Epistemic logic is a subfield of philosophical logic concerned withlogical approaches to knowledge, belief, and related notions. Thoughany logic with an epistemic interpretation may be called anepistemic logic, the most widespread type of epistemic logicsin use at present are modal logics. Knowledge and belief arerepresented via the modal operatorsK andB, often witha subscript indicating the agent that holds the attitude. Formulas\(K_{a}\varphi\) and \(B_{a}\varphi\) are then read “agenta knows that phi” and “agenta believes thatphi”, respectively. Epistemic logic allows the formalexploration of the implications of epistemic principles. For example,the formula \(K_{a}\varphi\rightarrow\varphi\) states that what isknown is true, while \(K_{a}\varphi\rightarrow K_{a}K_{a}\varphi\)states that what is known is known to be known. The semantics ofepistemic logic are typically given in terms of possible worldsvia Kripke models such that the formula \(K_{a}\varphi\) isread to assert that \(\varphi\) is true in all worlds agentaconsiders epistemically possible relative to its current information.The central problems that have concerned epistemic logicians include,for example, determining which epistemic principles are mostappropriate for characterizing knowledge and belief, the logicalrelations between different conceptions of knowledge and belief, andthe epistemic features of groups of agents. Beyond philosophy proper,epistemic logic flourishes in theoretical computer science, AI,economics, and related fields.

• 1. Introduction
• 2. The Modal Approach to Knowledge
  • 2.1 The Formal Language of Epistemic Logic
  • 2.2 Higher-Order Attitudes
  • 2.3 The Partition Principle and Modal Semantics
  • 2.4 Kripke Models and The Indistinguishability Interpretation of Knowledge
  • 2.5 Epistemological Principles in Epistemic Logic
  • 2.6 Principles of Knowledge and Belief
• 3. Knowledge in Groups
  • 3.1 Multi-Agent Languages and Models
  • 3.2 Notions of Group Knowledge
• 4. Beyond Knowing That
• 5. Logical Omniscience
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Introduction

Aristotelian texts set the groundwork for discussions of the logic ofknowledge and belief, particularlyDe Sophisticis Elenchisas well as thePrior andPosterior Analytics. WhileAristotle addressed the four alethic modes of possibility, necessity,impossibility, and contingency, Buridan, Pseudo Scotus, Ockham, andRalph Strode, helped to extend Aristotle’s insights to epistemicthemes and problems (Boh 1993; Knuuttila 1993). During this period,Pseudo Scotus and William of Ockham supplemented Aristotle’sstudy of mental acts of cognition and volition (see Boh 1993: 130).Ivan Boh’s studies of the history of fourteenth and fifteenthcentury investigations into epistemic logic provide an excellentcoverage of the topic, especially hisEpistemic Logic in the LaterMiddle Ages (1993).

According to Boh, the English philosopher Ralph Strode formulated afully general system of propositional epistemic rules in hisinfluential 1387 bookConsequences (Boh 1993: 135).Strode’s presentation built on the earlier logical treatises ofOckham and Burley. Problems of epistemic logic were also discussedbetween the 1330s and 1360s by the so-called Oxford Calculators, mostprominently by William Heytesbury and Richard Kilvington. By thefifteenth century, Paul of Venice and other Italian philosophers alsoengaged in sophisticated reflection on the relationship betweenknowledge, truth, and ontology.

Discussions of epistemic logic during the medieval period share asimilar set of foundational assumptions with contemporary discussions.Most importantly, medieval philosophers explored the connectionbetween knowledge and veracity: If I knowp, thenp istrue. Furthermore, many medieval discussions begin with an assumptionsimilar to G.E. Moore’s observation that an epistemic agentcannot coherently assert “p but I do not believe (know)p”. Sentences of this form are generally referred to asMoore sentences.

Modern treatments of the logic of knowledge and belief grew out of thework of philosophers and logicians writing from 1948 through the1950s. Rudolf Carnap, Jerzy Łoś, Arthur Prior, NicholasRescher, G.H. von Wright, and others recognized that our discourseconcerning knowledge and belief admits of an axiomatic-deductivetreatment. Among the many important papers that appeared in the 1950s,von Wright’s seminal work (1951) is widely acknowledged ashaving initiated the formal study of epistemic logic as we know ittoday. Von Wright’s insights were extended by Jaakko Hintikka inhis bookKnowledge and Belief: An Introduction to the Logic of theTwo Notions (1962). Hintikka provided a way of interpretingepistemic concepts in terms of possible world semantics and as such ithas served as the foundational text for the study of epistemic logicever since.

In the 1980s and 1990s, epistemic logicians focused on the logicalproperties of systems containing groups of knowers and later still onthe epistemic features of so-called “multi-modal”contexts. Since the 1990s work indynamic epistemic logic has extended traditional epistemic logic by modeling the dynamicprocess of knowledge acquisition and belief revision. In the past twodecades, epistemic logic has come to comprise a broad set of formalapproaches to the interdisciplinary study of knowledge and belief.

Interest in epistemic logic extends well beyond philosophers. Recentdecades have seen a great deal of interdisciplinary attention toepistemic logic with economists and computer scientists activelydeveloping the field together with logicians and philosophers. In 1995two important books signaled the fertile interplay between computerscience and epistemic logic: Fagin, Halpern, Moses, and Vardi (1995)and Meyer and van der Hoek (1995). Work by computer scientists hasbecome increasingly central to epistemic logic in the interveningyears.

Among philosophers, there is increased attention to the interplaybetween these formal approaches and traditional epistemologicalproblems (see for example, van Benthem 2006; Hendricks & Symons2006; Stalnaker 2006; Artemov 2008; Holliday 2018; Baltag,Bezhanishvili, et al. 2019).

Several introductory texts on epistemic logic exist, e.g., Fagin etal. (1995), van Benthem (2011); van Ditmarsch, van der Hoek, and Kooi(2007); van Ditmarsch, Halpern, et al. (2015); Gochet and Gribomont(2006); and Meyer (2001) with Lenzen (1980) providing an overview ofearly developments.

2. The Modal Approach to Knowledge

Until relatively recently, epistemic logic focused almost exclusivelyon propositional knowledge. In cases of propositional knowledge, anagent or a group of agents bears the propositional attitude of knowingtowards some proposition. For example, when one says: “Zoe knowsthat there is a hen in the yard”, one asserts that Zoe is theagent who bears the propositional attitudeknowing towardsthe proposition expressed by the English sentence “there is ahen in the yard”. Now imagine that Zoe does not know whetherthere is a hen in the yard. For example, it might be the case that shehas no access to information about whether there is or is not a hen inthe yard. In this case her lack of information means that she willconsider two scenarios as being possible, one in which there is a henin the yard and one in which there is not.

Perhaps she has some practical decision that involves not only hensbut also the presence of frightening dogs in the yard. She might wishto feed the hens but will only do so if there is no dog in the yard.If she were ignorant of whether there is a dog in the yard, the numberof scenarios she must consider in her deliberations grows to four.Clearly, one needs to consider epistemic alternatives when one doesnot have complete information concerning the situations that arerelevant to one’s decisions. As we shall see below, possibleworlds semantics has provided a useful framework for understanding themanner in which agents can reason about epistemic alternatives.

While epistemic logicians had traditionally focused onknowingthat, one finds a range of other uses of knowledge in naturallanguage. As Y. Wang (2018b) points out, the expressionsknowinghow,knowing what,knowing why are very common,appearing almost just as frequently (sometimes more frequently) inspoken and written language asknowing that. Recentlynon-standard epistemic logics of such know-wh expressions have beendeveloped, to which we will come back inSection 4. In the remainder of this section, we will introduce the basics of theepistemic logic of knowing that.

2.1 The Formal Language of Epistemic Logic

Recent work in epistemic logic relies on a modal conception ofknowledge. In order to be clear about the role of modality inepistemic logic it is helpful to introduce the basic elements of themodern formalism. For the sake of simplicity we begin with the case ofknowledge and belief for a single agent, postponing consideration ofmultiple agents toSection 3,

A prototypical epistemic logic language is given by first fixing a setofpropositional variables \(p_{1}\), \(p_{2}\),…. Inapplications of epistemic logic, propositional variables are givenspecific interpretations: For example, \(p_{1}\) could be taken torepresent the proposition “there is a hen in the yard” and\(p_{2}\) the proposition “there is a dog in the yard”,etc. The propositional variables represent propositions which arerepresented in no finer detail in the formal language. As such, theyare therefore often referred to asatomic propositions orsimplyatoms. LetAtom denote the set of atomicpropositions.

Apart from the atomic propositions, epistemic logic supplements thelanguage of propositional logic with a modal operator, \(K_{a}\), forknowledge and \(B_{a}\), for belief.

and similarly

In many recent publications on epistemic logic, the full set offormulas in the language is given using a so-calledBackus-NaurForm (BNF). This is simply a notational technique derived fromcomputer science that provides a recursive definition of the formulasdeemed grammatically “correct”, i.e., the set ofwell-formed formulas:

It should be noted that the Greek letter \(\varphi\) stands for thesyntactic category of formula. So this definition says: anatomp is a formula; \(\neg\varphi\) is a formula if\(\varphi\) is a formula (read \(\neg\) as ‘it is not the casethat’); \((\varphi\wedge\varphi)\) is a formula whenever any twoformulas are connected by the \(\wedge\) symbol (read \(\wedge\) as‘and’); and \(K_{a}\varphi\) and \(B_{a}\varphi\) areformulas whenever \(\varphi\) is a formula (the readings wereindicated above). Note that in a non-BNF recursive specification ofthe language, the Greek variable \(\varphi\) would be used ametavariable ranging over formulas, and one would normallystate the clause for conjunctions as: \((\varphi \wedge \psi)\) is aformula whenever \(\varphi\) and \(\psi\) are formulas. But the BNFlet’s us get away with just using \(\varphi\) to get the sameeffect.

We will call this basic language that includes both aKnowledgeand aBelief operator, \(\mathcal{L}_{KB}\). As inpropositional logic, additional connectives are defined from \(\neg\)and \(\wedge\): Typical notation is ‘\(\vee\)’ for‘or’, ‘\(\rightarrow\)’ for ‘if…,then …’ and ‘\(\leftrightarrow\)’ for‘… if, and only if, …’. Also typically\(\top\) (‘top’) and \(\bot\) (‘bottom’) isused to denote the constantly true proposition and the constantlyfalse proposition, respectively.

As we shall see below, \(K_{a}\varphi\) is read as stating that\(\varphi\) holds inall of the worlds accessible toa. In this sense,K can be regarded as behavingsimilarly to the ‘box’ operator, \(\square\), often usedto denote necessity. In evaluating \(K_{a}\varphi\) at a possibleworldw, one is in effect evaluating auniversalquantification over all the worlds accessible fromw. Theuniversal quantifier \(\forall\) in first-order logic has theexistential quantifier \(\exists\) as itsdual: This meansthat the quantifiers are mutually definable by taking either\(\forall\) as primitive and defining \(\exists x\varphi\) as shortfor \(\neg\forall x\neg\varphi\) or by taking \(\exists\) as primitiveand defining \(\forall x\varphi\) as \(\neg\exists x\neg\varphi\). Inthe case of \(K_{a}\), it may be seen that the formula \(\negK_{a}\neg\varphi\) makes anexistential quantification: Itsays that thereexists an accessible world that satisfies\(\varphi\). In the literature, a dual operator for \(K_{a}\) is oftenintroduced. The typical notation for \(\neg K_{a}\neg\) includes\(\langle K_{a}\rangle\) and \(\widehat{K}_{a}\). This notation mimicsthe diamond-shape \(\lozenge\), which is the standard dual operator tothe box \(\square\), which in turn is standard notation for theuniversally quantifying modal operator (see the entry onmodal logic).

More expressive languages in epistemic logic involve the addition ofoperators for various notions of group knowledge (seeSection 3). For example, as we discuss below, thecommon knowledgeoperator and so-calleddynamic operators are importantadditions to the language of epistemic logic. Dynamic operators canindicate for example thetruthful public announcement of\(\varphi\): \([\varphi!]\). A formula \([\varphi!]\psi\) is read“if \(\varphi\) is truthfully announced to everybody, then afterthe announcement, \(\psi\) is the case”. The question of whatkinds of expressive power is added with the addition of operators is aresearch topic that is actively being investigated indynamic epistemic logic. So, for example, adding \([\varphi!]\) by itself to\(\mathcal{L}_{KB}\) doesnot add expressive power, but in alanguage that also includes common knowledge, it does (e.g., vanDitmarsch, van der Hoek, & Kooi 2007).

2.2 Higher-Order Attitudes

Notice that, for example, \(K_{a}K_{a}p\) is a formula in the languagewe introduced above. It states that agenta knows that agenta knows thatp is the case. Formula withnestedepistemic operators of this kind express ahigher-orderattitude: an attitude concerning the attitude of some agent.

Higher-order attitudes is a recurring theme in epistemic logic. Theaforementioned Moore sentences, e.g., \(B_{a}(p\wedge B_{a}\neg p)\)express a higher-order attitude. So do many of the epistemicprinciples discussed in the literature and below. Consider thefollowing prominent epistemic principle involving higher-orderknowledge: \(K_{a}\varphi\rightarrow K_{a}K_{a}\varphi\). Is itreasonable to require that knowledge satisfies this scheme, i.e., thatif somebody knows \(\varphi\), then they know that they know\(\varphi\)? In part, we might hesitate before accepting thisprinciple in virtue of the higher-order attitude involved. This is amatter of ongoing discussion in epistemic logic and epistemology.

2.3 The Partition Principle and Modal Semantics

The semantics of the formal language introduced above is generallypresented in terms of so-called possible worlds. In epistemic logicpossible worlds are interpreted as epistemic alternatives. Hintikkawas the first to explicitly articulate such an approach (1962). Thisis another central feature of his approach to epistemology whichcontinues to inform developments today. It may be stated, simplified,^[1] as follows:

The partition principle may be used to provide a semantics for theknowledge operator. Informally,

Here, agenta knows that \(\varphi\) just in case the agent hasinformation that rules out every possibility of error and rules outevery case where \(\neg\varphi\).

2.4 Kripke Models and The Indistinguishability Interpretation of Knowledge

Since the 1960sKripke models, defined below, have served asthe basis of the most widely used semantics for all varieties of modallogic. The use of Kripke models in the representation of epistemicconcepts involves taking a philosophical stance with respect to thoseconcepts. One widespread interpretation, especially in theoreticaleconomics and theoretical computer science, understands knowledge interms of informational indistinguishability between possible worlds.What we will refer to here as theindistinguishabilityinterpretation goes back to Lehmann (1984), and Aumann (1976) hasan equivalent partition-based framework.

As the indistinguishability interpretation concerns knowledge, but notbelief, we will be working with a language without belief operators.Therefore, let the language \(\mathcal{L}_{K}\) be given by theBackus-Naur form

As we shall see, the indistinguishability interpretation involves verystringent requirements in order for something to qualify as knowledge.We introduce it here for pedagogical purposes, putting the formaldetails of the interpretation in place so as to introduce and explainrelatively less extreme positions thereafter.

Consider again the case of Zoe, the hen, and the dog. The exampleinvolves two propositions, which we will identify with the formalatoms:

It is worth emphasizing that for the purposes of our formalization ofthis scenario, these two are theonly propositions ofinterest. We are restricting our attention to \(\Atom=\{p,q\}\). Inearly presentations of epistemic logic and in much of standardepistemic logic at present,all the atoms of interest areincluded from the outset. Obviously, this is an idealized scenario. Itis important to notice what this approach leaves out. Considerationsthat are not captured in this way include the appearance of novelatoms; the idea that other atomic propositions might be introduced atsome future state via some process of learning for example, or thequestion of an agent’s awareness of propositions; the scenarioin which an agent might be temporarilyunaware of some atomdue to some psychological or other factor (seeSection 5 for references to so-calledawareness logic). For now, themain point is that standard epistemic logic begins with the assumptionthat the setAtom exhausts the space of propositions for theagent.

With two atoms, there are four different ways a world couldconsistently be. We can depict each by a box:

The four boxes may be formally represented by a set\(W=\{w_{1},w_{2},w_{3},w_{4}\}\), typically called a set ofpossible worlds. Each world is furtherlabeled with the atoms true at that world. They are labeled by afunctionV, thevaluation. Thevaluation specifies which atoms are true at each world in thefollowing way: Given an atomp, \(V(p)\) is the subset ofworlds at whichp is true.^[2] That \(w_{1}\) is labeled withp andq thus means that\(w_{1}\in V(p)\) and \(w_{1}\in V(q)\). In the illustration,\(V(p)=\{w_{1},w_{2}\}\) and \(V(q)=\{w_{1},w_{3}\}\).

For presentational purposes, assume that there really is a hen in theyard, but no dog. Then \(w_{2}\) would represent theactual world of the model. In illustrations,the actual world is commonly highlighted:

Now, assume that the hen is always clucking, but that the dog neverbarks, and that although Zoe has acute hearing, she cannot see theyard. Then there are certain possible worlds that Zoe cannotdistinguish: possible ways things may be which she cannottell apart. For example, being in the world with only a hen \((p,\negq)\), Zoe cannot tell if she is in the world with both hen and dog\((p,q)\): her situation is such that Zoe is aware of two ways thingscould be but her information does not allow her to eliminateeither.

To illustrate that one possible world cannot be distinguished fromanother, an arrow is typically drawn from the former to thelatter:

Here, arrows represent abinary relation on possible worlds.In modal logic in general, it is referred to as theaccessibility relation. Under theindistinguishability interpretation of epistemic logic, it issometimes called theindistinguishabilityrelation. Formally, denote the relation \(R_{a}\), withthe subscript showing the relation belongs to agenta. Therelation is a subset of the set ofordered pairs of possibleworlds, \(\{(w,w')\colon w,w'\in W\}\). One worldw“points” to another \(w'\) if \((w,w')\in R_{a}\). In thiscase, \(w'\) is said to beaccessible(indistinguishable) fromw. In the literature, this isoften written \(wR_{a}w'\) or \(R_{a}ww'\). The notation‘\(w'\in R_{a}(w)\)’ is also common: the set \(R_{a}(w)\)is then the worlds accessible fromw, i.e.,

A final note: the set \(\{(w,w')\colon w,w'\in W\}\) is often written\(W\times W\), theCartesian product ofW withitself.

For \(R_{a}\) to faithfully represent a relation ofindistinguishability, what worlds should it relate? If Zoe was plungedin \(w_{1}\) for example, could she tell that she is not in \(w_{2}\)?No: the relation of indistinguishability issymmetric—if one cannot tella fromb,neither can one tellb froma. That a relation issymmetric is typically drawn by omitting arrow-heads altogether or byputting them in both directions:

Which of the remaining worlds are indistinguishable? Given that thehen is always clucking, Zoe has information that allows her todistinguish \(w_{1}\) and \(w_{2}\) from \(w_{3}\) and \(w_{4}\) andvice versa, cf. symmetry. Hence, no arrows between these. Theworlds \(w_{3}\) and \(w_{4}\) are indistinguishable. This brings usto the following representation:

Since no information will ever allow Zoe to distinguish something fromitself, any possible world is thus related to itself and theindistinguishability relation isreflexive:

The standard interpretation of the Zoe example in terms of a possibleworlds model is now complete. Before turning to a general presentationof the indistinguishability interpretation, let us look at what Zoeknows.

Recall the informal modal semantics of the knowledge operator fromabove:

To approach a formal definition, take ‘\(w\vDash\varphi\)’to mean that \(\varphi\) is true in worldw. Thus we can,define truth of \(K_{a}\varphi\) inw by

This definition states thata knows \(\varphi\) in worldw if, and only if, \(\varphi\) is the case in all the worlds\(w'\) whicha cannot distinguish fromw.

So, where does that leave Zoe? First off, the definition allows us toevaluate her knowledge in each of the worlds, but seeing as \(w_{2}\)is the actual world, it is the world of interest. Leta denoteZoe, here are some examples of what we can say about Zoe’sknowledge in \(w_{2}\):

1. \(w_{2}\vDash K_{a}p\). Zoe knows that the hen is in the yard asall the worlds indistinguishable from \(w_{2}\) that would be\(w_{1}\) and \(w_{2}\) makep true.
2. \(w_{2}\vDash\neg K_{a}q\). Zoe does not know the dog is in theyard, as one of the indistinguishable worlds in fact \(w_{2}\) itselfmakesq false.
3. \(w_{2}\vDash K_{a}K_{a}p\). Zoe knows that she knowspbecause
   1. \(w_{2}\vDash K_{a}p\) (cf. 1.) and
   2. \(w_{1}\vDash K_{a}p\).
4. \(w_{2}\vDash K_{a}\neg K_{a}q\). Zoe knows that she does not knowq because
   1. \(w_{2}\vDash\neg K_{a}q\) (cf. 2.) and
   2. \(w_{1}\vDash\neg K_{a}q\).

We could say a lot more about Zoe’s knowledge: every formula ofthe epistemic language without belief operators may be evaluated inthe model. It thus represents all Zoe’s higher-order informationabout her own knowledge of which points 3. and 4. are the firstexamples.

One last ingredient is required before we can state theindistinguishability interpretation in its full generality. In theexample above, it was shown that the indistinguishability relation wasbothsymmetric andreflexive. Formally, theseproperties may be defined as follows:

The missing ingredient is then the relational property oftransitivity. ‘Shorter than’ is an example of atransitive relation: Letx be shorter thany, and lety be shorter thanz. Thenx must be shorter thanz. So, given \(w_{1},w_{2}\) and \(w_{3}\), if the relationR holds between \(w_{1}\) and \(w_{2}\) and between \(w_{2}\)and \(w_{3}\), then the arrow between \(w_{1}\) and \(w_{3}\) is theconsequence of requiring the relation to be transitive:

Formally, transitivity is defined as follows:

A relation that is both reflexive, symmetric and transitive is calledanequivalence relation.

With all the components in place, let us now define the Kripkemodel:

In the definition, ‘\(\mathcal{P}(W)\)’ denotes thepowerset ofW: It consists of all the subsets ofW. Hence \(V(p)\), the valuation of atomp in the modelM, is some subset of the possible worlds: Those wherepis true. In this general definition,R can be any relation onW.

To specify which world is actual, one last parameter is added to themodel. When the actual world is specified a Kripke model is commonlycalledpointed:

Finally, we may formally define the semantics that was somewhatloosely expressed above. This is done by defining a relation betweenpointed Kripke models and the formulas of the formal language. Therelation is denoted ‘\(\vDash\)’ and is often called thesatisfaction relation.

The definition then goes as follows:

In full generality, the indistinguishability interpretation holds thatfor \(K_{a}\) to capture knowledge, the relation \(R_{a}\) must be anequivalence relation. A pointed Kripke model for which this issatisfied is often referred to as anepistemicstate. In epistemic states, the relation is denoted by atilde with a subscript: \(\sim_{a}\).

Given pointed Kripke models and the indistinguishabilityinterpretation, we have a semantic specification of one concept ofknowledge. With this approach, we can build models of situationsinvolving knowledge as we did with the toy example of Zoe and thehens. We can use these models to determine what the agent does or doesnot know. We also have the formal foundations in place to begin askingquestions concerning how the agent’s knowledge or uncertaintydevelops when it receivesnew information, a topic studied indynamic epistemic logic.

We may also ask more general questions concerning the concept ofknowledge modeled using pointed Kripke models withindistinguishability relations: Instead of looking at a particularmodel at the time and asking which formulas the model makes true, wecan ask which general principles all such models agree on.

2.5 Epistemological Principles in Epistemic Logic

Settling on the correct formal representation of knowledge involvesreflecting carefully on the epistemological principles to which one iscommitted. An uncontroversial example of such a principle which mostphilosophers will accept is veridicality:

In a formal context this principle can be understood to say that if\(\varphi\) is known then it should always be satisfied in one’smodels. If it turns out that some of one’s chosen models falsifythe veridicality principle, then most philosophers would simply deemthose models unacceptable.

Returning to pointed Kripke models, we can now ask which principlesthese models commit one to. In order to begin answering this question,we need to understand the most general features of our formalism. Thestrategy in modal logic in general (see Blackburn, de Rijke, &Venema 2001) is to abstract away from any given model’scontingent features. Contingent features would include, forexample, the specific number of worlds under consideration, thespecific valuation of the atoms, and the choice of an actual world. Inthis case, the only features that are not contingent are thoserequired by the general definition of a pointed Kripke model.

To abstract suitably, take a pointed Kripke model \((M,w)=(W,R,V,w)\).To determine whether the relation of this model is an equivalencerelation we only need to consider the worlds and the relation. Thepair of these elements constitute the fundamental level of the modeland is called theframe of the model:

Consider again the epistemic state for Zoe from above:

Several other models may be built on the same frame. The following aretwo examples:

With the notion of a frame, we may define the notion of validity ofinterest. It is the second term defined in the following:

The set of formulas valid on a class of frames \(\mathsf{F}\) iscalled thelogic of \(\mathsf{F}\). Denotethis logic that is, the set\(\{\varphi\in\mathcal{L}_{K}\colon\mathsf{F}\vDash\varphi\}\) by\(\Lambda_{\mathsf{F}}\). This is asemantic approach todefining logics, each just a set of formulas. One may also definelogicsproof-theoretically by defining a logic as the set offormulas provable in some system. With logics as just sets offormulas,soundness andcompleteness results maythen be expressed using set inclusion. To exemplify, let\(\mathsf{A}\) be a set of axioms and write\(\mathsf{A}\vdash\varphi\) when \(\varphi\) is provable from\(\mathsf{A}\) using some given set of deduction rules. Let theresulting logic the set of theorems be denoted\(\Lambda_{\mathsf{A}}\). It is the set of formulas from\(\mathcal{L}_{K}\) provable from \(\mathsf{A}\), i.e., the set\(\{\varphi\in\mathcal{L}_{K}\colon\mathsf{A}\vdash\varphi\}\). Thelogic \(\Lambda_{\mathsf{A}}\) is sound with respect to \(\mathsf{F}\)iff \(\Lambda_{\mathsf{A}}\subseteq\Lambda_{\mathsf{F}}\) and completewith respect to \(\mathsf{F}\) iff\(\Lambda_{\mathsf{F}}\subseteq\Lambda_{\mathsf{A}}\).^[3]

Returning to the indistinguishability interpretation of knowledge, wemay then seek to find the epistemological principles which theinterpretation is committed to. There is a trivial answer of littledirect interest: Let \(\mathsf{EQ}\) be the class of frames withequivalence relations. Then the logic of the indistinguishabilityinterpretation is the set of formulas of \(\mathcal{L}_{K}\) which arevalid over \(\mathsf{EQ}\), i.e., the set\(\Lambda_{\mathsf{EQ}}:=\{\varphi\in\mathcal{L}_{K}\colon\mathsf{EQ}\vDash\varphi\}\).Not very informative.

Taking anaxiomatic approach to specifying the logic,however, yields a presentation in terms of easy to grasp principles.To start with the simplest, then the principle T states that knowledgeisfactual: If the agent knows \(\varphi\), then \(\varphi\)must be true. The more cumbersome K states that if the agent knows animplication, then if the agent knows the antecedent, it also knows theconsequent. I.e., if we include the derivation rulemodusponens (from \(\varphi\rightarrow\psi\) and \(\varphi\), conclude\(\psi\)) as rule of our logic of knowledge, K states that knowledgeisclosed under implication. The principle B states that if\(\varphi\) is true, then the agent knows that it considers\(\varphi\) possible. Finally, 4 states that if the agent knows\(\varphi\), then it knows that it knows \(\varphi\). T, B and 4 inthe table below (the names are historical and not all meaningful).

In lieu of epistemological intuitions, we could discuss a concept ofknowledge by discussing these and other principles. Should we accept Tas a principle that knowledge follows? What about the others? Beforewe proceed, let us first make clear how the four above principlesrelate to the indistinguishability interpretation. To do so, we needthe notion of anormal modal logic. In the below definition,as in the above principles, we are technically usingformulaschemas. For example, in \(K_{a}\varphi\rightarrow\varphi\), the\(\varphi\) is a variable ranging over formulas in\(\mathcal{L}_{K}\). Thus, strictly speaking,\(K_{a}\varphi\rightarrow\varphi\) is not a formula, but ascheme for obtaining a formula. Amodal instance of\(K_{a}\varphi\rightarrow\varphi\) is then the formula obtained byletting \(\varphi\) be some concrete formula from \(\mathcal{L}_{K}\).For example, \(K_{a}p\rightarrow p\) and \(K_{a}(p\wedgeK_{a}q)\rightarrow(p\wedge K_{a}q)\) are both modal instances ofT.

There is a uniquesmallest normal modal logic (given the setAtom)that which contains exactly what is required by thedefinition andnothing more. It is often called theminimal normal modal logic and is denoted bythe boldfaceK (not to be confused with thenon-boldface K denoting the schema).

The logicK is just a set of formulas from\(\mathcal{L}_{K}\). I.e.,K \(\subseteq\mathcal{L}_{K}\). Points 1.4.gives a perspective on this set: They provide anaxiomatization. Often, as below, the schema K is referred toas an axiom, though really the instantiations of K are axioms.

ToK, we can add additional principles as axioms(axiom schemes) to obtain stronger logics (logics that have additionaltheorems: Logics \(\Lambda\) for whichK \(\subseteq\Lambda\)). Of immediate interestis the logic calledS5:

Here, then, is the relationship between the above four principles andthe indistinguishability interpretation:

What does this theorem tell us with respect to the principles ofknowledge, then? In one direction it tells us that if one accepts theindistinguishability interpretation, then one has implicitly acceptedthe principles K, T, B and 4 as reasonable for knowledge. In the otherdirection, it tells us that if one finds thatS5 isthe appropriate logic of knowledgeand one finds that pointedKripke models are the right way to semantically represent knowledge,then one must use an equivalence relation. Whether one shouldinterpret this relation in terms of indistinguishability, though, is amatter on which logic is silent.

In discussing principles for knowledge, it may be that some of thefour above seem acceptable, while others do not: One may disagree withthe acceptability of B and 4, say, while accepting K and T. Inunderstanding the relationship betweenS5 andequivalence relations, a more fine-grained perspective is beneficial:Theorem 1 may be chopped into smaller pieces reflecting thecontribution of the individual principles K, T, 4 and B to theequivalence requirementi.e., that the relation should be at the sametime reflexive, symmetric and transitive.

There are a number of insights to gain from Theorem 2. First, if onewants to useany type of Kripke model to capture knowledge,then one must accept K. Skipping some details, one must in fact acceptthe full logicK as this is the logic of the class ofall Kripke models (see, e.g., Blackburn, de Rijke, &Venema 2001).

Second, the theorem shows that there is an intimate relationshipbetween the individual epistemic principles and the properties on therelation. This, in turn, means that one, in general, may approach the“logic” in epistemic logic from two sides from intuitionsabout the accessibility relation or from intuitions about epistemicprinciples.

Several normal modal logical systems weaker thanS5have been suggested in the literature. Here, we specify the logics bythe set of their modal axioms. For example, the logicK is given by \(\{\text{K}\}\), whileS5 is given by\(\{\text{K},\text{T},\text{B},\text{4}\}\). To establishnomenclature, the following table contains a selection of principlesfrom the literature with the frame properties they characterize, cf.Aucher (2014) and Blackburn, de Rijke, & Venema (2001), on theline below them. The frame conditions are not all straightforward.

In Table 1, the subscript on \(R_{a}\) is omitted to ease readability,and so is the domain of quantificationW over which the worldsvariables \(x,y,z\) range.

Adding epistemic principles as axioms to the basic minimal normalmodal logicK yields new, normal modal logics. Aselection is:

Different axiomatic specifications may produce the same logic. Notice,e.g., that the table’s axiomatic specification\(\{\text{K},\text{T},\text{5}\}\) ofS5 does notmatch that given in the definition preceding Theorem 1,\(\{\text{K},\text{T},\text{B},\text{4}\}\). Note also, there is morethan one axiomatization ofS5: the axioms\(\{\text{K},\text{T},\text{5}\}\),\(\{\text{K},\text{T},\text{B},\text{4}\}\),\(\{\text{K},\text{D},\text{B},\text{4}\}\) and\(\{\text{K},\text{D},\text{B},\text{5}\}\) all give theS5 logic (e.g., Chellas 1980). An often seen variantis \(\{\text{K},\text{T},\text{4},\text{5}\}\). However, it isredundant to add it as all its instances can be proven from K, T and5. But as both 4 and 5 capture important epistemic principles (seeSection 2.6), 4 is often sometimes included for the sake of philosophicaltransparency. For more equivalences between modal logics, see, e.g.,the entry onmodal logic or Chellas (1980) or Blackburn, de Rijke, and Venema (2001).

Logics may be stronger or weaker than each other, and knowing theframe properties of their axioms may help us to understand theirrelationship. For example, as 4 is derivable from\(\{\text{K},\text{T},\text{5}\}\), all the theorems ofS4 are derivable inS5.S5 is thusat least as strong asS4. In fact,S5 is alsostrictlystronger: It can prove things whichS4cannot.

ThatS5 may be axiomatized both by\(\{\text{K},\text{T},\text{B},\text{4}\}\) and\(\{\text{K},\text{T},\text{5}\}\) may be seen through the frameproperties of the axioms: every reflexive and euclidean relation (Tand 5) is an equivalence relation (T, B, and 4). This also shows theredundancy of 4: If one has assumed a relation reflexive andeuclidean, then it adds nothing new to additionally assume it to betransitive. In general, having an understanding of the interplaybetween relational properties is of great aid in seeing relationshipsbetween modal logics. For example, noticing that every reflexiverelation is also serial means that all formulas valid on the class ofserial models are also valid on the class of reflexive models. Hence,every theorem ofD is thus a theorem ofT. HenceT is at least as strong asD (i.e., \(\textbf{D}\subseteq\textbf{T}\)). ThatT is also strictly stronger (not\(\textbf{T}\subseteq\textbf{D}\)) can be shown by finding a serial,non-reflexive model which does not satisfy some theorem ofT (for example \(K_{a}p\rightarrow p\)).

2.6 Principles of Knowledge and Belief

With the formal background of epistemic logic in place, it isstraightforward to slightly vary the framework in order to accommodatethe concept of belief. Return to the language \(\mathcal{L}_{KB}\) ofboth knowledge and belief:

To interpret knowledge and belief formulas together in pointed Kripkemodels, all that is needed is an additional relation between possibleworlds:

\(R_{K}\) is the relation for the knowledge operator and \(R_{B}\) therelation for the belief operator. The definition makes no furtherassumptions about their properties. In the figure below we provide anillustration, where the arrows are labeled in accordance with therelation they correspond to. The reflexive loop at \(w_{3}\) is alabel indicating that it belongs to both relations, i.e.,\((w_{3},w_{3})\in R_{K}\) and \((w_{3},w_{3})\in R_{B}\).

The satisfaction relation is defined as above, but with the obviouschanges for knowledge and belief:

The indistinguishability interpretation puts very strong requirementson the accessibility relation for knowledge. These have now beenstripped away and so has any commitment to the principles T, B, D, 4and 5. Taking Kripke models as basic semantics, we are still committedto K, though this principle is not unproblematic as we shall see belowin our discussion of the problem of logical omniscience.

Of the principles from Table 1, T, D, B, 4 and 5 have been discussedmost extensively in the literature on epistemic logic, both asprinciples for knowledge and as principles for belief. The principle Tfor knowledge

is broadly accepted. Knowledge is commonly taken to beveridical—only true propositions can be known. For,e.g., Hintikka (1962) and Fagin et al. (1995), the failure of T forbelief is the defining difference between the two notions.

Though belief is not commonly taken to be veridical, beliefs aretypically taken to beconsistent. I.e., agents are taken tonever believe the contradiction that is, any formulaequivalent with \((p\wedge\neg p)\) or \(\bot\), for short. Thatbeliefs should be consistent is then captured by the principle

The principle \(\neg B_{a}\bot\) is, on Kripke models, equivalent withthe principle D, \(B_{a}\varphi\rightarrow\widehat{B}_{a}\varphi\).Hence the validity of \(\neg B_{a}\bot\) requires serial frames.Witness, e.g., its failure in \(w_{1}\) above: As there are no worldsaccessible through \(R_{B}\),all accessible worlds satisfy\(\bot\). Hence \(w_{1}\) satisfies \(B_{a}\bot\), violatingconsistency. Notice also that \(\neg B_{a}\bot\) may be re-written to\(\widehat{B}_{a}\top\), which is true at a world just in case someworld is accessible through \(R_{B}\). Its validity thus ensuresseriality.

Notice that the veridicality of knowledge ensures its consistency: Anyreflexive frame is automatically serial. Hence accepting\(K_{a}\varphi\rightarrow\varphi\) implies accepting \(\negK_{a}\bot\).

Of the principles D, 4 and 5, the two latter have received so far themost attention, both for knowledge and for belief. They are commonlyinterpreted as governing ofprincipled access to own mentalstates. The 4 principles

are often referred to asprinciples of positiveintrospection, or for knowledge the‘KK’principle. Both principles are deemed acceptable by, e.g.,Hintikka (1962) on groundsdifferent from introspection. Heargues based on an autoepistemic analysis of knowledge, using anon-Kripkean possible worlds semantics calledmodel systems.Hintikka holds that when an agent commits to knowing \(\varphi\), theagent commits to holding the same attitude no matter what newinformation the agent will encounter in the future. This entails thatin all the agent’s epistemic alternatives—for Hintikka,all the model sets (partial descriptions of possible worlds) where theagent knows at least as much they now do&\madash the agent stillknows \(\varphi\). As \(K_{a}\varphi\) thus holds in all theagent’s epistemic alternatives, Hintikka concludes that\(K_{a}K_{a}\varphi\). Likewise, Hintikka endorses 4 for belief, butLenzen raises objections (Lenzen 1978: ch. 4).

Williamson argues against the general acceptability of the principle(Williamson 2000: ch. 5) for a concept of knowledge based on slightlyinexact observations, a so-calledmargin of error principle(see, e.g., Aucher 2014 for a short summary).

The 5 principles

are often referred to asprinciples of negativeintrospection. Negative introspection is quite controversial asit poses very high demands on knowledge and belief. The schema 5 maybe seen as aclosed world assumption (Hendricks 2005): Theagent has a complete overview of all the possible worlds and owninformation. If \(\neg\psi\) is considered possible(\(\widehat{K}_{a}\neg\psi\), i.e., \(\neg K_{a}\psi\)), then theagent knows it is considered possible (\(K_{a}\neg K_{a}\psi\)). Sucha closed world assumption is natural when constructing hyper-rationalagents in, e.g., computer science or game theory, where the agents areassumed to reason as hard as logically possible about their owninformation when making decisions.

Arguing against 5 is Hintikka (1962), using his conception ofepistemic alternatives. Having accepted T and 4 for knowledge, 5stands or falls with the assumption of a symmetric accessibilityrelation. But, Hintikka argues, the accessibility relation is notsymmetric: If the agent possesses some amount of information at modelset \(s_{1}\), then the model set \(s_{2}\) where the agent haslearned something more will be an epistemic alternative to \(s_{1}\).But \(s_{1}\) will not be an epistemic alternative to \(s_{2}\),because in \(s_{1}\), the agent does by hypothesis not know as much asit does in \(s_{2}\). Hence the relation is not symmetric, so 5 is nota principle of knowledge, on Hintikka’s account.

Given Hintikka’s non-standard semantics, it is a bit difficultto pin down whether he would accept a normal modal logic as the logicof knowledge and belief, but if so, thenS4 andKD4 would be the closest candidates (see Hendricks& Rendsvig 2018 for this point). By contrast, for knowledge vonKutschera argued forS4.4 (1976), Lenzen suggestedS4.2 (1978), van der Hoek argued forS4.3 (1993), and Fagin, Halpern, Moses, and Vardi(1995) and many others useS5 for knowledge andKD45 for belief.

Beyond principles governing knowledge and principles governing belief,one may also consider principles governing the interplay betweenknowledge and belief. Three principles of interest are

The principles KB1 and KB2 were introduced by Hintikka (1962), whoendorses only the former (see §3.7), noting that Plato is alsocommitted to KB1 inTheatetus. The first principle, KB1,captures the intuition that knowledge is a stronger notion thanbelief. The second—like 4 and 5—captures the idea that onehas privileged access to one’s own beliefs. The third, stemmingfrom Lenzen (1978), captures the notion that beliefs are held withsome kind of conviction: if something is believed, it is believed tobe known.

Though the interaction principles KB1—KB3 may look innocent ontheir own, they may lead to counterintuitive conclusions when combinedwith specific logics of knowledge and belief. First, Voorbraak (1993)shows that combining 5 for knowledge and D for belief with KB1,implies that

is a theorem of the resulting logic. Assuming that knowledge istruthful, this theorem entails that agents cannot believe to knowsomething which happens to be false.

If additionally KB3 is added, the notions of knowledge and beliefcollapse. I.e., it may be proven that\(B_{a}\varphi\rightarrow K_{a}\varphi\), which, in combination withKB1 entails that

Hence, the two notions have collapsed to one. This was stated in 1986,by Kraus and Lehmann.

If one is not interested in knowledge and belief collapsing, one mustthus give something up: One cannot have both 5 for knowledge, D forbelief and KB1 and KB3 governing their interaction. Again, resultsconcerning correspondence between principles and relational propertiesmay assist: In 1993, van der Hoek showed, based on a semantic analysisthat where the four principles are jointly sufficient for collapse,no subset of them is. Giving up any one principle will thuseliminate the collapse. Weakening KB1 to hold only for non-modalformulas is also sufficient to avoid collapse (cf. Halpern 1996).

For more on epistemic interaction principles, the principles .2, .3,.3.2. and .4, and relations to so-calledconditional beliefs,see Aucher (2014). For an introduction to conditional beliefs andrelations to several other types of knowledge from the philosophicalliterature, see Baltag and Smets (2008). The latter also includesdiscussion concerning the interdefinability of various notions, asdoes Halpern, Samet, and Segev (2009) for knowledge and(non-conditional) belief.

Note that although we mainly focus on the Kripke semantics in thisentry, there are other epistemic models for potentially weaker logic,such as neighborhood models (e.g., van Benthem et al. 2014) andtopological models (e.g., Baltag, Bezhanishvili, et al. 2019).

3. Knowledge in Groups

We human beings are preoccupied with the epistemic states of otheragents. In ordinary life, we reason with varying degrees of successabout what others know. We are especially concerned with what othersknow about us, and often specifically about what they know about whatwe know.

Epistemic logic can reveal interesting epistemic features of systemsinvolving groups of agents. In some cases, for example, emergentsocial phenomena depend on agents reasoning in particular ways aboutthe knowledge and beliefs of other agents. As we have seen,traditional systems of epistemic logic applied only to single-agentcases. However, they can be extended to groups or multi-agent systemsin a relatively straightforward manner.

As David Lewis noted in his bookConvention (1969) manyprominent features of social life depend on agents assuming that therules of some practice are matters ofcommon knowledge. Forexample, drivers know that a red traffic light indicates that theyshould stop at an intersection. However, for the convention of trafficlights to be in place at all, it is first necessary that drivers mustalso know that other drivers know thatred meansstop. In addition, drivers must also know that everyone knowsthat everyone knows that …. The conventional role of trafficlights relies on all drivers knowing that all drivers know the rule,that the rule is a piece ofcommon knowledge.

A variety of norms, social and linguistic practices, agentinteractions and games presuppose common knowledge, first formalizedby Aumann (1976) and with earliest epistemic logical treatments byLehmann (1984) and by Halpern and Moses (1984). In order to see howepistemic logic sheds light on these phenomena, it is necessary tointroduce a little more formalism. Following the standard treatment(see, e.g., Fagin et al. 1995), we can syntactically augment thelanguage of propositional logic withn knowledge operators, onefor each agent involved in the group of agents under consideration.The primary difference between the semantics given for a mono-agentand a multi-agent semantics is roughly thatn accessibilityrelations are introduced. A modal system forn agents isobtained by joining togethern modal logics where forsimplicity it may be assumed that the agents are homogenous in thesense that they may all be described by the same logical system. Theepistemic logical system forn agents consists ofncopies of a certain modal system. In such an extended epistemic logicit is possible to express that some agent in the group knows a certainfact that an agent knows that another agent knows a fact etc. It ispossible to develop the logic even further: Not only may an agent knowthat another agent knows a fact, but they may all know this factsimultaneously.

3.1 Multi-Agent Languages and Models

To represent knowledge for a set \(\mathcal{A}\) ofn agents,first let’s stipulate a language. Let \(\mathcal{L}_{Kn}\) begiven by theBackus-Naur form

To represent knowledge for alln agents jointly in pointedKripke models, all that is needed is to add suitably manyrelations:

To also incorporate beliefs, simply apply the same move as in thesingle agent case: augment the language and let there be two relationsfor each agent.

The definition uses a family of relations\(\{R_{i}\}_{i\in\mathcal{A}}\). In the literature, the same isdenoted \((W,R_{i},V,w)_{i\in\mathcal{A}}\). Alternatively,Ris taken to be a function sending agents to relations, i.e.,\(R:\mathcal{A\rightarrow}\mathcal{P}(W\times W)\). Then for each\(i\in\mathcal{A}\), \(R(i)\) is a relation onW, often denoted\(R_{i}\). These are stylistic choices.

When considering only a single agent, it is typically not relevant toinclude more worlds inW than there are possible valuations ofatoms. In multi-agent cases, this is not the case: to express thedifferent forms of available higher-order knowledge, many copies of“the same” world are needed. Let us exemplify for\(\mathcal{A}=\{a,b\}\), \(\Atom=\{p\}\) and each\(R_{i},i\in\mathcal{A},\) an equivalence relation. Let us representthat botha andb knowp, butb does notknow thata knowsp, i.e., \(K_{a}p\wedgeK_{b}p\wedge\neg K_{b}K_{a}p\). Then we need three worlds:

If we delete \(w_{1}\) and let \(w_{2}\) be the actual world, thena would lose knowledge inp: bothp worlds areneeded. In general, ifW is assumed to have any fixed, finitesize, there will be some higher-order information formula that cannotbe satisfied in it.

3.2 Notions of Group Knowledge

Multi-agent systems are interesting for other reasons than torepresent higher-order information. The individual agents’information may also be pooled to capture what the agents knowjointly, as group knowledge (see Baltag, Boddy, & Smets 2018 for arecent discussion). A standard notion is this style isdistributedknowledge: The knowledge the groupwould have if theagents share all their individual knowledge. To represent it, augmentthe language \(\mathcal{L}_{Kn}\) with operators

to make \(D_{G}\varphi\) a well-formed formula. Where\(G\subseteq\mathcal{A}\) is a group of agents, the formula\(D_{G}\varphi\) reads that it isdistributed knowledge in thegroupG that \(\varphi\).

To evaluate \(D_{G}\varphi\), we define a new relation from thosealready present in the model. The idea behind the definition is thatif some one agent has eliminated a world as an epistemic alternative,then so will the group. Define the relation as the intersection of theindividual agents’ relations:

In the three state model, \(R_{G}^{D}\) contains only the three loops.To evaluate a distributed knowledge formula, use the same form as forother modal operators:

It may be the case that some very knowing agent knows all that isdistributed knowledge inG, but it is not guaranteed. Tocapture that all the agents know \(\varphi\), we could use theconjunction of the formulas \(K_{i}\varphi\) for \(\in\mathcal{A}\),i.e., \(\bigwedge_{i\in\mathcal{A}}K_{i}\varphi\). This is awell-defined formula if \(\mathcal{A}\) is finite (which it typicallyis). If \(\mathcal{A}\) is not finite, then\(\bigwedge_{i\in\mathcal{A}}K_{i}\varphi\) is not a formula in\(\mathcal{L}_{Kn}\), as it has only finite conjunctions. As ashorthand for \(\bigwedge_{i\in\mathcal{A}}K_{i}\varphi\), it isstandard to introduce theeverybody knows operator,\(E_{G}\):

In the three world model, \(K_{a}p\wedge K_{b}p\), so\(E_{\{a,b\}}p\).

That everybody knows something does not mean that this knowledge isshared between the members of the group. The three world modelexemplifies this: Though \(E_{\{a,b\}}p\), it also the case that\(\neg K_{b}E_{\{a,b\}}p\).

To capture that there is no uncertainty in the group about \(\varphi\)norany higher-order uncertainty about \(\varphi\) beingknown by all agents, no formula in the language \(\mathcal{L}_{Kn}\)is enough. Consider the formula

where \(E_{G}^{k}\) is short fork iterations of the \(E_{G}\)operator. Then for no natural numberk will the formula\(E_{G}^{k}\varphi\) be enough: it could be the case thatbdoesn’t know it! To rectify this situation, one could try

but this is not a formula as \(\mathcal{L}_{Kn}\) only contains finiteconjunctions.

Hence, though the \(E_{G}\) operator is definable in the language\(\mathcal{L}_{Kn}\), a suitable notion ofcommon knowledgeis not. For that, we again need to define a new relation on our model.This time, we are interested in capturing that nobody considers\(\varphi\) epistemically possibleanywhere. To build therelation, we therefore first take the union the relations of all theagents inG, but this is not quite enough: to use the standardmodal semantic clause, we must also be able to reach all of the worldsin this relationin a single step. Hence, let

where \((\cdotp)^{*}\) is the operation of taking thereflecxivetransitive closure. IfR is a relation, then \((R)^{*}\)isR plus the reflexive arrow and all the pairs missing to makeR a transitive relation. Consider the three world model: Withthe relation \(\bigcup_{i\in\{a,b\}}R_{i}\), we can reach \(w_{3}\)from \(w_{1}\) in two steps, stopping over at \(w_{2}\). With\((\bigcup_{i\in\{a,b\}}R_{i})^{*}\), \(w_{3}\) is reachable in onestep: By the newly added transitive link from \(w_{1}\) to\(w_{3}\).

To represent common knowledge, augment theBackus-Naur formof \(\mathcal{L}_{Kn}\) with operators

to make \(C_{G}\varphi\) a well-formed formula. Evaluate such formulasby the semantic clause

Varying the properties of the accessibility relations\(R_{1},R_{2},\ldots,R_{n}\), as described above results in differentepistemic logics. For instance, systemK with commonknowledge is determined by all frames, while systemS4 with common knowledge is determined by allreflexive and transitive frames. Similar results can be obtained forthe remaining epistemic logics (Fagin et al. 1995, van Ditmarsch etal. 2007). For an informal discussion, consultthe entry on common knowledge.

4. Beyond Knowing That

So far, we have focused onknow-that, the standard notionanalyzed in epistemic logic. However, as mentioned earlier, in naturallanguage, we also express knowledge in terms of know-wh, i.e., theverb “know” followed by an embedded question, such as:

• Aliceknows whether the claim is true.

• Bobknows what the password is.

• Charlieknows how to prove the theorem.

• Daveknows why Charlieknows how to prove thetheorem.

Logical research on know-wh also dates back to Hintikka (1962), whodedicated a chapter to the formal account ofknow-who, andproposed to formalize otherknow-wh alike. According toHintikka, to know “who Mary is” is to know the answer tothe corresponding question “Who is Mary?”. Essentially,one must be able to identify the person denoted by the nameMary (supposing it is a proper name). Therefore, it is apiece ofde re knowledge, i.e., the knowledge of anobject. To render it formally, Hintikka treats “Bobknows who Mary is” as “there is a person such that Bobknows that this person is Mary”, which connectsknow-that with know-wh by using quantification over objects. Toexpress Hintikka’s formalization, we need to extend our basiclanguage of propositional modal logic tofirst-order modallogic, the modal logic with quantifiers and predicates. Asanother example, “Bob knows who murdered Dave” can beformalized in first-order epistemic logic as \(\exists x K_{\Bob}\Murder(x, \Dave)\). Note that the order of the quantifier and theknow-that modality is crucial to capturede re knowledge. Theswappedde dicto version \(K_{\Bob} \exists x\Murder(x,\Dave)\) merely says “Bob knows that Dave is murdered”.For Hintikka, it is also important how the agent is identified, cf.(Hintikka & Symons 2003).

Such treatments of know-wh are also supported by a large body ofresearch on the semantics of (embedded) questions in linguistics(e.g., Groenendijk & Stokhof 1982; Harrah 2002), and epistemologyof knowledge-wh (e.g., Stanley & Williamson 2001). Linguists havediscussed various readings of epistemic expressions with embeddedquestions, in particular the so-calledmention-some andmention-all interpretations. The mention-some interpretationrequires that one knows at least one (correct) answer in order topossess know-wh knowledge, while the mention-all interpretationrequires that one knows all the (correct) answers (cf. Groenendijk& Stokhof 1982). For example, a (strong) exhaustive mention-allreading for “Bob knows who came to the party” isformalized by \(\forall x (K_{\Bob} \Came(x) \lor K_{\Bob}\neg \Came(x))\) where \(\forall\) replaces \(\exists\) as the leadingquantifier, contra the previous examples. The interpretation ofknow-wh is also context-dependent in general, which introduces furthercomplications (e.g., Aloni 2001, 2018). Coming back to logic, Hintikka(1962) discussed the notion of consistency in such a logic of know-whoand various philosophical issues, such as trans-world identity. In hislatter work, Hintikka (2003) also made use of hisIndependentFriendly Logic to go beyond the first-order quantification, inorder to capture wh-knowledge about higher-order entities, such as“I know whom every young mother should trust” with theintention of “trustingher own mother”.

Although Hintikka gave a basic semantics of know-wh, the systematicalstudy of the epistemic logics of know-wh was not developed untilrecently. This is partly due to the underdevelopment of first-ordermodal logic, compared to its propositional brother (cf., Gochet &Gribomont 2006 for a survey on first-order epistemic logic). Variouswell-known systems of first-order modal logic do not enjoy thetechnically desirable properties of propositional modal logic, butsuffers incompleteness, undecidability and the failure ofinterpolation (cf., Braüner & Ghilardi 2007). In particular,it is very hard to find decidable fragments of first-order modal logicto make it computationally appealing (e.g., Hodkinson, Wolter, &Zakharyaschev 2000).

Nevertheless, there are works on clarifying thede re andde dicto distinctions in the setting of quantified epistemiclogic (e.g., Grove 1995; Corsi & Orlandelli 2013; Holliday &Perry 2014; Occhipinti Liberman & Rendsvig 2022), and a number ofapplication-driven frameworks based on first-order epistemic logic(e.g., Mika Cohen & Dam 2007; Kaneko & Nagashima 1996; Sturm,Wolter, & Zakharyaschev 2000; Belardinelli & Lomuscio 2011),although they are not directly about know-wh. As one of a fewexceptions directly related to know-wh based on quantified modallogic, Rendsvig (2012) discusses Frege’s Puzzle about Identityin a framework where agents may have knowledge-who about the referentsof proper names.

Another line of research directly related to know-who is based onterm-modal logic proposed by (Fitting, Thalmann, &Voronkov 2001), a variant of quantified modal logic where modalitiesare subscripted byterms instead of indexes. This allowsformulas such as \(\exists y \exists x K_{y} \Murder(x, \Dave)\)(“there is someone that knows who murdered Dave”), or\(\exists x K_{\Bob}( \Murder(x, \Dave) \wedge x = \Adam)\)(“Bob knows who murdered Dave and that is Adam”). Startingfrom (Kooi 2007), various versions of (dynamic) term-modal logic wereproposed to discuss know-who, such as Occhipinti Liberman, Achen,& Rendsvig 2020; and Y. Wang, Wei, & Seligman 2022. These arecloser to the original systems suggested by Hintikka (1962), where thenames of agents are also treated as terms. Term-modal logicformalization of know-who in social networks, and the dynamics thatinfluence it, is a main theme in Occhipinti Liberman & Rendsvig2022. A few decidable fragments of term-modal logics were discoveredin Orlandelli & Corsi 2017; Padmanabha & Ramanujam 2019a,2019b; and Occhipinti Liberman, Achen, and Rendsvig 2020, where arecent review of the term-modal logic literature can be found.

Concerning a different type of know-wh, some formal discussions onknow-how have appeared in the setting ofstrategy logics. Itis observed that in the setting ofEpistemic Alternating TemporalLogic, just combining “know” and “can”does not capture know-how (cf., Jamroga & van der Hoek 2004;Herzig 2015) butde dicto knowledge: I know there is some wayto guarantee \(\varphi\). Instead of explicitly using the quantifiers,people introduced alternative semantics for the coalition operators inepistemic variants of ATL (e.g., Jamroga & Ågotnes 2007;Maubert, Pinchinat, Schwarzentruber, & Stranieri 2020; Maubert,Murano, Pinchinat, Schwarzentruber, & Stranieri 2020).

Recently, aknow-wh first approach has emerged that treatsknow-wh as an independent modality, just as the modality of know-that(cf. the survey by Y. Wang 2018b). The general motivation of this lineof research is to focus on the logical behavior of each know-wh as aprimitive concept, rather than breaking it down in the syntaxinto quantifiers, modalities, and predicates in the full language ofpredicate modal logic. This approach is inspired by the early work ofPlaza 1989 and Ma & Guo 1983, where a \(Kv\) operator isintroduced to capture the notion of “knowing the value of aconstant”. Semantically, \(Kv d\) can be viewed as a“bundle” of \(\exists x K(d=x)\), which says the agentknows what the value of \(d\) is. In this line of research,various languages were proposed with both the know-that operator andthe know-wh operator of certain types, whose semantics is given by thefirst-order modal interpretation, as we described above for the \(Kv\)operator.

As an example of aknow-wh first framework, consider thefollowing epistemic language of (goal-directed) know-how and know-thatdiscussed by Li & Wang (2021a), where \(K\!h_i\varphi\) says thatagent \(i\)knows how to achieve the goal \(\varphi\).

The semantics is given over a type of Kripke models with bothepistemic relations \(\sim_i, i \in \mathcal{A}\) and labeled actionrelations \(\stackrel{a}{\rightarrow}, a \in \Act\), where\(\Act=\cup_{i\in \mathcal{A}}\Act_i\) is the union of the sets ofactions of each agent, which do not explicitly described in the formallanguage. Such a model takes the form \(\mathcal{M}= (W,\{\sim_i\}_{i\in \mathcal{A}}, \{\stackrel{a}{\rightarrow}\}_{a \in \Act},V)\). Thesatisfaction relation between a pointed Kripke model \(\mathcal{M},w\)and a \(K\!h_i\varphi\) formula is given by the following \(\exists xK\)-style schema:

A plan \(\pi\) can have different forms, such as a finite linearsequence of actions, a conditional strategy given the knowledge ofagents, or even a program with loops (cf., Li & Wang 2021b). Aplan isstrongly executable intuitively means the plan willnever get stuck and always terminates successfully.

To exemplify the framework, suppose that a patient is experiencingsome rare symptom \(p\). To know the cause (\(q\) or \(\neg q\)),Doctor 1 suggests the patient first take a special test (\(a\)), whichis only available at Doctor 1’s hospital, and send the resultsto Doctor 2 in another city, who is more experienced in examining theresults. Doctor 2 will then know the cause, depending on which,different medicines (\(b\) or \(c\)) should be subscribed to cure thepatient. The situation is depicted below as a model with bothepistemic relations (dotted lines labeled by \(1,2\); reflexive onesomitted) and action relations (solid lines labeled by \(a,b,c\)). Notethat only Doctor 1 can execute action \(a\), and only Doctor 2 canperform \(b\) or \(c\). After action \(a\), Doctor 2 knows whether sheis at \(w_3\) or at \(w_4\), but Doctor 1 is still uncertain aboutit.

Intuitively, Doctor 1 knows how to let Doctor 2 know how to cure thepatient (\(\neg p\)), although neither Doctor 1 nor Doctor 2 knows howto cure the patient alone. This can be expressed by \(\neg K\!h_1 \negp\land \neg K\!h_2 \neg p\land K\!h_1((K_2 q\lor K_2\neg q)\landK\!h_2 \neg p)\), which is true at \(w_1\) and \(w_2\), given a propernotion of plans.

Finally, logics of know-how may be obtained for this framework. Theaxioms depend on the specific assumptions made of the Kripke models,as in standard epistemic logic. For example, if we allowconditional knowledge-based plans and consider agents withperfect recall (as in Fervari, Herzig, Li, & Wang 2017),then the following axioms and rules on top of the S5 proof system for\(K_i\) are complete:

Whenever an agentc knows all of the formulas in a set\(\Gamma\) andA follows logically from \(\Gamma\), thenc also knowsA.

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• Fan, Jie, Davide Grossi, Barteld Kooi, Xingchi Su, and RinekeVerbrugge, 2020, “Commonly Knowingly Whether”, manuscript.
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