On the Logic of Theory Change:Extending the AGM Model

Eduardo Ferme

PhD thesisStockholm, Sweden - 2011

AbstractFerme, Eduardo. 2011. On the Logic of Theory Change: Extending the AGM Model. PhD Thesis inPhilosophy from the Royal Institute of Technology.

This thesis consists in six articles and a comprehensive summary. The pourpose of the summary is to introduce the AGM theory of belief change and to exemplify the

diversity and significance of the research that has been inspired by the AGM article in the last 25 years. Theresearch areas associated with AGM was divided in three parts: criticisms, where we discussed some of themore common criticisms of AGM. Extensions where the most common extensions and variations of AGMare presented and applications where we provided an overview of applications and connections with otherareas of research. Article I elaborates on the connection between partial meet contractions [AGM85] and kernel con-

tractions [Han94a] in belief change theory. Also both functions are equivalent in belief sets, there are notequivalent in belief bases. A way to define incision functions (used in kernel contractions) from selectionfunctions (used in partial meet contractions) and vice versa is presented. It is explained under which condi-tions there are exact correspondences between selection and incision functions so that the same contractionoperations can be obtained by using either of them.Article II proposes an axiomatic characterization for ensconcement-based contraction functions, belief

base functions proposed by Williams and relates this function with other kinds of base contraction functions. Article III adapts the Ferme and Hansson model of Shielded Contraction [FH01] as well as Hansson et

all Credibility-Limited Revision [HFCF01] for belief bases, to join two of the many variations of the AGMmodel [AGM85], i.e. those in which knowledge is represented through belief bases instead of logic theories,and those in which the object of the epistemic change does not get the priority over the existing informationas it is the case in the AGM model. Article IV introduces revision by comparison a refined method for changing beliefs by specifying

constraints on the relative plausibility of propositions. Like the earlier belief revision models, the methodproposed is a qualitative one, in the sense that no numbers are needed in order to specify the posteriorplausibility of the new information. The method uses reference beliefs in order to determine the degree ofentrenchment of the newly accepted piece of information. Two kinds of semantics for this idea are proposedand a logical characterization of the new model is given. Article V focuses on the extension of AGM that allows change for a belief base by a set of sentences

instead of a single sentence. In [FH94], Fuhrmann and Hansson presented an axiomatic for Multiple Con-traction and a construction based on the AGM Partial Meet Contraction. This essay proposes for their modelanother way to construct functions: Multiple Kernel Contraction, that is a modification of Kernel Contrac-tion, proposed by Hansson [Han94a] to construct classical AGM contractions and belief base contractions. Article VI relates AGM model with the DFT model proposed by Carlos Alchourron [Alc93]. Al-

chourron devoted his last years to the analysis of the notion of defeasible conditionalization. His definitionof the defeasible conditional is given in terms of strict implication operator and a modal operator f whichis interpreted as a revision function at the language level. This essay points out that this underlying revisionfunction is more general than AGM revision. In addition, a complete characterization of that more generalkind of revision that permits to unify models of revision given by other authors is given.

Keywords: Logic of Theory Change. AGM model. Belief Bases. Iterated Models. Multiple belief change.AGM and defeasible Logic.©2011 by Eduardo Ferme.

A Bruno,por darle sentido a las cosas ...

Why a PhD in Philosophy?I begun my research career (and also in the belief revision area) back in 1988. A group of philoso-phers and people from artificial intelligence - led by Carlos Alchourron and Adolfo Kvitca wascreated in order to work in deontic logic and logic of theory change. In particular, AI researcherswere interested in explaining how to update databases and look for general theories of revisiondeveloped by Alchourron. Raul Carnota introduced me to the group, as the advisor of my degreethesis.

The early days of the group were very difficult, as we did not have the same ontology, language,motivations and expectations as philosophers. It took a long time, but finally we understood eachother. By the end of 1994 I started my PhD in Computer Science under Alchourron supervision(until his death in 1996). Over time I started working with other philosophers and I discovered thatmany of the questions I had were the same, but many issues were new, interesting and challenging.And always there was something in the conversations, a face of the prism, that I was missing.

One day I read a notable paper from one of the fathers of artificial intelligence, John McCarthy,“What has AI in Common with Philosophy? and I felt it reflected my interests in all matters andI decided to study philosophy to “bridge the gap I had observed. Sven Ove Hansson offered me aPhD position in the Royal Institute of Technology and John Cantwell kindly accept to superviseme.

Today, several years later, I’m at the end of a stage. I have a new PhD thesis, also in beliefrevision, but adresses to other concerns, with another approach and with a wealth of courses thathelped me to see how much I did not know and that I continue not to know.

This thesis provides me a PhD in Philosophy. The rest of my life will decide if I will be aPhilosopher.

AcknowledgementsI want to thank specially my supervisor John Cantwell and my study director Sven Ove Hanssonfor their support. I am also indebted to the coauthors of the papers that compound this thesis.Thanks to Mauricio Reis for his comments on earlier versions of this introduction.

I want to thank all the philosophers who worked and shared with me all these years and forall that I learned with them: Carlos A., Sven Ove, John, David, Erik, Gladys, Carlos O., Sandra,Hans, Isaac, Peter, Sten, Wlodek, Krister, Horacio, Andre, Silvio, ...

In a personal note, I want to thank the continuous support of family, friends and colleagues. Toname some of them necessarily implies unfairly excluding others. But I know who they are and Iextend my gratitude to them. And they, I assume, also know.

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List of PapersI Marcelo Falappa, Eduardo Ferme and Gabrielle Kern-Isberner. “On the Logic of Theory

Change: Relations between Incision and Selection Functions”. In: G. Brewka, S. Corade-schi, A. Perini, and P. Traverso (eds.): Proceedings 17th European Conference on ArtificialIntelligence, ECAI06. pp. 402-406. 2006.

II Eduardo Ferme, Martın Krevneris and Mauricio Reis. “An axiomatic characterization ofensconcement−based contraction”. Journal of Logic and Computation. Oxford UniversityPress. 18(5):739-753. 2008.

III Eduardo Ferme, Juan Mikalef, and Jorge Taboada. “Credibility-limited Functions for BeliefBases”. Journal of Logic and Computation. Oxford University Press. 13(1): 99-110. 2003

IV Eduardo Ferme and Hans Rott. “Revision by Comparison”. Artificial Intelligence. ElsevierPublisher. 157(1-2):5-47. 2004.

V Eduardo Ferme, Karina Saez and Pablo Sanz. “Multiple Kernel Contraction”. Studia Logica.Kluwer Academic Publisher. 73(2):183-195. 2003.

VI Eduardo Ferme and Ricardo Rodrıguez. “DFT and Belief Revision’. Analisis FilosoficoXXVII(2): 373–393. 2006.

I am indebted to the coauthors of the papers for their kindly permission to include them in thedissertation.

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Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 The AGM Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1 Axiomatic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Relation between Revision and Contraction . . . . . . . . . . . . . . . . . . 62.3 Partial Meet functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Kernel / Safe Contraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.5 Epistemic Entrenchment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.6 Grove’s sphere-systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.7 The interconnection among the five presentations . . . . . . . . . . . . . . . 11

3 Criticism of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.1 The recovery postulate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 The success postulates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3 Are belief sets too large? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.4 Lack of information in the belief set . . . . . . . . . . . . . . . . . . . . . . 15

4 Ways to extend the AGM model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.1 Extended representations of belief states . . . . . . . . . . . . . . . . . . . . 154.2 Iterated change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.3 Changes in the strength of beliefs . . . . . . . . . . . . . . . . . . . . . . . . 244.4 Non-prioritized change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.5 Multiple change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5 Applications and connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.1 Update . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.2 Non-monotonic and defeasible logic . . . . . . . . . . . . . . . . . . . . . . 305.3 Modal and dynamic logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.4 Game theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.5 Argumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.6 Belief Change by translation between logics . . . . . . . . . . . . . . . . . 335.7 Truth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.8 Use of choice functions and related preference orderings . . . . . . . . . . 34

6 Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

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1 IntroductionThe logic of theory change became a major subject in philosophical logic and artificial intelligencein the middle of the 1980’s. The most important model that is now known as the AGM model ofbelief change, was proposed by Alchourron, Gardenfors and Makinson in [AGM85]. The AGMmodel is a formal framework to characterize the dynamics and state of belief of a rational agent.In the AGM framework, the beliefs are represented ideally by belief sets, which are deductivelyclosed sets of sentences. A change consists in adding or removing a specific sentence from a beliefset to obtain a new belief set. The AGM model has acquired the status of a standard model of beliefchange.

The influence of the AGM model is witnessed by how often other researchers refer to it. Theyearly number of citations of the AGM article recorded in the Web of Science database has beensteadily growing, from about 15 per year in the 1990’s to about 35 per year in the first few years ofthe new millennium and around 50 in the most recent years − a remarkable record for a logic paper.Its impact has been profound both among philosophers and in the artificial intelligence community[CR10].

There are at least three reasons for the lasting impact of the 1985 article. First it provided asimple input–output framework for modeling change. This framework is applicable to a wide rangeof important areas: human belief change, changes in databases, transformations of the scientificcorpus, changes in norms and norm systems, changes in preferences and attitudes, etc. Secondly,by combining classical logical tools with representations of choice in an innovative way (based onprevious developments in the logic of conditionals), the AGM article contributed to extending thescope of subject-matter for logic modeling. The influence of AGM in areas such as non-monotonicand defeasible logic has been decisive. Thirdly, the logical tools introduced in the 1985 articlehave been surprisingly fruitful and continue to give rise to new results and new formal systems.

The AGM model inspired many researchers to propose extensions and generalizations as wellas applications and connections with other fields.

Regarding extensions we can mention:

Extended representations of belief states

Belief Base Dynamics: Instead of belief sets, a belief base is a set of sentences that is not(except as a limiting case) closed under logical consequence. A belief base has a fun-damental property: it can distinguish between explicit beliefs (element of the beliefbase) and derived belief, i.e., elements that are logical consequence of the belief base,but that are not (explicitly) in the belief base. In order to represent real cognitive agentsbelief bases are a more suitable representation than belief sets.

Others: We can mention: the use of probabilistic and possibilistic models to represent be-liefs and belief sets. Ranking models to represent agent’s degree of belief. Extendingthe language to modal logic, conditional, etc.

Iterated change A drawback of AGM definition of revision is that the conditions for the iterationof the process are very weak, and this is caused by the lack of expressive power of belief

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sets. In order to ensure good properties for the iteration of the revision process, one needsa more complex structure. So shifting from belief sets to epistemic states was proposed byDarwiche and Pearl in [DP96]. In this framework, it is possible to define interesting iteratedrevision operators.

Changes in the strength of beliefs In order to represent iterated change we need a representationof the belief state that includes, in addition to the belief set, an ordering that expresses therelative plausibility or retractibility of the beliefs. In such a model, a general type of oper-ation can be introduced that consists in raising or lowering the position of a sentence in theordering. Such a change may or may not affect the belief set, but will always affect how thebelief state responds to new inputs.

Non-prioritized change The AGM model always accepts the new information (success condi-tion). This feature appears, in general, unrealistic, since rational agents, when confrontedwith information that contradicts previous beliefs, often reject it altogether or accept onlyparts of it. In non-prioritized revision functions, the success postulate is relaxed by weakerconditions, that do not accept the new information in certain cases.

Multiple change: In standard AGM the input is a single sentence. In multiple change the inputis a (possibly infinite) set of sentences. The generalization of revision to the multiple caseis direct, since the multiple revision by a set correspond to the (singleton) revision by theconjunction of the elements of the input, unless if the input is an infinite set of sentences.In what concerns multiple contraction things are not so clear. In the literature we can findthree different models of multiple contraction: package contraction [FH94] (all elements ofthe input set are retracted), choice contraction [FH94] (at least one of the elements of theinput set are retracted), and set contraction [Zha96] (the output of the contraction must beconsistent with the input set).

Among applications and connections we can mention:

Update These operators are intended to represent changes in beliefs that result from changes inthe objects of belief, whereas revision operators are suited to capture changes that reflectevolving knowledge about a static situation.

Non Monotonic and Defeasible Logic As pointed out in [MG91] the connection is manifestedat the level of general conditions of nonmonotonic inference operations compared to therevision function.

Description Logics Description logics have been successful in detecting incoherences indatabases, but provide little support for resolving these incoherences. Methodologies forbelief change can be used to improve their performance in that respect.

Modal and Dynamic Logics As pointed out in [LR99b] the main reason to investigate belief re-vision in modal logic is that the theories of belief change developed within the AGM tradi-tion are not logics in a strict sense, but rather informal axiomatic theories of belief change.

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Instead of characterizing the models of belief and belief change in a formalized object lan-guage, the AGM approach uses a natural language (ordinary mathematical English) to char-acterize the mathematical structures under study.

Other Connections Argumentation theory, Game Theory, Truth, Belief Change by Translationbetween Logic, Preference orderings, etc.

In this summary we will present the AGM models and a some of the alternative models andconnections proposed in the literature1. In teh Epilogue we sum up the contribution of this thesis.

2 The AGM ModelIn the AGM account an epistemic state is represented by a belief set K (also called “theory”),which is a set of propositions of a (at least propositional) language L closed under logical con-sequence Cn, where Cn is an operator that satisfies the basic Tarskians properties: inclusion(X ⊆ Cn(X)), idempotence (Cn(Cn(X)) = Cn(X)) and monotony (Cn(X) ⊆ Cn(Y) if X ⊆ Y),as well as supraclassicality, deduction and compactness. Consequently for every theory K we havethat: Cn(K) = K.

We will sometimes use Cn(p) for Cn(p), A ⊢ p for p ∈ Cn(A), ⊢ p for p ∈ Cn(∅), A /⊢ pfor p /∈ Cn(A), /⊢ p for p /∈ Cn(∅). The letters p, pi,q, . . . will be used to denote sentences. ⊺stands for an arbitrary tautology and for an arbitrary contradiction. A,Ai,B, . . . shall denotesubsets of sentences of L. K is reserved to represent a belief set. We shall denote the set of allmaximal consistent subsets of L by L ⊥⊥. We will use the expression possible world (or justworld) to designate an element of L ⊥⊥. Given a set of sentences R, the set consisting of all thepossible worlds that contain R is denoted by ∥R∥. The elements of ∥R∥ are the R-worlds. ∥p∥ is anabbreviation of ∥p∥ and the elements of ∥p∥ are the p-worlds.

Given a belief set, three basic epistemic attitudes are assumed: acceptation (when p ∈ K),rejection (when ¬p ∈ K) and indetermination (when neither p ∈ K nor ¬p ∈ K)2. The three basicoperations of the AGM model, that correspond with a change of epistemic attitude towards theinput sentence p, are the following:

Expansion: This operation is in charge of incorporating sentences in the original set, withouteliminating any sentence from it. It allows the passage from an epistemic state in which abelief is undetermined to another epistemic state in which the belief is accepted or rejected.

1The following summary was created borrowed my previous paper “Revision de Creencias” [Fer07] and my paperwith Sven Ove Hansson “AGM 25 Years: Twenty-five years in research in belief change” [FH10]. I want to thanksSven Ove for her kindly permission to include our material here.

2Ideally, K is consistent.

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Contraction: This operation eliminates sentences from the original set without incorporating anynew ones. It allows the passage from an epistemic state in which a belief is accepted orrejected to another epistemic state in which the belief is undetermined.

Revision: This operation incorporates a sentence in the original set, but it can eliminate somebeliefs in order to preserve consistency of the revised set. It allows the passage from anepistemic state in which a belief is accepted (rejected) to another state in which the belief isrejected (accepted).

In the AGM model the changes are performed following the following rationality criteria[Gar88, Dal88]: Primacy of new information: the new information is always accepted. Consis-tency: the new epistemic state must be consistent if possible. Minimal loss of previous beliefs(informational economy): the attempt to retain as much of the old beliefs as possible. Adequacy ofrepresentation (categorical matching): the revised knowledge should have the same representationas the old knowledge. Fairness: If there are many epistemic states candidates for the outcome of abelief change, then one of then should not be arbritrarely choosen.

AGM has been characterized in at least five different equivalent ways: axiomatic[Gar82, AGM85], partial meet functions [AM82, AGM85], epistemic entrenchment[Gar88, GM88], safe/kernel contraction [AM85, Han94a] and Grove’s sphere-systems [Gro88]. Inthis section we summaryze these five presentation.

2.1 AxiomaticIn this subsection we present the AGM functions through a set of postulates that determine thebehavior of a change function, i.e., a set of conditions or constraints that change functions mustsatisfy.

The AGM axioms for contraction are:

Closure K−p is a belief set whenever K is a belief set.Success If /⊢ p , then K−p /⊢ p.Inclusion K−p ⊆ K.Vacuity If K /⊢ p, then K ⊆ K−p.Extensionality If ⊢ p↔ q then K−p = K−q.Recovery K ⊆ (K−p) + p.

Closure says that the outcome of a change performed in a theory must be a theory. Successsays that a contraction of K to exclude p does in fact give up p, unless p is a tautology (due toclosure, K−p includes all the tautologies). Inclusion says that when we contract K by p we alwaysobtain a subset of K, i.e., no proposition is added. In the limiting case that K /⊢ p, vacuity says thatnothing needs to be done to eliminate p from K. Extensionality says that contracting by logically

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equivalent sentences must yield the same result. The postulate of recovery says that when wecontract a theory to get rid of p, and then add p back again to the result of the contraction, werecover the initial theory. The postulates listed above are called the basic AGM (or Gardenfors)postulates. In addition to them, we have the following postulate for contraction by a conjunction.

Conjunctive factoring K−(p ∧ q) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

K−p, orK−q, orK−p ∩ K−q

Conjunctive factoring means that if we wish to contract the belief set by a conjunction and thereexists some preference between the conjuncts, then this contraction is equivalent to contractionby the non-preferred conjuncts. In the case of indifference among the conjuncts, the outcomeof contracting by the conjunction equals the intersection of the outcomes of contractions by theconjuncts.

The AGM axioms for revision are:

Closure: K∗p is a belief set whenever K is a belief set.Success: p ∈ K∗p.Inclusion: K∗p ⊆ K+p.Vacuity: If K /⊢ ¬p then K∗p = K+p.Consistency: /⊢ ¬p then K∗p /⊢⊥.Extensionality: If ⊢ p↔ q then K∗p = K∗q.

where closure and extensionality are similar to the corresponding postulates in the theory of con-traction. Success gives a tacit priority to the incoming information by saying that the new proposi-tion must be part of the transformed theory. Inclusion and vacuity say that when the new input doesnot contradict the background theory there is no reason to eliminate previous beliefs and thereforerevision should go by expansion. Consistency guarantees that the new theory K∗p must be consis-tent unless (due to success) in the case where p is itself inconsistent. As in contraction, we have asupplementary postulate for disjunctions

Disjunctive factoring K∗(p ∨ q) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

K∗p, orK∗q, orK∗p ∩ K∗q

The intuition behind this postulate is that if we wish to revise by a disjunction and there is somepreference between the disjuncts, then this revision is equivalent to revising by the preferred one.In the case of indifference, revising by the disjunction returns the beliefs that are common to theoutcomes of revising by each member of the disjunction.

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2.2 Relation between Revision and ContractionWe have seen that contraction and revision are characterized by two different sets of postulates.These postulates are independent in the sense that the postulates of revision do not refer to con-traction and vice versa. However, it is possible to define revision functions in terms of contractionfunctions, and vice versa by means of the following identities:

• Levi’s Identity: K∗p = (K−¬p)+p.

• Harper’s Identity: K−p = K ∩ K∗¬p.

Given these two identities, it can be proven that if a contraction operator satisfies the basiccontraction postulates then the revision operator obtained by applying Levi’s identity satisfies thebasic revision postulates. Moreover if the contraction operator also satisfy conjunctive factoringthen the revision operator obtained by applying Levi’s identity also satisfies disjunctive factoring.

In an analogous way, if a revision operator satisfies the basic revision postulates then the con-traction operator obtained by applying Harper’s identity satisfies the basic contraction postulates.Moreover if the revision operator also satisfy disjunctive factoring then the revision operator ob-tained by applying Harper’s identity also satisfies conjunctive factoring.

The Levi (Harper) identity allows us to use a contraction (revision) function as a primitive, andtreat revision (contraction) as defined in terms of contraction (revision).

2.3 Partial Meet functionsAccording to the informational economy criterion, the contraction function must retain as large asubset of K as possible. The sets that satisfy this property can be identified as follows:

Definition 1 [AM81] Let K be a belief set and p a sentence. The set K⊥p (K remainder p) is theset of sets such that H ∈ K⊥p if and only if:

⎧⎪⎪⎪⎨⎪⎪⎪⎩

H ⊆ KH /⊢ pThere is no set H′ such that H ⊂ H′ ⊆ K and H′ /⊢ p

K⊥p is called a remainder set and its elements are the remainders of K by p.In order to construct a contraction function, we need to make a selection among the remainder

sets.

Definition 2 [AGM85] Let K be a belief set. A selection function for K is a function γ such thatfor all sentences p:

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(i) If K⊥p is non-empty, then γ(K⊥p) is a non-empty subset of K⊥p.(ii) If K⊥p is empty, then γ(K⊥p) = K.

γ is relational if and only if there is a relation ⊑ such that for all sentences p, if K⊥p is non-empty,then

γ(K⊥p) = B ∈ K⊥p ∣ C ⊑ B for all C ∈ K⊥p

γ is transitively relational if and only if this hold for some transitive relation ⊑.

Definition 3 [AGM85] Let K be a belief set and γ a selection function for k. The partial meetcontraction on K that is generated by γ is the operation ∼γ such that for all sentences p:

K ∼γ p = ∩γ(K⊥p)

An operation − on K is a partial meet contraction if and only if there is a selection function γ forK such that for all sentences p ∶ K−p = K ∼γ p.

Furthermore, − is (transitively) relational if and only if it can be generated from a (transitively)relational selection function.

By means of the Levy identity we can construct the revision function based on remainder sets.

Definition 4 [AGM85] The operator ∗ on a belief set K is an operator of partial meet revisionif and only if there is some operator − of partial meet contraction on K such that for all sentences p

K∗p = (K−¬p)+p

Furthermore, ∗ is (transitively) relational if and only if this hold for some − that is (transitively)relational.

2.4 Kernel / Safe ContractionSafe Contraction [AM85] and its generalisation Kernel Contraction [Han94a] are based on a selec-tion among the sentences of a belief set K that contribute effectively to imply p; and how to usethis selection in contracting by p.

Definition 5 [Han94a] Let K be a belief set and p a sentence. Then K ⊥⊥ p is the set such thatA ∈ K ⊥⊥ p if and only if:

⎧⎪⎪⎪⎨⎪⎪⎪⎩

A ⊆ KA ⊢ pIf B ⊂ A then B /⊢ p

K ⊥⊥ p is called the kernel set of K with respect to p and its elements are the p-kernels of K

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In order to contract p from K, we have to give up sentences in each p-kernel, otherwise thesentence p would continue being implied. On the other hand, due to the minimality criterion,we only must discard sentences that are included in on or more elements of the kernel set. Theremaining problem is how to choose these sentences. This is done by incision functions.

Definition 6 [Han94a] An incision function σ for K is a function such that for all sentences p:

σ(K ⊥⊥ p) ⊆ ⋃(K ⊥⊥ p)If ∅ /= A ∈ K ⊥⊥ p, then A ∩σ(K ⊥⊥ p) /= ∅

So, incision functions cut into each p-kernel, removing at least one sentence. Since all p-kernels are minimal subsets implying p, from the resulting sets it is no longer possible to derive p.Hence, incision functions may be used to derive contraction operations.

Definition 7 Given a belief set K, a sentence p and an incision function σ for K, the kernelcontraction of K by p, denoted by K−σp, is defined as:

K−σp = Cn(K ∖σ(K ⊥⊥ p))

That is, K−σp can be obtained by erasing from K the sentences cut out by σ3.The next step is to introduce constraints on the incision function in order to select which sen-

tences we want to discard from each p-kernel set. Alchourron and Makinson [AM85, AM86]defined safe contraction. In this contraction, the belief set K is ordered according to a relation ∠.q∠δ means that δ should be retained rather that q if we have to give up one of them, and we saythat “q is less safe that δ”. The ordering ∠ helps us to choose which element to remove from eachkernel. The remaining beliefs are safe and can be used to determine the safe contraction of a beliefset K by p (modulo ∠). ∠ must be an acyclic, irreflexive and asymmetric relation. Alchourronand Makinson referred to this relation as a “hierarchy”. ∠ is virtually connected over K if andonly if for all p, q, δ ∈ K: if p∠q then either p∠δ or δ∠q. ∠ is regular if and only if it satisfiescontinuing-up (If p∠q and q ⊢ δ, then p∠δ ) and continuing-down (Ifp ⊢ q and q∠q, then p∠δ ).

Definition 8 [AM85] Any sentence q in a belief set K is safe with respect to p if and only if q isnot minimal under ∠ with respect to the elements of any A ∈ K ⊥⊥ p. The set of all safe sentencesof K respect to p is denoted by K/p.

Using the set K/p we can define a contraction function:

Definition 9 [AM85] Let K be a belief set, p a sentence and ∠ a regular and virtually connectedhierarchy. K ∼ p is a safe contraction, based on a regular and virtually connected hierarchy ∠, ifand only if:

K ∼ p = Cn(K/p)3We close by Cn the result of cutting K in order to the outcome be a belief set. This special kind of kernel

contraction has been called smooth [Han94a].

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2.5 Epistemic EntrenchmentEpistemic entrenchment which was introduced in [Gar84], [GM88] and [Gar88] is a binaryrelation ≤ on the sentences in the belief set K such that in contraction, giving up beliefs with lowerentrenchment is preferred to giving up those with higher entrenchment. Gardenfors proposed thefollowing set of axioms:

(EE1) Transitivity If p ≤K q and q ≤K δ, then p ≤K δ.(EE2) Dominance If p ⊢ q, then p ≤K q.(EE3) Conjunctiveness p ≤K (p ∧ q) or q ≤K (p ∧ q).(EE4) Minimality If K /⊢ , then p ∉ K if and only if p ≤K q for all q.(EE5) Maximality If q ≤K p for all q, then ⊢ p.

It follows from (EE1) – (EE3) that an epistemic entrenchment is a complete pre-order overL. Gardenfors and Makinson [GM88] investigated the connections between orders of epistemicentrenchment and contraction functions. The two are connected by the following equivalences,where we write p <K q when p ≤K q and q /≤K p:

(C ≤) p ≤K q if and only if p ∉ K−(p ∧ q) or ⊢ (p ∧ q).

(−G) q ∈ K−p if and only if q ∈ K and either ⊢ p or p <K (p ∨ q).

We can define entrenchment-based revision via the Levi identity. However, it is also possibleto define entrenchment-based revision directly from an entrenchment ordering, by means of thefollowing equivalences [LR91, Rot91, HFCF01]:

(C ≤∗) p ≤K q if and only if: If p ∈ K∗¬(p ∧ q) then q ∈ K∗¬(p ∧ q).

(∗EBR) q ∈ K∗p if and only if either (p→ ¬q) <K (p→ q) or p ⊢ .

Alternatively, a contraction operator can be based on entrenchment as follows:

K ÷ p = q ∈ K ∣ q < p

This is severe withdrawal (also called mild contraction or Rott’s contraction). It was axiomatizedindependently by Pagnucco and Rott [RP99] and by Ferme and Rodriguez [FR98a]. Arlo-Costaand Levi have analyzed it in terms of minimal loss of informational value [ACL06]. It has beenshown to satisfy the implausible postulate of expulsiveness. (If ⊬ p and ⊬ q, then either p ∉ K ÷ qor q ∉ K ÷ p) [Han99b]. However, Lindstrom and Rabinowicz have proposed that it can be used inthe following way:

“One would like to say that the truth lies somewhere in between the two ex-tremes: the original proposal [severe withdrawal] and Grove’s definition [Gardenfors’entrenchment-based contraction] seem to give us the lower and upper limits for con-traction.” [LR91, pp. 115]

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This has been called Lindstrom’s and Rabinowicz’s interpolation thesis [Rot00a].

2.6 Grove’s sphere-systemsThe last construction is based on possible worlds models. A proposition (set of possible worlds)can represent either a belief set or an input sentence. The belief set K can be replaced, as a beliefstate representation, by ∥K∥ = X ∣ K ⊆ X ∈ L ⊥⊥ , which is the set of worlds that containK. Similarly, each sentence p can be represented by the set ∥p∥ = ∥Cn(p)∥. The expansionoutcome K + p will then be represented by the set ∥K + p∥ = ∥K∥∩ ∥p∥, and a contraction outcomeK − p by some superset ∥K − p∥ of ∥K∥ that includes at least one ¬p-world. The revision outcomeK ∗ p is done by some (at least one) p-worlds. The selection of the worlds in the change functionsis by means of a propositional selection function. Formally:

Definition 10 [Han99b] Let X be a proposition. A propositional selection function for X is afunction f such that for all sentences p:

1. f (∥p∥) ⊆ ∥p∥

2. If ∥p∥ ≠ ∅ then f (∥p∥) ≠ ∅.

3. If X ∩ ∥p∥ ≠ ∅, then f (∥p∥) = X ∩ ∥p∥.

Definition 11 Let X be a proposition.

• An operator − is a propositional contraction operator for X if and only if there is a proposi-tional selection function f for X such that for all p, X − ∥p∥ = X ∪ f (∥¬p∥).

• An operator ∗ is a propositional revision operator for X if and only if there is a propositionalselection function f for X such that for all p, X ∗ ∥p∥ = f (∥p∥).

The Grove’s sphere-system makes use of a system of concentric spheres around the proposition.Intuitively, each sphere represents a degree of closeness or similarity to ∥K∥. In contraction by p,the closest ¬p-worlds are added to ∥K∥.

Definition 12 [Gro88] $ is a system of spheres if and only if it satisfies:$1 ∅ /= $ ⊆ P(L ⊥⊥),$2 ∩$ ∈ $,$3 If G, G′ ∈ $, then G ⊆ G′ or G′ ⊆ G,$4 ∪$ ∈ $,$5 If ∥p∥ ∩ (∪$) /= ∅, then S p ∈ $ and S p ∩ ∥p∥ /= ∅, and$6 L ⊥⊥∈ $,

where S p = ⋂G ∈ $ ∣ G ∩ ∥p∥ /= ∅4.

4i.e., the smallest sphere that contains p-worlds.

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Definition 13 [Gro88] A propositional selection function f for a proposition X is sphere-based ifand only if there is a system of spheres $ such that for all p: If ∥p∥ /= ∅, then f (∥p∥) = S p ∩ ∥p∥.

• A propositional contraction operator is sphere-based if and only if it is based on a sphere-based propositional selection function.

• A propositional revision operator is sphere-based if and only if it is based on a sphere-basedpropositional selection function.

in other words, given a sphere system S centered on ∥K∥, a sphere-based contraction on K isobtained by adding to ∥K∥ all the ¬p-worlds in the smallest sphere S that has any ¬p-world. Inthe same way, a sphere-based revision on K is obtained by all the p-worlds in the smallest sphereS that has any p-world.

Alternatively a sphere system can be characterized by a total preorder between worlds, i.e. areflexive, transitive and total relation on W . Let ⪯ be such a total preorder, ≺ the correspondingstrict relation, and ≪ the relation such that µ ≪ φ if and only if µ ≺ φ and there is no ϕ such thatµ ≺ ϕ ≺ φ. Then µ ≪ φ signifies that the smallest sphere containing µ is one step smaller than thesmallest sphere containing φ.

2.7 The interconnection among the five presentationsOne of the major achievements of the AGM model is that the five presentations are equivalent, aswe can see in the following theorems:

Theorem 1 Let K be a belief set and − an operator on K. Then the following conditions areequivalent:

1. − satisfies closure, inclusion, vacuity, success, extensionality and recovery.

2. − is a partial meet contraction function.

3. − is a smooth kernel contraction function.

4. There exists a propositional selection function f such that for all p, f (p) ⊆ ∥p∥ and K−p =Th(∥K∥ ∪ f (¬p)).

Theorem 2 Let K be a belief set and − an operator on K. Then the following conditions areequivalent:

1. − satisfies closure, inclusion, vacuity, success, extensionality, recovery, , and conjunctivefactoring.

2. − is a transitively relational partial meet contraction function.

3. − is a safe contraction function, based on a regular and virtually connected hierarchy ∠.

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4. − is a Gardenfors entrenchment-based contraction based on a relation ≤K defined from − bycondition (C ≤) and ≤K satisfies the (EE1)–(EE5) entrenchment postulates.

5. There exists a propositional sphere-based selection function f such that for all p, K−p =Th(∥K∥ ∪ f (¬p)).

3 Criticism of the modelMuch of the critical discussion on the AGM model has referred either to the postulates for partialmeet contraction and revision (Sections 3.1-3.2) or to various aspects of the use of belief sets torepresent belief states (Sections 3.3-3.4).

For additional elaborations on philosophical issues relating to the AGM model, see also[Han03] and [Rot00b].

3.1 The recovery postulateBy far the most criticized of the postulates is one of the basic postulates for contraction:

Recovery: K ⊆ (K − p) + p.

Recovery is based on the intuition that “it is reasonable to require that we get all of the beliefs [...]back again after first contracting and then expanding with respect to the same belief” [Gar82].However, counter-examples have been constructed in which the recovery postulate seems to giverise to implausible results.

Example 1 [Han93a] “I believe that “Cleopatra had a son” (p) and that “Cleopatra had adaughter” (q), and thus also “Cleopatra had a child” (p ∨ q, briefly r). Then I receive infor-mation that makes me give up my belief in r, and contract my belief set accordingly, forming K−r.Soon afterwards I learn from a reliable source that “Cleopatra had a child”. It seems perfectlyreasonable for me to then add r (i.e. p∨ q ) to my set of beliefs without also reintroducing either por q.”

Example 2 [Han93a]5 “I previously entertained the two beliefs, “x is divisible by 2” (p) and “xis divisible by 6” (q). When I received new information that induced me to give up the first of thesebeliefs (p), the second (q) had to go as well (since p would otherwise follow from q).I then received new information that made me accept the belief “x is divisible by 8.” (s). Since pfollows from s, (K−p) + p is a subset of (K−p) + s, then by recovery I obtain that “x is divisibleby 24” (r), contrary to the intuition.”

In a retort, Makinson [Mak97a, p. 478] noted that “as soon as contraction makes use of thenotion “y is believed only because of x’, we run into counterexamples to recovery”. He argued thatthis is provoked by the use of a justificatory structure that is not represented in the belief set andthat, without this structure, recovery can be accepted6. In [Han99a], Hansson replied that “Actual

5‘We use here the modified version introduced in [FR98b] in order to eliminate psychological aspects of Hansson’sexample.

6or, in Makinson’s words, it can be accepted in a “naked” theory.

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human beliefs always have such a justificatory structure (...). It is difficult if not impossible to findexamples about which we can have intuitions, and in which the belief set is not associated with ajustificatory structure that guides our intuitions. Against this background, it is not surprising that,as Makinson says, recovery “appears to be free of intuitive counterexamples” (...). It also seemsto be free of confirming examples of this kind.”

Glaister argued that the problem exhibited in this and other counter-examples dissolves if wepay sufficient attention to exactly what is to be contracted. In this case, he claims, the contraction ismore accurately represented by a multiple contraction by the set ¬p∨q, p∨¬q, p∨q than by p∨q[Gla00]. Niederee [Nie91] found several unintuitive properties that follow from recovery: Let K bea belief set and p ∈ K. Then, regardless of whether or not q is in K, recovery together with closureimplies that: (1) p → q ∈ K−(p ∨ q), (2) p ∈ (K−(p ∨ q)) + q, and (3) ¬q ∈ (K−(p ∨ q)) + ¬p.Ferme [Fer01] analyzed recovery in the five AGM presentations and show how the intuitions ornon-intuitions that surround recovery appear or disappear in each of them and consequently, thestatus of recovery turns out to differ substantially among the five approaches.

Belief base models do not in general satisfy recovery. This is often seen as one of their majorattractions. (See Section 4.1.) In Makinson’s terminology, operations that satisfy the other fivebasic contraction postulates but not Recovery are called withdrawals [Mak87].

3.2 The success postulatesPartial meet revision satisfies the following postulate:

Revision success: p ∈ K ∗ p

Several authors have found this to be an implausible feature of belief revision, even if p is not acontradiction. Hence Cross and Thomason pointed out that a system obeying this postulate

“is totally trusting at each stage about the input information; it is willing to give upwhatever elements of the background theory must be abandoned to render it consistentwith the new information. Once this information has been incorporated, however, it isat once as susceptible to revision as anything else in the current theory.

Such a rule of revision seems to place an inordinate value on novelty, and its behaviourtowards what it learns seems capricious” [CT92].

Similarly, one of the AGM postulates for partial meet contraction:

Contraction success: If /⊢ p , then K−p /⊢ p.

has been contested on the ground that we should “allow a reasoner to refuse the withdrawal ofp not only in the case where p is a logical truth. There may well be other sentences (‘necessarytruth’) which are of topmost importance for him” [Rot92, p.54]. Both with respect to revisionsuccess and contraction success, a common strategy among critics has been to construct AGM-style operations that do not always give primacy to the new information. (See Section 4.4)

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3.3 Are belief sets too large?

Belief sets have been criticized for being too extensive in two important respects that are both prob-lematic from the viewpoint of cognitive realism: their logical closure and their infinite structure.

The use of a logically closed belief set to represent the belief state has important implications.In particular it means that all beliefs are treated as if they have independent status. Suppose thatyou believe that you have your keys in your pocket (p). It follows that you also believe that eitheryou have your keys in your pocket or the archbishop of York is a quranist muslim (p∨q). However,p ∨ q has no independent standing; it is in the belief set “just because” q is there. Therefore, ifyou give up your belief in p we should expect p ∨ q to be lost directly, without the need for anymechanism to deselect it. In the AGM framework, however, “merely derived” beliefs such asp ∨ q have the same status as independently justified beliefs such as p. Belief base models (to bediscussed in subsection 4.1) have largely been constructed in order to make this distinction.

Makinson [Mak85, page 357] pointed out that “in general, neither p ∨ q nor p ∨ ¬q shouldbe retained in the process of eliminating p from K, unless there is “some reason” in K for theircontinued presence”. The concept can be extended to in general, no q should be retained in theprocess of eliminating p from K, unless there is “some reason” in K for their continued presence.This condition was explored by Fuhrmann [Fuh91] and gives rise to the filtering condition:

“If q has been retracted from a base B in order to bar derivations of p from B, then thecontraction of Cn(B) by p should not contain any sentences which were in Cn(B) “justbecause” q was in Cn(B).”

The filtering condition is a different notion of minimal change from that of recovery, sincep→ q maybe is in K “just because” q is in K.

The logical closure of belief sets is also problematic from another point of view. In a studyof the philosophical foundations of AGM, Rott pointed out that the theory is unrealistic in itsassumption that epistemic agents are “ideally competent regarding matters of logic. They shouldaccept all the consequences of the beliefs they hold (that is, their set of beliefs should be logicallyclosed), and they should rigorously see to it that their beliefs are consistent” [Rot00b]. In the samearticle he argued that AGM is not based on a principle of minimal change, something that has oftenbeen taken for granted.

Since actual human agents have finite minds, a good case can be made that a cognitively realis-tic model of belief change should be finitistic, and this in two senses. First, both the original beliefset and the belief sets that result from a contraction should be finite-based, i.e. obtainable as thelogical closure of some finite set. Secondly, the output set, i.e. the class of belief sets obtainableby contraction from the original belief set (X ∣ (∃p)(X = K ÷ p) should be finite [Han93c]. Par-tial meet contraction does not in general satisfy either of these two finitistic criteria. This has ledto the development of finitistic models such as belief base models and specified meet contraction[Han08].

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3.4 Lack of information in the belief set

Belief sets have been criticized not only for being too large but also for lacking important informa-tion.

Most importantly, AGM contraction or revision in its original form is a “one shot” operation.After contracting K by p with the operation ∼γ we obtain a new belief set K ∼γ p but we do not ob-tain a new selection function to be used in further operations on this new belief set. In other wordsthe original AGM framework does not satisfy the principle of categorial matching, according towhich the representation of a belief state after a change should have the same format (and containthe same types of information) as the representation of the belief state before the change [GR93].In studies of iterated revision, various ways to extend the belief state representation to solve thisproblem have been investigated. (See Section 5.)

The lack of modal and conditional sentences in belief sets has often been pointed out, butattempts to include them have given rise to severe difficulties. The same applies to introspectivebeliefs, i.e. the agent’s beliefs about her own belief state [FH99b]. The inclusion of sentencesreferring to preferences and norms in belief sets has been somewhat more successful (See subection5.8).

4 Ways to extend the AGM model

4.1 Extended representations of belief states

The AGM model is a simple and elegant representation of quite complex phenomena. Obviously,the trade-off between simplicity and relevance can be made differently. Many of the modificationsof the framework that have been proposed consist in extensions of the belief state representationthat make it contain more information in addition to what is contained in the belief set.

Belief bases

It was understood from the beginning that the use of logically closed sets of sentences to representbelief states is not cognitively realistic. In an article published in 1985 Makinson pointed out that“in real life, when we perform a contraction or derogation, we never do it to the theory itself (inthe sense of a set of propositions closed under consequence) but rather on some finite or recursiveor at least recursively enumerable base for the theory” [Mak85, p. 357]. Important distinctionscan be introduced through the use of belief bases. A belief base is a set B of sentences suchthat a sentence p is believed if and only if p ∈ Cn(B). Operations on belief bases have beenextensively investigated [Dal88, Fer92, Fuh88, Fuh91, Han89, Han91a, Han92b, Han94b, Neb89,Rot00a, Was00].

There are two major interpretations of belief bases. One of them, supported by Dalal [Dal88],uses belief bases as mere expressive devices; hence if Cn(B1) = Cn(B2) then B1 and B2 representthe same belief state and yield the same outcome under all operations of change. This is called the

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principle of “irrelevance of syntax”7.The other, more common approach treats inclusion in the belief base as epistemically sig-

nificant. The belief base contains those sentences that have an epistemic standing of their own.Suppose that the belief set contains the sentence s, “Shakespeare wrote Hamlet”. Due to logi-cal closure it then also contains the sentence s ∨ d, “Either Shakespeare wrote Hamlet or CharlesDickens wrote Hamlet”. The latter sentence is a “mere logical consequence” that should have nostanding of its own [Han06]. In this approach, belief bases increase the expressive power of thebelief representation, since two belief bases with the same logical closure can represent differentways to hold the same beliefs. Since the two belief bases p,q and p, p ↔ q have the samelogical closure, they are “statically equivalent”, i.e. they generate the same belief set. However,they are not “dynamically equivalent” since they behave differently under operations of change;revision by ¬p will presumably result in ¬p,q respectively ¬p, p ↔ q, and these belief basesgenerate different belief sets [Han92a].

In this model, changes are made on the belief base, and the merely derived sentences cannotsurvive when the basis of their derivation is lost. This principle was very precisely expressed bythe filtering condition (cf subsection 3.3).

An input-driven operation on a belief base B gives rise to a base-generated operation ′ onthe belief set K = Cn(B), such that K ′ p = Cn(B p) for all p. Partial meet contraction andrevision can be straightforwardly transferred to belief bases. Axiomatic characterizations havebeen obtained of these operations on belief bases [Han92a, Han93b] and of the base-generatedoperations that they give rise to on belief sets [Han93c]. Kernel contraction, which can be usedon belief sets as an alternative characterization of partial meet contraction (cf. Subsection 2.4)turns out to be a more general operation than partial meet contraction when applied to belief bases[Han94a]. In [FFKI06] is made precise under which conditions there are an exact correspondencebetween selection functions on remainder sets and incision functions on kernel sets.

One of the most important advantages of belief bases is that they make it possible to distinguishbetween different inconsistent belief states. This feature can be used to construct two substantiallydifferent types of revision operators based on contraction, depending on whether the negation ofthe added sentence is contracted before or after its addition:

B ∗ p = B ÷ ¬p + p (internal revision, Levi identity) [AGM85]

B ∗ p = B + p ÷ ¬p (external revision, reversed Levi identity) [Han93c]

The second of these options is not viable for a belief set, since if ¬p ∈ K then the first of itstwo suboperations results in the set K + p that contains the whole language and therefore removesall traces of the original belief set. The distinction between different inconsistent belief bases alsomakes it possible to construct meaningful operators of consolidation, i.e. removal of inconsistency.

The recovery postulate does not hold for partial meet contraction on belief bases. This hasbeen referred to as a major advantage of the belief base approach [Han91b]. Johnson and Shapirohave investigated conditions under which recovery, or closely related properties, hold in belief basecontraction, and argued for the plausibility of some of these conditions [JS05].

7For a discussion see [Neb92].

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Nebel has proposed belief base operations in which a complete, reflexive and transitive relationover the elements of the belief base is used to prioritize among its elements [Neb92]. This approachwas further developed by Weydert who also related it to the AGM postulates [Wey92].

Contrary to selection functions, epistemic entrenchment cannot be straightforwardly trans-ferred to a belief base framework. However, Williams introduced ensconcement relations, a relatedtype of transitive and connective relation on belief bases, and a corresponding class of contractionoperators, ensconcement-based contraction [Wil92]. She also showed that an ensconcement-basedcontraction − can be related to an AGM contraction ÷ by the formula B − p = (Cn(B) ÷ p) ∩ B[Wil94]. In [FKR08] ensconcement-based contraction was analysed and an axciomatic character-ization was given.

Di Giusto and Governatori have developed an approach in which the elements of the belief baseare divided into two categories, facts and rules. Facts are removed if necessary to accommodatenew facts. Rules are not removed but can instead be changed. Hence, suppose that the belief basecontains the fact a&b and the two rules a → c and b → c. After revision by the new fact ¬c, anew belief base is obtained that contains the facts a&b and ¬c and the two rules (a&¬b) → c and(b&¬a)→ c [DG99].

Bochman has developed a theory of belief revision in which an epistemic state is representedby a triple ⟨S ,<, l⟩, where S is a set of objects called admissible belief states, < a strict preferencerelation on these states, and l a function that assigns a (logically closed) belief set to each elementof S . One and the same belief set may be assigned to several elements of S . This structure sharesmany features with belief bases [Boc01].

It is commonly assumed that the belief base approach corresponds to foundationalist epistemol-ogy, whereas the original AGM framework that applies operations of change directly to the beliefset represents a coherentist view of belief change. Gardenfors has provided the most thoroughjustification of this interpretation [Gar90]. Del Val claimed that the two approaches are equivalent[Val97]. Doyle accepted Gardenfors’s analysis of the relationship between the belief set/belief baseand coherentism/foundationalism distinctions. However, he argued that the fundamental concernfor conservatism that Gardenfors appealed to in his defence of coherentism applies equally to thefoundations approach [Doy92]. A more radical criticism was ventured in [HO99] where it wasargued that the original AGM approach is incompatible with important characteristics of coher-entism. In [Han00] it was claimed that the application of partial meet contraction to belief basescomes much closer to expressing coherentist intuitions than their application directly to belief sets.In the framework proposed there, a belief base B is assigned to the belief set K. Then a logicallyclosed subset K′ of K is coherent if and only if there is some sentence p such that K′ = Cn(B ∼γ p).

Probability and plausibility

The AGM model, and other logical approaches to belief revision represent features of doxasticbehaviour that differ from those represented by probabilistic models. The degrees of belief rep-resented for instance by entrenchment relations do not coincide with probabilities [Rot09a]. Itseems difficult to construct a reasonably manageable model that covers both the logic-related andthe probabilistic properties of belief change. (Problems connected with the lottery and prefaceparadoxes have a major role in creating this difficulty [Kyb61, Mak65]).

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However, some authors have explored the interrelations between the two types of models. Lind-strom and Rabinowicz showed how belief revision can be connected with accounts of conditionalprobability that allow the condition to have probability zero [LR89]. Makinson further investigatesthis and other connections between the two framweorks [Mak10]. Insights from AGM can beused as an impetus for considering “revisionary” accounts of conditional probability, i.e. accountsin which p(q, r), the probability of q given r, is not defined in the standard way. (According tothe standard definition, p(q, r) is equal to p(q&r)/p(r) when p(r) ≠ 0, and otherwise undefined.)Furthermore, the notion of non-prioritized revision that has been developed in the AGM tradition(revision not satisfying p ∈ K ∗ p for all p, see Section 4.4) can be usefully transferred to a prob-abilistic context. There it corresponds to “vacuous” conditionalizing when the condition is toounbelievable to be taken seriously, i.e. p(q, r) = p(q) when r is highly unlikely. Makinson alsodiscusses “hyper-revisionary” probabilistic conditionalization, in which the fact that something be-lieved to be very improbable actually happens is taken as a reason to believe that the probabilitywas underestimated. There is an analogy between hyper-revisionary conditionalization and beliefrevision that violates the AGM postulate that if K + p is logically consistent, then K ∗ p = K + p.Such violations would be justified if K and p are epistemically but not logically incompatible.

In order to investigate the relationship between AGM and Bayesian conditionalization, Bo-nanno introduced what he called the qualitative Bayes rule, namely that

“... if at a state the information received is consistent with the initial beliefs – in thesense that there are states that were considered possible initially and are compatiblewith the information – then the states that are considered possible according to therevised beliefs are precisely those states.” [Bon05].

Bonanno constructed and characterized a model of belief revision that satisfies this condition. Itcomplies with the AGM postulates for partial meet revision.

Friedman and Halpern have developed a model based on a notion of plausibility that is a gen-eralization of probability. Instead of assigning to each set A of sentences a number p(A) in [0,1],representing its probability, they assign to it an element Pl(A) of a partially ordered set. Pl(A) iscalled the “plausiblity” of A. If Pl(A) ≤ Pl(B) then B is at least as plausible as A. A sentence p isbelieved if and only if p is more plausible than ¬p. Changes in belief take the form of changes inthe plausibility ordering. Conditions on such changes have been identified that produce a revisionoperator that is essentially equivalent with partial meet revision [FH97, FH99c].

Several other authors have presented probability-based and plausibility-based belief revisionmodels that have close connections with the AGM model [Auc07, DP91, AC01, ACP05].

Ranking models

In Spohn’s ranking theory of belief change a belief state is represented by a ranking function κ thatassigns a non-negative real number to each possible world w, representing the agent’s degree ofdisbelief in w [Spo83, Spo88, Spo09]. A sentence p is is assigned the value κ(p) = minκ(w) ∣p holds in w. Furthermore, p is believed if and only if κ(¬p) > 0, i.e., if and only if every¬p-world is disbelieved to a non-zero degree. The conditional rank of q given p is κ(q ∣ p) =κ(p&q) − κ(p). For any sentence p and number x, the p→ x-conditionalization of κ is defined by:

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κp→x(q) = minκ(q ∣ p), κ(q ∣ ¬p) + x

Contractions, expansions and revisions can all be represented as conditionalizations (depending onthe numerical values involved). In addition, other operations such as the strengthening or weak-ening of beliefs already held are straightforwardly representable in this framework. Importantresults on belief revision based on ranking functions, including an axiomatic representation thatclarifies their relationship to AGM operations, have been reported by Hild and Spohn [HS08]. Ageneralization of Spohn’s ranking functions has been proposed by Weydert [Wey94].

Extensions of the language

Belief revision theory has primarily been concerned with belief states and inputs expressed in termsof classical sentential (truth-functional) logic. The inclusion of non-truth functional expressionsinto the language has interesting and often surprisingly drastic effects.

Among the several formal interpretations of non-truthfunctional conditionals, such as coun-terfactuals, one is particularly well suited to the formal framework of belief revision, namely theso-called Ramsey test. It is based on a suggestion by Ramsey that has been further developed byRobert Stalnaker and others [Sta68, pp 98-112]. The basic idea is that “if p then q” is taken to bebelieved if and only if q would be believed after revising the present belief state by p. Let p qdenote “if p then q”, or more precisely: “if p were the case, then q would be the case”. The Ramseytest says:

p q holds if and only if q ∈ K ∗ p

In order to treat conditional statements like p q on par with statements about actual facts, theywill have to be included in the belief set when they are assented to by the agent, thus:

p q ∈ K if and only if q ∈ K ∗ p

However, inclusion in the belief set of conditionals that satisfy the Ramsey test will require radicalchanges in the logic of belief change. As one example of this, contraction cannot then satisfy theinclusion postulate (K ÷ p ⊆ K). The reason for this is that contraction typically provides supportfor conditional sentences that were not supported by the original belief state. Hence, if I give upmy belief that John is mentally retarded, then I gain support for the conditional sentence “If Johnhas lived 30 years in London, then John understands the English language” [Han92c].

A famous impossibility theorem by Gardenfors shows that the Ramsey test is incompatiblewith a set of plausible postulates for revision [Gar86]. The crucial part of the proof consists inshowing that the Ramsey test implies the following monotonicity condition:8

If K ⊆ K′ then K ∗ p ⊆ K′ ∗ p

8The proof is straightforward: Let K ⊆ K′ and q ∈ K ∗ p. The Ramsey test yields p q ∈ K, then K ⊆ K′ yieldsp q ∈ K′, and finally one more application of the Ramsey test yields q ∈ K′ ∗ p.

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This condition is incompatible with the AGM postulates for revision, and it is also easily shownto be implausible. Let K be a belief set in which you know nothing specific about Ellen and K′

one in which you know that she is a lesbian. Let p denote that she is married and q that she has ahusband. Then we can have K ⊆ K′ but q ∈ K ∗ p and q ∉ K′ ∗ p.

Several solutions to the impossibility theorem have been put forward. One option investigatedby Rott and others is to reject the Ramsey test as a criterion for the validity of conditional sentences[Rot86] Levi accepts the test as a criterion of validity but denies that such conditional sentencesshould be included in the belief set when they are valid. In his view, such conditional statementslack truth values and should therefore not be included in belief sets [Lev88]. This, of course,blocks the impossibility result. Levi and Arlo-Costa have investigated a weaker version of theRamsey test that is not blocked by Gardenfors’s result and is also compatible with the AGM model[AC95, ACL96].

In a somewhat similar vein, Lindstrom and Rabinowicz have proposed that a conditional sen-tence expresses a determinate proposition about the world only relative to the subject’s belief state.Given a conditional statement p q and a belief set K, there is some sentence rK

pq such thatp q holds in the belief state represented by K if and only if rK

pq ∈ K. In this way we can havethe Ramsey test in the form

rKpq ∈ K if and only if q ∈ K ∗ p,

that is not blocked by the impossibility result [LR98, LR92].Yet another option is to accept both the Ramsey test and the inclusion of conditional sentences

into the belief set. Then belief sets containing will behave very differently under operationsof change than the common AGM belief sets, and the standard AGM postulates will not hold[Han92c, Rot89]. Not even the simple operation of expansion can be retained. Suppose that youhave no idea about John’s profession, but then “expand” your belief set by the belief that he isa taxi-driver. You will then lose the conditional belief that if John goes home by taxi every day,then he is a rich man – hence this is not an expansion after all [Han92c]. As was noted by Rott,“[e]xpansions are not the right method to ‘add’ new sentences if the underlying language containsconditionals which are interpreted by the Ramsey test” [Rot89].

Ryan and Schobbens have related the Ramsey test to update [KM92] rather than revision andfound the test to be compatible and indeed closely connected with update operators [RS97].

Kern-Isberner has proposed a framework for revision that is based on a conditional valuationfunction that assigns (numerical) values to both non-conditional and conditional sentences. Inthis framework – which differs from AGM in important respects – conditional sentences can beelements of belief sets, and revisions can be performed with conditional sentences as inputs [KI04].A partly similar approach to the same issues has been developed by Weydert [Wey05].

The inclusion of modal sentences in belief sets has been investigated by Fuhrmann. Let pdenote that p is possible. The following, seemingly reasonable definition:

p ∈ K if and only if ¬p ∉ K

gives rise to problems similar to those exhibited in Gardenfors’s theorem, and essentially the sametypes of solutions have been discussed [Fuh89].

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Lindstrom and Rabinowicz have investigated the inclusion into a belief revision framework ofintrospective beliefs, i.e. allowing for Bp ∈ K, where Bp denotes “I believe p”. Paradoxical resultsnot unsimilar to those for conditionals are obtained in this case as well [LR99a]. Similar resultswere obtained by Friedman and Halpern [FH99b].

Dupin de Saint-Cyr and Lang introduced temporally labelled sentences into belief revision andproposed a belief change operator, called belief extrapolation, in which predictions are based oninitial observations and a principle of minimal change [DL02]. Bonanno has developed logics thatcontain both a next-time temporal operator and a belief operator. The basic postulates of AGMrevision are satisfied, and a strong version of the logic also satisfies the supplementary postulates[Bon07a, Bon07b].

Booth and Richter have developed a model of fuzzy revision on belief bases. In this model,both the elements of the belief base and the input formulas come attached with a numerical degree(whose precise interpretation is left open). They showed that partial meet operations on beliefbases can be faithfully extended to this fuzzy framework [BR01].

Finally, Fuhrmann has generalized partial meet operations to arbitrary collections of (not nec-essarily linguistic) items that have a dependency structure satisfying the Armstrong axioms fordependency structures in database relationships [Fuh96, Fuh97c].

Change in norms, preferences, goals, and desires

Norms: The AGM model was partly the outcome of attempts to formalize changes in norms[AM81]. In spite of this, authors who tried to apply the AGM model to normative change havefound the model to be in need of rather extensive modifications to make it suitable for that purpose.

Boella, Pigozzi, and van der Torre analyzed normative change in a framework where a normsystem is represented by a set of pairs of formulas. The pair ⟨p,q⟩ should be read “if p, then it isobligatory that q”. In this framework, however, postulates for norm contraction and revision thatare close analogues of the AGM postulates give rise to inconsistency[BPvdT09].

Governatori and Rotolo proposed a model for changes in legislation that among several otheraspects also includes an explicit representation of time. Such a model can account for phenomenasuch as retroactivity that are difficult to deal with in an input-assimilating framework such as AGM[GR10].

Hansson and Makinson investigated the relationship between changes and applications of anorms system. In order to apply a norm system with conflicting norms to a particular situation,some of the norms may have to be ignored. Although these norms will remain intact for futuresituations, the problem of how to prioritize among conflicting items is similar to the selection ofsentences for removal in belief contraction [HM97].

Preferences: A model of changes in preferences can be obtained by replacing the standardpropositional language in AGM by a language consisting of sentences of the form p ≥ q (“p is atleast as good as q”) and their truth-functional combinations. The acquisition of a new preferencetakes the form of revision by such a preference sentence. The adjustments of the original preferencestate that are needed to maintain consistency in such revisions can be modelled by partial meetcontraction. However, some modifications of the AGM model seem to be necessary in order toobtain a realistic model of preference change [GYH09, Han95, Lv08].

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Goals and desires: Desires are often allowed to be contradictory, since they need not be activelypursued by the agent. Likewise, we often have goals that are difficult or impossible to combine.Intention selection is a process aimed at removing such contradictions, to end up with a consistentset of intentions [Pag06]. Paglieri and Castelfranchi have proposed a model of Data-oriented BeliefRevision (DBR) in which significant attention is paid to the mutual influences between beliefs andgoals [Pag05]. In one direction, beliefs support and regulate goal processing and the transitionfrom desires to intentions [CDG95, Cas96, Cas97, CP07]. In the other direction, goals affect beliefrevision, by determining the relevance (usefulness to the agent’s current purposes) and likability(capability of fulfilling the agent’s goal) of potential beliefs. Relevance increases the likelihoodthat a potential belief will be considered as a candidate for belief, whereas likeability increases itschances of being actually believed, once considered. Boella, da Costa Pereira, Pigozzi, Tettamanzi,and van der Torre have also analyzed the role of goals in belief revision [BdCPP+10]. Their modelis similar to DBR in its selection criteria, but it puts more emphasis on avoiding influence on beliefsof wishful thinking.

4.2 Iterated changeAn AGM contraction or revision takes us from a belief set to a new belief set. In doing this,it makes use of a selection mechanism such as a selection function or an entrenchment relation.However, it does not provide a new selection mechanism to be used for further changes of the newbelief set. The problem of constructing models that allow for iterated change is probably the moststudied problem in the literature on belief change.

Revising epistemic states

In order to solve the problem of iterated change we need a belief state representation that containsmore information than the belief set, so that it can guide additional changes. Furthermore, theoperation of change has to yield a complete such belief state representation as its outcome, notmerely a new belief set. There are several ways to represent such an extended epistemic state.The most common of these is a preorder on the set of possible worlds, or equivalently a completesphere system (cf. Section 2.6). The belief set can be inferred from this preorder; it is simplythe intersection of the worlds in the highest equivalance class (innermost sphere). An operation ofchange gives rise to a new preorder (sphere system), from which the new belief set can be inferred,and which can in its turn be subject to further changes, etc.

The most influential formulation of this approach is due to Darwiche and Pearl [DP96]. Topresent it, let Ψ be the current belief state of the agent, and ⪯Ψ the preorder that represents it.Similarly, let Ψ p be the belief state obtained after revising by p, and ⪯Ψp the preorder thatrepresents it. (Following tradition we will focus on revision; relatively little has been written oniterated contraction.)

Darwiche and Pearl proposed the following conditions for iteration (µ and φ are possibleworlds):

(DP1): For any µ ⊧ p and φ ⊧ p, µ ⪯Ψ φ iff µ ⪯Ψp φ.

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(DP2): For any µ ⊧ ¬p and φ ⊧ ¬p, µ ⪯Ψ φ iff µ ⪯Ψp φ.(DP3): For any µ ⊧ p and φ ⊧ ¬p, if µ ≺Ψ φ, then µ ≺Ψp φ.(DP4): For any µ ⊧ p and φ ⊧ ¬p, if µ ⪯Ψ φ, then µ ⪯Ψp φ.

According to (DP1), the order among the p-worlds remains unchanged after revision by p . Ac-cording to (DP2) the order among the ¬p-worlds remains unchanged after revision by p . (DP3)says that if a p-world is strictly preferred to a ¬p-world, then that strict preference is maintainedafter revision by p. (DP4) says that if a p-world is weakly preferred to a ¬p-world, then that weakpreference is maintained after revision by p.

These conditions have been shown to correspond to the following postulates for iterated revi-sion: [DP96, Theorem 13.]

(DP1) If q ⊢ p, then (Ψ p) q = Ψ q.(DP2) If q ⊢ ¬p, then (Ψ p) q = Ψ q.(DP3) If Ψ q ⊢ p, then (Ψ p) q ⊢ p.(DP4) If Ψ q /⊢ ¬p, then (Ψ p) q /⊢ ¬p

These four postulates have become the benchmark for iterated revision, and new proposals arealmost invariably compared to them. However, Jin and Thielscher [JT07] and Booth and Meyer[BM06] have pointed out that these postulates are too permissive since they do not rule out op-erators by which all newly acquired information is given up as soon as an agent learns a factthat contradicts some of its current beliefs. To avoid this they proposed the following additionalcondition:

(Ind): For any µ ⊧ p and φ ⊧ ¬p, if µ ⪯Ψ φ, then µ ≺Ψp φ.

Major classes of iterable operators

In a model of iterated belief revision there may be more than one way to arrive at one and the samebelief set. Does it make any difference for further changes how we arrive at it? We can divideiterable operators into three classes according to their ability to remember and to take the revisionhistory into account :

Operators without memory: In this case, each belief set is revised in a predetermined way,independently of how it was obtained, i.e.:

If Ψ p and Υ p have the same belief set, then so have Ψ p r and Υ p r.

Full meet revision is a trivial example of an iterable operator without memory. Areces and Becherhave analyzed this class of operators [AB01].

Operators with full memory: In this case the full history of changes is conserved, so thatrollbacks of previous changes are possible. Operators with full memory have been proposed, for

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example, by Brewka [Bre91], by Lehmann [Leh95], and by Konieczny and Perez [KP00]. Falappa,Garcıa, Kern-Isberner and Simari proposed another type of revision in which discarded beliefs canbe reused [FGKIS10].

Operators with partial memory: In this case it makes a difference for future revisions how abelief set was arrived at, but the information remembered is not sufficient to identify the previousstates. Most of the proposed iterable revision operators are of this type. In a recent review Rottrecognized three major types of iterable revision operators. They all have partial memory [Rot09b]:

Conservative revision, originally called natural revision, has been studied by Boutilier[Bou93, Bou96] and Rott [Rot03]. This operation is conservative in the sense that it onlymakes the minimal changes of the preorder that are needed to accept the input. In revisionby p, the maximal p-worlds are moved to the top of the preorder which is otherwise leftunchanged. The main characteristic of this operator is:

(Nat): If µ ∉ [Ψ p] and φ ∉ [Ψ p], then µ ⪯Ψ φ iff µ ⪯Ψp φ.

Moderate revision, also called lexicographic revision, was originally studied by [Nay94] andby Nayak, Pagnucco and Peppas [NPP03]. When revising by p it rearranges the preorder byputting the p-worlds at top (but conserving their relative order) and the ¬p-worlds at bottom(but conserving their relative order). It has the following property.

(Lex): If µ ⊧ p and φ ⊧ ¬p, then µ ≺Ψp φ.

Radical revision is similar to moderate revision, but it differs in making the new belief irrevo-cable, i.e., impossible to remove. Segerberg proposed this type of revision and characterizedit axiomatically [Seg98]. It is further investigated in [Fer00]. In radical revision by p, the rel-ative order of the p-worlds is retained whereas the ¬p-worlds are removed from the preorder,thus becoming inaccessible. The main characteristic of this operator is:

(Irr): [(Ψ p) ¬p] = ∅.

Delgrande, Dubois and Lang argue that since revision assumes a static world, there is no reasonwhy the outcome of an iterated revision should depend on the order of the inputs. Therefore, theypropose that iterated revision should take the form of prioritized merging, a special case of multiplerevision [DDL06]. Several other types of iterable operators have been proposed, see for instance[BMW06, Bon09b, Can97, Joh06, KI08, Nit08].

4.3 Changes in the strength of beliefsAs explained above, in order to represent iterated change we need a representation of the beliefstate that includes, in addition to the belief set, an ordering that expresses the relative plausibilityor retractibility of the beliefs. In such a model, a general type of operation can be introduced that

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consists in raising or lowering the position of a sentence in the ordering. Such a change may ormay not affect the belief set, but will always affect how the belief state responds to new inputs.

One important class of such operators are the improvement operators investigated by SebastienKonieczny and Ramon Pino Perez [KP08]. Suppose that the agent receives an input p that is notin the original belief set. With an improvement operator the plausibility of p in the belief stateis improved, which means that some or all of the p-worlds are moved to a higher position in thepreorder. This does not necessarily lead to a change in the belief set, but it faciliates the acceptanceof p in later, additional operations. Hence we can have p ∉ K p but p ∈ K p p. It followsthat improvement operators do not satisy the success postulate, and consequently they are a type ofnon-prioritized revision operators.The key condition for improvement operators is the following:

(Imp): Let µ ⊧ p and φ ⊧ ¬p. If φ≪Ψ µ, then µ ⪯Ψp φ.

In quantitative theories of belief change, such as probabilistic and ranking theories (Sections 4.1and 4.1), the degree of acceptance of each sentence is represented by a numerical value. (Thisapplies both to the input sentence and the elements of the original belief set.) However, the meaningof these numbers is not entirely clear (especially not for non-probabilistic functions), and realagents are notoriously bad at reasoning with them [KST82]. These difficulties are largely avoidedif a preorder is used instead of a numerical representation. Comparative degrees of belief can thenbe specified by taking certain beliefs as points of reference. The operation of change will adjust theposition of an input sentence in a ordering to be the same as that of a reference sentence. Such anoperator requires two sentences as inputs: the sentence to be adjusted and the reference sentenceto which it will be adjusted. Since two sentences are involved, Hans Rott called such operatorstwo-dimensional [Rot09b].

This kind of operation was studied by John Cantwell. He also introduced the operations ofraising and lowering, whereby the degree of plausibility required for a sentence to be included intothe belief set is changed in either direction, without any change in the relative positions of any ofthe sentences. [Can97].

Ferme and Rott proposed the operation of revision by comparison. In the intended case, theinput sentence p is accepted to the same degree as a previously believed sentence q. However, ifthe negation of the input sentence p is more plausible than the reference sentence q, then q will beremoved from the outcome [FR04]. Therefore, revision by comparison violates the DP postulates(in particular DP2 since it collapses distinctions between some ¬p-worlds). Rott has proposeda variant, bounded revision, that captures the spirit of revision by comparison and also satisfiesthe DP postulates [Rot10]. As Rott pointed out, revision by comparison reduces the number ofequivalent classes in the preorder, whereas bounded revision increases it.

4.4 Non-prioritized changeIn AGM revision, new information has primacy. This is mirrored in the Success postulate forrevision. At each stage the system has total trust in the input information, and previous beliefsare discarded whenever that is needed to consistently incorporate the new information. This is anunrealistic feature since in real life, cognitive agents sometimes do not accept the new information

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that they receive. There may be different reasons for this: the new information contradicts highlyentrenched previous beliefs, the reliability of the source of the information is doubtful, or theinformation is too complex to be accepted more than partially.

Revision operators that violate the success postulate were called non-prioritized belief revisionfunctions. Several authors proposed different kinds of such functions. We can classify them in twodifferent taxonomies, according to: 1) The process of revision and 2) the outcome of revision.

Taxonomy based on how to revise

We can classify the different models of belief revision according to the process that the revisionfunction involves. The origin of this taxonomy is due to Hansson [Han97b]. We make minimalchanges on it:

Integrated Revision: It consists in revising (or updating) the belief set in one single step.Example of models that satisfy this condition: AGM revision[AGM85], Updating[KM92].

Decision + Revision: It consists in a first step where it is decided if the input p is fully accepted,partially accepted or rejected and, in a second step, if p is not rejected, in which the beliefset is revised by p or by the chosen part of p.Example of models that satisfy this condition: Screened Revision[Mak97b], SelectiveRevision[FH99a].

Integrated Choice: It consists in choosing among the originally believed sentences and the inputp in one integrated step.Example of models that satisfy this condition: Credibility-limited revision based on Epis-temic Entrenchment and Possible world approach of Credibility-limited revision [HFCF01],Schlechta and Rabinowicz [Sch97, Rab95] revision, revision by comparison [FR04], im-provement operators [KP08].

Contraction + Expansion: In these models revising consists in first contracting the belief stateand then expanding the new state by the new belief.Example of models that satisfy this condition: AGM revision via the Levi identity, internalrevision of belief bases [Han93b].

Expansion + Contraction: It consists in adding the new sentence to the corpus of belief and thencontracting by the negation of the input sentence.Example of models that satisfy this condition: External revision [Han93b].

Expansion + Consolidation: It consists in adding a new sentence to the corpus of belief and thenregaining consistency.Example of models that satisfy this condition: Semi-Revision [Han97a].

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Taxonomy based in the outcome of the revision

We can classify the different models of belief revision according to the outcome of the beliefrevision process. This outcome can be reflected in the status of the revised belief, the change ofthe original belief set and the changes made in the epistemic state (that can be conditioned futurechanges). The origin of this taxonomy is [Fer99]:

Non-Indifferent Revision Functions: Basically, the new status of the input would be acceptedor rejected i.e, the agent is force to assume a strong epistemic attitude regarding the inputsentence and cannot manifest ignorance about it. In symbols p ∈ Kp or ¬p ∈ Kp.Example of models that satisfy this condition: AGM revision, Updating, Screened Revi-sion, Credibility-limited revision, Irrevocable Belief Revision, Internal Revision.

All: The new input is accepted without any constraint: p ∈ Kp.Example of models that satisfy this condition: AGM revision, internal revision, updating.

All or Nothing: Or the new input is accepted, or no change is made in the belief set: p ∈ Kp orKp = K.Example of models that satisfy this condition: Screened revision, Credibility-limited re-vision, Improvement operator (in one single step).

All or Inconsistency: The new input is accepted, but under certain conditions the negation of theinput still survive to the revision process: p ∈ Kp or Kp = K.Example of models that satisfy this condition: AGM revision revised by an inconsistentsentence, Irrevocable belief revision, Revision by Comparison in the Collapsed Case.

All or Less: Either the input is accepted or the input induces a contraction process: p ∈ Kp orKp ⊆ K.Example of models that satisfy this condition: Semi-revision, Revision by Comparison.

Proxy success: The input is not accepted, however other belief (typically weaker or a partial partof the input)is accepted: There is some q such that K∗p ⊢ q,⊢ p→ q and K∗p = K∗q.Example of models that satisfy this condition: Selective revision, Schlechta and Rabinow-icz revision, Lin Revision [Lin96].

Improvement: This case occurs only in models with iteration, if the input is not accepted, improveits possibility to be accepted in later revisions: p ∈ K ... p.Example of models that satisfy this condition: Improvement Operators.

The success postulate for contraction requires that all non-tautological beliefs are retractable.This is not a fully realistic requirement, since actual agents are known to have beliefs of a non-logical nature that nothing can bring them to give up. In shielded contraction, the success postulatedoes not hold in general; some non-tautological beliefs are shielded from contraction and cannot begiven up. An operator − of shielded contraction can be based on an ordinary contraction operator÷ and a set R of retractable sentences, so that if p ∈ R, then K − p = K ÷ p and otherwise, K − p = K.This construction can be further specified by adding various requirements on the structure of R. Ithas close connections with credibility-limited revision [FH01, Mak97b, FMT03].

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4.5 Multiple changeIn the original AGM model the input is a single sentence. This is a limitation of the framework,since agents often simultaneously receive more than one piece of information. In models of multi-ple change, the input is a (possibly infinite) set of sentences.

There are fundamental differences between iterated and (simultaneous) multiple belief changes,i.e. K ∗ p∗q is not in general identical to K ∗p,q. Good reasons have been given why no two ofthe four contractions K÷ p÷q, K÷q÷ p, K÷p,q, and K÷(p∨q) should be expected to coincidein general [FH94]. However, it has been argued that the order-dependence of iterated operationsis in itself problematic since ideally we should treat the pieces of information that we receive onan equal footing, independently of the order in which they arrive. This can be taken as a reason tohave higher expectations on the rational re-constructibility of multiple than of iterated operations[Han10].

The generalization of revision to the multiple case is simple if the input is finite: Revision bythe set p1, ...pn corresponds to revision by the conjunction of its elements, p1&...&pn. It is muchless obvious how to generalize contraction.

It has been found that there are fundamental differences between iterated contraction and mul-tiple contraction. The revisions of a belief set by a sentence p and then by a sentence q are by nomeans identical to the revision simultaneously by the set p,q. Furhmann and Hansson [FH94]showed that the contraction by p,q it is also different than the contraction by p ∨ q, than thecontraction by p ∧ q and by the iterated contraction.

Fuhrmann and Hansson identified two types of multiple contraction. In package contractionall members of an input set A are removed, hence (K ÷ A)∩ A = ∅. In choice contraction it is onlyrequired that at least one member of the input set be removed, i.e. A ⊈ K ÷ A [FH94]. DongmoZhang has added a third option, set contraction. Its purpose is not to remove the input but tomake the outcome compatible with the input. Hence, the outcome K ⊖ A satisfies the property(K ⊖ A) ∪ A ⊬ ⊥ rather than traditional Success postulates for contraction. [Zha96] Fuhrmann andHansson characterized partial meet multiple contraction in terms of axioms.

Theorem 3 An operator ÷ for a set A is an operator of partial meet package contraction if andonly if it satisfies the following conditions:

P-success If B ∩Cn(∅) = ∅ then B ∩Cn(A ÷ B) = ∅.P-inclusion A ÷ B ⊆ A.P-relevance If q ∈ A and q ∉ A ÷ B, then there exists a set C, such that A ÷ B ⊆ C ⊆ A and

B ∩Cn(C) = ∅ but B ∩Cn(C ∪ q) ≠ ∅.P-uniformity If every subset A′ of A implies some element of B if and only if A′ implies some

element of C, then A ÷ B = A ÷C.

Theorem 4 An operator ÷ for a set A is an operator of partial meet choice contraction if and onlyif it satisfies the following conditions:

C-success If B ⊈ Cn(∅) then B ⊈ Cn(A ÷ B).C-inclusion A ÷ B ⊆ A.C-relevance If q ∈ A and q ∉ A ÷ B, then there exists a set C, such that A ÷ B ⊆ C ⊆ A, and

B ⊈ Cn(C) but B ⊆ Cn(C ∪ q).

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C-uniformity If for every subset A′ of A it follows that B ⊆ Cn(A′) if and only if C ⊆ Cn(A′),then A ÷ B = A ÷C.

Most of the major AGM-related contraction operators have been generalized to multiple con-traction: multiple partial meet contraction [Han89, FH94, Li98], multiple kernel contraction[FSS03], multiple specified meet contraction [Han10], and a multiple version of Grove’s spheresystem. [RF10, FR10]. Spohn has proposed a ranking-theoretic account of multiple package con-traction [Spo10]. Fuhrmann has investigated operations such as the subtraction p − q that assertsp with the exception of what q says, and the merge p q that extracts the maximized consistentcontent from p and q jointly[Fuh97a, Fuh97b]. Finally, Zhang has investigated the combinationof iterated and multiple contraction, represented by series of contractions such as K ÷ A1 ÷ ...An,where each Ak is a set of sentences [Zha04].

5 Applications and connectionsThe AGM model has turned out to have a surprising number of connections with other areas ofresearch, some of which will be mentioned here.

5.1 UpdateIn 1992, Katsuno and Mendelzon presented a type of operator of change that they called update.[KM92] Whereas revision operators are intended to capture the change yielded by evolving knowl-edge about a static situation, update operators are intrended to mirror the change in knowledgeproduced by an evolving situation.

“We make a fundamental distinction between two kinds of modifications to aknowledge base. The first one, update consists of bringing the knowledge base upto date, when the world described by it changes. . . . The second kind of modification,revision, is used when we are obtaining new information about a static world. . . . Weclaim the AGM postulates describe only revisions.”

The difference is captured in the following example [Bec99, adapted from Katsuno andMendelzon’s original one] :

Example 3 Suppose that each day I either have no breakfast at all or I have coffee and toast.Suppose you are now informed that I had coffee at breakfast. How should you incorporate thisinformation into your knowledge? You could take it as an indication that I had coffee and toast,with which moreover it is consistent, so you expand your knowledge. This is what AGM revisionsanctions for the example. Another way to look at it, is to perform a case analysis over what youknow. There are just two possibilities that are consistent with your knowledge: either

(1) I have coffee and toast, or(2) I have neither coffee nor toast.

Suppose (1). Finding that I had a coffee is perfectly reasonable with this case. Let us say that the

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outcome of case (1) is the scenario described by case (1) itself. Now suppose (2). Finding thatI actually had a coffee conflicts with the case. You are obliged to jump to the “closest” scenarioto case (2) that accommodates the information. For instance, it could be that I woke up late andleft having no breakfast; but at the bus stop I bought just a coffee from a vendor. From this caseanalysis you conclude that definitely I had a coffee but that nothing can be said about me havingtoast. This is the type of change dictated by update.

In their paper, Katsuno and Mendelzon compared update with AGM revision and noted someinteresting formal differences. One example is that the AGM postulate Vacuity (If K ⊬ ¬p thenK + p ⊆ K ∗ p) does not hold for updates. Update and its realtion with revision has been furtherstudied by Becher [Bec99] and others.

5.2 Non-monotonic and defeasible logic

In spite of the differencdes between belief revision and non-monotonic logic are [Gar91] it ispossible to translate concepts, models, and results between these areas [MG91]. Non-monotonicreasoning can be expressed by an inference operator ∣∼ such that A ∣∼ p denotes that A is a goodenough reason to believe that p or that is p a plausible consequence of A. However, contrary to Cn,C does not satisfy monotony (If A ⊆ B then Cn(A) ⊆ Cn(B)). Given a belief set K representing thebackground beliefs we can translate formulas between the two frameworks as follows:

p ∣∼ q if and only if q ∈ K ∗ p

Belief revision and non-monotonic logic have become increasingly interconnected. Satoh provideda model of belief change (“minimal revision”) that satisfies the AGM postulates except vacuityand conjunctive overlap. It can be interpreted as a non-monotonic reasoning operator [Sat88].Lindstrom [Lin91] and Rott [Rot93] have shown that belief revision and non-monotonic logic areclosely connected through their reconstructibility in terms of choice functions satisfying variousrationality postulates. Other contributions in this tradition are [Mak90, Rot91]. Billington et al fur-ther clarified the relationships between AGM revision and non-monotonic inference [BAGM99].

In the last years of his life Alchourron published a series of articles on the logic of defeasibleconditionals [Alc93, Alc95, Alc96]. He proposed that conditional constructions in ordinary lan-guage can often be understood as saying that an antecedent p together with a set of assumptionsis a sufficient condition for the consequent q. Such conditionals can be represented by a formula◻( f (p) → q), where f (p) is a function that takes us from p to the conjunction of p and its pre-suppositions. The connection between this approach and AGM was initially somewhat unclear[BFL+99], but in [FR06] axioms were given that relate it to a generalized version of AGM revi-sion for an implicit underlying belief set K. This shows that there are close connections betweenAlchourron’s approach and Pagnucco’s concept of abductive expansion which is also closely asso-ciated with non-monotonic inference [Pag96].

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5.3 Modal and dynamic logics

We can distinguish between three ways to integrate belief revision with a modal logic.9 First, onecan add epistemic or doxastic modal operators to make it explicit in the logical language that thebelief set consists of beliefs. Secondly, one can formalize the execution of expansion, revision, andcontraction with dynamic modal operators, similar to those for program execution. Thirdly, onecan add both epistemic and dynamic modal operators, to integrate belief and belief change in onelanguage; this has been done in the dynamic doxastic logic by Segerberg and collaborators andlargely separately from that, later in dynamic epistemic logic.

A reason to investigate belief revision in modal logic is that the theories of belief change de-veloped within the AGM tradition are not logics in a strict sense, but rather informal axiomatictheories of belief change. Instead of characterizing the models of belief and belief change in aformalized object language, the AGM approach uses a natural language (ordinary mathematicalEnglish) to characterize the mathematical structures under study [LR99b]. The approaches to bementioned here “internalize” the operations of belief change into the object language.

Explicit belief operators: An early approach was closely connected to non-prioritized beliefrevision. Let Bp denote that p is believed by the agent. Then ◻Bp can denote that p is necessarilybelieved, i.e., it is believed and no amount of epistemic input can change this. ¬Bp means thatit is possible to arrive at some state of belief in which p is not believed. ◻ B signifies that it ispossible to arrive at some state in which p is believed and can after that no longer be disbelieved,etc. Depending on the details of the revision process, this can be shown to give rise to either anS4.2 or an S4 logic for the modal operator [Han94b].

Dynamic modalities for belief revision: In various publications, van Benthem, de Rijke, andFuhrmann [vB89, Fuh91, dR94, vB94] introduced an “update logic” including the following nota-tion:

[÷p]q (q holds after contraction by p)[∗p]q (q holds after revision by p)[+p]q (q holds after expansion by p)This update logic can be seen as a precursor of subsequent treatments of belief change in

dynamic logic.Dynamic doxastic logic: DDL extends propositional logical theories of formulas with both

Hintikka-style doxastic operators [Hin62] as well as dynamic modal operators for belief change[Seg95a, Seg95b, Seg96]. It was defined by Krister Segerberg as a logical framework for reasoningabout doxastic change. The basic DDL represents an agent that has opinions about the externalworld and an ability to change these opinions in the light of new information. Such an agent isnon-introspective in the sense that it lacks opinions about its own belief states, for example B[∗p]qis not a well-formed formula. Lindstrom and Rabinowicz extended this model in order to includesuch formulas [LR99b]. The result allows not only introspective agents but also iterated change,which has been studied by John Cantwell [Can97].

Dynamic Epistemic Logic: DEL also studies changes in information and investigates actionswith epistemic impact on agents [Pla89, BMS98, vDvdHK07]. Like DDL it has epistemic oper-

9This section relies heavily on personal communication from Hans van Ditmarsch.

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ators for belief or for knowledge and also dynamic operators for changes in belief or knowledge.The best-studied dynamics is that of the public announcement of a formula φ. This can—withmany reservations—be seen as a kind of belief expansion with φ. One such reservation is that thepostulate of success is not necessarily satisfied: after public announcement of φ it need not be thecase that Bφ is true, i.e., that φ is believed. The standard counterexample is the Moore-sentencep ∧ ¬Bp [Moo42, Hin62]. Clearly, B(p ∧ ¬Bp) is inconsistent for standard notions of knowl-edge and belief. In DEL, the Moore-sentence is just one example of an unsuccessful update, see[vDK06].

Belief revision in the typical AGM sense is more problematic. If the agent believes in p (i.e.,Bp is true), then a public announcement of ¬p will make her or his beliefs inconsistent. To modelbelief revision in DEL, we need Kripke models where knowledge, belief, and degrees of belief (orconditional belief) can all be encoded. The way to do this is to add plausibility relations to theKripke models, and to identify belief in φ with truth of φ in the most plausible of the epistemicallyaccessible states.

For example, suppose that two states s and t are both considered possible by an agent, but (s)heconsiders s to be more plausible than t. Furthermore suppose that p holds in s but not in t. Then theagent believes that p. Belief revision with ¬p revises the plausibilities such that t becomes moreplausible than s. Now, the agent believes that p is false: B¬p. So we have Bp ∧ [∗¬p]B¬p, where[∗¬p] is not a ‘hard’ (i.e., truthful) public announcement but a tentative or ‘soft’ (i.e., preferencechanging) public announcement. An alternative belief revision mechanism in the DEL setting iswhen the state s is eliminated from consideration (a ‘hard’ update, as for the execution of publicannouncements), after which the t is the most plausible (namely the only remaining) state. Again,the agent believes that p is false. These issues were addressed by Aucher [Auc03], van Benthem[vB07], van Ditmarsch and Labuschagne [vD05, vDL07], and Baltag and Smets [BS06] (withmany follow-up papers). Conditional reasoning and reasoning with different degrees of belief canalso be modelled in such settings.

5.4 Game theoryBelief change and game theory are related in several ways. Booth and Meyer [BM10], investigatedequilibria in belief merging. The key idea is that a social belief removal function can be a minimalchange in the AGM sense. Two classes of removal functions for agents have been studied: basicand hyperregular removal. The former has been axiomatically characterized by Booth [BCMG10].Zhang studied bargaining situations from another viewpoint, proposing a logical axiomatization ofbargaining solutions that is based on postulates from AGM and game theory [Zha10].

The classical analysis of centipedes in the game theoretical literature (by Aumann, Binmoreand others) shows that adequate models should be able to represent belief-contravening hypothesesand therefore also the notion of supposition. Samet [Sam96] proposed that hypothetical knowledgecan be represented by the operator: KH

i (E) = ⋃ P ∈ Πi: Ti(P, H) ⊆ E, where Πi is a partition celland Ti is a transformation function mapping partition cells and hypotheses to partition cells. Thistransformation function has the flavour of belief change but it is constrained by properties notused in the literature on belief change. Arlo-Costa and Bicchieri proposed an operator of this sortthat satisfies some of the AGM properties. They developed models in which the condition that

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all players are disposed to behave rationally at all nodes is both necessary and sufficient for themto play the backward induction solution in centipede games. This result was obtained withoutassuming that rationality is commonly known (as is in [Aum92]) or commonly hypothesized bythe players (as in [Sam96]).

5.5 ArgumentationFalappa, Kern-Isberner and Simari [FKIS09] have argued that belief revision and argumentationtheory are complementary approaches. By combining the two, the variety and complexity of rea-soning processes is better accounted for than if only one of them is used. In [FKIS02], theycombine ATMS (Assumption-based truth-maintenance systems) [dK86] with belief revision andpropose a system that uses argumentative structures in the form of explanations for non-prioritizedrevisions of a belief base B. In this model, an epistemic input is composed of a sentence p anda set A of reasons to believe it. A partial acceptance revision operator is constructed such that Ais initially accepted, which leads to the creation of B ∪ A as a (possibly inconsistent) intermediatebelief base from which inconsistencies are removed, giving rise to a consistent revised belief baseB A. The operator is an operator of external revision in the sense explained in Section 4.1.

Paglieri and Castelfranchi proposed Data-oriented Belief Revision (DBR) as an alternativeto AGM [Cas97, Pag06]. This model combines belief revision with argumentation, followingToulmin’s account of argumentation. The application of DBR to argumentation is primarily in-tended to highlight structural communalities between arguments and belief-supporting networks.[PC04, PC06]. The model contains two basic informational categories, data and beliefs. Contraryto beliefs, information items are allowed to be contradictory. When a belief is abandoned, this doesnot entail removal of the corresponding information from the agent’s memory, i.e. disbelieving isnot forgetting.

5.6 Belief Change by translation between logicsThe AGM paradigm can be extended to non-classical logics by translating the source logic intoclassical logic, performing the change and then translating back. This method was proposed inde-pendently by two groups. Gabbay, Rodrigues and Russo [GRR99, GRR08] respectively Coniglioand Carnielli [CC02]. The latter defined logics as two-sorted first-order structures, and argued thatthis broad definition encompasses a wide class of logics with theoretical interest as well as interestfrom the point of view of applications. The language, concepts and methods of model theory canbe used to describe the relationships between logics through morphisms of structures called trans-fers. They define a model of belief change, called Wide Belief Revision Systems, to define beliefrevision for non-standard logic.

5.7 TruthIn eliminative induction a number of possible hypotheses concerning some state of affairs arepresumed, and rivals are progressively eliminated by new evidence. The process is an idealization,since in practice no closed set of initial theories is usually available. Kelly found similarities

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between the process of eliminative induction and theory change, and proposed the use of beliefrevision to arrive at informative, true, empirical belief on the basis of increasing information. Heanalyzed several algorithms of iterated revision (see Section 5), assuming that learning is based onthe outcomes of sequential experiments [Kel98].

5.8 Use of choice functions and related preference orderingsThe classical theory of rational choice was developed by mathematical economists [Arr51, Arr70,Sen71]. It has occupied a central role in the philosophy of the social sciences for almost a century.

Choice rationality is concerned with how to choose rationally among a set of alternatives.Formal requirements are imposed on choices from different, overlapping alternative sets. Thestandard presumption is that selection functions are rationalizable in terms of some underlyingbinary preference relation. Formally, this is represented in structures ⟨Ω,E , f ⟩ consisting of a setof alternatives Ω, a collection E of subsets of Ω (representing possible choice sets) and a functionf from E into the set of subsets of Ω, representing choices made. The main objective of rationalchoice theory is to investigate the conditions under which the function f can be rationalized by atotal pre-order R on Ω in the sense that, for every E ∈ E , f (E) coincides with the best elements ofE relative to R. However, as discussed by Sen [Sen97], social norms and menu dependence canmake rationalizability in terms of an underlying preference relation impossible.

The structures studied in rational choice theory have a close connection with the selectionfunctions employed in belief revision. In his book [Rot01], Rott relates belief revision, non-monotonic reasoning and rational choice, and shows how standard postulates of belief change andnon-monotonic reasoning correspond to the constraints of classical theories of rational choice. Ac-cording to Rott, these connections constitute an important bridge between practical and theoreticalrationality.

Olsson [Ols03] conceded that Rott’s work is indisputable as a formal achievement, but puts hisphilosophical conclusions in question. According to Olsson, Rott has not discovered any surprisingconnections between revision and choice. Instead he has reconnected the AGM theory with itsroots in counterfactual reasoning and rational choice.

Further studies in this area have been reported by Arlo-Costa and Pedersen [ACP10] and byBonanno [Bon09a].

6 EpilogueIn this thesis I had two pourposes. On one hand, I have tried to reflect the impact of the AGMmodel, not only in the Philosophy community, but also in other areas, like Artificial Intelligenceor Game Theory. The summary is devoted to this pourpose by exemplifying the diversity andsignificance of the research that has been inspired by the AGM theory in the last 25 years.

On the other hand, the paper that compound the thesis want to reflect my proper work in thearea. Since in each one of the papers the contribution is explicited, I have tried to do not focuss thesummary in them, in order to provide a balanced view of 25 years of research in belief revision.

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