ ) � K ..�� as desired.

(ii) Let..� be a severe withdrawal function (i.e., satisfies (

..�1) – (..�4),

(..�6), (

..�7a) and (..�8)) and letS be obtained from

..� by (DefS from.�) or,

equivalently, (DefS from..�). We have to verify that (a)

..�0 obtained fromSusing (Def ..� from S) is identical to..� and (b) thatS satisfies the conditionsfor a system of spheres (i.e., (S1) – (S4)).

We prove (b) first as part of it will be useful in shortening theproof of (a).(b) We verify thatS is indeed a system of spheres centred on[K].(S1) The nestedness of spheres follows directly from (DefS from ..�) and

Lemma 2(i).(S2) That [K] is a sphere follows via (DefS from ..�) and Lemma 0(v))

setting� � > (or any 2 L such that ) since by (..�3) and (..�2) we haveK ..�> = K. That[K] is the�-minimal sphere then follows by (DefS from..�) using (..�2) and Lemma 0(v)).

(S3) ThatML is a sphere follows directly by our constructions as weinclude it as a sphere.

(S4) Let 6` :�. We need to show that there is a sphereU 2 S suchthat U \ [�] 6= ; andV \ [�] 6= ; implies U � V for all V 2 S. We

46

show thatU = [K ..�:�] satisfies this condition. Since6` :�, then by (..�4):� 62 K ..�:� so clearly[K ..�:�] \ [�] 6= ;. Now suppose forreductiothere

is someV 2 S such thatV \ [�] 6= ; andU 6� V (i.e., V � U by (S1)which has been shown above to hold). That is, by (DefS from ..�), there issome 2 L such that[K ..� ] \ [�] 6= ; and [K ..� ] � [K ..�:�]. Since[K ..� ]\ [�] 6= ;, then:� 62 K ..� . It follows by (..�9) thatK ..� � K ..�:�or, in other words, by Lemma 0(v)) [K ..�:�] � [K ..� ] contradicting theabove.

The proof of (S4) actually shows that, for6` :�, cS(�) = [K ..�:�] (or,equivalently, that6` � impliescS(:�) = [K ..��]) which can be convenientlyused in the proof of (a).

(a) Whenever �, thenK ..�� = K by (..�2) and (..�3). Also, cS(:�) =[K] by (Def cS) and consequentlyK ..�0� = K by (Def ..� from S) and Lem-ma 0(i). HenceK ..�� = K ..�0� = K.

Consider, then, the case where6` �.Left to right. Suppose 2 K ..��. We need to show 2 K ..�0�. Now

clearly [K ..��] � [ ]. By following the proof of (S4) we get [K ..��] isa sphere andcS(:�) = [K ..��]. Therefore,cS(:�) � [ ]. Hence 2th(cS(:�)) and by Lemma 0(iv)) 2 K ..�0� by (Def ..� from S) as desired.

Right to left. (The proof follows essentially be reversing that for the pre-vious case.) Suppose 2 K ..�0�. We need to show 2 K ..��. Since 2K ..�0�, then 2 th(cS(:�)) by (Def ..� from S). Therefore,cS(:�) � [ ].Now cS(:�) = [K ..��] according to the proof of (S4). Hence[K ..��] � [ ]and thus 2 K ..�� as desired. 2LEMMA 16. If ..� is a severe withdrawal function, then the two conditions(Def� from .�) and(Def� from ..�) are equivalent.

Proof. We have to show that� =2 K ..�(� ^ ) or ` (� ^ )holds just in case � =2 K ..� or ` holds. To show that the former implies the latter, let� =2 K .�(� ^ ) or` (� ^ ) and assume that6` . Hence6` (� ^ ). Then� =2 K .�(� ^ ).Since6` , we getK ..� � K ..�(�^ ), by (..�7a). So since� =2 K ..�(�^ ),� =2 K ..� , as desired.

For the converse, let� =2 K .� or ` . From the former, we know that6` �, by (..�1). Now if ` , then� =2 K ..�� = K ..�(�^ ), by (..�4) and (..�6).So let 6` and thus� =2 K ..� . Assume for reductio that� 2 K ..�(� ^ ).Then by (

..�8c)K ..�(� ^ ) � K ..� . But then, since� =2 K ..� , we get that� =2 K ..�(� ^ ), and we have a contradiction. Hence� =2 K ..�(� ^ ), asdesired. 2

47

OBSERVATION 17. Let.� and

..� be corresponding AGM contraction andsevere withdrawal functions either via (Def

..� from.�) or via (Def

.� from..�).

Then .� and ..� lead to identical entrenchment relations, via (Def� from .�)and (Def� from ..�).

Proof. Let .� and ..� be corresponding AGM contraction and severe with-drawal functions either via (Def..�from .�) or (Def .�from ..�). It follows fromObservation 10 that it does not matter which of these definitions we apply.

Let� be the epistemic entrenchment relation that arises from.� via (Def� from .�) and�0 be the epistemic entrenchment relation arising from..� via(Def� from ..�).

We first show� � implies� �0 . Suppose� � . Now� 62 K .�(� ^ ) or ` � ^ by (Def� from.�). In the former case (and assuming6`

otherwise the result is trivial)� 62 K ..� by (Def..� from

.�). In the lattercase, surely . In either case,� �0 by (Def� from

..�).We now show� �0 implies� � . Suppose� �0 . Then� 62 K ..�

or ` by (Def� from ..�). If ` �, then the former case is not possible by( ..�1) and the latter case gives� ^ whereby� � follows by (Def�from .�).

Now let 6` �. Consider first the case where� 62 K ..� . By ( ..�4) � 62K ..��. Now it follows by Lemma 2(ii) that� 62 K ..�(� ^ ). Equivalently,:(� ^ )! � 62 K ..�(� ^ ) and consequently, by the Deduction Theorem,� 62 Cn(K ..�(� ^ ) [ f:(� ^ )g). Therefore� 62 K \Cn(K ..�(� ^ ) [f:(� ^ )g) and by (Def .� from ..�) � 62 K .�(� ^ ) whereby� � follows by (Def� from .�). Consider now the case where . Then 2K .�(� ^ ) by ( .�1) which we know to hold by Observation 9 and therefore� 62 K .�(� ^ ) (

.�4) (again, this holds by Observation 9). (Def� from.�)

now gives� � as desired. 2OBSERVATION 19. (i) If � satisfies (E1) – (E5), then the function..�obtained from� by (Def ..� from�) is a severe withdrawal function.(ii) If ..� is a severe withdrawal function, then..� can be represented as anentrenchment-based withdrawal where the relation� on which ..� is basedis obtained by(Def� from .�) (or equivalently, by(Def� from ..�)), and�satisfies (E1) – (E5).

Proof.(i) Assume that� satisfies (E1) – (E5) and letK ..�� = K \ f : � < g

when� 2 K and 6` �, andK ..�� = K otherwise. We have to verify that..�satisfies the postulates for severe withdrawals.

(..�1) LetK ..�� ` . We want to show that 2 K ..��. The case whereK ..�� = K is trivial, sinceK is a theory. So let� 2 K and 6` �. By compact-

48

ness, there are�1; : : : ; �n 2 K ..�� � K such that�1^: : :^�n ` . SinceKis a theory,�1^: : :^�n and are inK. So it remains to show that� < . Byrepeated application of (E3), there is ani such that�i � �1 ^ : : :^�n. Since�i is in K ..��, we have� < �i. Hence, by the transitivity condition (E1),� < �1 ^ : : : ^ �n. But�1 ^ : : : ^ �n ` , so by (E2),�1 ^ : : : ^ �n � .Hence, by (E1) again,� < , so is inK ..��.

( ..�2) and (..�3) are immediate from (Def..� from�).( ..�4) Assume for reductio that6` � and� 2 K ..��. By the latter and (Def..�

from�), we get� 2 K. So by (Def..� from�) again,� < �, that is� � �and� 6� � which is impossible.

( ..�6) If Cn(�) = Cn( ), then� 2 K iff 2 K, and 6` � iff 6` . Itremains to show that� < � iff < �, for all �. But this follows from� � and � �, which is implied by (E2), and transitivity, (E1).

(..�7a) Let 6` �, and thus6` (� ^ ). If � ^ =2 K, thenK ..�� � K =K ..�(� ^ ) by (Def

..� from �). If � ^ 2 K, and thus� 2 K, we needto show that� < � implies � ^ < � for all �. But from (E2), we get� ^ � �, so the claim follows by transitivity, (E1).

( ..�8) Let � =2 K ..�(� ^ ). Hence6` �, by (E1), and also6` (� ^ ). If� =2 K, soK ..�(� ^ ) � K = K ..�� by (Def ..� from �). So let� 2 K.HenceK ..�(� ^ ) 6= K, so� ^ 2 K. Hence� 62 K ..�(� ^ ) means that� ^ 6< �. Now assume that� 2 K ..�(� ^ ), i.e.,� ^ < �. We needto show that� < �. But since� ^ � �, by (E2),� ^ 6< � means that� � � ^ . From this and� ^ < �, we get by transitivity (E1) that� < �and therefore 2 K ..�� by (Def ..� from�), as desired.

(ii) Assume that..� satisfies (..�1) – (..�4), (..�6), (..�7a), (..�8), and let� � if and only if � =2 K ..� or ` . (That is, we use Lemma 16 and basethe following on (Def� from

..�) rather than directly on (Def� from.�).)

We have to verify (a) that the withdrawal function..�0 obtained from� withthe help of (Def..� from �) is identical with ..�, and (b) that� satisfies thedefining conditions for epistemic entrenchment.

(a) Using the definition (Def..� from�) we get that 2 K ..�0� iff 2 K and� � =2 K or ` � or� <

which means, by the definition (Def� from ..�), that(�) 2 K and� � =2 K or ` � or(� =2 K ..� or ` ) and 2 K ..�� and 6` �

First we show that 2 K ..�� implies 2 K ..�0�. Suppose that 2 K ..��holds. If ` � or � =2 K, then by (

..�3) and (..�2) K ..�� = K, so 2 K

and 2 K ..�0� by the upper line of(�). So let 6` � and� 2 K. We have 2 K ..�� � K, by (..�2). For the lower line of(�), it remains to show that

49

either` or � =2 K ..� . If 6` , then, we need to show that� =2 K ..� . Butthis follows from 2 K ..�� by Lemma 2(iii) (Expulsiveness).

For the converse, we show that 2 K ..�0� implies 2 K ..��. So let(�) be given. From the upper line, we get 2 K ..�� with the help of (..�3).So suppose that the lower line is true. But this line containsas a conjunct 2 K ..��, which is just what we set out to prove.

(b) Finally we show that� indeed satisfies (E1) – (E5).(E1) Let� � and � �, that is,� =2 K ..� or` , and also =2 K ..��

or ` � by (Def� from ..�). We need to show that� � �, i.e.� =2 K ..�� or` �. Assume that6` �. Then =2 K ..��, and hence, by (..�1), 6` . So wealso have� =2 K ..� . We conclude from =2 K ..�� with the help of (..�9) thatK ..�� � K ..� . Since� =2 K ..� , we finally get� =2 K ..��, as desired.

(E2) Let� ` . In order to see that� � , we need to show that� =2K ..� or ` . Assume6` . Then by (..�4) =2 K ..� . Hence by (..�1),� =2 K ..� , as desired.(E3) In order to see that either� � �^ or � � ^ , we need to show

that either� =2 K ..�(� ^ ) or ` � ^ , or =2 K ..�(� ^ ) or ` � ^ .Assume that6` �^ . Then by (

..�4),�^ =2 K ..�(�^ ), so by (..�1) in fact

either� =2 K ..�(� ^ ) or =2 K ..�(� ^ ).(E4) Assume thatK 6= L. We need to show that� =2 K just in case� �

is true for every 2 L. The latter condition means, by (Def� from ..�), that� =2 K ..� or ` , for every 2 LWe know from (

..�2) that� =2 K is sufficient for this condition. To show that� =2 K is also necessary, observe that the condition entails that� =2 K ..�?.SinceK 6= L, we know that? =2 K. So by (

..�3),K ..�? = K. So� =2 K, asdesired.

(E5) Assume that � � for all 2 L. This means, by (Def� from ..�),that either =2 K ..�� for all 2 L or ` �. The former cannot be, however,since if is in Cn(;), it will be in K ..�� no matter whatK ..�� looks like, by( ..�1). Hence �. 2LEMMA 20. For any entrenchment relation�with respect toK, the systemof spheresS(�) satisfies conditions (S1) – (S4) with respect to[K].

Proof. Let� be an entrenchment relation. We show thatS(�) is indeeda system of spheres centred on[K].

(S1) By the connectedness of� (which follows from (E1)–(E3) — see [9,Lemma 3(i) p. 189]) we have that either� � or � � for �; 2 L. Itfollows that eitherf� : � < �g � f� : < �g or f� : < �g � f� :� < �g. Denoting[f� : � < �g] by S� and[f� : < �g] by S as in (DefS from �), it follows by Lemma 0(v) and the fact the cuts are theories thatS� � S or S � S� for S�; S 2 S(�).

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(S2) If K 6= L, then? 62 K sinceK is a belief set. It follows by (E4)that f� : ? < �g = K. ThereforeS? = [f� : ? < �g] = [K] and, by(Def S from �), [K] 2 S(�). Now suppose for reductio that there is someS 2 S(�) such thatS � [K]. That is[f� : < �g] � [f� : ? < �g]. ByLemma 0(iv) and(i) (for cuts are theories — see [29, p. 159])f� : ? < �g �f� : < �g. That is there is some� 2 f� : < �g but � 62 f� : ? < �g.Therefore < � but? 6< �. By the connectedness of� we have� � ? and,by transitivity of� (E1), < ?. This contradicts (E2) (for? ` ). Hence nosuchS� exists and[K] is the�-minimum sphere ofS(�). OtherwiseK = Land; 2 S by definition.

(S3) Take the cutS>. Now f : > < g = ; by (E5). SoS> = [f :> < g] = [;] =ML.(S4) Let� 2 L and 6` :�. It remains to show thatcS(�) = [f : :� < g] for all �.It follows from the compactness ofCn and from (E1) – (E3) that the setf : :� < g does not entail:� (see [29, proof of Lemma 5, p. 161]), so[f : :� < g] intersects[�]. It remains to show that everyS in S which

is a proper subset of[f : :� < g] does not intersect[�]. Suppose thatS in S is a proper subset of[f : :� < g]. Then there is a� such thatS = [f : � < g] andf : � < g is a proper superset off : :� < g.Let � be inf : � < g � f : :� < g. Since� =2 f : :� < g and�is connected,� � :�. But then, since� < �, we can conclude with the helpof (E1) that� < :� as well. Thus:� 2 f : � < g, so[f : � < g] doesnot intersect[�]. We conclude that indeedcS(�) = [f : :� < g]. 2OBSERVATION 21. For any entrenchment relation�, the AGM contrac-tions and the severe withdrawals generated from� andS(�) are identical,i.e.,C(S(�)) = C(�) andW(S(�)) =W(�).

Proof. Let� be an entrenchment relation. We know from Lemma 20 thatS(�) is indeed a system of spheres centred on[K].It remains to show that forS = S(�) generated by (DefS from �) it

holds that(i) f 2 K : � < � _ g = th([K] [ fS(:�))(ii) f 2 K : � < g = th(cS(:�))We begin by showing(ii) : Using the proof of the Lemma 20, the part

concerningS4, we know that for every�, the smallest sphere intersecting[:�], that iscS(:�), is identical with the set[f : � < g]. Moreover, sincef : � < g is a theory (see [29, p. 159]), by Lemma 0(i), th(cS(:�) =f : � < g. Furthermore, we concludefS(:�) = cS(:�) \ [:�] = [f :� < g [ f:�g].

51

Knowing this, we have to show for(i) thatth([K] [ [f : � < g [ f:�g]) = f 2 K : � < � _ gTo show that the left-hand-side is included in the right-hand-side, let� 2m for all m 2 [K][ [f : � < g[f:�g]. We need to show that� 2 K and� < � _ �. But we know from the assumption that� 2 m for all m 2 [K].

SinceK is a theory,� 2 K. Moreover,� 2 m for all m 2 [f : � < g [ f:�g]. Hence� _ � 2 m for all m 2 [f : � < g]. Hence, by thecompleteness ofCn, f : � < g ` � _ �. Sincef : � < g is a theory(see [29, p. 159]), we get that� _ � 2 f : � < g, that is,� < � _ �.

To show conversely that the right-hand-side is included in the left-hand-side, let� 2 K and� < �_�. Clearly� 2 m for all m 2 [K], since� 2 K.It remains to show that� 2 m for all m 2 [f : � < g [ f:�g]. Let suchanm be given. Since� < � _ �, we know that fromm 2 [f : � < g]we can infer that� _ � 2 m. But also:� 2 m (since� 62 f : � < g— see [29, proof of Lemma 5, p. 161]). So, sincem is a theory,� 2 m. Wehave shown that� 2m for all m in [K] [ [f : � < g [ f:�g]. But by thedefinition ofth, this just means that� is in th([K][ [f : � < g [ f:�g]),as desired. 2LEMMA 22. For any system of spheresS with respect to[K], the entrench-ment relationE(S) satisfies conditions (E1) – (E5) with respect toK.

Proof. LetS be a system of spheres centred on[K]. We show that� is anepistemic entrenchment relation onK.

(E1) Let� � and � �. By (Def0 � from S) we have thatcS(: ) 6�[�] and cS(:�) 6� [ ]. It follows that cS(: ) \ [:�] 6= ; and cS(:�) \[: ] 6= ;. Consequently, from the latter,cS(: ) � cS(:�). ThereforecS(:�) \ [:�] 6= ;. HencecS(:�) 6� [�] and, by (Def0 � from S) � � � asdesired.

(E2) Let � ` . It follows that [�] � [ ]. Suppose forreductio thatcS(: ) � [�]. It follows that cS(: ) � [�] which contradicts the defini-tion of cS(: ). HencecS(: ) 6� [�] as required.

(E3) Suppose� 6� � ^ . We need to show that � � ^ . From oursupposition an (Def0 � from S) it follows that cS(:(� ^ )) � [�]. Now[:(�^ )] = [:�_: ] = [:�][ [: ]. ThereforecS(:(�^ ))\ [: ] 6= ;.ConsequentlycS(:(� ^ )) 6� [ ] and hence by (Def0 � from S) we have � � ^ as desired.

(E4) LetK 6= LWe need to show� � for every 2 L iff � 62 K.Left to right. we shall prove the contrapositive. Let� 2 K. We need to

show� 6� for some 2 L. By (Def0 � from S) this amounts to showingcS(: ) � [�] for some 2 L. Consider � : . cS(::�) = cS(�) = [K]

52

by (S2) and since� 2 K. Since� 2 K it follows that[K] � [�] and thereforecS(�) � [�] as desired.Right to left. We shall prove the contrapositive. Let� � for some 2L. By (Def0 � from S), there is some 2 L such thatcS(: ) � [�].

Consequently by (S2), [K] � cS(: ) � [�]. Hence� 2 K as required.(E5) Let 6` �. Again, we show this by considering the contrapositive. We

need to show that 6� � for some 2 S. That is, by (Def0 � from S),we need to showcS(:�) � [ ] for some 2 L. take � >. ClearlycS(:�) � [>] as desired. 2OBSERVATION 23. Let� be an entrenchment relation andS a system ofspheres. Then

(i) E(S(�)) =�.(ii) S(E(S)) is the topological closure of the trimming ofS, i.e.,(St)cl.Proof. (i) Let� be an entrenchment relation.Furthermore, let�0 be E(S(�)). ThatS(�) is a system of spheres was

proved in Observation 21(i) and that�0= E(S(�)) is an epistemic entrench-ment relation subsequently follows by the proof of Observation 24 below.

Now � �0 iff cS(: ) 6� [�] iff by (Def 0 � from S). This is the caseiff [f� : < �g] 6� [�] by (Def S from �) (see beginning of the proof ofObservation 21(i)) which holds ifff� : < �g 6` �. This holds iff 6< � iff(sincef� : < �g is a theory (see [29, proof of Lemma 5, p. 161])) whichholds iff � � by the connectedness of� which follows from (E1)–(E3)(see [9, Lemma 3(i) p. 189]) as desired.(ii) LetS be a system of spheres. Furthermore, letS 0 beS(E(S)). ThatE(S)is indeed an epistemic entrenchment relation has been shownin Lemma 22and thatS(E(S)) is a system of spheres follows by the proof of Observa-tion 21(i).

Now S is in S 0 iff there is some� such thatS = [f : � < g], by(Def S from �). This holds, by (Def0 � from S), iff there is some� suchthatS = [f : cS(:�) � [ ]g]. Now this holds iff there is some� such thatS = [f : 2 th(cS(:�))g], Simplifying, this is the case iff there is some� such thatS = [th(cS(:�))] as desired.

Now the trimming ofS consists exactly of all the sets of the formcS(�),and the operation of taking the topological closure is just the one that takesevery setX of worlds to [th(X)]. Thus we have proved thatS 0 is the topo-logical closure of the trimming ofS. 2OBSERVATION 24. For any system of spheresS, the AGM contractionsand the severe withdrawals generated fromS and E(S) are identical, i.e.,C(E(S)) = C(S) andW(E(S)) =W(S).

53

Proof. Let S be a system of spheres centred on[K], and�= E(S) asdefined by (Def0 � from S). Notice that since� is connected (see [9, Lem-ma 3(i) p. 189]), we have that� < if and only if cS(:�) � [ ].

Lemma 22 showed that� is indeed an epistemic entrenchment relationwith respect toK.

For the limiting case where � all contractions and withdrawals withrespect to� are set toK, so we assume in the following that6` �.

Since�= E(S) is an entrenchment relation, we know from Observation21 thatC(E(S)) = C(S(E(S))). But since by Observation 23,S(E(S)) =(St)cl, we conclude with Lemma 12 thatC(E(S)) = C((St)cl) = C(S).

Precisely the same argument shows thatW(E(S)) =W(S). 2B. Twelve Methods of Withdrawing a Belief �

In this appendix we contrast various methods for withdrawalof a belief�from a belief setK currently found in the literature. We consider the principalcase where6` � as the majority of these methods satisfy the failure property(i.e.,` � impliesK .�� = K).

The following table lists a number of proposals together with an indicationof which:�-worlds and which�-worlds are contained in[K .��].S� refers tothe smallest sphere intersecting� (i.e.,cS(�)). [K] is, of course, the smallest(innermost) sphere.[K .��] . . . within [:�] . . . within [�]

1. AGM (trans. rel.) partial meet [1] [:�] \ S� [K]2. Severe withdrawal [Section 3] [:�] \ S� [�] \ S�3. AGM maxichoice [1] single:�-world [K]4. Saturatable set [18, 14]) single:�-world someX s.th.[K] � X � [�]5. Partial meet of saturatable sets [14] [:�] \ S� someX s.th.[K] � X � [�]6. Iron-fisted withdrawal [Section 7] [:�] \ S� [�]7. Levi – damped type 1 [20] [:�] \ S� [�] \ S2288. Cantwell fallback-based [4] [:�] \ S� [�] \ Si for somei 2 f1; : : : ; ng9. Systematic withdrawal [26] [:�] \ S� [�] \ S��1

10. Lindstrom and Rabinowicz [22] [:�] \ S� someX s.th.[K] � X � [�] \ S�11. Semi-contraction [6] [:�] \ S� someX s.th.[K] � X � [�] \ S�2912. Nayak [p.c.] [:�] \ S� [�] \ (S� � S��1)30

54

The diagrams on the following pages illustrate typical situations for thesevarious proposals. Note that Figures 1 and 2 appeared in Section 3.

55

ML

[K]

[¬ φ]

Figure 3. Maxichoice — pure minimalchange (wrt�).

ML

[K]

[¬ φ]

Figure 4. Saturatable set (no recovery).

ML

[K]

[¬ φ]

Figure 5. Partial meet of saturatable sets.

ML

[K]

[¬ φ]

Figure 6. “Iron-fisted” withdrawal — mini-mal revision equivalent withdrawal

ML

[K]

[¬ φ]

Figure 7. Levi Contraction via damped infor-mational value of type 1.

ML

[K]

[¬ φ]

Figure 8. Cantwell “fallback-based”.

56

ML

[K]

[¬ φ]

Figure 9. Meyeret al.systematic withdrawal.

ML

[K]

[¬ φ]

Figure 10. Lindstrom and Rabinowicz (Inter-polation).

ML

[K]

[¬ φ]

[ψ]

Figure 11. Ferme and Rodriguez semi-contraction.

ML

[K]

[¬ φ]

Figure 12. Nayak (personal communication).

57

C. Interrelationship Between Methods

u��������������������������

Nayak (p.c.)

u@@@@@@@@@PM saturatable

u Saturatable

uIron-fisted

uLindstrom-Rabinowicz

uCantwellu u �������������AGM PMC u�����Levi d.i.v.1

u@@@@@Meyer et al.

uPPPPPPPPPPPPPSevereuMaxichoice

6more general

?less general

Notes1 In the AGM literature these two terms have a precise meaning,differentiating importantforms of belief removal, which we shall introduce later. Forthe time being, however, we deferto the term contraction when referring to any operation removing beliefs from an agent’sepistemic state.2 Levi [18] refers to this ascoerced contraction(as distinct fromuncoerced contractionwhich refers to belief removal for purposes described in what follows).3 Even more so if one considers the rather trivial nature of AGMexpansion.4 This generalisation allows us to retain the spirit of the original while not being tied downto loaded terms such as ‘information’.

585 A contraction ofK by � 2 L in the AGM framework ismaxichoiceif it leads to amaximal subset ofK that does not imply�. It is partial meetif it is the intersection of selectmaxichoice contractions.6 A revision function defined from a maxichoice AGM contraction function via the LeviIdentity [8, p. 69] (K � � = Cn(K .�(:�) [ f�g)) always returns a belief state which ismaximally consistent and therefore has an opinion as to the truth or falsity of every sentencein the object language.7 More formally,K0 2 K?� if and only if (i) K0 � K; (ii) � 62 Cn(K0); and,(iii) foranyK00 such thatK0 � K00 � K, � 2 Cn(K00).8 An AGM full meet contraction may be constructed fromK asK .�� = T(K?�).9 Strictly speaking, Grove’s [12] construction deals solelywith syntactic, rather than seman-tic, objects; maximally consistent sets of sentences take the place of worlds. It can be thoughtof as furnishing a semantics in so far as it provides a “picture” for the belief change process.10 In traditional AGM terminology,K .�� 2 K?� whereK?� is the set of maximalsubsets ofK failing to imply �. The connection has been established by Grove [12].11 Recently, Ferme and Rodriguez [7] have also, independently, proposed an axiomatisationfor severe withdrawal using the postulate (

..�9) — see Section 6.12 Technically, this can be viewed as the dual of the Grove ordering. See Gardenfors [8,Section 4.8].13 It is easy to show that a necessary and sufficient condition for (Def

.� from�) to generatemaxichoice contractions is that the entrenchment relationsatisfies either� < or < ( !�), for all sentences� and in L. See Rott [31, Chapter 8].14 Or among those for nonmonotonic consequence relations. A nonmonotonic consequencerelation j� can be defined from a revision operator� and a belief set (or rather, expectationset)K by putting� j� iff 2 K � � [10]. The parameterK is here left implicit.15 However, via the so-called Harper Identity —K .�� = K \ K � :� — Recovery canbe linked with success for revision operations and reflexivity for nonmonotonic consequencerelations.16 Cut: If � j� and� ^ j��, then� j��.Cumulative Monotony: If� j� and� j��, then� ^ j��.Or: If � j�� and j��, then� _ j��.Rational Monotony: If� j6� : and� j��, then� ^ j��.For a detailed discussion, we refer the reader to Makinson’s[24] survey.17 Rott [31] considers a slightly different set of postulates in an attempt to remove refer-ence to the underlying logic. We shall remain closer in spirit to the AGM as this additionalgenerality does not affect our aims in this paper.18 We are grateful to Sven Ove Hansson for highlighting this property.19 In fact, Makinson’s [23] result regarding the maximality ofAGM contraction functionsbears this out also.20 That is,X is in S iff it is a fixed point in the operation taking every model setY toS f[K .� ] : Y 6� [ ]g21 For some purposes it is convenient to rephrase this definition in terms of contractions ofconjuncts as follows.

(Def0 S from.�) X� = � Sf[K .�� ^ ] : 2 Lg whenever 6` �[K] otherwise22 In model theoretic terms, the spheres of the Lewis-Grove constructions are��-elementary

but not�-elementary (see [3, p. 141]). If, however, their second construction is applied tosevere withdrawals, the resulting spheres turn out to be�-elementary.23 Similarly all constructions of systems of spheres from someentrenchment relation�(which will use “cuts” or “up-sets” with respect to�, see Section 12 below) yield�-elementaryspheres.24 Priestet al. [28] have pointed to an error in Grove’s [12, Theorem 1] proofverifying thatthe revision postulate analogues of (

.�7) and (.�8) are satisfied by a revision function derived

59

from a system of spheres. They demonstrate one way to fix Grove’s proof. Alternatively, theysuggest that every sphere inML be required to be elementary. In any case, Grove’s statementof the result is not at fault.25 A generalisation of this theorem for relations of epistemicentrenchment with incompa-rabilities (and ones that need not satisfy Minimality and Maximality) is given in Rott [30,Theorem 2].26 Under the same assumptions it also reduces to what might be called the standard defini-tion in the literature (cf. Gardenfors [8, pp. 95–96]):

(Def00 � from S) � � iff cS(:�) � cS(: ):27 We change the notation of Kaluzhny and Lehmann in order to avoid confusion with thenotation used in this paper.30 S2 refers to the second smallest sphere, that is, that sphereX such that[K] � X andX � Y for all Y 6= [K].30 S��1 refers to the sphere immediately smaller thanS�, that is, that sphereX such thatX � S� andY � X for all Y � S�.30 This is for reasonable semi-contraction functions (see [6,Section 5]). Otherwise we have,within [�], someX s.th.[K] � X �ML.

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2. C. E. Alchourron and D. Makinson. The logic of theory change: Contraction functionsand their associated revision functions.Theoria, 48:14–37, 1982.

3. J. L. Bell and A. B. Slomson.Models and Ultraproducts. North-Holland PublishingCompany, 1969.

4. J. Cantwell. Some logics of iterated belief change. Studia Logica, 1998 (to appear).5. P. M. Cohn.Universal Algebra. Harper & Row, 1965.6. E. Ferme and R. O. Rodriguez. On the logic of theory change: Axiomatic characterisa-

tion and construction for semi-contraction functions. Unpublished manuscript, Univer-sidad de Buenos Aires, 1997.

7. E. Ferme and R. O. Rodriguez. A brief note about the Rott contraction. Unpublishedmanuscript, Universidad de Buenos Aires, 1997.

8. P. Gardenfors.Knowledge in Flux: Modeling the Dynamics of Epistemic States. BradfordBooks, MIT Press, Cambridge Massachusetts, 1988.

9. P. Gardenfors and D. Makinson. Revisions of knowledge systems using epistemicentrenchment. InProceedings of the Second Conference on Theoretical Aspectof Rea-soning About Knowledge, pages 83–96, 1988.

10. P. Gardenfors and D. Makinson. Nonmonotonic inferencebased on expectations.Artifi-cial Intelligence, 65:197-245, 1994.

11. P. Gardenfors and H. Rott. Belief revision. In D. M. Gabbay, C. J. Hogger, and J. A.Robinson, editors,Handbook of Logic in Artificial Intelligence and Logic ProgrammingVolume IV: Epistemic and Temporal Reasoning, pages 35–132. Oxford University Press,1995.

12. A. Grove. Two modellings for theory change.Journal of Philosophical Logic, 17:157–170, 1988.

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Notre Dame Journal of Formal Logic, 36(1), 1995. Also appears as Uppsala Prints andPreprints in Philosophy number 1993-6, Uppsala University, 1993.

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17. Y. Kaluzhny and D. Lehmann. Deductive nonmonotonic inference operations: Antitonicrepresentations.Journal of Logic and Computation, 5(1):111–122, 1995.

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20. I. Levi. Contraction and information value. Unpublished manuscript, Columbia Univer-sity, March 1997.

21. D. Lewis.Counterfactuals. Basil Blackwell, Oxford, 1986.22. S. Lindstrom and W. Rabinowicz. Epistemic entrenchment with incomparabilities and

relational belief revision. In A. Fuhrmann and M. Morreau (eds).,The logic of theorychange. Springer-Verlag, LNAI 465, Berlin, pp. 93-126, 1991.

23. D. Makinson. On the status of the postulate of recovery inthe logic of theory change.Journal of Philosophical Logic, 16:383–394, 1987.

24. D. Makinson. General patterns in nonmonotonic reasoning. In D. M. Gabbay, C. J.Hogger, and J. A. Robinson, editors,Handbook of Logic in Artificial Intelligence andLogic Programming Volume III: Nonmonotonic Reasoning and Uncertain Reasoning,pages 35–110. Oxford University Press, 1994.

25. D. Makinson. On the force of some apparent counterexamples to recovery. InE. G. Valdes, W. Krawietz, G. H. von Wright and R. Zimmerling., editors,Festschriftfor Carlos E. Alchourron and Eugenio Bulygin, Duncker and Humblot, Berlin, 1997.

26. T. A. Meyer, W. A. Labuschagne and J. Heidema. A semantic weakening of the recoverypostulate. Unpublished manuscript, University of South Africa, 1997.

27. M. Pagnucco.The Role of Abductive Reasoning Within the Process of BeliefRevision.PhD thesis, Department of Computer Science, University of Sydney, February 1996.

28. G. Priest, T. J. Surendonk and K. Tanaka. An error in Grove’s Proof. Technical ReportTR-ARP-07-96. Automated Reasoning Project, Australian National University, 1996.

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31. H. Rott. Making Up One’s Mind: Foundations, Coherence, Nonmonotonicity. Habili-tationsschrift, Philosophische Fakultat, UniversitatKonstanz, October 1996. Also to bepublished by Oxford University Press, 1998.