Epistemic Survey

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Epistemic Logic and Logical Omniscience:

A SurveyKwang Mong Sim*Department of Computer Science and Engineering, Ho Sin-Hang Building,The Chinese University of Hong Kong, Shatin, NT, Hong Kong

This survey brings together a collection of epistemic logics and discusses their approachesin alleviating the logical omniscience problem. Of particular note is the logic ofimplicit

and explicit belief. Explicit belief refers to information actively held by an agent, whileimplicit belief refers to the logical consequence of explicit belief. Ramifications of Lev-esques logic include nonstandard epistemic logic and the logics of awareness and localreasoning. Models of nonstandard epistemic logic are defined with respect to nonstandardproportional logic to weaken its semantics. In the logic of awareness, an agent can onlybelieve a concept that it is aware of. Closely related to awareness are S-1 and S-3 epistemicoperators which can be used to model skeptical and credulous agents. The logic of localreasoning provides a semantics for representing the fact that agents can have differentclusters of beliefs which may contradict each other. Other variations include epistemic

structures which are generalizations of the logic of local reasoning and fusion epistemicmodels which provide an account that agents can combine information conjunctively ordisjunctively. Another closely related approach is the logic of explicit propostions whichcaptures the insight that agents can hold beliefs independently without putting themtogether. 1997 John Wiley & Sons, Inc.

I. INTRODUCTION

Reasoning about knowledge and belief were first studied in philosophyunder the topics ofepistemic (knowledge) and doxastic (belief) logic.1 Epistemiclogic and doxastic logic are ramifications of modal logic.2 They are primarilylogics that involve notions such as knowing that and believing that1 andthe formal logical analysis of reasoning about these notions. The concepts ofknowledge and belief are related but different; an agent cannotknow a fact thatis false but the agent can believe it.3 Since the logics discussed in this articleaddress issues of representing information in the knowledge bases of intelligentagents which is typically not required to be true, the term belief seems to be

*E-mail: [email protected]; Tel: (31) 43 388-2023; Fax: (31) 43 325-2392.

INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, VOL. 12, 5781 (1997) 1997 John Wiley & Sons, Inc. CCC 0884-8173/97/010025-25


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more appropriate here. However, for the ease of exposition, this article prefersto use the two terms quite interchangeably throughout the discussion.

In addition, it is noteworthy to mention that aside from its long standingtradition in philosophy, reasoning about knowledge has also been investigatedand applied in other disciplines such as game theory, distributed systems, andartificial intelligence.3,4 In game theory, reasoning about knowledge is crucialin decision making because each person (or player) affects everybody else.5,6

Therefore, one cannot make decisions without some knowledge or beliefs aboutwhat others will act or do. Reasoning about knowledge may provide usefulinsights in understanding and designing protocols in distributed systems.79 Forinstance, in a distributed system, one can view communication as a transformationof the systems state of knowledge. The analysis and design of distributed proto-cols may be carried out by reasoning about the state of knowledge of groups ofprocessors when the system goes through an execution.

A. Reasoning About Knowledge in Artificial Intelligence

In artificial intelligence, research in reasoning about knowledge10,11 focuseson designing intelligent agents that can reason about the current state of theworld, beliefs of other agents beliefs and its own beliefs. The motivation forreasoning about knowledge in artificial intelligence,12 stems from areas such asplanning, nonmonotonic reasoning, and natural language processing. For exam-ple, in multiagent planning an agent may need to predict how other agentswill react in order to construct any complex plan. In multiagent nonmonotonicreasoning, an agent may need to reason about other agents default beliefs.13 Incomputational linguistics, the system needs to take into account the mentalstate of the person it is communicating with when interpreting and generatingutterances. All these require the agent to know about the beliefs of other agents.

The literature in reasoning about knowledge in artificial intelligence formsa rich collection and expositions of (some representative examples of) theselogics occupy the main bulk of this article. However, to set the stage for laterdiscussions it seems prudent to review the classicalapproach for epistemic reason-ing based on possible worlds.

II. CLASSICAL EPISTEMIC LOGIC

The idea of applying the possible-worlds semantics to model knowledge andbelief was originally due to Hintikka.1 The intuitive idea of the possible-worldsparadigm is to acknowledge in the semantics that things might have happeneddifferently from the way they did in fact happen. Thus, besides the true stateof affairs (actual world), there are other possible states of affairs. Under this

interpretation, an agent is said to know (or believe) a fact if it is true in all thestates that it considers possible. The language that is employed to formalize theseideas is typically some variations of propositional modal logic.14,15 Such a languagePL uses a set P of primitive letters (atomic sentences . . . , p, q, r, . . .) torepresent knowledge in a world. Formulas from PL are formed by letters in P


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EPISTEMIC LOGIC AND LOGICAL OMNISCIENCE 59

Table I. Characteristic axioms and rules of inference of knowledge andbelief.

Axioms

Name Axiom Schema Restriction on Ri

K Ki Ki( ) Ki noneT Ki reflexiveD Ki Ki serial4 Ki KiKi transitive5 Ki Ki Ki euclidean

Rules of Inference

Name Rules

MP (Modus Ponen) From and infer RN (Necessitation) From infer K

and logical connectives such as and , together with the modal operatorsK1 , . . . , Kn . Thus, it follows that if and are formulas of PL, then so are, , and Ki (for i 1, . . . , n where Ki reads agent i knows ).

To give semantics to sentences in PL, a Kripke16 structure is usually em-ployed. A Kripke Structure M for a system of n agents is a tuple S, ,R1 , . . . , Rn where S is a set of possible worlds or states and associates eachworld in S with a truth assignment. That is, for each state w S, (w)(p) is amapping from a primitive letter p P to true, false. Each Ri is a binary(accessibility) relation on S that captures the possibility relation according toagent i. Thus, wRiw1 holds if agent i considers world w1 possible when in worldw. In addition, a relation can be defined, where M, w means is trueat M, w or holds at M, w. Some of the clauses defined using are

as follows:

4,14,15

(1) M, w p iff(w)(p) true.(2) M, w iff M, w .(3) M, w if both M, w and M, w .(4) M, w Ki iff M, w1 for all w1 such that wRiw1 .

Clause (1) simply states that is used to define the semantics of primitivesentences and (2) and (3) are just recursive clauses for and . M, w Kimeans that agent i knows a formula at world w in a structure M, if is truein all worlds w1 that agent i considers possible in world w. Notions of validityand satisfiability are defined in the usual sense as in Ref. 2.

A. Axiom Systems for Knowledge and BeliefIt has been noted14,15,17 that by putting different constraints on the accessibility

relation Ri , different properties of knowledge can be captured. Some of thecommon characteristics ofRi are best summarized in Table I. The axiom schemasK, T, 4, and 5 (shown in Table I) are sometimes called the distribution axion, the


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knowledge axiom, the positive introspection axiom, and the negtive introspectionaxiom, respectively.14,15 While the distribution axiom states that knowledge isclosed under implication, the knowledge axiom says that an agent can only knowtrue facts (it is interesting to note that the knowledge axiom is sometime usedto differentiate knowledge and belief). The positive and negative introspectionaxiom state that an agent knows (believes) what it knows (believes) and whatit does not know (believe), respectively. An agent capable of both positive andnegative introspection is said to be fully introspective. The standard logic ofknowledge S5 usually contains the schema K, T, 4, and 5 together with allinstances of propositional tautologies while the standard logic of belief is obtainedby replacing the axiom Twith the axiom D. In addition, most logics of knowledgeand belief usually have the two rules of inference: modus ponen (MP) andknowledge generalization or rule of necessitation (RN) (see Table I).

Although the possible-worlds approach can be easily altered to representdifferent properties of knowledge and belief, it suffers from what Hintikka18

termed the logical omniscience problem.

III. THE LOGICAL OMNISCIENCE PROBLEM

Logical omniscience requires an agent to know all logical consequences ofits beliefs (that is, the set of beliefs held by the agent is closed under implication)and all valid sentences (including tautologies). The various problems associatedwith logical omniscience are described below.

Consequential closure. If an agent knows a set of formulas and if logicallyimplies a formula then the agent will also know (that is, if and areboth true in every world then must also be true in every world). The problemwith closure under implication is that it does not consider what an agent believes

directly but what the world would be like if what it believed were true.19Irrelevant beliefs. Under the possible-worlds interpretation, a valid sentence

is one that is true in every world that the agent considers possible. This will alsomean that the agent will need to know all tautologies in addition to its activebeliefs. For instance, aside from what it already knows, the agent should alsoknow tautologies such as it is raining in Boston at this instant or it is notrainingin Boston at this instant. The problem is that the possible-worlds approachdoes not distinguish between an agents active beliefs and tautologies that areirrelevant to the agents beliefs. One way to block irrelevant beliefs is to onlyconsider beliefs that are actively held by the agent.19 This may be achievedby not assigning truth values to sentences that are not relevant to the agentsbelief set.

Inconsistent beliefs. Additionally, an agent cannot believe both a sentence

and its negation without believing every sentence. This follows from thefact that none of the possible worlds is consistent with ( ) [since ( ) is false in every world]. One of the possible solutions is to allow worlds thatsupport both the truth and the falsity of a sentence and thus allowing inconsistentbeliefs to be satisfiable.


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Computational intractability. Computationally, these problems result in in-tractability because the agent is required to compute all logical consequences ofits beliefs. In practice, resource-bounded agents usually do not have enough timeor memory to derive an explicitrepresentation of each fact. Hintikka18 remarkedthat the possible-worlds semantics is unsuitable for modeling human reasoningbecause humans do not seem to be logically omniscient.

A. Approaches to Mitigate Logical Omniscience

It has been noted that logical omniscience is perhaps one of the most hotlycontested topics in reasoning about knowledge.12 In the last decade, there hasbeen a resurgence of work (including some best-article prizes at IJCAI andAAAI conferences) to mitigate the logical omniscience problem. They areroughly classified in Ref. 3 under the syntactic approach, the semantic (or Mon-tagueScott) approach, the impossible-worlds approach and the nonstandard

propositional logic approach.Syntactic approach. One method for distinguishing beliefs is by examiningtheir syntactic shapes. Levesque19 called this the syntactic approach. In this ap-proach, an epistemic model is a structure consisting of an explicit set of sentences(that is not necessarily closed under logical consequence); each formula that anagent believes is a member of this set. For instance, under this interpretation,an agent may believe p q without believing p q (r r). This approachwas pursued by Eberle,20 Moore and Hendrix.21 Another variant of this approachis considered by Konolige22 who represents an agents beliefs by a set of basefacts that is closed under a set of deduction rules. Under this approach, logicalomniscience is avoided by allowing the set of deduction rules to be incomplete.

MontagueScott approach. Another approach, due to Montague23,24 andScott25 represents an agents beliefs by a set of sets (or clusters) of possible

worlds. A MontagueScott model is sometimes referred to as a minimal (orneighborhood) model.2 A brief exposition of the MontagueScott model is givenin Section IV-B. In this approach every proposition is identified with a set ofpossible worlds; an agent knows a proposition if is true in at least one ofthe sets of worlds that it considers possible. Some of the problems of logicalomniscience are avoided because each cluster is not closed under its superset.Vardis26 epistemic structures pursues such an approach. Other variants includescluser models (the logic of local reasoning17,27) and Vardis fusion epistemic mod-els.28 A description of epistemic structures, the logic of local reaosning, and fusionepistemic models will be given in Sections IV-B.1, IV-B.2, and IV-B.3, respec-tively.

Impossible-worlds approach. Still another approach called the impossible-worlds approach,2933 augments possible worlds with impossible worlds. An impos-

sible world is a world in which valid formulas are not necessarily true or in whichinconsistent formulas may be true31 (where valid and inconsistent refer to logicalvalidity and inconsistency in the sense of classical logic). However, impossibleworlds are worlds that are admitted only as epistemic alternatives but are notlogically possible.18 Thus, in this approach an agent may not know all tautologies


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of classical logic because there may be impossible worlds that it considers possiblewhere these tautologies fail to be true. A variant of this approach is Levesques 19

logic of implicit and explicit belief (a detailed description will be given in Sec.IV-A.1).

Nonstandard logic approach. Finally, another approach which is based ona nonstandard logic is proposed by Fagin, Halpern, and Vardi.34 In this approach,possible worlds are models of a nonstandard propositional logic (NPL). A slightlymore detailed description of NPL will be given in Section IV-A.5.

IV. EPISTEMIC LOGICS IN ARTIFICIAL INTELLIGENCE

In the design of intelligent systems, a logic that is used to model finite agentsoperating in a real-time environment should take into account their limitedreasoning capabilities. This section describes some of the research in artificialintelligence that deals with the knowledge representation and reasoning of non-omniscient(resource-bounded) agents. It elaborates some of the approaches foralleviating the problem of logical omniscience (mentioned in Sec. III-A). Asmentioned earlier, the literature in reasoning about knowledge form a very largecollection and space limitation precludes introducing all of them here. Thus, thelogics that are presented in this article clearly reflects my own bias. However, itserves the emphasis to mention that for ease of exposition, this article adoptsthe position that classes of epistemic logics are identified by the structure oftheir semantic models (that is either the Kripke structure or the MontagueScott structure).

A. Epistemic Logic Based on Kripke Structure

Although Levesques19 logic of implicit and explicit belief is a variant of theimpossible-worlds approach, the semantic model is essentially a Kripke structure.

This is also the case for Fagin, Halpern, and Vardis34 nonstandard epistemiclogic (Sec. IV-A.5), where a Kripke structure is used to give semantics to anonstandard modal propositional logic rather than a model propositional logic.In addition, there were several outgrowths of Levesques logic that employedsome variant of the Kripke structure. Among them were the logic of awareness(Sec. IV-A.3), the CadoliSchaerf epistemic model (Sec. IV-A.4) and Lakemey-ers35 extension to Levesques model (Sec. IV-A.2).

1. Levesques Logic of Implicit and Explicit Belief

In his model, Levesque defined explicit belief as those sentences activelyheld by an agent and implicit belief as logical consequence of its explicit belief.His idea of avoiding some aspects of logical omniscience is to relax the condition

that worlds have to be complete and consistent. This is achieved by decouplingthe semantics of a formula with two independent notions of truth support andfalsity support. In this approach, a sentence in a given situation can be truesupported, false supported, both true and false supported, and neither true norfalse supported.


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Thus, a situation semantics36 rather than a possible-worlds semantics isemployed. The main deviation from a possible-worlds semantics is that in-complete situations (where a sentence can be neither true or false) and inco-herent situations (where a sentence can be both true and false) are allowed. Acomplete situation (or possible world) is one that is not incoherent and not incom-plete.

An epistemic model for implicit and explicit belief is a tuple ML S, ,T, F where S is a set of all situations and is any subset of S. T and F arefunctions from P (an enumerably infinite set of atomic sentences) to P(S), suchthat for any sentence p P, T(p) is those situations that support the truth of

p and F(p) is those that support the falsity of p. Thus, a situation is incoherentif it is in the set T(p) F(p). Consequently, there are two support relationsT and F between situations and formulas from a propositional language PL.PL is a language that is formed using P and the standard connectives , , ,, together with two epistemic operators L and B (representing implicit belief

and explicit belief, respectively); the only restriction is that sentences containingB or L cannot occur within the scope of the two epistemic operators. Thus,for all PL, ML , s T means that s supports the truth of and ML , s F means that s supports the falsity of . In Ref. 19, the following semanticsis defined:

(L-pT) ML , s T p iffs T(p)

(L-pF) ML , s F p iffs F(p)

(L-T) ML , s T iffML, s F

(L-F) ML , s F iffML , s T

(L-T) ML , s T iffML , s T and ML , s T

(L-F) ML , s F iffML , s F or ML , s F

(L-BT) ML , s T B iffML , tT for all t

(L-BF) ML , s F B iffML , s T B.

(L-LT) ML , s T L iffML , tT for all tW()

(L-LF) ML , s F L iffML , s T L.

In (L-LT), W() is the set of complete situations where each possible worldw W() is compatible with some situation s in . A possible world w iscompatible with another situation s if every sentence that is true (respectively,false) in s is also true (respectively, false) in w. More formally, for each s S

the function W is defined s follows:W(s) w S for every p P,

(i) w is a member of exactly one of T(p) and F(p)(ii) ifs T(p) then w T(p)

(iii) ifs F(p) then w F(p)


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Figure 1. Avoiding consequential closure.

Also, for any subset of S, W() s W(s). Thus, W() is simply a set ofpossible worlds. Since implicit belief is defined with respect to a set of possibleworlds, it is clear that the L operator is closed under implication. Also, alltautologies and valid sentences are implicitly believed. In Levesques logic, aformula is valid (written ) if it is true in every complete situation (possibleworld) s in every model ML . A sentence is true or is satisfied at a situation sif s T holds.

While implicit belief suffers from logical omniscience, explicit belief definedin (L-BT) and (L-BF) seems to have nonomniscient properties. In particular,the following sets of sentences given in (LO1), (LO2), (LO3), and (LO4) are satis-fiable.

(LO1) B B( ) B(LO2) B B() B(LO3) B( )(LO4) B B( ( ))

Avoiding consequential closure. In the classical possible-worlds semantics,the assumption that ( ) is logically valid means that is true in everyinterpretation (possible worlds) in which is true. Hintikkas18 original idea to

solve the logical omniscience prolem is to allow epistemic alternatives (or worlds)in which is false when both ( ) and are true. Levesques notion ofexplicit belief admits such epistemic alternatives in that (LO1) is satisfiable (Fig.1). In Figure 1, s T B and s T B( ) but s T B. Thus, explicit beliefis not closed under implication. It is noted that the situation s3 (shown in Fig.


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Figure 2. Admitting inconsistent beliefs.

1) is an incoherent situation. The fact that (LO1) is satisfiable is due to thepresence of some incoherent situations.

Admitting inconsistent beliefs. Satisfiability of (LO2) means that an agentcan explicitly believe inconsistent beliefs such as B B and without believingevery sentence (this is illustrated in Fig. 2). In Figure 2, s T B and s T B()but s T B. However, inconsistent beliefs are only possible if every visiblesituation is incoherent.

Blocking tautologies and closure under valid implications. (LO3) states thattautologies are not explicitly believed, while (LO4) says explicit belief is notclosed under valid implication, thus, even though ( ) is valid,B B( ( )) is not. (LO3) is illustrated in Figure 3. The satisfiabilityof (LO3) and (LO4) stems from the presence of incomplete situations. In Figure3 for instance, neither the truth nor the falsity of is supported in s3 . Thus, itfollows that S T B( ).

Proof theory for explicit belief. Beside having nonomniscient properties, aproof theory is required to reason about explicit belief. The proof theory forexplicit belief rests heavily on the tautological entailment in relevance logic.37 Infact, the idea of a situation bears strong similarities to the notion of set-ups inrelevance logic. Levesques contribution was to make the connection betweenexplicit belief and tautological entailment by proving that (B B) iff

entails . This means that the logic of explicit belief contains relevance logic asa subpart and thus the constraints of explicit belief can be characterized by aset of axioms that characterize tautological entailment.

Comments. In this connection, agents in Levesques logic are committed tobelieve all sentences that are tautologically entailed by its knowledge base. This


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Figure 3. Blocking tautologies.

means that they are perfect reasoners in relevance logic, as Vardi28 has pointedout. On this account, it seems that the logic of implicit and explicit belief ispsychologically implausible for modeling human reasoning since it is no moreplausible to expect people to reason perfectly in relevance logic than in classicallogic. In addition, Levesques logic is rather restrictive because it can only beapplied to a single agent environment. This issue has been taken up by Faginand Halpern17,27 in their logic of awareness and logic of local reasoning.

2. Logic of (Nested) Implicit and Explicit Belief

In his model, Lakemeyer35 had employed not one but two accessibilityrelations. A Lakemeyer model for implicit and explicit belief is a tuple MLL S,T, F, R, R where S, T, and F are defined as in Section IV-A.1 and R andR are binary relations on S. R is used when an agent is confirming a beliefwhile R is used to determine a disbelief. These two accessibility relationscapture the intuition that in a given situation, an agent uses different sets ofpossible situations to confirm or disconfirm a belief. However, in a possibleworld, R and R coincide.

DEFINITION 1. For all w W() and for all situations s S: wRs wRs.

The language PL that is employed is similar to those in Section IV-A.1 exceptthat nesting of modal operators is allowed. In this case sentences that do notcontain B and L are called objective sentences and sentences that occur in thescope of a B or L are called subjective sentences. However, there is a constraint


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stating that no L may occur within the scope of a B. Thus, for all PL thefollowing semantics was defined:35

(LL-Tp) MLL, s T p iffs T(p)

(LL-Fp) MLL, s F p iffs F(p)

(LL-T) MLL, s T iffMLL, s F

(LL-F) MLL, s F iffMLL, s T

(LL-T) MLL, s T iffMLL, s T and MLL, s T

(LL-F) MLL, s F iffMLL, s F or MLL, s F

(LL-TB) MLL, s T B iff for all t sRtMLL, tT

(LL-FB) MLL, s F B iff for some t sRt MLL, tT

(LL-TL) MLL, s T L iff for all tW() sRtMLL, tT

(LL-FL) MLL, s F L iffMLL, s T L.

The definitions of (LL-Tp), (LL-Fp), (LL-T), (LL-F), (LL-F), and (LL-F)are similar to those in Section IV-A.1. The only difference is that in (LL-TB),B is true supported at s if all situations accessible from s via R support thetruth of. For (LL-FB), the situation s supports the falsity of B just in case ifthere is a situation that is accessible from s via R which does not support thetruth of.

Nested beliefs. For explicit belief, the properties (LO1), (LO2), (LO3),and (LO4) (presented in Section IV-A.1) are still retained. Furthermore, theseproperties also hold for nested sentences within the limit of the language above.

For instance,

B(B

B

) is satisfiable.

3. Logic of Awareness

Other researchers beside Lakemeyer, such as Fagin and Halpern, 17,27 havealso extended Levesques logic. In their logic of awareness, Fagin and Halperndeal with both the issues of nested beliefs and multiagent reasoning. The languagethat is employed is a propositional language PL, built from a countably infiniteset P of atomic letters p1 , p2 , p3 , . . . and the usual logical connectives , ,together with two sets of modal operators L1 , . . . , Ln and B1 , . . . , Bn . Sincethe logic of awareness deals with multiagent reasoning, a set A of n agents isassumed and i is an index referring to agents in A. Thus, for i 1, . . . , n, Bireads agent i explicitly believes and Li means agent i implicitly believes

. Additionally, arbitrary nesting of the Li and BK in formulas are also permittedsince nested beliefs among different agents are allowed in the logic of awareness.

A Kripke structure for awareness17,27 is a tuple MA (S, A1 , . . . , An ,R1 , . . . , Rn , ) where S is a set of possible worlds, is a truth assignment(: P true, false), R1 , . . . , Rn are serial, transitive, and euclidean relations


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on S and Ai is a function associating each possible world with a set of primitiveformulas P , where P is the set of all primitive sentences. That is, Ai(s) isthe set of primitive formulas that agent i is aware of at state s. Two supportrelations T and F are defined for each set of formula . This places arestriction such that in every state only the formulas in are defined with eithertrue or false. T and F together with the Ai(s) function provide some effectof partial situations from the perspective of agent i. For instance, a sentence may be true at state s, but agent i may not be aware of at s (that is Ai(s)). Thus, for all PL the following semantics for the logic of awarenessis defined:17,27

(A-pT) MA , s T p iff(s, p) true and p

(A-pF) MA , s F p iff(s, p) false and p

(A-p) MA , s p iff(s, p) true

(A-T) MA, s T iffMA, s

F

(A-F) MA , s F iffMA , s T

(A-) MA , s iffMA, s

(A-T) MA, s T iffMA , s T and MA , s T

(A-F) MA , s F iffMA , s F or MA , s F

(A-) MA , s iffMA, s and MA, s

(A-BiT) MA, s T Bi iffMA, tAi(s)T for all tsuch that (s, t)Ri

(A-BiF) MA , s F Bi iffMA , t

A

i(s)

F for some tsuch that (s, t)Ri

(A-Bi) MA , s Bi iffMA, tPT Bi, (P is the set of all primitive letters.)

(A-LiT) MA , s T Li iffMA , tT for all tsuch that (s, t)Ri

(A-LiF) MA, s F Li iffMA, tF for some tsuch that (s, t)Ri

(A-Li) MA, s Li iffMA , t for all tsuch that (s, t)Ri

Clauses (A-pT) and (A-pF) give the property that for each set of primitive sen-tences :

(i) MA , s T MA , s (ii) MA , s F MA , s

Both (i) and (ii) are proven in Ref. 27. Clauses (A-) and (A-) are just recursive

definitions of (A-p). In clause (A-BiT), MA , s T Bi means that agent i at states explicitly believes relative to , if is true relative to Ai(s) at all thestates accessible from s. It also implies that Bi is true at s. This is because MA ,

s T Bi MA , s Bi follows from property (i) above. In the case of MA,s Bi, agent i explicitly believes relative to P, where as for MA, s T Bi


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Figure 4. Awareness.

agent i explicitly believes relative to some subset ofP. Implicit belief is definedin clause (A-Li) does not take the awareness function into account. Thus, Li istrue at s if is true at all states accessible from s, regardless of whether agent

i is aware of. In addition, a sentence is valid ifMA , s for all models MAand all states s; is satisfiable if is not valid.

Example. An example to illustrate implicit and explicit belief in the contextof awareness is shown in Figure 4. Let , be a subset of P. That is, forevery state, letters in are defined with either true or false. A further restrictionis provided by Ai(s), which can be thought of as a syntactic filter. In this example,Ai(s) . Thus, in Figure 4, agent i explicitly believes relative to at state

s because in every state accessible from s (that is s1 , s2 , and s3) is true relativeto Ai(s) . Although is true in s1 , s2 , and s3 , agent i does not explicitlybelieve because it is not aware of [that is, is not in Ai(s)]. Thus, inFigure 4, it follows that MA, s T Bi and MA , s F Bi and, consequently, MA ,

s T Bi Bi. However, agent i implicitly believes since it is true in allaccessible states and the awareness function is not taken into account. Similarly

agent i also believes implicitly.Avoiding logical omniscience. Some of properties of Levesques explicit

belief are present in the logic of awareness. For instance (LO3) and (LO4) arestill satisfiable. Further, the example in Figure 4 shows that formula such asBi(Bk Bk) (which is in the form of (LO3) but with arbitrary nesting of


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beliefs) is satisfiable. However, since Fagin and Halpern dispense the use ofincoherent situations, an agent cannot hold inconsistent beliefs and explicit beliefis closed under implication in this logic. Thus, both (LO1) and (LO2) are notsatisfiable in the logic of awareness.

4. CadoliSchaerf Episemic Model

In the previous section, the awareness function was used as a syntactic filter.The approach adopted by Cadoli and Schaerf38 is somewhat similar. A set ofcountably infinite atomic letters P p1 , p2 , p3 , . . . is assumed but with thefollowing interpretation:

DEFINITION 2:38 (1-, 2-, 3-interpretation).

(1) A 3-interpretation of P is a truth assignment I3 which maps every letter p P to

true, false, T(2) A 2-interpretation of P is a truth assignment I2 which maps every letter p P totrue, false

(3) A 1-interpretation ofP is a truth assignmentI1 which maps every letter p P to

A generalized notion of 1- and 3- interpretation can be defined by restrictingthe possibility of mapping some subsets of primitive letters S to and T, whereP S, giving the following definition:

DEFINITION 3:38 (S-3, S-1 interpretation).

(1) An S-3 interpretation of P is a truth assignment I3s which maps every letter of Sto true, false and P/S to true, false, T where P S. Thus, for every letter p P there are three possibilities:

(i) I3s(p) 1, I3s(p) 0(ii) I3s(p) 0, I

3s(p) 1

(iii) I3s(p) 1, I3s(p) 1 iff p P/S

(2) An S-1 interpretation of P is a truth assignment I1s which maps every letter of Sto true, false and P/S to where P S. Thus, for every letter p P there arethree possibilities:

(i) I1s(p) 1, I1s(p) 0 iff p S

(ii) I1s(p) 0, I1s(p) 1 iff p S

(iii) I1s(p) 0, I1s(p) 0 iff p P/S

Modal Extensions to Definition 3 are given as follows:S-3-Kripke interpretation. An S-3-Kripke model is a tuple M (Sit,

R, I3s), where Sit is a set of situations, R is an accessibility relations on Sit, andI3s is an S-3-interpretation for every situation in Sit.

S-1-Kripke interpretation. An S-1-Kripke model is a tuple M (Sit, R, I1s)where Sit is a set of situations, R is an accessibility relations on Sit, and I1s is anS-1-interpretation for every situation in Sit.

Cadoli and Schaerf proposed that the ideas of S-3 and S-1 Kripke interpreta-tion can also be used to model nonomniscient agents.


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Modeling Resource-bounded agents. A CadoliSchaerf model for epistemicreasoning is a tuple MCS Sit, V, R where Sit is a set of situations and R is areflexive, transitive, and euclidean relation on Sit. V is a valuation mappingsituations to truth assignments (V: Sit I3s, I1s, I2). A situation with an S-3interpretation (respectively, S-1 interpretation) is called an S-3 (respectively, S-1) situation while a possible world has a 2-interpretation. The set of S-1 situations,S-3 situations, and possible worlds are denoted S-1(Sit), S-3(Sit), and W(Sit),respectively. In addition, a propositional language PL (built from P togetherwith the usual connectives and , and two modal operators K1s and K3s) isassumed. Thus, for all PL the following semantics was defined:38

(CS-p) MCS, s p iffV(s)(p) 1, where p is atomic

(Cs-) MCS, s iffV(s)() 1

(CS-p) and (CS-) can be expanded as follows (sentences of the form

may also be inductively defined in the usual fashion):

(CS-I3s) MCS, s p iffs S-3(Sit) and I3s(s)(p) 1

(CS-I3s) MCS, s iffs S-3(Sit) and I3s(s)() 1

(CS-I1s) MCS, s p iffs S-1(Sit) and I1s(s)(p) 1

(CS-I1s) MCS, s iffs S-1(Sit) and I1s(s)() 1

(CS-I2) MCS, s p iffsW(Sit) and I2(s)(p) 1

(CS-I2) MCS, s iffsW(Sit) and I2(s)() 1

The modal operator K3s is defined over a set of S-3 situations, while the K1s modal

operator is taken over a set of S-1 situations as stated below.(CS-K3s) MCS, s K3s ifft S-3(Sit) sRit MCS, t

(CS-K3s) MCS, s K3s ifft S-3(Sit) sRit MCS, t



K3s was designed as a filter for selecting complete but not necessarily coherentsituations while K1s was meant to select situations that are coherent but notnecessarily complete. In this regard, the CadoliSchaerf model can be viewedas an extension of Levesque model. For K3s, it is easily checked that bothB B( ) B and B B() B are satisfiable. However,

B( ) and B B ( ( )) are both unsatisfiable because forevery sentence PL in an S-3 situation, at least one of and must betrue. The converse holds for the K1s operator. Additionally, it is noteworthy tomention that the K1s and the K

3s operator can be used to model skeptical agents

and credulous agents, respectively.


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72 SIM

5. Nonstandard Epistemic Logic

Levesques notion of an incoherent situation has been criticized as unintu-

itive.28 Fagin, Halpern, and Vardi34 argue that in the real world formulas canonly either be true or false. In fact, their nonstandard epistemic logic ( NE) wasproposed as an alternative to Levesques model to achieve the same results butwithout using incoherent and incomplete situations. Fagin et al.s approach isbased on a nonstandard propositional logic (NPL). The semantics of NPL isdefined such that each world w is associated with a sister world w*. While thetruth of a formula is determined by w, the semantics of negated formulas is given by w*. Thus, is true in w if is not true in w*. However, if w w*then the definition of negation is as the usual sense and for each world w,w** w. This approach which was due to Routley and Routley39 has a similareffect of decoupling the truth values of a formula and its negation (which wasone of the novel features of Levesques model). The difference is that in thisapproach possible worlds are used as models of nonstandard propositional logic

and this leads to the idea of a nonstandard Kripke structure.A nonstandard Kripke structure for a system of n agents is a tuple MNE

S, , R1 , . . . , Rn , * where S is a set of possible states or worlds, is a truthassignment to the primitive sentences for each world s S and R1 , . . . , Rn arebinary relations on S and * is defined as above. A propositional language PL asdefined in Section IV-A.3 is assumed. Thus, for all PL the following semanticsfor an NE model is defined:

(NE-p) MNE, s p iff(s)(p) true

(NE-) MNE, s , iffMNE, s *

(NE-) MNE, s iffMNE, s and MNE, s

(NE-K) MNE, s Ki iffMNE, t for all tsuch that (s, t)Ri

The definitions of (NE-p), (NE-), and (NE-K) are similar to those in thestandard Kripke structure. The main deviation is the definition of negation(NE-). For instance, ifKi holds at s then MNE, s * Ki which means thatthere is a world t that is accessible from s* (instead of s) such that MNE, t .

B. Epistemic Logic Based on Minimal (MontagueScott) Models

Throughout the literature in epistemic logic, there were several epistemicmodels that were based on minimal models. A minimal (MontagueScott) model2

is a structure M W, N, P where W is a set of possible worlds, N is a functon

assigning to each possible world s W a collection of sets of possible world (N:W P(P(W))). I is an intensional assignment which associates each atomicsentence p P with a set of possible worlds (I: P P(W))). In this case of anarbitrary sentence , its truth set is expressed as M w w . Thus, thefollowing semantics can be defined:


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(MS-p) M, w p iffw I(p)

(MS-) M, w , iffM, w

(MS-) M, w iff both M, w and M, w

(MS-K) M, w K iffMN(w)

Clause (MS-p) states that p is true at a world w if w is in the set I(p). (MS-)and (MS-) are recursive clauses for and . The definition of necessity isgiven in (MS-K) which says that is necessary if its truth set M is among thecollection of subsets of W at w.

Reasoning about knowledge. There are various interpretations for epistemicmodels that are based on minimal models. Vardis26 epistemic structures (Sec.IV-B.1) are closest to the definitions given above. Other variants include Faginand Halperns17,27 logic of local reasoning (Sec. IV-B.2) which employs the notionof belief cells and Vardis28 fusion epistemic models (Sec. IV-B.3) constructed

from world fusion.30

However, it may be interesting to note that Delgrandes40

logic of explicit propositions (Sec. IV-B.4) adopts two different interpretations;the first is close to the original MontagueScott semantics and the second isbased on the belief cell approach.

1. Epistemic Structures

Vardis26 Epistemic Structures (ES) are based on the MontagueScott modelwhere epistemic notions are modeled by epistemic sets. An epistemic set is a setof sentences that an agent believes or knows. In this approach, each sentence is associated with a proposition (or an intension) which is a set of worlds inwhich is satisfied.

An epistemic structure for a system of n agents is a tuple MES S,

N1 , . . . , Nn , I where S is a set of possible states or worlds, I is an intensionalassignment which associates each atomic sentence p P with a set of possibleworlds. Intensional assignments can also be extended to any arbitrary sentence PL (a propositional language), that is I() t MES, t . N1 , . . . , Nnare epistemic assignments, where Ni assigns to agent i an epistemic set (a set ofsets of possible worlds). The following semantics for an epistemic structureis defined:26

(ES-p) MES, s p where p P, ifs I(p)

(ES-) MES, s , ifMES, s

(ES-) MES, s iffMES, s and MES, s

(ES-E ) MES, s

Ei

iffI(

)

Ni(s)The definitions in (ES-p), (ES-), and (ES-) follow directly from those in (MS-

p), (MS-) and (MS-), respectively. In the case of (ES-E ) agent i believes (orknows) a sentence just in case if I() is in Ni(s) (that is, there is a set T Ni(s) such that t T MES, t ).


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74 SIM

Avoiding logical omniscience. (LO1), (LO2), and (LO3) are satisfiable inES but (LO4) is not. The fact that B B( ) B is satisfiable is becausethe two beliefs B and B( ) may be expressed by different sets of worldssay w1 and w2. Thus, the two propositions w1 and w2 may be necessary ata world w (where Ni(w) w1, w2), but their intersection w1 w2 may not be necessary at w. The same reason holds for the satisfiability of B B() B because B and B() may be characterized by different sets ofworlds. To see that B( ) is satisfiable, consider an ES structure with justone world w and Ni(w) . Thus, at w, there is no necessary proposition, sothat valid sentences such as B( ) do not hold. However, B B( ( )) is not satisfiable because of the fundamental inference rule from infer B B in the neighborhood semantics. Since ( )is valid, it follows that B B ( ( )).

Constraints on Ni(s). In addition, Vardi also mentions that conditions canbe placed on Ni(s) to express different modes of reasoning. For instance, by

imposing that Ni(s) is nonempty, one obtains the property that an agent doesnot believe sentences such as ( ). Since ( ) is false in every world,it corresponds to the empty set. Another possible restriction on Ni(s) is to requirethat sets in Ni(s) are closed under supersets [that is if U Ni(s) and V Uthen V Ni(s)]. This means that if an agent believes a sentence ( ), thenit also believes any sentence that is more general than ( ) (for exampleB( ) B). In fact, by imposing on Ni(s) the two conditions mentionedabove together with another condition S Ni(s), one obtains the logic of lo-cal reasoning.

2. Logic of Local Reasoning

The motivationfor local reasoning is that humans do not focus their attentionon all issues simultaneously. Thus, if an agent believes a sentence then it issaid that it believes in a certain frame of mind. In the logic for local reasoning,an agent is viewed as perceiving different frames of mind (or society ofmind41,42); each frame (or cluster or belief cell) is modeled with a different setof possible worlds.

The epistemic model for local reasoning can be thought of as a special caseof the minimal model described in chapter 7 of ref. 2. A model for local reasoningin a system of n agents is a tuple MLR S, C1 , . . . , Cn , where S is a set ofpossible states or worlds, is a truth assignment to the primitive sentences p P for each world s S, and Ci(s) is a nonempty set of nonempty subsets of S.Thus, Ci(s) F1 , . . . , Fk means that in a world s, agent i can believe differentsets of possible worlds (frames of mind) represented by F1 , . . . , Fk . A proposi-

tional language PL similar to that defined in Section IV-A.3 is assumed. Thus,for all PL the following semantics for local reasoning is defined:27

(LR-p) MLR, s p (where p is a primitive letter) iff(s, p) true

(LR-) MLR, s , iffMLR, s


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(LR-) MLR, s iffMLR, s and MLR, s

(LR-B) MLR, s

Bi iff

F

Ci(s) such that

t

F

MLR, t

.(LR-L) MLR, s Li iffMLR, t for all tFCi(s) F.

Two modal operators B and L are defined, respectively, for local(or explicit)belief and implicit belief. Bi is interpreted as agent i believes in some frameof mind. This kind of belief is called weak belief.17 Li is interpreted as agent iimplicitly believes as a result of putting together the information of its variousframes of mind. This is illustrated in Figure 5, in which frame 2 is a frame wherethe agent combines the information from frame 1 and frame 3. The shaded circlesrepresent the set of worlds that are common in both frame 1 and frame 3. Theexample in Figure 5 also shows that by intersecting sets of possible worlds,information from different frames can be used to eliminate possibilities. It isinteresting to note that implicit belief in the logic of local reasoning correspondsto the smallest proposition [consisting of the set of worlds that are members ofevery proposition in N(w)] in the original neighborhood semantics mentionedin Section IV-B. An even stronger notion called strong belief (Si) is defined inRef. 17; a strong belief is one that is true in every frame of mind:

(LR-S) MLR, s Si iff MLR, t for all F Ci(s) and all t F.

Avoiding Logical Omniscience. In this approach, local belief is not closedunder implication. Thus, B B( ) B is satisfiable because an agentmay believe in one frame of mind and in another, but it may not be ina frame of mind to bring these facts together. An agent can also hold inconsistentbeliefs such as B B. This follows from the fact that an agent may believe in one frame of mind and believes in another as illustrated in Figure 5.

However, B( ) and B B ( ( )) are both not satisfiable.In the logic of local reasoning, agents are perfect reasoners within each frameof mind. As such, an agents set of beliefs are closed under valid implication andall valid formulas are believed.

3. Fusion Epistemic Models

A Nonstandard (NS) fusion epistemic model28 for a system of n agents is atuple MNS S, N1 , . . . , Nn , where S is a set of nonstandard worlds.30 isan intensional assignment which associates every atomic sentence p P with aset of nonstandard worlds (that is : P 2S) and a propositional language PL(formed from the usual connectives together with Bi) is assumed. N1 , . . . , Nn

are epistemic assignments, where Ni assigns to agent i a nonstandard worldexpression (that is N: A S E, where A is a set of agents and E is a class ofnonstandard world expressions). These nonstandard world expressions can beconstructed by the two world-fusion operations: schematization

and superposi-

tion

30 and E is closed under the following conditions:


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76 SIM

Figure 5. Transforming local (explicit) belief to implicit belief.

(1) E S

(2) If sj S for all j in an index set I, then sj E(3) If sj S for all j in an index set I, then

sj E

Schematized and superposed belief. Schematization combines worlds con-junctively, while superposition fuses worlds together disjunctively. For instance,


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p obtains in the schematization of two worlds s1 and s2 (that is s1

s2) if p holdsin both s1 and s2 . It is noted that schematized worlds may be underdeterminedand thus neither p or p may hold in s

1

s2, while superposed worlds may be

overdetermined and both p and p may hold in s1 s2 . Based on these ideas,Vardi defined two notions of belief; one with respect to schematized worlds(called schematized belief) and the other is taken over superposed worlds (called

superposed belief). Formal definitions for schematized belief and superposedbelief are given in (NS-Sch) and (NS-Sup), respectively. Thus, for all PLthe following semantics for a fusion epistemic model is defined: 28

(NS-p) MNS, s p where p P, ifs(p)

(NS-) MNS, s ifMNS, s

(NS-) MNS, s iffMNS, s and MNS, s

(NS-Sch) If N(i, s)

sj then MNS, s Bi ifMNS, sj for all jJ

(NS-Sup) If N(i, s) sj then MNS, s Bi ifMNS, sj for some jJ

Since a schematizied belief is considered over a cluster of schematized worlds,it may be viewed asa semantic counterpart of the K1s epistemic operator. Similarly,superposed belief may be considered as a semantic counterpart of the K3s operator.In addition, it is noted that both fusion epistemic models and epistemic structurescan be viewed as outgrowths of the logic of local reasoning. While Vardisepistemic structures are generalizations of the logic of local reasoning, fusionepistemic models provide an account that agents can combine information con-

junctively or disjunctively.

4. Logic of Explicit Propositions

The treatment of explicit belief in Lakemeyer and Levesques model wasbeing criticized for being inadequate because of the validity of the collectiveconjunction axiom (B12, in Sec. IV-A.1). (B12) says that believing two sentences and , is equivalent to believing their conjunction . That is B B B( ) and B( ) B B are valid. Although B( ) B Bseems plausible B B B( ) seems unintuitive. For instance, the factthat an agent believes that 3 is a prime number and penguins cannot flydoes not necessarily mean that it should also believe 3 is a prime numberand penguins cannot fly since it may not connect the two sentences together.Furthermore, Delgrande40 remarked that (B12) is valid because there is no wayof imposing that B B does not imply B( ) by using the standard notionof an accessibility relation. This is because the truth of B and B are determined

by the same set of situations. He suggested that explicit belief should be individu-ally characterized by a set of (possibly different) situations by defining theirsemantics relative to a minimal model.

A minimal model for explicit propositions is a tuple MEP S, NT , NF , T,F where S is a set of all situations. T and F are defined as in Section IV-A.1.


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78 SIM

The deviation from the LevesqueLakemeyer model is the use of NT andNF in place of R and R. NT expresses those propositions that an agentbelieves while N

Fare used to specify those propositions which are not believed.

A propositional language PL as defined in Sections IV-A.1 and IV-A.2 is as-sumed. Thus, for all PL the following semantics for explicit propositionsis defined:40

(EP-Tp) MEP, s T p iffs T(p)

(EP-Fp) MEP, s F p iffs F(p)

(EP-T) MEP, s T iffMEP, s F

(EP-F) MEP, s F iffMEP, s T

(EP-T) MEP, s T iffMEP, s T and MEP, s T

(EP-F) MEP, s F iffMEP, s F or MEP, s F

As mentioned before the truth and falsity of explicit propositions have twointerpretations. The definitions based on intensions are given in (EP-TB) and(EP-FB), while those in (EP-TB) and (EP-FB) are similar to the belief cellapproach adopted in the logic of local reasoning.

(EP-TB) MEP, s T B iffMs NT(s), where Ms tMEP, tT

(EP-FB) MEP, s F B iffMs NF(s)

(EP-TB) MEP, s T B iff there is some CNT(s) such that MEP, tT

for all t C

(EP-FB) MEP, s F B iff there do not exist CNF(s), such that MEP, tT

for all tC

Avoiding logical omniscience: Although, B B( ) B, B B() B, and B( ) are satisfiable, the fundamental inference rulefrom infer B B in the neighborhood semantics still presents a problem.If two propositions and have the same set of supporting situations, then anagent who believes must also believe .

V. DISCUSSION

Levesques model is an imporant contribution to epistemic logic because itis one of the first attempts in knowledge representation to solve the logical

omniscience problem. Several models were built based on Levesques model toimprove some aspects of it. This has taken several directions. The issues whichare of interest include: multiagent reasoning, nested beliefs, avoiding the use ofincoherent (and incomplete) situations, and disallowing the collective conjunc-tive axiom.


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Multiagent reasoning and nested beliefs. Lakemeyer has extended Lev-esques model by allowing nesting of beliefs. An interesting feature of Lakemey-ers logic is that belief and disbelief are consider over different sets of situationsvia two separate accessibility relations. Additionally, Fagin and Halperns logicof awareness was proposed to address both the issues of introspection andmultiagent reasoning. However, the logic of awareness does not retain all theproperties of explicit belief because agents cannot hold inconsistent beliefs sinceincoherent situations are not allowed.

Avoiding the use of incoherent (and incomplete) situations. Several re-searchers such as Fagin, Halpern, and Vardi28,34 have argued strongly against theused of incoherent situations. They maintain that in the real world formulas canonly either be true or false. In fact, in most of their epistemic logics, such as thelogic of awareness, the logic of local reasoning and nonstandard epistemic logic,incoherent and incomplete situations were not allowed. The semantics of non-standard epistemic logic is still defined in terms of classical worlds, although each

world w is associated with a dual w*. This approach seems to provide the effectof decoupling the truth and falsity of a formula. The logic of awareness dispenseswith the use of both incoherent and incomplete situations but the awarenessfunction at each state seems to provide an effect that is similar to an incompletesituation. In the logic of local reasoning an agent can also hold inconsistentbeliefs such as B B. This is because an agent may believe a sentence inone frame of mind and believe its negations in another. However, a belief suchas B( ) is not satisfiable.

Disallowing the collective conjunctive axiom. Degrande40 remarked that thetreatment of explicit belief in Lakemeyer and Levesques model is inadequatebecause it does not capture the insight that agents can hold beliefs independentlywithout putting them together. In particular the validity of the collective conjunc-tion axiom is viewed as problematic for the reasons that were given in Section

IV-B.4. Additionally, it is noted that validity of the collective conjunction axiomwill also mean that B( ) B B( ) is valid. Thus, either anagents beliefs are closed under logical implication or every situation that itconsiders possible is incoherent. The issues of the collective conjunctive axiomhave been taken up by Delgrande40 who discards the notion of an accessibilityrelation by defining explicit belief relative to a minimal model. Since differentexplicit propositions may be characerized by different sets of situations, thecollective conjunctive axiom is not valid in Delgrandes model.

Despite the above criticisms, it seems noteworthy to mention that Levesquesnotion of explicit belief does account for the fact that some sentences lie outsidethe scope of an agents beliefs. Further, he seems to have anticipated criticismsabout incoherent situations, and argued in Ref. 19 that even though incoherentsituations cannot be realized, they can be imagined by an agent. In other

words, inconsistency can still arise because of an agents epistemic short-comings even though the world is inherently consistent from an ontologicalperspective. If situations (or worlds) are to be thought of as epistemic alternatives,then there is no reason to assume that an agent will not receive inconsistent infor-mation.


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80 SIM

VI. CONCLUSION

In this survey, several logics for reasoning about knowledge and belief were

reviewed. Although these different models appear to have surface dissimilarities,a closer examination shows that they have strong resemblance. Further, giventhe current proliferation of epistemic logics, it seems prudent to establish compar-isons among the various approaches and to consider the issue of unificationsamong them. One of the issues that is of theoretical interest is to formulate aunified framework that harmonizes the ideas in these logics. This issue is ad-dressed in Refs. 43 and 44.

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