Embed Size (px)
Citation preview
Interactive MistakesA Theory of Belief Revision with False Messages
Antoine Billot, PSE and U. Paris 2 (Paris, France)�
Jean-Christophe Vergnaud, CES-CNRS and U. Paris 1 (Paris, France)y
Bernard Walliser, PSE (Paris, France)z
April 7, 2009
Abstract
The aim of this paper is to propose a formal treatment of any kind of message and ofits consequences in terms of belief revision. An epistemic logic framework is used wheresyntax and semantics are linked. A message is modelled as beliefs by a set of propositionsor a structure of possible worlds. It is characterized by its content and its status, the lastbeing expressed through an auxiliary language that can express informational mistakes.Therevision process leading from the initial to the �nal belief is based, on one side, on a dynamicmodus-ponens-like axiom and, on the other side, on a product-rule mixing up initial belief andmessage structures. Moreover, we prove that an "accuracy" order which allows to comparebeliefs is preserved under belief revision.Journal of Economic Literature Classi�cation Number: C72, D82
1 Introduction
These days, no one can deny that a substantial part of the current �nancial information ismistaken. For instance, credit rating agencies provide ratings which are de�nitively less truthfulthan what people think. Flawed information occurs in many other empirical situations. Oneparticular aspect of this issue is that of secret information de�ned as information provided toone particular agent while the others think nothing happens. For instance, an insider tradingcan be viewed as an unfair use of secret information. The same can be said for misstatementsby insured agents or when crooked employers simultaneously o¤er the same reputed �exclusive�contract to several employees. In medecine, for the psychological sake of their patients, it happensthat physicians withdraw negative aspects of diagnosis. All these examples share in common thefact that errors are induced by wrong messages which are not interpreted as such by receivers.
Standard models like agency ones consider incomplete and imperfect information but neglectthe possibility for the agents to have mistaken beliefs and information. The main aim of thispaper is then to propose a formal treatment of a message and of its consequences in terms of beliefrevision whatever its quality, true or false. For that purpose, we use logical methods to studybelief structure and revision in a multiplayer setting. We model the initial situation through a
�[email protected]@[email protected]
1
complete and consistent set of propositions in the weak multi-modal version of the logic KD. Thisset expresses various facts about the world, but it also expresses what di¤erent agents believeabout the world as well as agents�higher order beliefs about one another�s beliefs. Ultimately,the agents update their beliefs after they receive some messsage, i.e. a new information.
Concerning the structure of a message, two aspects can be distinguished: its content andits status. The content expresses what is transmitted by the message. The status indicates towhom the message is sent and what the agents know about its di¤usion. To represent these twodimensions of the message, we introduce an auxiliary language in which the new information isexpressed. The auxiliary language starts with a �nite collection of propositions such that theconjunction of every pair of distinct propositions is inconsistent and such that the disjunction ofall propositions is a tautology. Then it recursively builds more complex propositions out of thepropositions using propositional connectives and belief operators in the usual way. A messagestructure is then de�ned as a new, complete and consistent set. Therefore, a list of messages canbe constructed including all the traditional messages as special cases (public, private, secret).For instance, a public message sent by a forecasting institute consists in a content (�the oilprice will increase in the next days�) that is commonly learnt. A private message sent by a guruconsists in a content (�a new technology is about to be launched�) that is learnt by few agents(his clients), the others only knowing that the message was sent. A secret message sent by aninformed manager consists in a content (�the company is going to be taken-over bid�) that isreceived by few agents, the others not knowing that a message was sent.
The model we suggest explains how to take an initial belief and a message structure expressingwhat information the agents learn, and then to derive a �nal belief. This last describes how allagents update their beliefs and higher order beliefs conditional on the new information describedby the message structure. In order to achieve this, we impose two axioms. The �rst axiom, NoCon�ict, implies that no agent learns a proposition that he initially believed to be false. Thesecond axiom, Belief and Message Inference, describes how agents combine the information thatthey initially detained with the information that they receive to form new beliefs. For instance,it implies that they can combine old and new information via modus ponens. There are threetypes of belief operators, Bi (for initial belief), Bi (for belief imposed by the message structure),and B�i (for �nal belief). The updated belief is set up by �nding the unique complete andconsistent set in a new language containing the operators B�i with respect to the constraintsimposed by Belief and Message Inference and No Con�ict. The �rst theorem of the paper showsthat the �nal set is unique given the initial one and the message so that we impose a well de�nedbelief revision rule. The second theorem develops a semantic counterpart to this sort of beliefrevision by means of the Cartesian product of the two worlds spaces respectively associated tothe initial belief and the message.
Indeed, there already exists a usual framework to deal with belief and information based onpartitional information structures. More precisely, in Aumann (1976), initial belief is modelledthanks to a prior probability distribution and a partition of possible worlds and new informationcorresponds also to a partition de�ned on the same worlds. Then, updated belief proceeds fromthe intersection of both partitions. Even though it is only semantical, such a framework issu¢ cient for public or private messages, but not for secret messages.1 For instance, in case of
1Note that a previous departure from Aumann consists in the unawareness approach (Dekel; Lipman andRustichini (1998), Meier, Schipper and Heifetz, (2006). However, the kind of informational failure we considercannot be mixed up with unawareness since we stay within a Kripke structure while unawareness suggests thatthere exist some totally unaccessible worlds.
2
secret information, the informed agent takes into account the world where he knows this newinformation while uninformed agents unforesee it. Hence, secret messages require to introducenew worlds in the �nal beliefs. And this can be achieved thanks to our Cartesian product ofworld spaces being the counterpart of multiagent belief revision.
The main attempt to design a belief revision rule in epistemic logics starts with Alchourron-Gärdenfors-Makinson (1985) and is called the AGM system. It concerns only a single agent,but it is adapted to the case where the message contradicts the initial belief. Note that theepistemic operators are not speci�cally introduced. More recently, three main papers can beconsidered relevant for our purpose. We present them according to their level of proximity withour model. Bonanno (2005) still considers one player but, exactly as we do, this agent is endowedwith three syntactical operators for initial belief, message and �nal belief. However, contraryto us, he restricts semantics to a unique world space. Board (2004) also restricts semantics toa unique world space but he considers several players with an in�nity of syntactical operatorsparametrized by a proposition, i.e. the content of an implicit message. His axiomatics is thesame than ours. But two main di¤erences emerge. First, his model is more general since Boardallows for contradictions between the content of the message and the initial belief. Second, thismodel is unable to study message structures in general. For instance, it cannot deal with privatemessages. The closest framework to ours is Baltag and alii (1998) even if the syntax is expressedin a dynamic logics set-up. It is less general than ours since it deals only with material messages.Moreover, it does not formally identify the structure of a message and, consequently, a taxonomyof messages.
An important fallout of the previous results is to study how a comparison order of beliefs canremain under belief revision. In this context, we use a generalization of �partition re�nement�2
as in Billot, Vergnaud and Walliser (2009) called �accuracy�. More precisely, for a given initialbelief, if a message is more accurate than another, the same holds for the corresponding �nalbeliefs. Such a result is typically original since it cannot proceed from Board (2004) and Baltagand alii (1998) models
The plan is the following. Section 2 presents a motivating example based onMadame Bovarystory that we follow all along. In Section 3, the syntactical and semantical de�nitions of ahierarchical belief structure are recalled. In Section 4, a message is de�ned within the sameformal belief structure but interpreted in a di¤erent way. In Section 5, axioms are providedfor multiplayer belief revision and the two main theorems are established. In Section 6, thetransmission of accuracy orders from messages to �nal beliefs is analyzed.
2 Motivating Example
The literature is full of psychological situations in which characters have crossed beliefs oneabout another. All along the novels, messages are sent and received which lead to revise beliefs.For instance, in the famous French novel Madame Bovary (1856), Gustave Flaubert sets uphis scenario around Emma�s adultery. Since an adultery is not a public event in the Frenchsociety of the XIXth century, crossed beliefs are quite natural, especially those of Emma and herhusband Charles. Let us imagine the following Flaubert style example.
Just after Emma�s �rst love a¤air with Rodolphe, the initial beliefs are the following. At the�rst level, Emma knows that she is unfaithful and her husband does not know but he has some
2A partition is �ner than another when the cells of the former are included in the cells of the latter.
3
doubts. At the second level, Emma believes either that Charles knows she is unfaithful or thatCharles does not know and Charles obviously knows that his wife knows the truth, whatever itis. At the third level, Emma thinks that Charles thinks that she knows the truth and Charlesthinks that Emma thinks that he believes she is faithful.
Suppose now that a message occurs... Charles learns the truth from Félicité, the homemaid.Suppose moreover that Emma, being hidden behind the door, hears Félicité at the very momentshe speaks to Charles. Note that the message received by Charles is a �rst level one since itdeals with the material situation, namely the fact that Emma is adultery. Moreover Charlesthinks that the message he received is secret because he meets Félicité alone. Of course Emmaknows its content which then becomes shared. In return, the message received by Emma is asecond level one since it deals with Charles�beliefs. Moreover, it is secret since Charles is notaware that his wife is behind the door.
Such a story does not lead to an immediate belief revision. Actually, the intuition workswith di¢ culty (try yourself!) even though it follows the natural language and then is syntactical.Furthermore, there exists no revision method in the literature that is able to deal with such asituation characterized by the following features: beliefs and messages are syntactical; messagesinduce a simultaneous revision of all players� beliefs; the content of messages is material orepistemic; the status of messages is of any kind, especially secret; erroneous beliefs must betaken into account.
For clarity, the Bovary example will be formally treated all along the paper. The �nal beliefwill �rst be computed in a semantical framework then translated into syntax. The results arethe following. At the �rst two levels, Emma and Charles know and know that the other knowsthe truth. At the third level, Emma knows that Charles knows that she knows the thruth. ButCharles still believes that Emma believes that he believes she is faithful. Finally, at the fourthlevel, Emma knows that Charles knows that Emma believes either that Charles knows the truthor that Charles doubts while Charles believes that Emma believes that he believes that shebelieves she is faithful.
3 Initial Belief
3.1 Syntax
We describe here the physical universe as well as players�belief about it by means of propositionsbelonging to a particular language.
Let I be a �nite set of players i. A multiplayer language is de�ned according to threecomponents:
(i) primitive propositions typically labeled p, q... that describe the common physical envi-ronment faced by the players and form a �nite nonempty set P,
(ii) propositional connectors, i.e. negation :, conjunction ^ , disjunction _, material impli-cation !,
(iii) epistemic operators, denoted (Bi)i2I , that describe players�beliefs about physical envi-ronment and about other players�beliefs.
The multiplayer language L is a set of well-formed formulas typically labeled ' or ... which
4
is induced by three requirements:8<:if p 2 P, then p 2 L,if '; 2 L, then :', : ;' ^ ;'! ;' _ 2 L,if ' 2 L, then 8i 2 I, Bi' 2 L.
(1)
Especially, L0 � L is the subset of all propositions without belief operator.A SYntactic Structure (SYS) is a complete list of propositions which are true in the actual
universe. It combines two classes of propositions: the tautological ones and the contingent ones.First, consider the subset T � L of propositions that are necessarily true. They are obtained
by three axioms and two inference rules.3
A1: All tautologies of propositional calculus.
A2 (Logical Omniscience): Bi' ^Bi ('! )! Bi .
A3 (Weak Consistency): :Bi?.
R1 (Modus Ponens): From ' and '! , infer .
R2 (Necessitation Rule): From ', infer Bi'.
Second, for a subset C of L, let us de�ne two constraints:
C1 (No Contradiction): If ' 2 C, then :' =2 C.
C2 (Completeness): If ' =2 C, then :' 2 C.
De�nition 1 : A SYntactic Structure (SYS) is a subset K of L which satis�es R1, C1 andC2, with T � K.
Note that we do not assume the usual axioms of system S5 for epistemic operators, that isVeridicity (or Truth), Positive and Negative Introspection.
Bovary Example: In order to describe the Bovary belief-language, we need two operators, BE(Emma believes...) and BC (Charles believes...). The only primitive proposition is denotedp (Emma is unfaithful). The SYS KBovary contains the following contingent propositionsat the three �rst levels:
physical environment
0-level p
Emma�s beliefs
1-level BEp
2-level BE:BC:p, :BEBCp, :BE:BCp3-level BE (BCBEp _ ((BC (BEp _BE:p) ^ :BCBEp ^ :BCBE:p))
Charles�beliefs
1-level :BCp, :BC:p2-level :BCBEp, :BCBE:p, BC ((p! BEp) ^ (:p! BE:p))3-level BCBEBC:p
3The propositions ' 2 T are provable and sometimes denoted ` ' in the literature.
5
3.2 Semantics
To represent the actual universe in semantics, we consider a model à la Kripke composed ofpossible worlds and accessibility relations between them. More precisely, we de�ne a SEmanticStructure (SES)4 as a list of its components plus a condition of Connectedness:
De�nition 2 : A SEmantic Structure (SES) for a set I of players is de�ned as a 5-uple H =(W; (Hi)i2I ; S;H0;w) where:
(i) W is a set of mutually exclusive possible worlds denoted w,(ii) Hi is an accessibility relation from W toward 2W n;,(iii) S is a set of mutually exclusive states of nature s,(iv) H0 is the accessibility relation of nature de�ned from W toward S,(v) w is the actual world
such that:(Connectedness) In the actual world w, for each world w, there exists a �nite sequence of
players i1; :::; in and a �nite sequence of worlds w0; :::; wn such that w0 = w, wn = w and forall n � k � 1, wk 2 Hik (wk�1).
A (possible) world is a full description of the physical environment and of the players beliefsabout it. The physical part called nature is represented by a state of nature. The psychical partrepresenting the beliefs of the players about the physical world are embodied in the accessibibilityrelations. They gather the worlds which are considered undiscernable by a given player in a givenworld. Connectedness means that the set of possible worlds does not include inaccessible worldsfrom the actual one.
Bovary Example: In order to describe the Bovary semantical structure, we just introduce twostates of nature, p (Emma is unfaithful) and :p (Emma is faithful). The SES HBovarycorresponds to the following graph.5 (Figure1(structureinitiale).pdf)
3.3 Transcription rules
To link syntax to semantics we need some transcription rules which allow to translate anysyntactical statement into a semantical one. Consider a SES H and de�ne an interpretationfunction � as a mapping which associates to each state s 2 S a truth assignment of the primitivepropositions, i.e. � (s) : p 2 P ! ftrue; falseg.6 The truth value of a formula ' in someparticular world w is written: (H; w) j= ', which means that ' is true in world w within theSES H. Let j'j be the �eld of the formula ', i.e. the set of worlds where ' is true. The following
4 In epistemic logics, a SES corresponds to a connected pointed belief model.5The actual world w = w0 is represented by a square and the other worlds (w1; :::; w7) by a circle. All of
them are numbered as indexes. Besides, the state of each world appears inside. Emma�s accessibility relation isrepresented by a full line while Charles�is a doted line.
6The usual valuation function v is a composition of the accessibility relation of the nature H0 with the in-terpretation function �: it associates, for each world, a truth assignment of the primitive propositions: v(w) =�(H0(w)).
6
rules express the truth assignment for any formula:8>>>><>>>>:(H; w) j= p i¤ � �H0 (w) (p) = true,(H; w) j= :' i¤ (H; w) 2 ',(H; w) j= ' ^ i¤ (H; w) j= ' and (H; w) j= ,�(H; w) j= Bi' i¤ (H; w0) j= ' for all w0 such that w0 2 Hi (w) ,i.e. (H; w) j= Bi' i¤Hi (w) � j'j .
(2)
A SYS K is associated with a SES H by the following bridge principle:
' 2 K i¤ (H;w) j= '. (3)
The SYS K corresponds to all formulas which are true in the actual world w. The Kripkerepresentation theorem asserts that it is always possible to associate a set of SESs H to a SYSK (van Benthem, 2000). All of these sets are bisimilar. (see section 6.1)
Bovary Example: .For instance, let us check if (HBovary;w) j= BCBEBC:p. First, note)that, on Figure 1, j:pj = fw5; w6; w7g. Then, jBC:pj = fw1; w4; w5; w7g, jBEBC:pj =fw3; w4; w6g. Finally, jBCBEBC:pj = fw0 = w; w3; w4; w6g.
7
4 Message
4.1 Syntax
Informally, a message is de�ned by two components. Its content describes the informationreceived by each player (eventually di¤erent from one player to another). Its status describeswhat the players know about the di¤usion of the information among players. Formally, from thepoint of view of the modeler, a message can be represented by a SYS in an auxiliary language(or a message language) denoted L. It is formed of three components:
(i) the set of primitive propositions,M = fm0; :::;ml; :::;mLg, which is such that:(i-a) for all l, ml 2 L,(i-b) for all l 6= l0, ml ^ml0 =? and(i-c) m0 _m1 _ ::: _mL = >,7
(ii) the usual propositional connectors,(iii) the epistemic operators denoted
�Bi�i2I . that describe players learning about the mes-
sage and about what the other players learn.
De�nition 3 : A message structure is a SYS K de�ned on the language L.
More precisely, the content of a message received by an agent is a formula m in L. It isobtained as a combination of primitive messages ml which jointly express the maximal degreeof resolution of the message. The status of the message is given by all formulas containing atleast one epistemic operator, contained in LnL0. For instance, the formula BiBjm expressesthat player i learns that player j learns the content m.
Bovary Example: The syntactical structure of the message KBovary carried by Félicité is basedon the following three �rst levels. The message m0 stands for p and m1 for :p:
physical environment
0-level m0 = p
Emma�s beliefs
1-level BEm0
2-level BEBCm0
3-level BEBC�:BEm0 ^ :BEm1
�Charles�beliefs
1-level BCm0
2-level BC�:BEm0 ^ :BEm1
�3-level BCBE
�:BCm0 ^ :BCm1
�7This condition on the set of primitive propositions for the auxiliary language L seems more demanding than
the corresponding condition for the set of primitive propositions for the language L. It is not true. Indeed, the setof primitive propositions is arbitrary and it is always possible to use a set which satis�es conditions (i-a) and (i-b).We introduce these conditions because the description of the content of a message is de�nitely more intuitive.
8
4.2 Semantics
The SYS K associated to the message has a semantical counterpart given by a SES H =(W;
�H i
�i2I ; S;H0;w). The set of states of nature S = fs0; :::; sl; :::; sLg is such that the
interpretation function � is de�ned as � (sl) [ml] = true, while � (sl) [ml0 ] = false, for l 6= l0.Hence, each state of nature in S corresponds to a primitive proposition inM. For a syntacticalcontent m in L and L0, denote M = jmj � W its semantical content in the initial SES andM = jmj �W its semantical content in the auxiliary SES H.
Bovary Example: The semantical structure of the message is given by the following graph:(Figure2(message).pdf)
4.3 Types of Message
Four di¤erent standard types of message are usually considered, according to their di¤erentstatus. They are informally de�ned in syntax and formally de�ned in semantics.
In a public message, it is commonly learned that each player receives the content m0. It isde�ned by the SES Hpub: �
W = fwg ; w 2M0,8i 2 I, H i (w) = fwg .
In a private message, one player i receives a message of contentm0 while other players receivenothing, and this is commonly learned. It is de�ned by the SES Hpri:8<:
W = fw; w1; :::; wl; :::; wLg ; w 2M0; 8l 6= 0, wl 2M l,H i (w) = fwg ; 8l 6= 0, H i (wl) = fwlg and,8j 6= i, 8l 6= 0, Hj (w) = Hj (wl) = fw; w1; :::; wLg .
In a secret message, one player i receives a message of content m0, the other players learnthat nothing was learned and the �rst player learns it. It is de�ned by the SES Hsec:8<:
W = fw; w0; w1; :::; wLg ; w; w0 2M0; 8l 6= 0, wl 2M l,H i (w) = fwg ; 8l 6= 0, H i (w0) = H i (wl) = fw0; w1; :::; wLg and,8j 6= i, 8l 6= 0, Hj (w) = Hj (w0) = Hj (wl) = fw0; w1; :::; wLg .
Finally, in a null message, it is commonly learned that nobody learns any content. It isde�ned by the SES Hnul:�
W = fw; w1; :::; wNg ; w 2M0; 8l 6= 0, wl 2M l,8i 2 I;8l 6= 0; H i (w) = H i (wl) = fw; w1; :::; wLg .
But all possible types of message are not described within these four basic types.Bovary example: The message cannot be reduced to one simple type. From the point
of view of Charles, the message concerning the physical environment (Emma is unfaithful) iswrongly believed secret.
9
5 Multiplayer Belief Revision
5.1 Syntactic Foundations
At this point, we need to determine the structure of the �nal belief, that is the result of thecombination of an initial belief with a message. We assume that the �nal belief is of the sametype than the initial one. However, an intermediate structure is required to treat simultaneouslyboth initial and �nal propositions. Then, belief revision needs a particular language which canexpress the way the agents combine the initial belief with a message in order to get the �nalbelief.
Denote L� the language in which the �nal belief is expressed. L� is de�ned according to:(i) the primitive propositions P which are the same in L� than in L,(ii) the epistemic operators which are denoted B�i .Denote L =
�L;L;L�
�the language which extends L [ L [ L�:�if '; 2 L, then :', : and (' ^ ) 2 L,if ' 2 L, then 8i 2 I, B�i' 2 L.
(4)
This language allows simultaneously to deal with propositions belonging to the initial belief,to the message and to the �nal belief as well as propositions of the �nal belief over initial beliefor message. However, this language contains propositions of the form B�iBj' or B
�iBj but not
of the form BiB�j' or BiB
�j . Hence, even if there is no explicit dynamics in this language, this
asymmetry translates a certain kind of memory but prevents from any expectation.We de�ne a Multioperator SYS (denoted MSYS ) by extension of a SYS. We consider �rst
a subset T � L of tautological propositions that are obtained by axioms A1-A3 (for the threeoperators Bi, Bi, B�i ) and inference rules R1 and R2 [from ' 2 L (resp. L (resp. L)), inferBi' (resp. Bi' (resp. B�i'))]. Then we consider a subset C � L of contingent propositions,with T � K, that are true in the actual universe and which are submitted to Modus Ponens R1,No Contradiction C1 and Completeness C2 (cf. §3.1). The MSYS is a subset K of L whichsatis�es R1, C1 and C2 and contains T [ C.
The MSYS K embodies the whole belief revision process. It allows to recover each speci�cstructure by projection: K\L is a SYS in L interpreted as the initial belief K, K\L is a SYS inL interpreted as the message K and K\L� is a SYS in L� which will be interpreted as the �nalbelief K�.
In order to characterize the revision process within a syntax, we introduce axioms whichrule the way to go through the initial belief toward the �nal one. More precisely, we impose tothe MSYS K the two following axioms which are considered common belief according to beliefoperators B�i :
A4 (No Con�ict): Bim! :Bi:m.
No Con�ict means that the message does not contradict the initial belief. For instance,it forbids Bjm ^ Bj:m, and B�i
�Bjm ^Bj:m
�... No Con�ict prevents all revision to be a
recti�cation.
A5 (Belief and Message Inference): For any ' in K\L and any in K\L,
^ml2M(Bi:ml _Bi:ml _Bi (ml �! ') _Bi (ml �! )) ! B�i (' _ ) .
10
This axiom has meaningful consequences. Consider, for instance, the case where player ilearns the true content m0: Bim0 Then, it follows by No Con�ict: :Bi:m0. Belief and MessageInference becomes:
(Bi (m0 �! ') _Bi (m0 �! )) ! B�i (' _ ) .
Here, the axiom means that an agent learns ' or if and only if ' or are believed to beentailed by m0.
Two sub-axioms can be deduced in restricting respectively the revision process to initialbelief, on one side, and to message, on the other side.
Proposition 1 : Belief and Message Inference A5 implies
_m2L0Bim ^Bi (m �! ')! B�i'. (A5a (Belief Inference))
_m2L0�Bim ^Bi (m �! )
�! B�i . (A5b (Message Inference))
Proof. Consider now that is a tautology, hence,
^ml2M(Bi:ml _Bi:ml _ (Bi (ml �! ')) ! B�i (') .
Consider moreover that the message the player i learns is the basic message ms. It follows thatBims and, by No Con�ict, :Bi:ms: For l 6= s, the terms in the conjonction are tautologies(since Bi:ml is true) and for l = s, the term reduces to Bi (ms �! ') (since Bi:ms _Bi:ms
is a contradiction). It follows that:
Bims ^Bi (ms �! ')! B�i'.
It is easy to prove that the same relation applies to any message m.Belief Inference describes precisely the belief revision process as some kind of Modus Ponens:
A proposition is �nally believed i¤ the player accepts some message and believes initially thatthis message entails the proposition. Message Inference means that a proposition is �nallybelieved i¤ the player who already knows some message learns that this message entails thatproposition. Note that, when m is a tautology, this axiom moreover entails that any propositioninitially believed and any proposition which is learnt is �nally believed.
The following theorem ensures the unicity of the MSYS K stemmed from the SYSs K andK and the axioms.
Theorem 1 : Consider two MSYSs K =�K;K;K�
�and K0=
�K0;K0;K0�
�satisfying No Con-
�ict A4 and Belief and Message Inference A5. If K = K0 and K = K0 then K� = K0�.
Proof is in Appendix A.Observe that there exists a unique �nal belief K� associated to K. Hence, this theorem
establishes that the two axioms of No Con�ict and Belief and Message Inference are su¢ cientto characterize the revision process as unique.
11
Bovary Example: At this step, we just give an example of the way the conditions work. Letsprove that B�EB
�CB
�Ep holds. By Belief Inference A5a,
�BCm0 ^BC (m0 �! BEp)
��!
B�CBEp. Since A5a is common belief, B�E
��BCm0 ^BC (m0 �! BEp)
��! B�CBEp
�.
By Message Inference A5b with tautology as message, BEBCm0 �! B�EBCm0 andsince by hypothesis BEBCm0 holds, then B�EBCm0 is also true. By A5a with tautol-ogy as message, BEBC (m0 �! BEp) �! B�EBC (m0 �! BEp) and since by hypothesisBEBC (m0 �! BEp) holds, then B�EBC (m0 �! BEp) is also true. Therefore by ModusPonens R1, B�EB
�CBEp. Finally, once more by A5a , B
�EB
�C (BEp �! B�Ep), and by R1,
we get the result.
5.2 Semantic Rule
Now, we search for the semantical revision rule which transcribes the syntactical revision axioms,that is A4 and A5.
Let us combine an initial SES
H = (W; (Hi)i2I ; S;H0;w)
and an auxiliary SESH = (W;
�H i
�i2I ; S;H0;w)
in order to get a �nal product structure denoted H�:
H� = H �H =�W �; (H�
i )i2I ; S�;H�
0 ;w�� .
De�nition 4 (Product Structure) : From an initial belief SES H and a message SES H,the product structure H� is generated as follows:
Step 1: De�ne fH� = �fW �; (H�i )i2I ; S
�;H�0 ;w
��such that
(i) fW � =W �W ,(ii) 8 (w;w) 2 fW �, 8i 2 I, H�
i (w;w) =Sl=0;1;::;L
�(Hi (w) \Ml)�
�H i (w) \M l
��,
(iii) S� = S,(iv) 8 (w;w) 2 fW �, H�
0 (w;w) = H0 (w),(v) w� = (w;w).
Step 2: H� is the connected part of fH�.In the product structure, each �nal world is obtained as a Cartesian product of an initial
world and a message world. For any �nal world, its accessibility domain proceeds from theelimination of all initial and message worlds which contradict the same basic message. The setof states of nature remains. The state of nature associated to each initial world is kept in thecorresponding �nal world. The �nal actual world is just a combination of the initial and messageactual worlds.
One can observe that H� is a SES as soon as the accessibility relations H�i (w;w) are non
empty.
12
5.3 Representation Theorem
The last step is to make the link between syntactical revision and semantical revision.The following representation theorem can then be proved:
Theorem 2 : Consider two SESs H and H and let the SYSs K and K be their syntacticalcounterpart.
(1) If H� = H � H is a SES, then there exists a MSYS K =�K;K;K�
�which satis�es No
Con�ict A4 and Belief and Message Inference A5 and such that K� is the syntactical counterpartof H�.
(2) If there exists a MSYS K =�K;K;K�
�which satis�es No Con�ict A4 and Belief and
Message Inference A5, then H� = H �H is a SES and is a semantical counterpart of K�.
Proof is in Appendix B.
This representation theorem holds for every kind of message, say public, private, secret andso on... It solves the question of private or secret messages in a multiplayer setting which wasout of reach of standard models such as Board�s and Baltag�s. Moreover, the semantical revisionrule associated to the axioms appears as a Cartesian product and is very easy to use. Hence, inorder to revise, the best is to work �rst in semantics and then to transcribe the result in syntax.
Bovary Example: We apply the product structure operation as follows. We consider all cou-ples of worlds obtained by combining a world of the initial belief and a world of the message.We �rst de�ne the accessibility relations stemming from the actual �nal world. We thendo the same from the worlds which were accessible from the actual world. We proceedsequentially until no new world can be reached. The result is the �nal belief structuregiven in (Figure3(intermedstructure�nale).pdf). We can deduce from the previous graphthe syntactical belief composed of the following propositions at the �rst four levels.
physical environment
0-level p
Emma�s beliefs
1-level B�Ep2-level B�EB
�Cp
3-level B�EB�CB
�Ep
4-level B�EB�C (B
�E (B
�Cp _ (:B�Cp ^B�C:p)))
Charles�beliefs
1-level B�Cp2-level B�CB
�Ep
3-level B�CB�EB
�C:p
4-level B�CB�EB
�CB
�E:p
13
6 Preservation of accuracy order
6.1 De�nition of accuracy order
Whatever the message, it is generally assumed that the players learn something in the re-vising process. More precisely, the players �nal belief embody more information than theirinitial beliefs. To explicit formally this intuition, we need a criterion for comparing beliefsstructure. In Billot, Vergnaud and Walliser, (2009) we study an accuracy order on beliefs insyntax as well as in semantics. For our purpose, we only need the semantics and we restrainour attention to the case of �nite world spaces. Let two SESs H = (W; (Hi)i2I ; S;H0;w) andH0 = (W 0; (H 0
i)i2I ; S;H00;w
0) where W , W 0 are �nite, and a binary relation R �W 0�W whichconnects worlds of the two structures that share some features in common. A relation R satis�esthe following properties:
Actual-world Equivalence (AE) : (w0;w) 2 R.
Material Preservation (MP) : If (w0; w) 2 R, H 00(w
0) = H0(w).
Left Surjectivity (LS) : 8i 2 I, if (w0; w) 2 R, then 8 ew 2 Hi(w), 9 ew0 2 H 0i(w
0) such that( ew0; ew) 2 R.
Right Surjectivity for player i (RSi) : If (w0; w) 2 R, 8 ew 2 Hi(w); 9 ew0 2 H 0i(w
0) such that( ew0; ew) 2 R.
AE requires the two actual worlds to be related since we compare beliefs in these worlds. MPensures that the material environment is the same in two corresponding worlds. LS expressesthat all worlds which are accessible in the second structure are connected to some accessibleworlds of the �rst. But some accessible worlds of the �rst structure may have disappeared. RSiis the converse axiom which works for player i only.
De�nition 5 (J-more accurate PBM) : A SES H is J-more accurate than a SES H0 ifthere exists a binary relation R which satis�es AE, MP, LS and RSi for all i 2 InJ .
This de�nition (see Billot, Vergnaud and Walliser 2009 for detailed discussion) expresses thata subset J of player learned directly (they received some original content) while the InJ playersonly learned indirectly (they only learned that the others learned). Direct learning correspondformally to the shrinking of the accessibility domains. In the partitional case where two SESshare the same world spaces, �J-accuracy�means that the J players have �ner partitions. Thespecial case obtained when J = ; corresponds to bisimilarity of two SESs, i.e: the case wherethe two SESs represent the same propositions.
Bovary example: The �nal belief H� in (�g. 3) can be simpli�ed by bisimilarity. Indeed,remark that the worlds w10 and w20 are equivalent to w1 and w2 (they have the same accessibilitydomains). Therefore, we can consider a simpli�ed structure H�� in (�g. 4) thanks to a relation R
de�ned as follows: R =�(w�0; w
��0 ) ; (w
�1; w
��1 ) ;
�w�10 ; w
��1
�; (w�2; w
��2 ) ;
�w�20 ; w
��2
�; (w�4; w
��3 ) ; (w
�3; w
��4 ) ;
(w�5; w��5 ) ; (w
�6; w
��6 ) ; (w
�7; w
��7 ) ; (w
�8; w
��8 )
�Note that R satis�es AE, MP, LS, RSEmma and RSCharles.
14
6.2 Transmission of accuracy order
This accuracy order can be applied to the message structures. We can check that the publicmessage, and in fact any message, are �I-more accurate� than the null message. The privatemessage and the secret message for player i are �fig-more accurate�than the null message Notealso that the public message is �In fig-more accurate� than the private message. Comparinga private message and a secret message requires the accuracy order to be re�ned in order tointroduce a third category of players, i.e. those who are ignorant of what happens.
We study now how the accuracy order on message is translated through the revision process.Intuitively, we expect that the accuracy order is preserved on �nal beliefs. The following propo-sition shows that it is indeed the case.
Proposition 2 : Consider an initial belief SES H and two message SESs H and H0 such thatH is J-more accurate than H0: Let suppose that H� = H � H and H�0 = H � H0 are two �nalbelief SESs. Then H� is J-more accurate than H�0:
Proof is in Appendix C.
As a corollary, revising by any message leads to a �nal belief which is �I-more accurate�thanthe initial belief as expected. Especially, revising by a null message leads to a �nal belief whichis bisimilar to the initial belief.
Note that the symmetric result does not hold. For a given message, the accuracy order oninitial beliefs may not be preserved by revision.
Bovary example: We can observe that the messageHBovary in (�g. 2) is fEmma;Charlesg-more accurate than the null messageH0null . Consider the binary relationR = f(w0; w0) ; (w0; w1) ; (w0; w2) ; (w01; w3)g.It satis�esAE, MP and LS. We can check that the �nal beliefH�� in (�g. 4) is fEmma;Charlesg-more accurate than the initial belief HBovary in (�g. 1). Consider the binary relation R =f(w0; w��0 ) ; (w0; w��1 ) ; (w0; w��2 ) ; (w1; w��3 ) ; (w2; w��4 ) ; (w3; w��5 ) ; (w4; w��6 ) ; (w5; w��7 ) ; (w7; w��8 )g.It satis�es AE, MP and LS.
7 Acknowledgements
We thank A. Baltag, G. Bonanno, I. Gilboa, J. Lang, J.M. Tallon, J. van Benthem and S. Zamirfor helpful remarks and comments (especially during conferences and seminars). We also thankL. Ménager for her help in designing the �gures.
8 References
Alchourron, C.E., Gardenfors, P. and Makinson, D. (1985): �On the logic of theorychange: partial meet contraction and revision functions,� Journal of Symbolic Logic, 50(2),510-530.
Aucher, G. (2008): �Internal models and private multi-agent belief revision,� in Interna-tional Joint Conference on Autonomous Agents and Multiagent Systems (AAMAS 2008).
Aumann, R.J. (1999): �Interactive epistemology II: probability,� International Journal ofGame Theory 28, 301-314.
15
Baltag, A., Moss, L. and Solecki, S (1998): �The logic of public announcements, com-mon knowledge, and private suspicions,�Proceedings of the 7th conference on Theoretical aspectsof rationality and knowledge.
Billot, A., Vergnaud, J.C.. and Walliser, B (2009): �When do we know more?,�mimeo.
Board, O. (2004): �Dynamic interactive epistemology,�Games and Economic Behavior 49,49-80.
Bonanno, G. (2005): �A simple modal logic for belief revision,�Synthese, 147 (2), 193-228.Dekel, E.,Lipman, B. and Rustichini, A (1998): �Standard State-Space Models Preclude
Unawareness,�Econometrica, 66 (1), 159-173.Gossner, O. (2000): �Comparison of information structures,�Games and Economic Behav-
ior 30, 44-63.Meier, M.,Schipper, B.C.. and Heifetz, A (2006): �Interactive Unawareness,�Journal
of Economic Theory, 130 , 78-94.van Benthem, J. (2000): �Update delights,�mimeo.
9 Appendix A
Theorem 1 : Consider two MSYSs K =�K;K;K�
�and K0=
�K0;K0;K0�
�satisfying No Con-
�ict A4 and Belief and Message Inference A5. If K = K0 and K = K0 then K� = K0�.Proof. : De�ne the depth of a formula ' in L, denoted d('), as the number of modaloperators B�i (and not Bi) that are hierarchically used. It is de�ned recursively by d(') = 0if ' 2 L [ L, d(:') = d('), d(' ^ ) = max(d('); d( )) and d(B�i') = d(') + 1. Weare going to prove that K = K0. We proceed by induction on the depth of proposition inL. Let L� be the set of propositions of L whose depth is inferior or equal to �. Consider�rst propositions of depth 0. Note that L0 is the closure of L [ L with respect to thepropositionnal connectors. Remark that K \
�L [ L
�= K [ K = K0 [ K0 = K0 \
�L [ L
�.
Since K and K0 satisfy R1, K \ L0 = K0 \ L0. Consider now L1 and propositions of theform B�i (' _ ) where ' 2 L and 2 L. By A5, B�i (' _ ) is equivalent to a propositionof L0, it follows that B�i (' _ ) 2 K i¤B�i (' _ ) 2 K0. Therefore K \ L1 = K0 \ L1.Suppose that K \ L� = K0 \ L�, for � � 1. Consider a proposition ' of depth � + 1:it contains a set f 1; 2:::g of well-formed formulas of depth 1 that begins with a modaloperator B�i . By A5, each i is equivalent to a proposition
0i of L0. If in the proposition
', we replace all the i by the 0i, we obtain a proposition '
0 of depth �. By A5 and A2,' 2 K i¤ '0 2 K as well as ' 2 K0 i¤ '0 2 K0. Since '0 2 K i¤ '0 2 K0, ' 2 K i¤ ' 2 K0and thus K \ L�+1 = K0 \ L�+1.
10 Appendix B
Theorem 2 : Consider two SESs H and H and let the SYSs K and K be their syntacticalcounterpart.(1) If H� = H � H is a SES, then there exists a MSYS K =
�K;K;K�
�which satis�es
No Con�ict A4 and Belief and Message Inference A5 and such that K� is the syntacticalcounterpart of H�.
16
(2) If there exists a MSYS K =�K;K;K�
�which satis�es No Con�ict A4 and Belief and
Message Inference A5, then H� = H �H is a SES and is a semantical counterpart of K�.
Proof. : (1) Let de�ne the semantic structure H as follows:
H =�W;W;W �; (Hi)i2I ;
�Hi�i2I ; (H
�i )i2I ; S; S; S
�;H0; H0;H�0 ;w
��
with(i) Hi is an extended accessibility relation from W [W � toward 2W n;, with Hi (w;w) =
Hi (w),(ii) Hi is an extended accessibility relation from W [W � toward 2W n;, with Hi (w;w) =
Hi (w),(iii)H0 is an extended accessibility relation of nature fromW[W � toward S, withH0 (w;w) =
H0 (w).The truth value of a formula ' in some particular world (w;w) is written: (H; (w;w)) j= ',
which means that ' is true in (w;w) within the structure H. The following valuation rulesexpress the truth assignment in W � for any formula:8>>>>>>>><>>>>>>>>:
(H; (w;w)) j= p i¤ (H�; (w;w)) j= p,(H; (w;w)) j= :' i¤ (H; (w;w)) 2 ',(H; (w;w)) j= ' ^ i¤ (H; (w;w)) j= ' and (H; (w;w)) j= ,8>><>>:for ' 2 L, (H; (w;w)) j= B�i'
i¤ (H; (w0; w0)) j= ' for all (w0; w0) s.t. (w0; w0) 2 H�i (w;w) ,
for ' 2 L, (H; (w;w)) j= ' i¤ (H; w) j= ',for ' 2 L, (H; (w;w)) j= ' i¤
�H; w
�j= '.
We denote j'j� the set f(w;w) 2W � s.t. (H; (w;w)) j= 'g, j'j the set fw 2W s.t. (H; w) j= 'gand j'j the set
�w 2W s.t.
�H; w
�j= '
. LetK be the syntactical counterpart ofH: K = f' 2 L s.t. (H; (w;w)) j= 'g.
Step 1: We show that, if H� is a SES, then K is a MSYS. Remark that T =T [T [T �.Hence, T � K. It is obvious from the validation rules that K satis�es C1, C2 and R1.
Step 2: We show now that, if H� is a SES, K satis�es A4: ^m2L0�Bim! :Bi:m
�.
Consider any m 2 L0. There exists an index subset L0 � L such that m is equivalentto _l2L0ml. Consider any (w;w) 2 W � and any player i 2 I. We have that H�
i (w;w) 6= ;.Suppose �rst that (H; (w;w)) j= Bim, that is H i (w) �
Sl2L0M l. Then H�
i (w;w) 6= ; impliesthat Hi (w) \
Sl2L0Ml 6= ; and hence, (H; (w;w)) j= :Bi:m. Therefore, (H; (w;w)) j= Bim!
:Bi:m. Now, if (H; (w;w)) 2 Bim, then necessarily (H; (w;w)) j= Bim ! :Bi:m. Since itis true for all m 2 L0, the validation rules imply that (H; (w;w)) j= ^m2L0
�Bim! :Bi:m
�.
Since A4 holds in any world, it is common belief in the semantical counterpart K of H.Step 3: We show now that if H� is a SES, K satis�es A5: 8' 2 L;8 2 L,
^ml2M�Bi:ml _Bi:ml _Bi (ml �! ') _Bi (ml �! )
� ! B�i (' _ ) .
(�!) Consider any (w;w) 2W �. Suppose �rst that
(H; (w;w)) j= ^ml2M�Bi:ml _Bi:ml _Bi (ml �! ') _Bi (ml �! )
�.
The validation rules imply that for all ml 2M,
(H; (w;w)) j= Bi:ml _Bi:ml _Bi (ml �! ') _Bi (ml �! )
and therefore either:
17
� (H; (w;w)) j= Bi:ml, which implies that, for all w0 2 Hi (w), (H; w0) j= :ml: since byde�nition,
H�i (w;w) =
[l0=0;1;::;L
�(Hi (w) \Ml0)�
�H i (w) \M l0
��,
for all (w0; w0) 2 H�i (w;w), (H; (w0; w0)) j= :ml, and thus (H; (w;w)) j= B�i:ml, which
also implies that (H; (w;w)) j= B�i (ml �! (' _ )),
� or (H; (w;w)) j= Bi:ml, which implies that, for all w0 2 H i (w),�H; w0
�j= :ml, and thus
similarly to the preceeding case, (H; (w;w)) j= B�i (ml �! (' _ )),
� or (H; (w;w)) j= Bi (ml �! '), which implies that, for all w0 2 Hi (w), (H; w0) j=(ml �! '). Remark that the semantic rule implies that jml �! 'j� =
�jml �! 'j �W
�\
W � and therefore, for all (w0; w0) 2 H�i (w;w),H
�i (w;w) � jml �! 'j�. Thus, (H; (w;w)) j=
B�i (ml �! ') which in turn implies that (H; (w;w)) j= B�i (ml �! (' _ )),
� or (H; (w;w)) j= Bi (ml �! ), which implies that, for all w0 2 H i (w),�H; w0
�j=
(ml �! ). Thus, similarly to the preceeding case, (H; (w;w)) j= B�i (ml �! (' _ )).
Thus, for all ml 2 M, (H; (w;w)) j= B�i (ml �! (' _ )),and therefore (H; (w;w)) j=B�i (' _ ), which implies that
(H; (w;w)) j= ^ml2M�Bi:ml _Bi:ml _Bi (ml �! ') _Bi (ml �! )
��! B�i (' _ ) .
( �) Suppose now that (H; (w;w)) j= B�i (' _ ). Therefore, for all (w0; w0) 2 H�i (w;w),
(H; (w0; w0)) j= ' _ . Consider ml 2 M and let suppose that (H; (w;w)) 2�Bi:ml _Bi:ml
�.
Then, the two sets A = Hi (w) \Ml and B = Hi (w) \M l are both non-empty. The semanticrule implies that A � B = H�
i (w;w) \�Ml �M l
�. (H; (w;w)) j= B�i (' _ ) implies that
A�B � j' _ j�. The semantic rule implies that
j' _ j� = j'j� [ j j� =��j'j �W
�\W �� [ h�W � j j� \W �
i.
So, A � B ��j'j �W
�[�W � j j
�implies that A � j'j, and thus, for all w0 2 A,
(H; w0) j= ' or B � j j, and thus, for all w0 2 B,�H; w0
�j= . Hence, we have one of the two
conditions:
� for all (w0; w0) 2 H�i (w;w), either (H; w0) j= :ml or (H; w0) j= ', which implies that
(H; (w;w)) j= Bi (ml �! '),
� for all (w0; w0) 2 H�i (w;w), either
�H; w0
�j= :ml or
�H; w0
�j= , which implies that
(H; (w;w)) j= Bi (ml �! ).
Thus, we prove that if (H; (w;w)) 2�Bi:ml _Bi:ml
�then (H; (w;w)) j= Bi (ml �! ') _
Bi (ml �! ). Hence,
(H; (w;w)) j= B�i (' _ )!�Bi:ml _Bi:ml _Bi (ml �! ') _Bi (ml �! )
�.
On the whole, we have proved that
(H; (w;w)) j= ^ml2M�Bi:ml _Bi:ml _Bi (ml �! ') _Bi (ml �! )
� ! B�i (' _ ) .
18
Since A5 holds in any world, this axiom is common belief in the semantical counterpart Kof H.
Step 4: Note �nally that K� is the syntactical counterpart of H�. Actually, for all ' 2 L�,(H; (w;w)) j= ' i¤ (H�; (w;w)) j= '. Since K� = K\L�, then K� is the syntactical counterpartof H�.
(2) Suppose that K =�K;K;K�
�is a MSYS which satis�es A4 on one side, A5 on the other.
Suppose (ad absurdum) that H� is not a SES. Then, there exists at least a world (w;w) 2 W �
and a player i 2 I such that H�i (w;w) = ;. Then, it follows that, for any l 2 L = f0; :::; Lg, if�
H i (w) \M l
�6= ;, then (Hi (w) \Ml) = ;. Since
�M l
l=0;:::;L
and fMlgl=0;:::;L are respectivelypartitions of W and W , this means that there exist two nonempty index subsets L0, L
0 � L,such that L0 \L0 = ;, Hi (w) �
Sl2L0Ml and H i (w) �
Sl2L0M l. Let us note m = _
l2L0ml. Bytranscription, (H; w) j= Bi:m and
�H; w
�j= Bim. By connectedness of H�, there exists a �nite
sequence of players i1; :::; in and a �nite sequence of worlds (w1; w1) ; :::; (wn; wn) such that in = i,(w1; w1) = (w;w), (wn; wn) = (w;w) and, for all n � 1 � k � 1, (wk+1; wk+1) 2 H�
ik(wk; wk).
Note that, for all n� 1 � k � 1, wk+1 2 Hik (wk) and wk+1 2 H ik (wk).We check rigorously the proof for the two simplest cases, that is when n = 1 and n = 2, and
more informally for n > 2.When n = 1, we have (H;w) j= Bi:m and
�H;w
�j= Bim which means that Bi:m 2 K
and Bim 2 K, and thus Bi:m ^Bim 2 K which is a direct violation of A4.When n = 2, there exists ml such that w 2Ml, w 2M l, and thus (H; w) j= ml ^Bi:ml and�
H; w�j= ml ^Biml.
Therefore, (H;w) j= :Bi1:ml, (H;w) j= :Bi1: (ml ^Bi:ml),�H;w
�j= :Bi1:ml and�
H;w�j= :Bi1:
�ml ^Biml
�. Thus
:Bi1:ml ^ :Bi1:ml ^ :Bi1: (ml ^Bi:ml) ^ :Bi1:�ml ^Biml
�2 K
which is equivalent to
:�Bi1:ml _Bi1:ml _Bi1 (ml �! :Bi:ml) _Bi1
�ml �! :Biml
��2 K
i.e.
:�Bi1:ml _Bi1:ml _Bi1 (ml �! (:ml _ :Bi:ml))_
Bi1�ml �!
�:ml _ :Biml
�� �2 K.
Since, for all l0 6= l, Bi1 (ml0 �! (:ml _ :Bi:ml)) 2 KandBi1�ml0 �!
�:ml _ :Biml
��2 K,
then
_ml0:�Bi1:ml0 _Bi1:ml0 _Bi1 (ml0 �! (:ml _ :Bi:ml))_
Bi1�ml0 �!
�:ml _ :Biml
�� �2 K
and A5 implies that :B�i1�(:ml _ :Bi:ml) _
�:ml _ :Biml
��2 K which is equivalent to
:B�i1:�ml ^Bi:ml ^Biml
�2 K. Otherwise, by A4, B�i1:
�Bi:ml ^Biml
�2 K. Besides,
since :�Bi:ml ^Biml
�! :
�ml ^Bi:ml ^Biml
�is a tautology, then
B�i1�:�Bi:ml ^Biml
�! :
�ml ^Bi:ml ^Biml
��2 K.
Finally, by A2, we get: B�i1:�ml ^Bi:ml ^Biml
�2 K which is a contradiction.
19
For n > 2, there exists a sequence l1; :::; ln�1 2 L such that
(H;w) j= :Bi1:�ml1 ^ :Bi2:(:::: ^ :Bin�1:
�mln�1 ^Bi:m
��)
and �H;w
�j= :Bi1:
�ml1 ^ :Bi2:(:::: ^ :Bin�1:
�mln�1 ^Bim
��).
Similarly, according to A5, we can prove �rst that
:B�i1:�ml1 ^ :Bi2:(:::: ^ :Bin�1:
�mln�1 ^Bi:m
�)^
:Bi2:(:::: ^ :Bin�1:�mln�1 ^Bim
�)
�2 K
then that
:B�i1:�ml1 ^ :B�i2:(ml2 ^ :Bi3:(:::::Bin�1:
�mln�1 ^Bi:m
�)^
:Bi3:(:::: ^ :Bin�1:�mln�1 ^Bim
�))
�2 K
until that :B�i1:�ml1 ^ :B�i2:(:::: ^ :B
�in�1:
�mln�1 ^Bi:m ^Bim
�)�2 K which can be
rewritten as:
:B�i1�:ml1 _B�i2(:::: _B
�in�1:
�mln�1 ^Bi:m ^Bim
�)�2 K.
Otherwise, by A4, B�i1 :::B�in�1:
�Bi:m ^Bim
�2 K. By using the rule of classical logic and
A2, we can successively prove that
B�i1 :::B�in�1:
�mln�1 ^Bi:m ^Bim
�2 K,
B�i1 :::B�in�2
�:mln�2 _B�in�1:
�mln�1 ^Bi:m ^Bim
��2 K,
until B�i1
�:ml1 _B�i2(:::: _B
�in�2
�:mln�2 _B�in�1:
�mln�1 ^Bi:m ^Bim
���2 K which
yields a contradiction. Since H� is a SES, as proved just above, we can construct a semanticstructure H as in the proof of (1). The syntactical counterpart of H is a MSYS K0=
�K0;K0;K0�
�such that
� K0 = K since K0 and K are the syntactical counterpart of H,
� K0 = K since K0 and K are the syntactical counterpart of H,
� K0� is the syntactical counterpart of H�,
� K0 satis�es A4 and A5.
By Theorem 1, K0 = K, and thus K� is the syntactical counterpart of H�.
20
11 Appendix C
Proposition 1 : Consider an initial belief SES H and two message SESs H and H0 such thatH is J-more accurate than H0: Let suppose that H� = H � H and H�0 = H � H0 are two�nal belief SESs. Then H� is J-more accurate than H�0:
Proof. Let H be J-more accurate than H0 with R a binary relation which satis�es AE,MP, LS and RSi for all i 2 InJ . Let de�ne a binary relation R� � W �0 � W � as follows:((w0; w0) ; (w;w)) 2 R� i¤ w = w0 and (w0; w) 2 R. Let us prove that R� satis�es AE, MP, LSand RSi for all i 2 InJ .
AE: Since w� = (w;w), w�0 = (w;w0) and (w0;w) 2 R, (w�0;w�) 2 R� which proves AE:MP: Let ((w0; w0) ; (w;w)) 2 R�, then w = w0 which implies that H
�00 (w
0; w0) = H0(w0) =
H0(w) = H�0 (w;w), which proves MP.
LS: Let i 2 I, ((w0; w0) ; (w;w)) 2 R� and let consider� ew; ew� 2 H
�i ((w;w)). Then by
de�nition, ew 2 Hi (w) = Hi (w0) and ew 2 H i (w). Thus 9 ew0 2 H 0
i(w0) such that
� ew0; ew� 2 R.There exists l such that ew 2 M l. Therefore, by AE ew0 2 M 0
l. Then� ew; ew0� 2 H�0
i ((w0; w0))
and�� ew; ew0� ;� ew; ew�� 2 R�, which proves LS.RSi for all i 2 InJ : Let i 2 InJ , ((w0; w0) ; (w;w)) 2 R� and let consider
� ew0; ew0� 2H
�0i ((w
0; w0)). Then by de�nition, ew0 2 H 0i (w
0) = Hi (w) and ew0 2 H 0i (w
0). Thus 9 ew 2 H i(w0)
such that� ew0; ew� 2 R. There exists l such that ew0 2 M l. Therefore, by AE ew 2 M l. Then� ew; ew� 2 H�i ((w;w)) and
�� ew; ew0� ;� ew; ew�� 2 R�, which proves RSi.
21
