In the logic of belief revision, a belief state (or database) isrepresented by a set of sentences. The major operations of change arethose consisting in the introduction or removal of abelief-representing sentence. In both cases, changes affecting othersentences may be needed (for instance in order to retainconsistency). Rationality postulates for such operations have beenproposed, and representation theorems have been obtained thatcharacterize specific types of operations in terms of thesepostulates.

In the dominant belief revision theory, the so-called AGM model, theset representing the belief state is assumed to be a logically closedset of sentences (abelief set). In an alternative approach,the corresponding set is not logically closed (abeliefbase). One of most debated topics in belief revision theory isthe recovery postulate, according to which all the original beliefsare regained if one of them is first removed and then reinserted. Therecovery postulate holds in the AGM model but not in closely relatedmodels employing belief bases. Another much discussed topic is howrepeated changes can be adequately represented.

• Introduction
  • 1.1 History
  • 1.2 The representation of beliefs and changes
  • 1.3 Formal preliminaries
• 2. Contraction
  • 2.1 Partial meet contraction
  • 2.2 Entrenchment-based contraction
  • 2.3 Recovery and its avoidance
• 3. Revision
• 4. Possible world modelling
• 5. Belief bases
  • 5.1 Increased expressive power
  • 5.2 Belief base contraction
  • 5.3 Belief base revision
  • 5.4 Connections between belief bases and belief sets
• 6. Other operations
  • 6.1 Update
  • 6.2 Consolidation
  • 6.3 Semi-revision
  • 6.4 Selective revision
  • 6.5 Shielded contraction
  • 6.6 Replacement
  • 6.7 Multiple contraction and revision
  • 6.8 Indeterministic belief change
  • 6.9 Operations for an extended language
  • 6.10 Changes in the strength of beliefs
  • 6.11 Changes in norms and preferences
• 7. Repeated change
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

Introduction

1.1 History

Belief revision (belief change, belief dynamics) is a young field ofresearch that has been recognized as a subject of its own since themiddle of the 1980s. The new subject grew out of two convergingresearch traditions.

One of these emerged in computer science. Since the beginning ofcomputing, programmers have developed procedures by which databasescan be updated. The development of Artificial Intelligence inspiredcomputer scientists to construct more sophisticated models of databaseupdating. The truth maintenance systems developed by Jon Doyle (1979)were important in this development. One of the most significanttheoretical contributions was a 1983 paper by Ronald Fagin, JeffreyUllman and Moshe Vardi, in which they introduced the notion ofdatabase priorities.

The second of these two research traditions is philosophical. In awide sense, belief change has been a subject of philosophicalreflection since antiquity. In the twentieth century, philosophershave discussed the mechanisms by which scientific theories develop,and they have proposed criteria of rationality for revisions ofprobability assignments. Beginning in the 1970s a more focuseddiscussion has taken place on the requirements of rational beliefchange. Two milestones can be pointed out. The first was a series ofstudies conducted by Isaac Levi in the 1970s (Levi 1977, 1980). Leviposed many of the problems that have since then been majorconcerns in this field of research. He also provided much of the basicformal framework. William Harper's (1977) work from the same periodhas also had a lasting influence.

The next milestone was the AGM model, so called after its threeoriginators, Carlos Alchourrón, Peter Gärdenfors, andDavid Makinson (1985). Alchourrón and Makinson had previouslycooperated in studies of changes in legal codes (Alchourrónand Makinson 1981, 1982). Gärdenfors's early work was concernedwith the connections between belief change and conditionalsentences (Gärdenfors 1978, 1981). With combined forces the threewrote a paper that provided a new, much more general and versatileformal framework for studies of belief change. (On the history oftheir joint work, see Makinson 2003 and Gärdenfors 2011.) Sincethe paper was published in theJournal of Symbolic Logic in1985, its major concepts and constructions have been the subject ofsignificant elaboration and development. The AGM model anddevelopments that have grown out of it still form the core of beliefrevision theory.

1.2 The representation of beliefs and changes

In the AGM model and most other models of belief change, beliefs arerepresented by sentences in some formal language. Sentences do notcapture all aspects of belief, but they are the best general-purposerepresentation that is presently available.

The beliefs held by an agent are represented by a set of suchbelief-representing sentences. It is usually assumed that this set isclosed under logical consequence, i.e., every sentence that followslogically from this set is already in the set. This is clearly anunrealistic idealization, since it means that the agent is taken to be“logically omniscient.” However, it is a usefulidealization since it simplifies the logical treatment; indeed, itseems difficult to obtain an interesting formal treatment without it.In logic, logically closed sets are called “theories”. Informal epistemology they are also called “corpora”,“knowledge sets”, or (more commonly) “beliefsets”.

Isaac Levi (1991) has clarified the nature of this idealization bypointing out that a belief set consists of the sentences that someoneiscommitted tobelieve, not those that she actuallybelieves in. According to Levi, we are doxastically committed tobelieve in all the logical consequences of our beliefs, but typicallyour performance does not live up to this commitment. The belief set isthe set of the agent's epistemic commitments, and therefore largerthan the set of her actually held belief.

In the AGM framework, there are three types of belief change. Incontraction, a specified sentencep is removed, i.e.,a belief setK is superseded by another belief setK÷p that is a subset ofK notcontainingp. Inexpansion a sentencep isadded toK, and nothing is removed, i.e.K isreplaced by a setK+p that is the smallest logicallyclosed set that contains bothK andp. Inrevision a sentencep is added toK, and atthe same time other sentences are removed if this is needed to ensurethat the resulting belief setK*p is consistent.

It is important to note the specific character of these models. Theyareinput-assimilating. This means that the object of change,the belief set, is exposed to an input, and is changed as a result ofthis. No explicit representation of time is included. Instead, thecharacteristic mathematical constituent is a function that, to eachpair of a belief set and an input, assigns a new belief set.

1.3 Formal preliminaries

The belief-representing sentences form a language. (As is usual inlogic, the language is identified with the set of all sentences itcontains.) Sentences, i.e. elements of this language, will berepresented by lowercase letters (p,q…) andsets of sentences by capital letters. This language contains the usualtruth-functional connectives: negation (¬), conjunction (&),disjunction (∨), implication (→), and equivalence(↔). ⊥ denotes an arbitrary contradiction(“falsum”), and ⊤ an arbitrary tautology.

To express the logic, a Tarskianconsequence operator willbe used. Intuitively speaking, for any setA of sentences,Cn(A) is the set of logical consequences ofA. Moreformally, a consequence operation on a given language is a function Cnfrom sets of sentences to sets of sentences. It satisfies the followingthree conditions:

Inclusion:A⊆Cn(A)

Monotony: IfA⊆B, thenCn(A) ⊆ Cn(B)

Iteration: Cn(A) = Cn(Cn(A))

Cn is assumed to be supraclassical, i.e. ifp can bederived fromA by classical truth-functional logic, thenp ∈ Cn(A).A is a belief set if andonly ifA = Cn(A). In what follows,K willdenote a belief set.X ⊢p is an alternative notation forp ∈Cn(X), andX ⊬p forp ∉ Cn(X). Cn(∅) is theset of tautologies.

The expansion ofK by a sentencep, i.e. theoperation that just addsp and removes nothing, is denotedK+p and defined as follows:K+p =Cn(K∪{p}).

2. Contraction

2.1 Partial meet contraction

The outcome of contractingK byp should be asubset ofK that does not implyp, but from which noelements ofK have been unnecessarily removed. Therefore, itis of interest to consider the inclusion-maximal subsets ofKthat do not implyp.

For any setA and sentencep theremaindersetA⊥p (“A remainderp”) is the set of inclusion-maximal subsets ofA that do not implyp. In other words, a setB is an element ofA⊥p if and only ifit it is a subset ofA that does not implyp, andthere is no setB′ not implyingp such thatB⊂B′⊆A. The elements ofA⊥p are called “remainders”.

If the operation of contraction uncompromisingly minimizesinformation loss, thenK÷p will be one of theremainders. However, this construction can be shown to have implausibleproperties. A more reasonable recipe for contraction is to letK÷p be the intersection of some of theremainders. This ispartial meet contraction, the majorinnovation in the classic 1985 paper by Carlos Alchourrón, PeterGärdenfors and David Makinson. An operator of partial meetcontraction employs aselection function that selects the“best” elements ofK⊥p. Moreprecisely, a selection function forK is a functionγ such that ifK⊥p is non-empty, thenγ(K⊥p) is a non-empty subset ofK⊥p. In the limiting case whenK⊥p is empty, thenγ(K⊥p) is defined to be equal to{K}.

The outcome of the partial meet contraction is equal to theintersection of the set of selected elements ofK⊥p, i.e.K÷p =∩γ(K⊥p).

The special case when for all sentencesp,γ(K⊥p) has exactly one element is calledmaxichoice contraction. The special case whenγ(K⊥p) =K⊥pwheneverK⊥p is non-empty is calledfullmeet contraction.

Partial meet contraction of a belief set satisfies six postulatesthat are called the basic Gärdenfors postulates (or basic AGMpostulates). To begin with, when a belief setK is contractedby a sentencep, the outcome should be logically closed.

Closure:K÷p =Cn(K÷p)

Contraction should be successful, i.e.,K÷pshould not implyp (or not containp, which is thesame thing if Closure is satisfied). However, it would be too much torequire thatp ∉ Cn(K÷p) for all sentencesp, since it cannot hold ifp is atautology. The success postulate has to be conditional onpnot being logically true.

Success: Ifp ∉ Cn(∅), thenp ∉ Cn(K÷p).

The Success postulate has been put in doubt since there may besentences other than tautologies that an epistemic agent may refuse towithdraw. (On operations that do not satisfy Success, see Section6.5.)

The contracted set should be a subset of the original one:

Inclusion:K÷p ⊆K

If the sentence to be contracted is not included in the originalbelief set, then contraction by that sentence involves no change atall. Such contractions are idle (vacuous) operations, and they shouldleave the original set unchanged.

Vacuity: Ifp ∉ Cn(K), thenK÷p =K.

Logically equivalent sentences should be treated alike incontraction:

Extensionality: Ifp↔q∈ Cn(∅), thenK÷p =K÷q.

Extensionality guarantees that the logic of contraction isextensional in the sense of allowing logically equivalent sentences tobe freely substituted for each other.

Belief contraction should not only be successful, it should also beminimal in the sense of leading to the loss of as few previousbeliefs as possible. The epistemic agent should give up beliefs onlywhen forced to do so, and should then give up as few of them aspossible. This is ensured by the following postulate:

Recovery:K ⊆(K÷p)+p

According to Recovery, so much is retained afterp has beenremoved that everything can be recovered by reinclusion (throughexpansion) ofp.

One of the central results of the AGM model is a representationtheorem for partial meet contraction. According to this theorem, anoperator ÷ is an operator of partial meet contraction for abelief setK if and only if it satisfies the sixabove-mentioned postulates, namely Closure, Success, Inclusion,Vacuity, Extensionality, and Recovery.

A selection function for a belief setK should, for allsentencesp, select those elements ofK⊥p that are “best”, or most worthretaining. However, the definition of a selection function is verygeneral, and allows for quite disorderly selection patterns. Anorderly selection function should always choose the best element(s) ofthe remainder set according to some well-behaved preferencerelation. A selection function γ for a belief setK isrelational if and only if there is a binary relationR such that for all sentencesp, ifK⊥p is non-empty, thenγ(K⊥p) = {B ∈K⊥p |CRBfor allC ∈K⊥p}. Furthermore, ifR is transitive (i.e., it satisfies: IfARB andBRC, thenARC), then γ and the partialmeet contraction that it gives rise to aretransitivelyrelational.

In order to characterize transitively relational partial meetcontraction, postulates are needed that refer to contraction byconjunctions.

In order to give up a conjunctionp&q, theagent must relinquish either her belief inp or her belief inq (or both). Suppose that contracting byp&q leads to loss of the belief inp,i.e., thatp ∉K÷(p&q). It can be expectedthat in this case contraction byp&q should leadto the loss of all beliefs that would have been lost in order tocontract byp. Another way to express this is that everythingthat is retained inK÷(p&q) isalso retained inK÷p:

Conjunctive inclusion: Ifp ∉K÷(p&q), thenK÷(p&q) ⊆K÷p.

Another fairly reasonable principle for contraction by conjunctionsis that whatever can withstand both contraction byp andcontraction byq can also withstand contraction byp&q. In other words, whatever is an element ofbothK÷p andK÷q isalso an element ofK÷(p&q).

Conjunctive overlap:(K÷p) ∩ (K÷q)⊆K÷(p&q)

Conjunctive overlap and Conjunctive inclusion are commonly calledGärdenfors's supplementary postulates for beliefcontraction. An operation ÷ forK is a transitivelyrelational partial meet contraction if and only if it satisfies thesix basic postulates and in addition both Conjunctive overlap andConjunctive inclusion.

2.2 Entrenchment-based contraction

When forced to give up previous beliefs, the epistemic agent shouldgive up beliefs that have as little explanatory power and overallinformational value as possible. As an example of this, in the choicebetween giving up beliefs in natural laws and beliefs in singlefactual statements, beliefs in the natural laws, that have much higherexplanatory power, should in general be retained. This was the basicidea behind Peter Gärdenfors's proposal that contraction ofbeliefs should be ruled by a binary relation,epistemicentrenchment. (Gärdenfors 1988, Gärdenfors and Makinson1988) To say of two elementsp andq of the beliefset that “q is more entrenched thanp”means thatq is more useful in inquiry or deliberation, orhas more “epistemic value” thanp. In beliefcontraction, the beliefs with the lowest entrenchment should be theones that are most readily given up.

The following symbols will be used for epistemic entrenchment:

p≤q :p is at most as entrenched asq.

p<q :p is less entrenched thanq((p≤q)&¬(q≤p))).

p≡q :p andq are equallyentrenched((p≤q)&(q≤p)).

Gärdenfors has proposed the following five postulates forepistemic entrenchment, that are often referred to as the standardpostulates for entrenchment:

Transitivity: Ifp≤q andq≤r, thenp≤r.

Dominance: Ifp ⊢q, thenp≤q.

Conjunctiveness: Eitherp≤(p&q) orq≤(p&q).

Minimality: If the belief setK is consistent,thenp ∉K if and only ifp≤qfor allq.

Maximality: Ifq≤p for allq,thenp ∈ Cn(∅).

It follows from the first three of these postulates that anentrenchment relation satisfies connectivity, i.e. it holds for allp andq that eitherp≤q orq≤p.

An entrenchment relation ≤ gives rise to an operator ÷ ofentrenchment-based contraction according to the followingdefinition:

q ∈K÷p if and onlyifq ∈K and eitherp <(p ∨q) orp ∈Cn(∅).

Entrenchment-based contraction has been shown to coincide exactlywith transitively relational partial meet contraction.

2.3 Recovery and its avoidance

Recovery is the most debated postulate of belief change. It is easyto find examples that seem to validate Recovery. A person who firstloses and then regains her belief that she has a dollar in her pocketseems to return to her previous state of belief. However, otherexamples can also be presented, in which Recovery yields implausibleresults. The following are two of the examples that have been offeredto show that Recovery does not hold in general:

1. I have read in a book about Cleopatra that she had both a son and adaughter. My set of beliefs therefore contains bothp andq, wherep denotes that Cleopatra had a son andq that she had a daughter. I then learn from a knowledgeablefriend that the book is in fact a historical novel. After that Icontractp∨q from myset of beliefs, i.e., I do not any longer believe that Cleopatra had achild. Soon after that, however, I learn from a reliable source thatCleopatra had a child. It seems perfectly reasonable for me to then addp∨q to my set ofbeliefs without also reintroducing eitherp orq.This contradicts Recovery.

2. 2. I previously entertained the two beliefs “George is acriminal” (p) and “George is a massmurderer” (q). When I received information that inducedme to give up the first of these beliefs (p), the second(q) had to go as well (sincep would otherwise followfromq).

   I then I received new information that made me accept the belief“George is a shoplifter” (r). The resulting newbelief set is the expansion ofK÷p byr, (K÷p)+r . Sincep follows fromr,(K÷p)+p is a subset of(K÷p)+r. By recovery,(K÷p)+p includesq, fromwhich follows that (K÷p)+r includesq.

   Thus, since I previously believed George to be a mass murderer, itfollows from Recovery that I cannot any longer believe him to be ashoplifter without believing him to be a mass murderer.

Due to the problematic nature of this postulate, it should beinteresting to find intuitively less controversial postulates thatprevent unnecessary losses in contraction. The following is anattempt to do that:

Core-retainment: Ifq ∈Kandq ∉K÷p, then there is abelief setK′ such thatK′ ⊆K and thatp ∉K′ butp∈K′+q.

Core-retainment requires of an excluded sentenceq that itin some way contributes to the fact thatK impliesp.It gives the impression of being weaker and more plausible thanRecovery. However, it has been shown that if an operator ÷ for abelief setK satisfies Core-retainment, then it satisfiesRecovery.

Attempts have been made to construct operations of contraction onbelief sets that do not satisfy Recovery. Arguably the most plausibleof these constructions is the operation ofsevere withdrawalthat has been thoroughly investigated by Hans Rott and Maurice Pagnucco(2000). It canbe constructed from an operation of epistemic entrenchment by modifyingthe definition as follows:

q ∈K÷p if and onlyifq ∈K and eitherp <qorp ∈ Cn(∅).

Severe withdrawal has interesting features, but it also has thefollowing property:

Expulsiveness: Ifp ∉Cn(∅) andq ∉Cn(∅) then eitherp∉K÷q orq∉K÷p.

This is a highly implausible property of belief contraction, sinceit does not allow unrelated beliefs to be undisturbed by each other'scontraction. Consider a scholar who believes that her car is parked infront of the house. She also believes that Shakespeare wrote theTempest. It should be possible for her to give up the first of thesebeliefs while retaining the second. She should also be able to give upthe second without giving up the first. Expulsiveness does not allowthis. The construction of a plausible operation of contraction forbelief sets that does not satisfy Recovery is still an open issue.

3. Revision

The two major tasks of a revision operator * are (1) to add the newbeliefp to the belief setK, and (2) to ensure thatthe resulting belief setK*p is consistent (unlessp is inconsistent). The first task can be accomplished byexpansion byp. The second can be accomplished by priorcontraction by its negation ¬p. If a belief set does notimply ¬p, thenp can be added to it without lossof consistency. This composition of suboperations gives rise to thefollowing definition of a revision operator (Gärdenfors 1981, Levi 1977):

Levi identity:K*p =(K÷¬p)+p.

If ÷ is partial meet contraction, then the operator * that isdefined in this way ispartial meet revision. It is thestandard operation of revision in the AGM model.

Partial meet revision has been axiomatically characterized. Anoperator * is an operator of partial meet revision if and only if itsatisfies the following six postulates:

Closure:K*p =Cn(K*p)

Success:p ∈K*p

Inclusion:K*p ⊆K+p

Vacuity: If ¬p ∉ K, thenK*p =K+p.

Consistency:K*p is consistent ifp is consistent.

Extensionality: If (p ↔q) ∈Cn(∅), thenK*p =K*q.

These six postulates are commonly called thebasicGärdenfors postulates for revision. In addition, twosupplementary postulates are part of the standard repertoire:

Superexpansion:K*(p&q)⊆ (K*p)+q

Subexpansion: If ¬q ∉ Cn(K*p), then(K*p)+q ⊆K*(p&q).

These postulates are closely related to the supplementary postulatesfor contraction. Let * be the partial meet revision defined from thepartial meet contraction ÷ via the Levi identity. Then *satisfies superexpansion if and only if ÷ satisfies conjunctiveoverlap. Furthermore, * satisfies subexpansion if and only if ÷satisfies conjunctive inclusion.

4. Possible world modelling

Alternative models of belief states can be constructed out of sets ofpossible worlds (Grove 1988). In logical parlance, by apossibleworld is meant a maximal consistent subset of the language. By aproposition is meant a set of possible worlds. There is aone-to-one correspondence between propositions and belief sets. Eachbelief set can be represented by the proposition (set of possibleworlds) that consists of those possible worlds that contain the beliefset in question.

For any setA of sentences, let [A] denote the setof possible worlds that containA as a subset, and similarlyfor any sentencep let [p] be the set of possibleworlds that containp as an element. IfA isinconsistent, then [A] = ∅. Otherwise, [A] isa non-empty set of possible worlds. (It is assumed that ∩∅is equal to the whole language.) IfK is a belief set, then∩[K] =K.

The propositional account provides an intuitively clear picture ofsome aspects of belief change. It is convenient to use a geometricalsurface to represent the set of possible worlds. InDiagram 1,every point on the rectangle's surface represents a possible world. Thecircle marked [K] represents those possible worlds in whichall sentences inK are true, i.e., the set [K] ofpossible worlds. The area marked [p] represents those possibleworlds in which the sentencep is true.

Diagram 1. Revision of K by p.

InDiagram 1, [K] and [p] have anon-empty intersection, which means thatK is compatible withp. The revision ofK byp is therefore notbelief-contravening. Its outcome is obtained by giving up thoseelements of [K] that are incompatible withp. Inother words, the result of revising [K] by [p] shouldbe equal to [K]∩[p].

If [K] and [p] do not intersect, then the outcomeof the revision must be sought outside of [K], but it shouldnevertheless be a subset of [p]. In general:

The outcome of revising [K] by [p] is a subset of[p] that is

(1) non-empty if [p] is non-empty

(2) equal to [K]∩[p] if[K]∩[p] is non-empty

This simple rule for revision can be shown tocorrespond exactlyto partial meet revision.

The revised belief state should not differ more from the originalbelief state [K] than what is motivated by [p]. Thiscan be achieved by requiring that the outcome of revising [K]by [p] consists of those elements of [p] that are asclose as possible to [K]. For that purpose, [K] canbe thought of as surrounded by a system of concentric spheres (just asin David Lewis's account of counterfactual conditionals). Each sphererepresents a degree of closeness or similarity to [K].

In this model, the outcome of revising [K] by [p]should be the intersection of [p] with the narrowest spherearound [K] that has a non-empty intersection with[p], as inDiagram 2. This construction was inventedby Adam Grove (1988), who also proved that such sphere-based revisioncorresponds exactly to transitively relational partial meetrevision. It follows of course that it corresponds exactly toentrenchment-based revision.

Diagram 2. Sphere-based revision of K by p.

Possible world models can also be used for contraction. Incontraction, a restriction on what worlds are “possible”(compatible with the agent's beliefs) is removed. Thus, the set ofpossibilities is enlarged, so that the contraction of [K] by[p] will result in a superset of [K]. Furthermore,the new possibilities should be worlds in whichp does nothold, i.e., they should be worlds in which ¬p holds. Inthe limiting case when [K] and [¬p] have anon-empty intersection, no enlargement of [K] is necessary tomake ¬p possible, and the original belief state willtherefore be unchanged. In summary, contraction should be performedaccording to the following rule:

The outcome of contracting [K] by [p] isthe union of [K] and a subset of [¬p] that is

(1) non-empty if [¬p] is non-empty

(2) equal to [K]∩[¬p] if[K]∩[¬p] is non-empty

Belief-contravening contraction is illustrated inDiagram3. Contraction performed according to this rule can be shown tocorrespond exactly to partial meet contraction. Furthermore,the special case when the whole of [¬p] is added to[K]corresponds exactly to full meet contraction. Theother extreme case, when only one element of [¬p] (a“point” on the surface) is added to [K]corresponds exactly to maxichoice contraction. Thus, inmaxichoice contraction byp only one possible way in whichp can be false (¬p can be true) is added.

Diagram 3. Contraction of K by p.

Grove's sphere systems can also be used for contraction. Insphere-based contraction byp,, those elements of [¬p] areadded that belong to the closest sphere around [K] that has anon-empty intersection with [¬p]. The procedure is showninDiagram 4. Sphere-based contractioncorrespondsexactly to transitively relational partial meet contraction.

Diagram 4. Sphere-based contraction of K by p.

5. Belief bases

5.1 Increased expressive power

In the approaches discussed above, all beliefs in the belief set aretreated equally in the sense that they are all taken seriously asbeliefs in their own right. However, due to logical closure the beliefset contains many elements that are not really worth to be takenseriously. Hence, suppose that the belief set contains the sentencep, “Shakespeare wrote Hamlet”. Due to logicalclosure it then also contains the sentencep ∨q, “Either Shakespeare wroteHamlet or Charles Dickens wrote Hamlet”. The latter sentence is a“mere logical consequence” that should have no standing ofits own.

Belief bases have been introduced to capture this feature of thestructure of human beliefs. A belief base is a set of sentences thatis not (except as a limiting case) closed under logicalconsequence. Its elements represent beliefs that are heldindependently of any other belief or set of beliefs. Those elements ofthe belief set that are not in the belief base are “merelyderived”, i.e., they have no independent standing.

Changes are performed on the belief base. The underlying intuitionis that the merely derived beliefs are not worth retaining for theirown sake. If one of them loses the support that it had in basicbeliefs, then it will be automatically discarded.

For every belief baseA, there is a belief set Cn(A)that represents the beliefs held according toA. On the otherhand, one and the same belief set can be represented by differentbelief bases. In this sense, belief bases have more expressive powerthan belief sets. As an example, the two belief bases {p,q} and {p,p ↔q} have thesame logical closure. They are thereforestaticallyequivalent, in the sense of representing the same beliefs. On theother hand, the following example shows that they are notdynamically equivalent in the sense of behaving in the sameway under operations of change. They can be taken to representdifferent ways of holding the same beliefs.

Letp denote that the Liberal Party will support theproposal to subsidize the steel industry, and letq denotethat Ms. Smith, who is a liberal MP, will vote in favour of thatproposal.

Abe has the basic beliefsp andq, whereas Bob hasthe basic beliefsp andp ↔q. Thus,their beliefs (on the belief set level) with respect top andq are the same.

Both Abe and Bob receive and accept the information thatpis wrong, and they both revise their belief states to include the newbelief that ¬p. After that, Abe has the basic beliefs¬p andq, whereas Bob has the basic beliefs¬p andp ↔q. Now, their beliefsets are no longer the same. Abe believes thatq whereas Bobbelieves that ¬q.

(In belief set models, cases like these are taken care of byassuming that although Abe's and Bob's belief states are represented bythe same belief set, this belief set is associated with differentselection mechanisms in the two cases. Abe has a selection mechanismthat gives priority toq overp ↔q,whereas Bob's selection mechanism has the opposite priorities.)

There is only one inconsistent belief set (logically closedinconsistent set), namely the whole language. On the other hand thereare, in any non-trivial logic, many different inconsistent beliefbases. Therefore, belief bases make it possible to distinguish betweendifferent inconsistent belief states.

In belief revision theory it has mostly been taken for granted thatbelief sets correspond to a coherentist epistemology, whereas beliefbases represent foundationalism. However, the logical relationshipsamong the elements of a logically closed set do not adequatelyrepresent epistemic coherence. Although coherentists typically claimthatall beliefs contribute to the justification of otherbeliefs, they hardly mean this to apply to merely derived beliefs suchas “either Paris or Rome is the capital of France”, thatone believes only because one believes Paris to be the capital ofFrance. Therefore, the distinction between operations on belief basesand operations on belief sets should not be equated with that betweenfoundationalism and coherentism.

5.2 Belief base contraction

Partial meet contraction, as defined in Section 2.1, is equallyapplicable to belief bases. Note thatA⊥p is theset of maximal subsets ofA that do not implyp, itis not sufficient that they do not containp. Hence{p ∨q,p↔q} ⊥p = {{p ∨q}, {p ↔q}}.

Most of the basic postulates for partial meet contraction on beliefsets hold for belief bases as well. However, Recovery does not hold forpartial meet contraction of belief bases. This can be seen from thefollowing example (that is adopted from Isaac Levi (2004) who used it forother purposes:

Let the belief setK include both a belief that the coinwas tossed (c) and a belief that it landed heads (h).The epistemic agent wishes to consider whether on the supposition thatthe coin had been tossed, it would have landed heads. In order to dothat, it would seem reasonable to removec from the belief setand then reinsert it, i.e. to perform the series of operationsK÷c+c.

(1) If partial meet contraction is performed directly on the beliefset, then it follows from Recovery thath ∈K÷c+c, i.e.h comes backwithc. This is contrary to reasonable intuitions.

(2) If partial meet contraction is instead performed on a belief baseforK, then recovery can be avoided. Let the belief base be{p₁,…p_n,c,h}, where the background beliefsp₁,…p_n are unrelated toc andh, whereash logically impliesc. ThenK =Cn({p₁,…p_n,c,h}). Sinceh impliesc, it will have to gowhenc is removed, so thatK÷c =Cn({p₁,…p_n}). Whenc is reinserted, the outcome is(K÷c)+c =Cn({p₁,…p_n,c}) that does not containh, as desired.

The following representation theorem has been obtained for partialmeet contraction on belief bases (Hansson 1999). An operator ÷is an operator of partial meet contraction for a setA if andonly if it satisfies the following four postulates:

Success: Ifp ∉ Cn(∅), thenp ∉ Cn(A÷p).

Inclusion:A÷p ⊆A

Relevance: Ifq ∈A andq ∉A÷p, then there is a setA′ such thatA÷p ⊆A′ ⊆Aand thatp ∉ Cn(A′) butp ∈ Cn(A′∪{q }).

Uniformity: If it holds for all subsetsA′ofA thatp ∈ Cn(A′) if and onlyifq ∈ Cn(A′), thenA÷p =A÷q.

The Relevance postulate has much the same function as Recovery hasfor belief sets, namely to prevent unnecessary losses of beliefs.

An alternative approach to contraction of belief bases has beenproposed under the namekernel contraction. For any sentencep, ap-kernel is a minimalp-implying set,i.e. a set that impliesp but has no proper subset thatimpliesp. A contraction operation ÷ can be based onthe simple principle that nop-kernel should be included inA÷p. This can be obtained with an incisionfunction, a function that selects at least one element from eachp-kernel for removal. An operation that removes exactly thoseelements that are selected for removal by an incision function iscalled an operation of kernel contraction. It turns out that allpartial meet contractions on belief bases are kernel contractions, butthe converse relationship does not hold, i.e. there are kernelcontractions that are not partial meet contractions. In other words,kernel contraction is a generalization of partial meet contraction.

5.3 Belief base revision

The expansion operator for belief sets,K+p =Cn(K∪{p}), was constructed to ensure that theoutcome is logically closed. This is not desirable for belief bases,and therefore expansion on belief bases must be different fromexpansion on belief sets. For any belief baseA and sentencep,A +′p, the(non-closing)expansion ofA byp is the setA∪{p}.

Just like the corresponding operators for belief sets, revisionoperators for belief bases can be constructed out of twosuboperations: expansion byp and contraction by¬p. According to the Levi identity (A*p= (A÷ ¬p)+′p), thecontractive suboperation should take place first. Alternatively, thetwo suboperations may take place in reverse order,A*p = (A+′p) ÷¬p. This latter possibility does not exist for beliefsets. IfK∪{p} is inconsistent, thenK+p is always the same (namely identical to thewhole language) independently of the identity ofK and ofp, so that all distinctions are lost. For belief bases, thislimitation is not present, and thus there are two distinct ways tobase revision on contraction and expansion:

Internal revision: A*p = (A ÷¬p) +′p

External revision: A*p = (A +′p) ÷ ¬p

Intuitively, external revision byp is revision with anintermediate inconsistent state in which bothp and¬p are believed, whereas internal revision has anintermediate non-committed state in which neitherpnor ¬p is believed. External and internal revision differin their logical properties, and neither of them can be subsumed underthe other.

5.4 Connections between belief bases and belief sets

An operator of contraction on a belief base gives rise to an operatorof contraction on its corresponding belief set. LetA be abelief base andK = Cn(A) its corresponding beliefset. Furthermore, let − be an operator of contraction onA. It gives rise to an operator ÷ ofbase-generated contraction onK, such that for allsentencesp:K÷p =Cn(A−p). Base-generated contraction has beenaxiomatically characterized. An operator ÷ on a consistentbelief setK is generated by an operator of partial meetcontraction for some finite base forK if and only if itsatisfies the following eight postulates:

Closure:K÷p =Cn(K÷p)

Success: Ifp ∉ Cn(∅), thenp ∉ Cn(K÷p).

Inclusion:K÷p ⊆K

Vacuity: Ifp ∉ Cn(K), thenK÷p =K.

Extensionality: Ifp↔q ∈Cn(∅), thenK÷p =K÷q.

Finitude: There is a finite setX such that forevery sentencep,K÷p =Cn(X′) for someX′ ⊆X.

Symmetry: If it holds for allr thatK÷r ⊢p if and only ifK÷r ⊢q, thenK÷p =K÷q.

Conservativity: IfK÷q is not a subset ofK÷p, then there is somer such thatK÷p ⊆K÷r ⊬p andK÷r ∪K÷q ⊢p.

Operators of revision on a belief set can be base-generated in thesame sense as operators of contraction.

6. Other operations

The AGM framework has been extended in many ways. Some of theseextensions have taken the form of the introducing of new types ofoperators in addition to the three standard types in AGM, namelycontraction, expansion, and revision.

6.1 Update

There are two types of reasons why the epistemic agent may wish toadd new information to the belief set. One is that she has received newinformation about the world, the other that the world has changed. Itis common to reserve the term “revision” for the first ofthese types, and use the term “updating” for the second.The logic of updating differs from that of revision. This can be seenfrom the following example:

To begin with, the agent knows that there is either a book on thetable (p) or a magazine on the table (q), but notboth.

Case 1: The agent is told that there is a book on the table. Sheconcludes that there is no magazine on the table. This is revision.

Case 2: The agent is told that after the first information wasgiven, a book has been put on the table. In this case she should notconclude that there is no magazine on the table. This is updating.

One useful approach to updating is to assign time indices to thesentences, as proposed by Katsuno and Mendelzon (1992). Then thebelief set will not consist of sentencesp but of pairs<p,t₁> of a sentencep anda point in timet₁, signifying thatpholds att₁. In the book-and-magazine example, lett₁ denote the point in time that the firststatement refers to, andt₂ the moment when thesecond information was given in Case 2. The original belief setcontained the pair <¬(p↔q),t₁>. (¬(p↔q) is the exclusive disjunction ofp andq.) Revision byp can be represented by theincorporation of <p,t₁>, andupdating byp by the incorporation of <p,t₂> into the belief set. It followsquite naturally that <¬q,t₁> isimplied by the revised belief set but not by the updated beliefset.

6.2 Consolidation

If a belief base is inconsistent, then it can be made consistent byremoving enough of its more dispensable elements. This operation iscalled consolidation. The consolidation of a belief baseA isdenotedA!. A plausible way to perform consolidation is tocontract by falsum (contradiction), i.e.A! =A÷⊥.

Unfortunately, this recipe for consolidation of inconsistent beliefbases does not have a plausible counterpart for inconsistent beliefsets. The reason is that since belief revision operates withinclassical logic, there is only one inconsistent belief set. Once aninconsistent belief set has been obtained, all distinctions have beenlost, and consolidation cannot restore them.

6.3 Semi-revision

By non-prioritized belief change is meant a process in which newinformation is received, and weighed against old information, with nospecial priority assigned to the new information due to its novelty. A(modified) revision operator that operates in this way is called asemi-revision operator. Semi-revision ofK by a sentencep can be denotedK?p. A sentencepthat contradicts previous beliefs is accepted only if it has moreepistemic value than the original beliefs that contradict it. In thatcase, enough of the previous sentences are deleted to make theresulting set consistent. Otherwise, the input is itself rejected.

One way to construct semi-revision on a belief baseA is tolet it consist of two suboperations:

(1) ExpandA byp.

(2) Restore consistency by giving up eitherp or someoriginal belief(s)

This amounts to defining semi-revision in terms of expansion andconsolidation:

A?p = (A +′p)!

This identity cannot be used for belief sets. Since all inconsistentbelief sets are identical, an operator ? such thatK?p = (K+p)! will have theextremely implausible property that if ¬p₁∈K₁ and ¬p₂ ∈K₂, thenK₁?p₁ =K₂?p₂. However, other ways toperform semirevision on belief sets have been proposed, in particular,the following two-step process:

(1) Decide whether the inputp should be accepted orrejected.

(2) Ifp was accepted, revise byp.

A simple way to apply this recipe is David Makinson's (1997)screened revision, in which there is a setX ofpotential core beliefs that are immune to revision. The belief setK should be revised by the input sentencep ifp is consistent with the setX∩K ofactual core beliefs, otherwise not. The second step of screenedrevision is a revision ofK byp, but with therestriction that no element ofX∩K is allowed tobe removed.

Another variant of the same recipe is calledcredibility-limitedrevision. It is based on the assumption that some inputs areaccepted, others not. Those that are accepted form the set C ofcredible sentences. Ifp ∈ C, thenK?p=K*p. Otherwise,K?p=K. This construction can be further specified by choosinga revision operator and assigning properties to the set C. A varietyof such constructions have been investigated (Hanssonetal. 2001).

6.4 Selective revision

Selective revision is a generalization of semi-revision. Insemi-revision, the input information is either rejected or fullyaccepted. In selective revision, it is possible for only a part of theinput information to be accepted. An operator o of selective revisioncan be constructed from a standard revision operator * and atransformation functionf from and to sentences:

Kp =K*f(p)

In the intended cases,f(p) does not contain anyinformation that is not contained inp. This is ensured iff(p) is a logical consequence ofp. Byadding further conditions onf, various additional propertiescan be obtained for the operator of selective revision.

6.5 Shielded contraction

In shielded contraction, the success postulate of contraction is notuniversally satisfied. This postulate requires that allnon-tautological beliefs are retractable. This is not a fully realisticrequirement, since actual agents are known to have beliefs of anon-logical nature that nothing can bring them to give up. In shieldedcontraction, some non-tautological beliefs cannot be given up; they areshielded from contraction. Shielded contraction can be based on anordinary contraction operator ÷ and a setR ofretractable sentences. Ifp ∈R, thenK−p =K÷p. Otherwise,K−p =K.

This construction can be further specified by adding variousrequirements on the structure ofR. Close connections havebeen shown to hold between shielded contraction and semi-revision.(Fermé and Hansson 2001)

6.6 Replacement

By replacement is meant an operation that replaces one sentence byanother in a belief set. It is an operation with two variables, suchthat [p/q] replacesp byq. Hence,K[p/q] is abelief set that containsq but notp.

Replacement aims both at the removal of a sentencep andthe addition of a sentenceq. This amounts to two successconditions,p∉K[p/q] andq∈K[p/q].However, these two conditions cannot both be satisfied withoutexception. First, just as for belief contraction, an exception must bemade from the first of them in the case whenp is a tautologyand so cannot be removed. Secondly, the two conditions are notcompatible in cases whenq logically impliesp. Thiscan be dealt with by extending the exception that is provided for inthe success postulate for contraction from cases whenp ∈Cn(∅) to cases whenp ∈Cn({q}). Thisgives rise to the following two success conditions:

Contractive success: Ifp ∉ Cn({q}), thenp ∉Cn(K[p/q]).

Revision success:q ∈K[p/q]

A replacement operation can be constructed by combining contractionand expansion. The outcomeK[p/q] ofreplacingp byq should be a consistent set thatcontainsq but notp. Thus a subset ofKneeds to be found, to whichq can be added withoutpbeing reintroduced. This is equivalent to requiring that this subset ofK does not implyq→p. Such a subset canbe obtained by contracting byq→p:

K[p/q] =K ÷(q→p)+q

Replacement can also be used as a kind of “Shefferstroke” for belief revision, i.e. as an operator in terms ofwhich the other operations can be defined. Contraction bypcan be defined asK[p/⊤] and revision byp asK[⊥/p], where ⊥ is falsum and ⊤ is tautology. Provided that the unperformable suboperation ofremoving the tautology ⊤ is treated as an empty suboperation, expansion byp can bedefined asK[⊤/p].

In both positions of the replacement operator, tautology gives riseto an empty suboperation. (Removal of ⊤is empty since ⊤ cannot be removed. Addition of ⊤ is empty since ⊤ is already in the belief set.) Therefore, it may be instructive toomit ⊤ in the operator. Then [p/ ] denotescontraction, [ /q] expansion, and[⊥/q] revision.

6.7 Multiple contraction and revision

By multiple contraction is meant the simultaneous contraction ofmore than one sentence. Similarly, multiple revision is revision bymore than one sentence.

There are two major types of multiple contraction. Inpackagecontraction, all elements of the input set are retracted: theyhave to go in a package. Inchoice contraction it suffices toremove at least one of them. Hence, the success conditions for the twotypes of multiple contraction are as follows:

Package success: IfB∩Cn(∅) = ∅thenB∩Cn(A÷B) = ∅

Choice success: IfB is not a subset of Cn(∅), thenB is not either a subset of Cn(A÷B).

Partial meet contraction and kernel contraction can both begeneralized fairly straight-forwardly to both package and choicevariants of multiple contraction.

The construction of package revision gives rise to an interestingextension of the notion of negation. The reason why contraction by¬p is performed as a suboperation of revision byp is thatp can be consistently added to a set if andonly if it does not imply ¬p. It turns out that (in alogic satisfying compactness) a setB can be consistentlyadded to another set if and only if this other set does not contain anyelement of the set ¬B, that is defined as follows:

Negation of a set: p ∈ ¬B if andonly ifp is either a negation of an element ofB∪{⊤} or a disjunction ofnegations of elements ofB∪{⊤}.

Therefore multiple revision can be defined from (package) multiplecontraction and expansion via a generalized Levi identity:

K*B =(K÷¬B)+B

Most of the major AGM-related contraction operators have beengeneralized to multiple contraction: multiple partial meet contraction(Hansson 1989, Fuhrmann and Hansson 1994, Li 1998), multiple kernelcontraction (Ferméet al. 2003), multiple specified meetcontraction (Hansson 2010), and a multiple version of Grove's spheresystem (Reis and Fermé 2011, Fermé and Reis 2011).

6.8 Indeterministic belief change

Most models of belief change aredeterministic in the sensethat given a belief set and an input, the resulting belief set iswell-determined. There is no scope for chance in selecting the newbelief set. Clearly, this is not a realistic feature, but it makes themodels much simpler and easier to handle, not least from acomputational point of view. Inindeterministic belief change,the subjection of a specified belief set to a specified input has morethan one admissible outcome.

Indeterministic operations can be constructed as sets of deterministicoperations. Hence, given three revision operators *, *′ and*′′ the set{*,*′,*′′}can be used as an indeterministic operator. The success condition issimply:

K{*, *′, *′′}p∈{K*p,K*′p,K*′′p}

Lindström and Rabinowicz (1991) constructed indeterministiccontraction by giving up the assumption that epistemic entrenchmentsatisfies connectedness. This resulted in Grove's sphere systems with“fallbacks” that are not linearly ordered but still allinclude the original belief set.

6.9 Operations for an extended language

Belief revision theory has been almost exclusively developed withinthe framework of classical sentential (truth-functional) logic. Theinclusion of non-truthfunctional expressions into the language hasinteresting and indeed drastic effects. In particular, this applies toconditional sentences.

Many types of conditional sentences, such as counterfactualconditionals, cannot be adequately expressed with truth-functionalimplication (material implication). Several formal interpretations ofconditional sentences have been proposed. One of them, namely theRamsey test, is particularly well suited to the formalframework of belief revision. It is based on a suggestion by F. P.Ramsey that has been further developed by Robert Stalnaker and others(Stalnaker 1968). The basic idea is that “ifp thenq” is taken to be believed if and only ifqwould be believed after revising the present belief state byp. Letp⇒q denote “ifpthenq”, or more precisely: “ifp werethe case, thenq would be the case”. In order to treatconditional statements on par with statements about actual facts, theywill have to be included in the belief set, thus:

TheRamsey test:(p⇒q)∈K if and only ifq∈K*p.

However, inclusion in the belief set of conditionals that satisfythe Ramsey test will require radical changes in the logic of beliefchange. As one example of this, contraction cannot then satisfy theinclusion postulate (K÷p ⊆K).The reason for this is that contraction typically provides support forconditional sentences that were not supported by the original beliefstate. This can be seen from the following example:

If I give up my belief that John is mentally retarded, thenI gain support for the conditional sentence “If John has lived 30years in London, then John understands the Englishlanguage.”

A famous impossibility theorem by Peter Gärdenfors (1986, 1987)shows that the Ramsey test is incompatible with a set of plausiblepostulates for revision. Several solutions to the impossibilitytheorem have been put forward. One option is to reject the Ramsey testas a criterion for the validity of conditional sentences (Rott1986). Another, proposed by Isaac Levi, is to accept the test as acriterion of validity but deny that such conditional sentences shouldbe included in the belief set when they are valid (Levi1988. Arló-Costa 1995. Arló-Costa and Levi1996). Several other proposals have been put forward. However, it isfair to say that operations of belief change have not yet beenconstructed that are generally recognized as able to deal adequatelywith conditional or modal sentences.

6.10 Changes in the strength of beliefs

An operation of change can raise or lower the position of a sentencein the ordering without affecting the belief set (but affecting howthe belief state responds to new inputs). An operatorofimprovement, as proposed by Konieczny and Pérez(2008) increases the plausibility of a sentencep by movingsome of thep-worlds to a higher position in the preferenceordering of worlds. Even if such a change does not lead topbecoming a belief, its acceptance in later, additional operations canbe facilitated.

An operation of change can be so constructed that it adjusts theposition of the input sentence in an ordering to be the same as thatof a reference sentence. This means that two sentences have to bespecified in the input: the (input) sentence to be moved and the(reference) sentence indicating its new position. Hans Rott (2007,Other Internet Resources) called such operators two-dimensional. JohnCantwell (1997) classified them asraisingorlowering, depending on the direction of change. (See alsoFermé and Rott 2004 and Rott 2009.)

6.11 Changes in norms and preferences

There are close parallels between changes in norms and changes inbeliefs. In order to apply a norm system with conflicting norms to aparticular situation, some of the norms may have to be ignored. Theproblem of how to prioritize among conflicting norms is similar to theselection of sentences for removal in belief contraction (Hansson andMakinson 1997). The AGM model was in fact partly the outcome ofattempts to formalize changes in norms rather thanbeliefs (Alchourrón and Makinson 1981). In spite of this,authors who apply the AGM model to norms have found it in need ofrather extensive modifications to make it suitable for thatpurpose. Hence, in their model of changes in legislation, Governatoriand Rotolo (2010) introduced an explicit representation of time inorder to account for phenomena such as retroactivity.

A model of changes in preferences can be obtained by replacing thestandard AGM language by sentences of the formp≥q (“p is at least as goodasq”) and their truth-functional combinations. Theadoption of a new preference can then take the form of revision bysuch a preference sentence. Partial meet contraction can be used, butsome modifications of the AGM model seem to be necessary inapplications to preferences (Hansson 1995, Lang and van der Torre2008, Grüne-Yanoff and Hansson 2009).

7. Repeated change

The preceding sections have only dealt with changes of one and thesame belief set or belief base. This is clearly a severe limitation. Arealistic model of belief change should allow for repeated (iterated)changes, such asK÷p÷q*r*s÷t… In other words, operators are needed that can contractor revise any belief set (belief base) by any sentence. Such operatorsare calledglobal, in contrast tolocal operatorsthat are defined only for a single specified set.

An obvious way to obtain a global operator ofpartial meetcontraction would be to use the same selection function for allsets to be contracted. However, this is not possible, due to the wayselection functions treat the empty set. The way selection functionshave been defined, ifA⊥p = ∅, thenγ(A⊥p ) = {A}. As a consequenceof this, if γ is a selection function forA, andA ≠B, then γ is not a selection functionforB. For letA⊥p =B⊥p = ∅. For γ to be a function itmust be the case thatγ(A⊥p) =γ(B⊥p).For γ to be a selectionfunction forA it must be case thatγ(A⊥p) = {A}, and in order for itto be a selection function forB it must be the case thatγ(B⊥p) = {B}. SinceA≠B, this is impossible.

Therefore, a global operator of partial meet contraction must applydifferent selection functions to different belief sets (belief bases).A convenient way to achieve this is to introduce two-place selectionfunctions. A two-place selection function γ has two argumentplaces, one for the belief set (belief base) and one for the remainderset. γ(A,A⊥p) is a subset ofA⊥p if the latter is non-empty, otherwise it isequal toA. Some formal results on global operations based ontwo-place selections functions have been obtained.

Epistemicentrenchment has an advantage that makesit one of the more promising approaches to repeated belief change,namely what may be called theself-sufficiency of entrenchmentorderings. Given an entrenchment ordering ≤ for a belief setK,K can be “recovered” from ≤ via thesimple identity:

K = {q |r<q forsomer}

Contractions and revisions can therefore be performed onentrenchment orderings rather than on belief sets. Thus, given a beliefsetK and an entrenchment ordering ≤ forK, whencontracting by a sentencep the outcome should be a newentrenchment ordering ≤′. It automatically provides a newbelief set, namely the set {q |r<′q for somer}. If the operationhas been successful, then this new belief set should not containp.

It has turned out to be quite difficult to construct a satisfactorymethod to contract or revise an entrenchment ordering. The mostinteresting proposals involve a rather radical departure from thetraditional AGM framework in which epistemic entrenchment is usuallydiscussed. Hence, Abhaya Nayak (1994) has proposed that not only thebelief states but also the inputs should be binary relations (thatsatisfy the standard entrenchment postulates exceptMinimality). Inputs of this type may be seen as“fragments” of belief states, to be incorporated into theprevious belief state. In the same vein, Eduardo Fermé and HansRott (2004) have investigated belief revision with inputs of the moregeneral form: “acceptq with a degree of plausibilitythat at least equals that ofp”. They call thisrevision by comparison. Belief states are represented byentrenchment orderings. Hence, from an entrenchment ordering ≤ andsuch a generalized input a new entrenchment ordering ≤′ isobtained that contains the information needed to construct the newbelief set.

As this example illustrates, many of the more recent developments inbelief change show a need to go beyond the expressive power of the AGMmodel, but that model is still the standard model to which all othermodels of belief change are compared.

Bibliography

Citations annotated in refer to books orarticles that are recommended for further introductory reading.

• Alchourrón, C.E., P. Gärdenfors, and D. Makinson,1985, “On the Logic of Theory Change: Partial Meet Contractionand Revision Functions”,Journal of Symbolic Logic, 50:510–530.
     
• Alchourrón, C.E. and D. Makinson, 1981, “Hierarchiesof Regulation and Their Logic”, in R. Hilpinen (ed.),New Studies in Deontic Logic, Dordrecht: D. Reidel PublishingCompany, pp. 125–148.
• Alchourrón, C.E. and D. Makinson, 1982, “On thelogic of theory change: Contraction functions and their associatedrevision functions”,Theoria, 48: 14–37.
• Arló-Costa, H., 1995, “Epistemic conditionals, snakesand stars”, in G. Crocco, L. Fariñas del Cerro, andA. Herzig (eds.), Conditionals: from Philosophy to ComputerScience,Studies in Logic and Computation (Volume 5),Oxford: Oxford University Press.
• Arló-Costa, H. and I. Levi, 1996, “Two notions ofepistemic validity”,Synthese, 109: 217–262.
• Cantwell, J., 1997, “On the logic of small changes inhypertheories”,Theoria, 63: 54–89.
• –––, 1999, “Some logics of iteratedrevision”,Studia Logica, 7: 49–84.
     
• Doyle, J., 1979, “A Truth Maintenance System”,Artificial Intelligence, 12: 231–272.
• Fagin, R., J.D. Ullman, and M.Y. Vardi, 1983, “On thesemantics of updates in databases”,Proceedings of SecondACM SIGACT-SIGMOD, pp. 352-365.
• Fermé, E. and S.O. Hansson, 2001, “ShieldedContraction”, pp. 85–107 in H. Rott and M.-A. Williams (eds.),Frontiers of Belief Revision, Dordrecht: Kluwer.
• –––, 2011, “AGM 25years. Twenty-Five Years of Research in BeliefChange”,Journal of Philosophical Logic, 40:295–331.
     
• Fermé, Eduardo and Maurício D.L. Reis, 2011,“System of Spheres-based MultipleContractions”,Journal of Philosophical Logic, inpress.
• Fermé, E. and H. Rott, 2004, “Revision bycomparison”,Artificial Intelligence, 157: 5–47.
• Fermé, E., K. Saez, and P. Sanz, 2003, “MultipleKernel Contraction”,Studia Logica, 73: 183–195.
• Fuhrmann, A. and S.O. Hansson, 1994, “A Survey of Multiple Contraction”,Journal of Logic, Language and Information, 3: 39–74.
     
• Gärdenfors, P., 1978, “Conditionals and Changes ofBelief”,Acta Philosophica Fennica, 30: 381–404.
• –––, 1981, “An Epistemic Approach toConditionals”,American Philosophical Quarterly, 18:203–211.
• –––, 1986, “Belief Revision and the RamseyTest for Conditionals”,Philosophical Review, 95:81–93.
     
• –––, 1987, “Variations on the Ramsey Test:More triviality results”,Studia Logica, 46:319–325.
• –––, 1988,Knowledge in Flux. Modeling theDynamics of Epistemic States, Cambridge, MA: MIT Press.
     
• –––, 2011, “Notes on the history of ideasbehind AGM”,Journal of Philosophical Logic, 40:115–120.
• Gärdenfors, P., and D. Makinson, 1988, “Revisions ofKnowledge Systems Using Epistemic Entrenchment”,SecondConference on Theoretical Aspects of Reasoning about Knowledge,pp. 83–95.
• Governatori, G. and A. Rotolo, 2010, “Changing legalsystems: legal abrogations and annulments in DefeasibleLogic”,Logic Journal of IGPL, 18: 157–194.
• Grove, A., 1988, “Two Modellings for Theory Change”,Journal of Philosophical Logic, 17: 157–170.
     
• Grüne-Yanoff, T. and S.O. Hansson, 2009, “From BeliefRevision to Preference Change”, inT. Grüne-Yanoff and S.O. Hansson (eds.),Preference Change:Approaches from Philosophy, Economics and Psychology,Berlin: Springer, pp. 159–184.
• Hansson, S.O., 1989, “New Operators for TheoryChange”,Theoria, 55: 114–132.
• –––, 1995, “Changes inPreference“,Theory and Decision, 38: 1–28.
• –––, 1999,A Textbook of Belief Dynamics. TheoryChange and Database Updating, Dordrecht: Kluwer.
     
• –––, 2003, “Ten Philosophical Problems in BeliefRevision”,Journal of Logic and Computation, 13: 37–49.
     
• –––, 2010, “Multiple and Iterated ContractionReduced to Single-Step Single-SentenceContraction”,Synthese, 173: 153–177.
• Hansson, S.O., Fermé, E., Cantwell, J., and Falappa, M.,2001, “Credibility-Limited Revision”,Journal ofSymbolic Logic, 66: 1581–1596.
     
• Hansson, S.O. and D. Makinson, 1997, “Applying Normativerules with restraint”, in M.L. Dalla Chiara,etal. (eds.),Logic and Scientific Method, Dordrecht: Kluwer AcademicPublishers, pp 313–332.
• Harper, W., 1977, “Rational Conceptual Change”,PSA1976, pp. 462–494.
• Katsuno, H., and A.O. Mendelzon, 1992, “On the Differencebetween Updating a Knowledge Base and Revising it”, inP. Gärdenfors (ed.),Belief Revision, Cambridge:Cambridge University Press pp. 183–203
• Konieczny, S. and R. P. Pérez, 2008, “ImprovementOperators“,Eleventh International Conferenceon Principles of Knowledge Representation and Reasoning(KR08), pp. 177–186.
• Lang, J. and L. van der Torre, 2008, “From Belief Change toPreference Change”, in M. Ghallab, C.D. Spyropoulos,N. Fakotakis, and N.M. Avouris (eds.),ECAI 2008 –Proceedings of the 18th European Conference on ArtificialIntelligence, Patras, Greece, July 21–25, 2008 (Frontiers inArtificial Intelligence and Applications: Volume 178),pp. 351–355.
• Levi, I., 1977, “Subjunctives, Dispositions andChances”,Synthese, 34: 423–455.
• –––, 1980,The Enterprise of Knowledge, Cambridge,MA: MIT Press.
• –––, 1988, “Iteration of conditionals and the Ramseytest”,Synthese, 76: 49–81.
• –––, 1991,The Fixation of Belief and Its Undoing,Cambridge, MA: Cambridge University Press.
• –––, 2004,Mild Contraction. Evaluating Loss ofInformation due to Loss of Belief, Oxford: Clarendon Press.
• Li, J., 1998, “A Note on Partial Meet PackageContraction”,Journal of Logic, Language andInformation, 7: 139–142.
• Lindström, S. and W. Rabinowicz, 1991, “Epistemicentrenchment with incomparabilities and relational beliefrevision”, in A. Fuhrmann and M. Morreau (eds.),The Logicof Theory Change, Berlin: Springer, pp. 93–126.
• Makinson, D., 1997, “Screened Revision”,Theoria, 63 (1–2): 14–23.
• –––, 2003, “Ways of doing logic: what was new aboutAGM 1985”,Journal of Logic and Computation, 13:5–15.
• Nayak, A. C., 1994, “Iterated Belief Change Based onEpistemic Entrenchment”,Erkenntnis, 41: 353–390.
• Reis, M.D.L. and E. Fermé, 2011, “Possible WorldsSemantics for Partial Meet Multiple Contraction”,Journal ofPhilosophical Logic, in press.
• Rott, H., 1986, “Ifs, though andbecause”,Erkenntnis, 25: 345–37.
• –––, 2001,Change, choice and inference. A study ofbelief revision and nonmonotonic reasoning, Oxford: ClarendonPress.
     
• –––, 2009, “Shifting Priorities: Simple Representationsfor Twenty-seven Iterated Theory Change Operators”, in D. Makinson, J. Malinowski and H. Wansing (eds.)TowardsMathematical Philosophy (Trends in Logic: Volume 28),Berlin: Springer, pp. 269–296.
• Rott, H. and M. Pagnucco, 2000, “Severe Withdrawal (andRecovery)”,Journal of Philosophical Logic, 29:501–547.
• Stalnaker, R., 1968, “A Theory of Conditionals”, inN. Rescher (ed.),Studies in Logical Theory, Oxford:Blackwell, pp. 98–112.

Academic Tools

	How to cite this entry.
	Preview the PDF version of this entry at theFriends of the SEP Society.
	Look up this entry topic at theIndiana Philosophy Ontology Project (InPhO).
	Enhanced bibliography for this entryatPhilPapers, with links to its database.

Other Internet Resources

• Rott, H., 2007, “Bounded Revision: Two Dimensional BeliefChange Between Conservative and Moderate Revision”, in T. Rønnow-Rasmussen, B. Petersson, J. Josefsson andD. Egonsson,Hommage à Wlodek, Philosophical Papers Dedicated to Wlodek Rabinowicz.

Related Entries

logic: conditionals |logic: non-monotonic |reasoning: defeasible